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Cosmology Corner

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June 10, 2013

Black Hole Singularities Revisited

In earlier blogs I stated my theory that the core of a black hole is not a singularity but rather an extremely dense mass of Planck-sized particles or strings, equal in number to the entropy of the black hole. The entropy has already been shown to be the surface area of the event horizon of a black hole in units of 4 square Plancks (Susskind redefines the Planck size to be twice its original value, so that the surface area of the event horizon in units of square Plancks is equal to the entropy). I have not reconsidered this position, and I shall not do so in the foreseeable future—which is defined as the length of time until someone comes up with a better theory.

In this blog I consider the implications of the present theory that the center of a black hole is a singularity of infinite density with a radius of zero. A point, in other words. Sadly, most physicists and cosmologists do not appear to have contemplated the implications of this theory, which are, as you shall see, formidable.


A singularity has an infinitely strong gravitational field  
Current theory about the singularity at the center of a black hole holds that its gravitational field is infinitely strong. Indeed, it must be so in order for the singularity to exist even theoretically. This idea of the mass of a body being concentrated in a point at the center of the body is not new. Isaac Newton used it in developing his theory of gravity over 300 years ago. In fact, the equation relating the radius and mass of a black hole can be derived directly from Newton’s equations of gravity and the escape velocity for a gravitational field. It does not require Einstein’s general theory of relativity for its derivation, but it was only after the publication of this theory in 1915 that the idea of real black holes began to be taken seriously by scientists.

Newton computed the gravitational field of the earth mathematically by assuming that all of earth’s mass was concentrated at a point in the center. This was only a mathematical convenience, and we know it’s not true. We can dig holes in the earth, which certainly appears to be rather solid. But nobody can see inside a black hole and nobody ever will. Therefore, the theory that there is an infinitely dense singularity lurking inside a black hole is not scientific because it fails the criterion of testability, which is crucial to any scientific theory.

But wait—it gets even worse. What about the infinitely strong gravitational field caused by the singularity? Since the time of Newton we have known that the strength of a gravitational field decreases as the square of the distance from the center of the body causing that field. But there is a big problem with an infinitely strong gravitational field. You can divide infinity by any number you wish to, and it will still be infinite. In other words, that infinitely strong gravitational field would extend throughout the observable universe. If this were really true, the very first black hole that was ever created would have caused the entire universe to implode and crush everything into oblivion. It is apparent by just looking around that this did not happen. Ergo, there can be no infinitely strong gravitational field lurking in the depths of a black hole. Period.


Problems with the compressibility of matter
The creation of a singularity such as the ones posited to be formed by the collapse of a massive star into a black hole requires that the matter in the star be compressed into nothing at all. Aside from the problem that a singularity cannot have entropy, and we know that black holes do have entropy, there is the problem of completely crushing matter into nothing at all. Is this even possible? Many scientists believe it’s impossible. For a black hole, Einstein’s theory “predicts infinitely strong gravitational fields and density. That’s nonsensical,” said Paulo Freire, an astrophysicist at the Max Planck Institute for Radioastronomy in Germany. (“Einstein Proved Right on Gravity—Again” by Gautam Naik, Wall Street Journal, April 25, 2013.)

There are numerous papers in recent years describing the close relationship between black holes and string theory, ideas that are important to the compressibility of matter. In a 1993 paper titled “Some speculations about black hole entropy in string theory,” Leonard Susskind wrote:

The last puzzle concerns the connection between the spectrum of black holes and that of unperturbed strings. In both cases that level density increases rapidly with mass. Furthermore, most of the spectrum of strings must actually be black holes since they lie within their Schwarzschild radii. Nevertheless I do not know of any speculation that the two spectra may really be the same.[3] In fact at first sight such a suggestion seems nonsensical. The level density of black holes grows like exp 4πM2 while that of strings is exponential in the first power of the mass. We shall see that this argument is wrong and that the two spectra, when properly interpreted, could easily be the same. . .


The finiteness of string theoretic loops is due to the extreme paucity of degrees of freedom at short distances. This lack of short distance structure is seen in several ways. . .


3) It appears to be impossible to force the dimensions of compact space dimensions to be smaller than a certain size of order ℓs.[7]


4) Following the progress of a string falling toward a horizon, an external observer fails to see the object Lorentz contract.[2] There appears to be a minimum longitudinal size that strings can occupy. Furthermore for non vanishing coupling there is a bound to the number of strings that can pass through a small region without inducing violent interactions.

All these facts point to a common conclusion. When we attempt to localize strings or parts of strings in distances much smaller than ℓs we discover a complete lack of local degrees of freedom. This strongly suggests that higher genus contributions to black hole entropy is finite and that[,] to an external observer, indefinite quantities of information can’t collect arbitrarily near the event horizon. What is desperately needed is a computation to confirm this.


The quantitys that Susskind refers to is the “string length scale,” a quantity close to or equal to the Planck distance, about 10-35 meters. He states that it may be impossible to compress things to a size smaller than the Planck length (a Planck volume, presumably). He also says that there “appears to be” a minimum length for strings, again implying incompressibility beyond the Planck scale.

Besides all this, Susskind says that trying to pass an arbitrarily large number of strings through a small region will induce “violent interactions.” It sounds as though pushing all those strings through a singularity may be a hard trick to pull off. (See the end of this blog for the references Susskind makes here.)

Then there is this from an article titled “Bekenstein-Hawking entropy” by Jacob D. Bekenstein, Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel [Jacob D. Bekenstein (2008), Scholarpedia, 3(10):7375]:

Strings in string theory have a variety of excitations, so there is a multitude of string states. Therefore, a string has entropy, which turns out to be proportional to its mass. This is quite in contrast with black hole entropy. However, an argument by Bowick, Smolin and Wijewardhana (1987) suggests that by adiabatically (i.e. sufficiently slowly) reducing the string coupling constant g, it is possible to shrink a black hole’s size as well as to reduce its mass (while keeping its entropy constant) until eventually it gets to be the size of the string length scale ℓs when the black hole should not be distinguishable from a string. At the corresponding value of g, string and black hole entropy are quite similar (see e.g. Zwiebach 2004). This has been taken to mean that there is a one-to-one correspondence between black hole and string states, where both entities have the same entropy (Susskind 1993). This picture has been corroborated in the context of five-dimensional extreme black holes (Strominger and Vafa 1996). Hence black hole entropy can be understood in terms of string entropy.


Note that the string length is believed to be the same, or nearly the same, as the Planck distance, about 1.6 x 10-35 meters. Bekenstein, one of the giants in the world of cosmologists—as is Susskind—describes the close relationship between string theory and black holes, at least so far as entropy is concerned. (Again, the references Bekenstein makes above will be found at the end of this blog.)

If strings prove to be the elemental building blocks of matter, as now seems probable, and strings cannot be compressed, then the core of a black hole must comprise strings all crowded together in a minimum amount of space. That is precisely what my theory proposes: one string per cubic Planck volume, each one corresponding to one bit of the entropy encoded in the event horizon of the black hole.


Conclusions
This evidence taken as a whole strongly implies that there can be no singularity lurking at the heart of a black hole. The size and mass of a black hole are related by the familiar relationship

R = 2MG/c2

where R is the radius of the event horizon (assuming a Schwarzschild black hole), M is the mass of the black hole (the amount of matter that went into it), G is the gravitational constant (derived originally by Isaac Newton in 1687), and c is the speed of light in a vacuum (299,792.458 kilometers per second). This equation tells us nothing at all about what lies inside the event horizon of a black hole; anything about that is speculation—but speculation has long fueled scientific progress. We can know only the external attributes of a black hole: size, mass, spin (if any), and electrical charge (if any). As for what is inside a black hole, one theory is as good as another.

I rest my case. For now.


References
(Numbers are from the quote by Leonard Susskind)

2. L. Susskind, “Strings, Black Holes and Lorentz Contraction,” Stanford University preprint SUITP-93-21

3. ’t Hooft has had the long standing belief that black holes are the extrapolation of elementary particles to high mass. See for example G. ’t Hooft,
Nucl. Phys. B 335, 138.

7. D.J. Gross and P. Mende,
Nucl Phys. B 303, 407.

Bekenstein, Jacob D. “Bekenstein-Hawking entropy.”
Scholarpedia, 3(10):7375

Bowick, M., Smolin, L. and Wijewardhana, L.C.R.:
Gen. Rel. Grav. 19, 113 (1987)

Strominger, A. and Wafa, C.:
Physics Letters B, 379, 99 (1996) (http://xxx.lanl.gov/abs/hep-th/9601029)

Susskind, L.: “Some speculations about black hole entropy in string theory,” unpublished (1993) (http://xxx.lanl.gov/abs/hep-th/9309145)

Zwiebach, B.:
A first course in string theory, Cambridge University Press, Cambridge (2004)


Note added on 16 September 2014
 
A New Theory of Black Hole Formation
  
My paper “
A New Model of Black Hole Formation” published in Progress in Physics, 2013 Volume 4, page 44, shows that once an event horizon has formed, the matter in the collapsing star will accrete to a shell just inside the event horizon, and the central portion will be empty. A rough mathematical development is included.

Additional evidence for this new theory comes from Yoshifumi Hyakutake of the College of Science at Ibaraki University (Japan) in a paper published on 
arXiv 1311 [hep-th] 29 Nov 2013 titled “Quantum Near Horizon Geometry of Black 0-Brane.” The applicable statement is near the end of Section 6 of the paper, where Hyatkutake writes: “From this we see that the gravitational force becomes repulsive near the horizon xH.” Hyakatuke’s paper also was published by the Oxford Journals Progress of Theoretical and Experimental Physics 14 March 2014.

This repulsive gravitational field is what causes the shell to form, which holds the matter taken inside the event horizon.

Now can I rest my case?

April 10, 2013

Is the Quantum Uncertainty Principle Real?

First, let me make it plain that I am not questioning the uncertainty principle as first described by Werner Heisenberg in 1927. This is a fundamental property of any dual wave/particle object and is formalized by a relationship between the standard deviation of the position of a particle, σ
x, and the standard deviation of the momentum, σp, of that particle, where x is the position and p is the momentum (the product of mass and velocity, a vector quantity):

σxσp ≥ ħ / 2

where ħ is the reduced Planck constant. If σx is reduced, σp must necessarily increase. There is no way to avoid this fundamental limitation on the measurement of position and momentum of quantum-scale (i.e., very small) objects.

What I am questioning here is a more general principle of uncertainty, one that has provoked much discussion over the years. In 1935, Einstein, Boris Podolsky, and Nathan Rosen published an article titled “Can Quantum-Mechanical Description of Physical Reality by Considered Complete?” This is known as the EPR paradox, wherein knowledge of a quantum state seems to be instantaneously transmitted over an indefinitely large distance—an obvious violation of the rules of the special theory of relativity. Einstein called this “spooky action at a distance.”

Paul Halpern, in his book
Edge of the Universe, describes the situation thusly:

A simple way of looking at the EPR paradox is to imagine two electrons emitted from the ground state of an atom. From the Pauli exclusion principle … we know that these electrons cannot have all the same quantum numbers and must have opposite values of the quantum parameter called spin. There are two types of spin for electrons, up and down. So we know if one is up the other is down, like kids on opposite ends of a seesaw. However, unless we take a measurement, quantum uncertainty informs us that they are in a mixed state and we don’t know which is up and which is down. This is hard to picture, but we can try to imagine a seesaw moving so blurrily fast that we don’t know which side is up.


Let’s assume that both electrons are released at once, in opposite directions. As they separate more and more, we still don’t know which electron has which spin. Now we take a measurement of one of the electron’s spin state. According to the Copenhagen interpretation, the electron being examined immediately collapses (with a 50-50 likelihood) into either an up or a down state. Right away we measure the other electron’s spin state, and unwaveringly it is the opposite of the first. How does the second electron instantly know what the first one has “decided”? Einstein, Podolsky, and Rosen thought they had slung an arrow into the heart of the theory, but quantum theory survived much stronger than ever after experiments showed that this is what actually would happen.


—Halpern, Paul. Edge of the Universe, page 177


I see this and similar situations as a philosophical question, not a scientific problem. What happens depends on whether you are an electron or an observer. First of all, the Pauli exclusion principle tells us that the two electrons are emitted with opposite spin states. They cannot exist even momentarily with the same spin state, because that would violate the Pauli exclusion principle. The problem is that we don’t know which one is the “up” electron and which is the “down” electron. We only know that they must be different.

Once we measure the spin state of one of the electrons, we immediately know what the spin state of the other electron is. We don’t even have to measure it. Of course it will be opposite to that of the first electron: the Pauli exclusion principle virtually dictates that it must be so. So there is no “spooky action at a distance.” How on earth any experiment could show that “this is what actually would happen,” as Halpern puts it, is beyond the pale. Any experiment is going to require measurements of some kind, and quantum theory already tells us that taking measurements affects the results of those measurements. In other words, we cannot carry out an experiment without tampering with the evidence, as it were.

This quantum uncertainty principle is a statement about observational reality, not physical reality. The so-called “mixed state” is simply a statistical uncertainty, one that can only be resolved by observation. Once one of the electrons has been observed, the uncertainty is instantly resolved and no “spooky action at a distance” is required. What “collapses” is the uncertainty on the part of the observer.

The famous example of “Shrödinger’s Cat” presents similar philosophical problems. There may not be any Pauli exclusion principle that says a cat cannot be both alive and dead, but there is certainly a reality test that excludes the possibility of being both alive and dead at the same time. There are no degrees of “deadness.” The poor cat must be either alive or dead, it cannot be both at the same time. Opening the box simply reveals the result of the experiment; it does not determine the result of the experiment. (And God help Shrödinger if PETA ever catches up to him! Oh, wait—he’s already dead, so they can’t touch him.)

So we are left with the rather unsatisfactory situation that the results of a theory (quantum mechanics) are produced by experiments designed to test the theory. No experiment, no result. If no one ever measures the spin state of one of the electrons emitted by the atom in the thought experiment proposed by Halpern, that state will never be determined. The poor thing is doomed to fly through space not knowing up from down. All because nobody bothered to make a measurement and cause its wave function to “collapse.” If this doesn’t sound like one of Alice’s “seven impossible things before breakfast,” then I don’t know what does.

March 22, 2013

Our Universe: How Old and How Big?

You see a lot of numbers bandied about on this subject. The “best” current figure seems to be 46 billion light-years as the size of our universe. The age of our universe, however, is evidently not in dispute. The current accepted age is 13.77 billion years, give or take several dozen million years. But you will find cosmologists talking about galaxies and other objects at distances like 16 billion light-years, or even more than that. I’ve got to wonder what on earth they are talking about. Have they ever seen any such objects? Just because their theory says they might exist doesn’t mean that they do exist. All of these people seem to miss the obvious: There is a not only a speed limit in our universe (the speed of light) but there is also a time-distance limit. It’s not hiding somewhere waiting to be discovered. It is already known, studied, and inspected inside and out. Our more fanciful cosmologists apparently just ignore it.


The light at the end of this tunnel
What am I writing about here? The cosmic microwave background (CMB) that’s what. It was discovered in 1964 and has been studied by satellites dedicated to just that task, the latest being the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck-ESA project. The CMB was generated by the surface of last scattering, a time when the budding universe became transparent to light waves (or photons = light particles). This has also been studied extensively. The oldest photons in the CMB are believed to have been generated when the universe was only 372,000 years old. It took another 115,000 years for all of the stored up photons to find their way to freedom. The CMB was found to have a perfect black body radiation curve, which sealed the fate of those who still argued against the Big Bang theory of the origin of our universe.

The CMB is all around us. The WMAP probe sent back data that, when analyzed and spurious items were eliminated (such as the anisotropy caused by the earth’s motion in the universe and the glare from our local galaxy, the Milky Way), revealed a nearly perfect, uniform glow—albeit a weak one. You see, on its nearly 13.77 billion year journey to reach our inquiring microwave radio detectors, the poor CMB was nearly redshifted out of existence. It started out on its journey with an apparent temperature of some 3,000 degrees Kelvin—that’s nearly as hot as our sun—but now glows feebly at a temperature of 2.725 degrees Kelvin. That’s almost absolute zero. It’s colder than liquid helium-4 (boils at 4.2 degrees Kelvin) and even helium-3 (boils at 3.2 degrees Kelvin). The redshift factor for the CMB is about 1090. By way of contrast, the greatest red shift ever measured for an observed galaxy (announced on November 16, 2012) is about 11.

And just how nearly perfect is the uniformity—or the isotropy, as cosmologists call it—of the CMB? Once the spurious items mentioned earlier have been eliminated the difference in apparent temperature between the coolest and warmest portions of the CMB is only two ten-thousandths of a degree Kelvin. That’s ±0.0001 degrees Kelvin. In fact, the difference is one part in 100,000, exactly the amount cosmologists had determined—in advance—that was required to form the seeds of the galaxies, which now fill the space around us as far as our telescopes can see. The farthest galaxy discovered last year is at a distance of about 13.3 billion light-years and is dated to about 420,000,000 years after the Big Bang. It has a redshift of 11, the highest value ever recorded.


WMAP+9-Yr+Image--Small
9-year WMAP image of background cosmic radiation (2012)

Planck+CMB+Smalli
2013 Planck — ESA image of cosmic radiation background

These pictures adequately show the slight anisotropies that exist in the CMB. You have to keep in mind that this is what the CMB looked like almost 14 billion years ago, not what it looks like today. Without a time machine, there is no way that we can see what it looks like now. We would have to travel almost 14 billion years into the future to see that. No way, José!

We are seeing what the CMB looked like about 372,000 years after the Big Bang, and it took the those photons almost 13.77 billion years to arrive at the earth and be detected by our microwave receivers. The recessional speed of the CMB as computed from the redshift is 99.997 percent of the speed of light. That is definitely moving right along. Compared with that, the recessional speed of the recently discovered farthest galaxy, with a redshift of 11, is “only” 96.95 percent of the speed of light. Oddly enough, the time at which galaxies are first believed to have formed is very close to the 420 million years that was evidently when the light from this farthest galaxy was emitted.

I will stick my neck all the way out here and make a prediction: No stellar object will ever be found that is farther away from us than the CMB (13.77 billion light-years). In order for such an object to be found it would have to have a redshift at least equal to that of the CMB, an obvious impossibility, since only a hot plasma is believed to have existed prior to the surface of last scattering—the source of the CMB. To have a redshift greater that the CMB would mean it was not part of our universe. (I except here the possible detection of neutrinos or gravity waves from the Big Bang itself, if and when they are found, since they would predate the CMB; but they are hardly “stellar objects.” And by “ever” I mean at least the next 100 years.)

Just remember—you saw it here first.

A star with a surface temperature of 30,000 degrees Kelvin—a very hot blue star—would have an apparent temperature of only 27 degrees Kelvin after a redshift of 1090 (like the CMB). You would need a very good radio telescope to spot a star that dim. Even a type 1a supernova, which has a peak luminosity about 5 billion times greater than that of our sun, would have a redshifted temperature of only some 1200 degrees Kelvin.1 A star at that temperature would appear as a deep red to the human eye. A good infra-red telescope could probably spot such a distant supernova, although it might prove difficult to identify since most supernovae are associated with galaxies, and no ordinary galaxy would be identifiable at such a huge value of the redshift (we’re still talking about the redshift of 1090 that applies to the CMB).

So the age of the CMB combined with its identification as being composed on the average of the photons that were emitted by the surface of last scattering 372,000 years after the Big Bang (in its original definition as the beginning of our universe) pins down the age of our universe at 13.77 ± 0.06 billion years. That’s a long time but a lot shorter than forever.


How big is our universe?
The answer to this question depends on what you mean by it. If you mean our observable universe, then the answer quite simply is that it has a radius of 13.77 billion light-years and a diameter twice that, or 27.54 billion light-years. Our observable universe, however, is seen through a time filter. Every object that we observe is as it was when the light by which we see it left that object. If you mean our universe as it is today, that is basically an unanswerable question. It is most certainly much larger than 27.5 billion light-years in diameter.

The answer to the question of how big our universe is today depends to a great extent on what is called the “scale factor” of the universe. Without going into details that are highly mathematical, this scale factor is affected by the size of the so-called cosmological constant, from Einstein’s equations, and which he called the greatest blunder in his life. It is now enjoying a renaissance of sorts. The idea here is that space has been expanding while the photons from distant objects have been traveling from where they began to where we are (the earth). This affects the total distance traversed by the photons, even though their apparent speed is only that of light. The bottom line is that with the best current estimates of the cosmological constant, and hence of the scale factor of our universe, photons from the CMB have covered a total distance of 46 billion light-years even though they appear to be only 13.77 billion years “old.” This makes the radius of the universe at this time to be 46 billion light-years, and hence the diameter to be 92 billion light-years. This scheme effectively triples the speed of light and then some.

You can believe whatever you want to about this. The truth is that no one really knows how big the universe actually is right now. All we do know is how big it appears to us: 27.54 billion light-years in diameter. How big it really is right now depends on too many assumptions and crudely measured variables to be reliable. Let’s just say it’s really, really big and let it go at that.


What is our “observable” universe?
You see a number of different definitions of the “observable universe.” To set the record straight, what I mean by our observable universe is what we can actually see, what we can actually observe. With our telescopes we can look backward in time to a point where our universe was very young: the time when the CMB was generated. Things that we might be able to observe someday but cannot now are not a part of this observable universe. There are objects that we can observe that have not yet been seen; new galaxies are being discovered every day. I am sure that the distant galaxy reported last November will not be the last such discovery. More galaxies (or other stellar objects, such as quasars) will surely be found, some of them even more distant than last November’s object. But the gains in that direction will become smaller and more incremental. The only way astronomers could see the galaxy discovered last fall was to take advantage of gravitational lensing: a cluster of three huge galaxies closer to the earth provided a gravitational “lens” that magnified the image of that far distant galaxy enough for a large telescope to gather in its light.

Technology does not stand still. There is an old joke that a computer is the only thing you can buy that will be obsolete before you can get it into the trunk of your car. Moore’s law holds that the power of computer chips doubles every 18 months. It’s held true for the past 30 years or so. So will the technology of astronomical observing also advance. In a few years astronomers will be reporting that they have imaged stellar objects that are completely invisible with today’s technology.

Some scientists state that the observable universe includes galaxies that are presently farther away from us than the event horizon which defines the “edge” of our present observable universe. I object to this term; it should be “potentially observable universe.” If something is observable that means you can see it now if you have the means and the desire to do so. If you can’t do that, it is not observable.

The “unobservable universe” would also include stellar objects at distances in the “now” universe that are so great that we shall never be able to see them because they are “running away from us” at a speed greater than the speed of light—because of the expansion of the universe (they cannot exceed the speed of light with respect to objects that are in their locality).

The outer limits of our observable universe will always be receding from us at the speed of light. Those objects that are closer than the CMB will always be receding as our universe expands by Hubble’s law. Their redshifts will gradually increase, but ever so slowly. It will require thousands of years to notice the change in the redshift of even the most distant objects we can detect. The temperature of the CMB will continue to decrease, although again it will take thousands of years for this to be noticeable. I would not hold my breath waiting for it to happen!


A conflict in our data
A possible contradiction in some of the data collected and interpreted by the WMAP scientists is evident from information contained on the NASA web pages for this project. The best estimate for the Hubble constant was reported by NASA on December 12, 2012, as 69.32 ± 0.80 km/sec/Mpc. Assuming a flat universe, which is a prediction that can be derived from Inflation theory, this new value of the Hubble constant yields an age for our universe of 14.11 ± 0.16 billion years. Yet the best estimates from WMAP measurements of the CMB yield an age of 13.77 ± 0.06 billion years. The latter age yields a value for the Hubble constant of 71.06 ± 0.30. So it appears that the best estimate of the age of our universe is somewhere between 13.77 and 14.11 billion years. If the latter figure is correct it is mostly likely owing to the value of the cosmological constant being somewhat larger than zero (Susskind says it is zero out to the 123rd decimal place and then becomes positive; this may be enough to tip the balance in favor of the 14.1-billion-year age for our universe.)

[Note added 3-29-13: Recent Planck-ESA news releases have stated that their results show the universe to be 80 to 100 million years older than “previously believed.” It is unclear whether this applies to the WMAP age of 13.77 billion years, since Planck-ESA states that this changes the age of the universe from 13.7 to 13.8 billion years. But the WMAP age already rounds to 13.8 billion; an addition of 90 million years (the average of the range given by Planck-ESA) would increase the WMAP age to 13.86 billion years. At the same time, Planck-ESA data give the value of the Hubble constant as 67 km/sec/Mpc. With the flat, Λ = 0 model, this value of Ho would compute to an age of 14.6 billion years. It is becoming obvious that the Λ = 0 model—known as Einstein-de Sitter space—is invalid. Current opinion among cosmologists holds that the universe is flat with Λ > 0.]


A strange way to begin
Here in a nutshell is a brief history of the beginning of our universe. It all began, most likely, with a tiny quantum of extremely dense material (and that’s putting it very mildly) about one Planck volume in size. A Planck volume (based on Susskind’s revision of the Planck length as given in his book The Cosmic Landscape) is 3.379 x 10-104 cubic meters. We assume that this embryonic universe began expanding at the speed of light instantly. Working Alan Guth’s figures backwards, this initial expansion covered the ground between the initial volume of 3.379 x 10-104 cubic meters and a volume of about 2.7 x 10-84 cubic meters, involving an increase in the diameter of the embryonic universe from 3.23 x 10-35 meters to 6.5 x 10-28 meters—a factor of about 8 x 1021 in volume. This initial expansion would have taken about 10-36 seconds, and the sudden increase in volume would have reduced the temperature of the nascent universe by a considerable amount from whatever it started out as. In his book Edge of the Universe Paul Halpern describes Guth’s reasoning as follows: “Through the mechanism of spontaneous symmetry-breaking and the production of a scalar field, Guth saw the opportunity to describe a phase-change for the primordial universe, analogous to supercooling.”

Halpern goes on to say:

Guth imagined that the symmetry-breaking process would occur through a kind of supercooling of the universe right after the Big Bang, in which it would be in a metastable state primed for transformation. It would be like a bottle of water left out on the frigid porch, with a temperature below freezing but still in the liquid state. As the universe continued to cool, patches of false vacuum would spontaneously lose their initial symmetry, decay into a lower energy state, and produce a scalar field.. The field—called an inflaton—would trigger a brief but explosive inflationary epoch.… During an interval of about 10-36 seconds—more than one quadrillion times faster than ultrashort laser pulses, some of the quickest measured events—the volume of space would increase by a factor of 1078. Imagine if a grain of sand suddenly blew up to become larger than the Milky Way [galaxy]; that gives you an idea of the colossal burst of expansion during that fleeting instant.


 —(Halpern, Edge of the Universe, page 58)

Halpern states earlier in his book that this Inflation resulted in a universe about the size of a baseball. I worked this figure backward to obtain the size of the universe before Inflation, and obtained the figures given above. Calculating the difference in size between one Planck volume and the result of the calculation, I obtained the time necessary to get to this point. Simple, but laborious, math.

Halpern goes on to write: “What happened right after the inflationary blast would set the course of all cosmic history. In a proposed process called ‘reheating,’ enormous, locked-up quantities of energy would flood space with massive amounts of particles. Theorists have estimated that some 1090 particles emerged during reheating. These would constitute the essence of the material from which the stars, planets, and everything around us would be forged over time. Thus, according to Guth’s theory, it was Inflation, not the initial Big Bang, that created the bulk of everything.” (Ibid., pages 58-59)

What any of this has to do with the supposed anisotropies of the embryonic universe and the subsequent isotropic appearance of the CMB, I haven’t a clue. But the explanation given here by Paul Halpern strikes me as a rather reasonable theory. You may draw your own conclusions.

After this explosive inflationary event, the universe continued expanding at the speed of light—that is, the outer boundary expanded at that rate and everything inside expanded in proportion to its distance from the edge of the nascent universe. Mind you this whole episode from the Big Bang to the end of Inflation required only some 10-32 seconds to complete (the first 10-36 seconds comprised only about one part in 104 of the total).2


Struggling to fit things the wrong way to
This inflationary theory has been fashioned so as to explain how the universe could have ended up in almost perfect isotropy no matter how it began. It is supposed to explain how the universe could start up totally disorganized, like a rocky sea coast, and end up smooth as silk. Any police detective could tell them that this is the wrong way to go about things. Work from the evidence, they would say. And the evidence is that the universe started out isotropic because that’s the way it ended up.3 Why complicate things by postulating ridiculous beginnings and then find a theory to explain them away?

So my theory is that the universe began as a smooth compact kernel that exploded equally in all directions and so produced an isotropic CMB with just enough variations caused by random quantum fluctuations (which can be observed in the CMB today) to create the world as we know it. A universe filled with galaxies and clusters of galaxies. And even a few curious mortals who dare to think about how it started, and who imagine that in the beginning there was light, action, and even a sort of rudimentary camera.

And so it all begins.…


Notes
1. Luminosity increases as the 4th power of the absolute temperature of a black body like a star. The temperature of the sun is about 5,000 degrees Kelvin; the original temperature of the CMB was about 3,000 degrees Kelvin. So a star 5 billion times as bright as the sun would be about 38.5 billion times brighter than the original CMB. The fourth root of 38.5 billion is 443, so the redshifted temperature of a type 1a supernova would be about 2.725 degrees Kelvin times 443, or roughly 1200 degrees Kelvin. This temperature corresponds to that of glowing red embers in a fireplace.

2. Note that if the universe had continued expanding at the speed of light during the 10-32 seconds of the inflationary period, it would have achieved a diameter of about 6 x 10-24 meters, and its volume would have been only some 2 x 10-70 cubic meters. Compared with the size of a baseball, which has a diameter of about 0.08 meters and a volume of about 2.7 x 10-4 cubic meters, that is minuscule. In other words, the Inflation proposed by Guth's theory is truly enormous for the amount of time involved: 10-32 seconds.

3. In fact, this is exactly what has been proposed by Misner et al in their book on gravitation, where they wrote: “One crucial assumption underlies the standard hot big-bang model: that the universe ‘began’ in a state of rapid expansion from a very nearly homogeneous, isotropic condition of infinite (or near infinite) density and pressure.” [Emphasis added.]


Bibliography
Halpern, Paul, PhD. Edge of the Universe. Hoboken, NJ: John Wiley & Sons, Inc. 2012.

Misner C. W., Thorne K. S., Wheeler J. A. Gravitation. New York: W. H. Freeman and Company, 1970.

NASA, WMAP Highlights, http://wmap.gsfc.nasa.gov/

Susskind, Leonard. The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. New York: Little, Brown and Company, 2005. Paperback edition, New York: Back Bay Books / Little, Brown and Company, 2006.

March 21, 2013

Black Hole Land

Black holes are among the least understood phenomena in our universe. First of all, no one has ever seen one. You can't see something from which no light is emitted or reflected. If you could see one, it would look like a hole in the universe. But you can't see one because it is surrounded by an event horizon that is glowing hot with objects that the black hole is swallowing. And even then you could only see the particles that were far enough away from the event horizon that their light was not redshifted so much as to make them invisible to the human eye.

But come, let us explore these mysterious objects some more.


Cosmic hide and go seek
I speak here of the root problem behind the phenomena we call “black holes.” They are enigmas wrapped in a bigger enigma. The fault of it all lies in quantum mechanics: the successful theory of the behavior of very small things, things even smaller than molecules and atoms. The world of subatomic particles.

Albert Einstein once famously said, “I refuse to believe that God plays at dice with the universe.” (He really did say that, although he repeated it so many times that at least half a dozen variations exist. This one is my favorite of all of them.) Sadly for him, he was wrong. His refusal to accept quantum mechanics—of which he was one of the founders—was the biggest blunder of his life. He thought his biggest mistake was the cosmological constant that he introduced to make the universe behave itself. But the cosmological constant has been reborn and is currently an accepted part of the big picture. Quantum mechanics, however, rolled over the poor old professor like a steamroller running over a marshmallow.

It is very hard for us to conceive what the world of the ultra-small is like. Imagine you are trying to track down an electron, for example. Just when you think you know where it is, you find you no longer know how fast it’s moving or where it’s going. So you glom onto that only to find that you have now lost the doggoned thing again. Now you have it spotted again, when it suddenly disappears and reappears a little way off.

Keeping track of particles in quantum mechanics is like nailing Jell-O to a wall. No wonder my old friend1 Richard Feynman once said, “It’s safe to say that nobody understands quantum mechanics.” Yet if there was anyone who did understand it, it would have been Feynman himself, who in 1999 was ranked number seven in a list of the top ten physicists of all time.

To illustrate the strangeness of quantum effects, I borrow an example from Leonard Susskind (The Black Hole War). Suppose you are driving your car when it suddenly dies. You are in a bowl-shaped valley encircled by a fairly high ridge. All you can do is coast to the bottom. You cannot get your car started again, so you walk off in search of help. Now imagine that you return sometime later only to find your car sitting on a road outside of the little valley where you left it. You know things cannot roll uphill, so how did it get there?

In quantum theory, it’s called tunneling. A particle enclosed by what seems an impenetrable barrier will suddenly appear outside the boundary. You see, the position of the particle is only a probability. There is a small, but finite, probability that it will suddenly jump to some other position regardless of any barrier that may lie in between. (It’s a good thing this is true because tunnel diode transistors depend on this effect.) The same thing applies to large objects like your car. But the probability that your car would suddenly appear outside of the little valley where you left it is vanishingly small. It could happen tomorrow, but more likely you would have to wait for a period of time many times longer than the age of the universe for it to happen. So tell me, does God play at dice with the universe, or what?

When thinking about the many problems associated with black holes, this little ditty often runs through my mind (with the usual melody):

            Where, oh where, has my little god gone?
            Where, oh where, can he be?

Swap just two letters from the original version and suddenly the entire meaning of the verses is changed. With black holes, much the same thing applies. To paraphrase the late Senator Dirksen, you add a billion kilograms here, a billion kilograms there, pretty soon it adds up to real mass. And a black hole has mass in abundance—a case of way too much of a good thing, one might say.

Yet, as I wrote earlier, one of the problems with black holes is how did they get there in the first place? Sure, we know where they allegedly come from: a white dwarf star that is too big for its own britches—in technical terms, one whose mass exceeds the Chandrasekhar limit (named after Subrahmanyan Chandrasekhar, an Indian-American astrophysicist, who predicted the effect in 1930). Stars that have a mass greater than about eight solar masses (the value is not known with accuracy) will eventually collapse to form a white dwarf star. If that white dwarf has a mass greater than 1.4 solar masses—the Chandrasekhar limit, whose value is fairly firm—the result will be a neutron star or a black hole. After a stupendous burst of energy called a supernova, the remains of a white dwarf with mass beyond the Chandrasekhar limit may collapse to nothing, the remaining mass being too great and packed too densely to avoid unlimited collapse.

And this is where the rub comes in. (You knew there was a catch, right?) This huge amount of material, with a mass generally several times greater than that of the sun, or even more than that, collapses to a point in space—a singularity. A point is, essentially, nothing at all. It is simply a position somewhere in space (we can worry about whether the term “somewhere in space” means anything later, but you know what I mean). It has no width. No height. No radius. No size. No nothing. Yet this little bit of nothing is supposed to be holding a huge amount of mass, a mass large enough to form a gravitation field from which not even light can escape. A black hole. In order to accomplish this feat the center of the black hole, the “singularity,” must have an infinite density. There is no other way to put so much stuff into a bag with no size to it.

Now, these theories have been developed with considerable effort by some of the greatest minds around. Who am I to argue with their math? I am sure that the equations they have developed after so much work are accurate. But do they represent reality, or do they simply give a mathematical explanation for something that cannot actually be comprehended?

These guys—no disrespect intended—are great physicists, and their mathematics is doubtless impeccable. I, however, practice gut physics, not mathematical physics. Their theories are based on mathematical constructs and basic theoretical considerations. Nothing wrong with that. But my theories are based on thought experiments, not mathematical legerdemain. I like things I can put my finger on, at least speaking figuratively. A point in space with infinite density is not among these things.

Recall the earlier description of watching an astronaut fall into a black hole. It seemed to an observer at a distance that it took forever for the poor astronaut to finish falling into the event horizon, the “surface” as it were of the black hole. If it took forever for this poor fellow to fall into the black hole, then it must have taken forever for the darn thing to have formed in the first place. After all, it had to “fall into” itself, did it not?

And just what happens to our poor fellow astronaut after he passes through the event horizon?

Allegedly, he does not notice anything. But his speed has been increasing to the point that he reaches the speed of light at the event horizon—after all, that is the escape velocity for a black hole. What happens then? Does he go on accelerating right on past the speed of light? Whoa! Give that guy a ticket for cosmic speeding. Say what you will, but a black hole is still in this universe, and the speed of light is still a cosmic speed limit faster than which nothing can go. There is no “faster” speed. The “cosmic” speed of light is infinite; only its projection on our four-dimensional space-time continuum is finite:

c = 299,792.46 km per second.

To those who would argue that the equations developed to describe black holes are accurate in picturing it as a huge mass concentrated in a singularity at the center, I would remind them that the gravitational field of the earth, from its surface to the farthest reaches of the universe (or infinity, whichever comes first!) can be accurately described by assuming that the entire mass of the earth is concentrated in a point located at the center of the earth.
2 Does this begin to sound familiar?

We know, of course, that the latter is not true. The earth is made up of rocks with an iron core. It is not a point with mass equal to the earth surrounded by a hollow shell we call the “earth’s surface.” Neither do I believe that a black hole is made of a huge mass concentrated in a point located at the center of the black hole. This is a mathematical fiction, convenient for describing the effects of a black hole but not representing any physical reality that we can “put our fingers on.”

To figure out what may have happened when a particular black hole was formed, let us go back to the point where the supernova is over and the remaining material is collapsing in upon itself, as the theory dictates. This remaining material has a certain mass, M, a mass that has already predetermined what the radius of the event horizon is going to be (we assume a non-rotating Schwarzschild black hole for the sake of argument here). As the radius of the collapsing material keeps shrinking, the speed at which the outer layers are moving toward the center will equal the speed of light at exactly the Schwarzschild radius—the radius of the event horizon. At this instant the event horizon will form, and the shrinking, collapsing mass will have to stop accelerating lest it violate the sanctity of the speed of light as a cosmic limit.

The remaining material inside the event horizon can continue to shrink, but the radius of the shrinking mass will decrease only at the speed of light. This will continue until such time as quantum-mechanical considerations call a halt to further compression of the matter from the fallen star. The end result will be a central kernel of material at an incredible—but finite—density surrounded by an impenetrable event horizon, from which even light cannot escape.


The birth of a black hole
But how does this happen? What is the process by which such a black hole kernel could be formed?

To start with, imagine the collapsing star as the speed of the shrinking outer edge attains the speed of light. Immediately an event horizon is formed, which surrounds the shrinking star. In this process, the Planck length becomes important. It is designated by the symbol ℓp and given by the formula

ℓp = (ħG/c
3)½

where ħ is the “reduced Planck constant,” G is the gravitational constant (first named by Newton), and c is the speed of light. The Planck length is extremely small and equal to 1.6162 x 10
-35 meters, which is about 1020 times as small as the diameter of a proton. This size can be and has been regarded as a quantum of space.

My theory is that there is some minimum amount of space required for the degenerated matter of a black hole kernel to exist in. If there is anything real about the matter, and hence the mass-energy, of the star that collapsed to form this black hole, it cannot just disappear down the “drain hole” of a singularity—a point of exit from our universe to who knows where, from which it could not possibly cause a gravitational field in our universe. It cannot leave a gravitational field behind as a sort of “ghost” as this would be an absurdity.

At this point I must digress to discuss the size and strength of very small particles or strings. Here is a quote from Susskind’s Cosmic Landscape:

Hadrons are small objects, typically about 100,000 times smaller than an atom. This makes them 10-13 centimeters in diameter. It takes an enormous force to bind quarks [the elements of hadrons] at such small separation. Hadronic strings, the rubber bands of my imagination, although microscopically small, are prodigiously strong. If you could find a way to attach one end of a meson (one kind of hadron) to a car and the other end to a crane, you could easily lift the car. Hadronic strings are not particularly small on the scale of today’s experiments. Modern accelerators are probing nature at scales from a hundred to a thousand times smaller. Just for comparison let me get ahead of the story and tell you what the strength of a string is in the modern reincarnation. In order to hold particles together at the Planck distance [about 10-35 meters], a string would have to be about 1040 times stronger than the hadronic strings; one of them could support a weight equal to the entire mass of our galaxy if we could somehow concentrate the galaxy at the surface of the earth.


—Susskind, Leonard. The Cosmic Landscape. Page 207 (paperback)


Physicists would have us believe that a string strong enough to hold up all by itself an entire galaxy in a gravitational field of one g could somehow be compressed into nothingness by the relatively puny gravitation field of a black hole? I think not. Such a string could be compressed into the space of a single Planck cube, but no more. So our black hole must have a core or kernel at least one Planck cube in size.

Such a black hole kernel would have to have the same amount of mass-energy that the original collapsing star possessed, even though the amount of space it occupies at the center of the black hole may be tiny and hence the density of the kernel material extremely large. But what information do we have that might indicate what the size of this kernel would be?

Leonard Susskind in his book The Black Hole War describes the event horizon of a black hole as containing all of the entropy—and hence, all of the information about the contents—of the black hole. He further states that this entropy will be contained bit-by-bit in little squares the size of Planck lengths occupying ¼ of the area of the event horizon. In other words, the entropy is contained in square Planck units, each 2.6121 x 10-70 square meters in size. That’s a rather small piece of real estate (don’t try to build a house on it!).

So, how many Planck squares will fit on the event horizon of a black hole roughly the size of a solar-mass with a radius of 3,000 meters? The area of a sphere of radius R is 4πR2, so our equation is:

Np = π(9 x 106) / 2.6121 x 10-70

Where Np is the number of Planck squares and the denominator is the size of one square Planck unit. This works out to 1.0824 x 1077 bits of information.


I propose here that the mass of the collapsing star stops shrinking when it occupies Np cubic Planck units at the center of the black hole. A cubic Planck unit has a size of 4.2217 x 10-105 cubic meters. Our black hole kernel will thus occupy Np times 4.2217 x 10-105 cubic meters in a spherical body at the center of the black hole, or 4.5697 x 10-28 cubic meters. The radius of a sphere of volume V is (0.75V/π) so our kernel has a radius of 4.7782 x 10-10 meters.

This is a very small sphere, comprising only about one part in 6.3 x 1012 of the radius of the event horizon. Yes, it is a tiny amount of space at the center of the black hole, and it will have an immense density. But there will be no singularity, and hence no escape of the collapsed material into some other universe—ideas that are repugnant to most physicists, and especially to me.

I have no idea whether this hypothesis, or theory, will prove to be the correct analysis of a black hole in terms of quantum gravity, since as of now this is still an unsolved problem of physics. And it is impossible to see inside a black hole so as to determine exactly what goes on in there.

Leonard Susskind in his book The Black Hole War makes an odd statement on page 235:

The paper that CGHS [an initialism for physicists Curt Callan, Steve Giddings, Jeff Harvey, and Andy Strominger] had written was an extremely elegant mathematical analysis of Hawking radiation, but somewhere in the analysis they had made a mistake, claiming that Quantum Mechanics eliminated the singularity, and with it the horizon.


—Susskind, The Black Hole War, page 235


I wish to point out that the event horizon of a black hole does not depend on the existence of a singularity at the center. An event horizon requires only the presence of a sufficient mass, M, within a radius given by the Schwarzschild radius, Rs:

R
s = 2MG/c2

Where G is the gravitational constant and the other terms have their usual meanings. This equation says nothing about the distribution of the mass, M, within the sphere of radius R
s. It defines the size and mass of a particular black hole but sheds no light whatsoever on the distribution of the matter within the sphere of a non-rotating, uncharged black hole. (Black holes that rotate or have a net electrical charge are slightly more complex than Schwarzschild black holes, but these differences do not affect my conclusions here.)

The inside of a black hole is still terra incognita.


The problem with singularities
But let’s return to the original hypothesis, which is that the matter comprising a black hole will collapse to a point in space—a singularity. There is a problem with this scenario as well. As the collapsing matter reaches a very small amount of space it runs head-on into quantum-mechanical effects. Physicists now generally recognize that space itself is quantized into Planck-sized chunks. A particle, no matter how tiny it may be, cannot move an arbitrarily small distance. It must “disappear” from the Planck-sized quantum of space that it occupied and “reappear” in the next one. In other words, the shortest distance something can move is one Planck length. And the shortest interval of time in which it can make such a move is one Planck-second (I use this term to replace the clumsy term “one Planck unit of time”), which is about 5.4 x 10-44 seconds).

Back in the 1960s I predicted that both space and time would eventually prove to be quantized. But this conclusion was based on “gut physics” and thought experiments, and of course I never published anything about it. Not having any authority in the field, I would have been laughed out of any scientific journal. It is merely a modicum of personal satisfaction that my private prediction subsequently became a reality.3

So our collapsing star runs smack into a quantum limitation long before it becomes a point singularity. The so-called singularity cannot be smaller than one Planck length in diameter. This makes the density of the central mass incredibly large—but it is still finite and the abhorrence of singularities shared by nature, Professor Einstein, and me is satisfied. Huge, but not infinite.

Why is the Planck-second the size that it is? Because it is the length of time it takes light to travel one Planck length, and nothing can move faster than light. So proclaimed the Master, and he has never been proved wrong about that. He may have goofed on quantum mechanics, but his rule about the inviolability of the speed of light has never been proved wrong, or even to have any exceptions. It is easy to see why this should be the case when examination of things from a frame of reference moving at the speed of light shows that the true speed of light is infinite, as I demonstrated earlier.

A photon of light arrives at its destination as soon as it starts. It cannot get there before it starts, because this would violate the direction of the time vector and make a photon into a time machine. Not likely!

In addition to these problems, there is yet another even more fundamental crisis: what happens to the information that goes into a black hole? Leonard Susskind fought a two-decade long “war” against Stephen Hawking, because Hawking insisted that information was destroyed when matter passed into a black hole. Susskind eventually showed that the information was still there, encoded as a holograph in the event horizon of the black hole. He also proposed that the universe itself is a hologram and that the holographic image was encoded in the event horizon at the edge of the universe. But we all know that the “hologram” that is produced by this universal holograph is very real. So also must be the hologram that is produced by the holograph in the event horizon of a black hole. If it represents anything, it cannot be a singularity which has swallowed whole everything that was cast into it. If the matter is gone, the information has gone with it. But Susskind proved that the information is preserved, which agrees with the fundamental laws of thermodynamics (conservation of mass-energy and the law of increasing entropy). All of this would seem to rule out a real singularity at the center of a black hole; it must be simply a mathematical convenience, like the one Newton used to prove his theory of gravity more than thee hundred years ago. 

As for black holes, it remains true that any distribution of the mass M that has three-dimensional symmetry around the center of the event horizon will account for the externally observed features of a black hole with a radius R that is half the diameter of the observed event horizon.

For a black hole with the same mass as the sun, the radius of the inner core is about 4.7782 x 10-10 m. This is very small indeed, but as Susskind would say, “it ain’t zero.” In terms of Planck units this is about 2.956 x 1025 Planck lengths. With this scenario, the collapsed matter (whatever it may be called in such a compressed state) forms a tiny inner core to the black hole, almost as close as one can get to a singularity without actually dipping one’s feet in that icy water.

Restating this theory: The remnant core of a black hole is equal to the number of surface entropy bits compressed into a sphere containing the same number of cubic Planck units located at the center of the black hole. There is a one-to-one correspondence between the entropy (or information bits) in the event horizon and the cubic Planck units that form the inner core of the black hole.


Notes
1. I call him a friend even though I met him only twice and spoke with him only briefly. He had the most engaging smile and made me feel at ease instantly. He was a giant in the field of physics, but he was also a helluva nice guy.

2. The great physicist Isaac Newton made this assumption in order to develop his theory of gravitation, which although superseded by general relativity is still an adequate description of what happens in relatively weak gravitational fields, such as that of the earth.

3. I also predicted about 1990 that eventually all galaxies would be found to have massive black holes at their centers. This would be especially true of the spiral galaxies, and the barred spirals would likely be found to have two massive black holes, rotating about a common center and carrying their galactic star systems with them. It is now widely accepted that (nearly) every galaxy contains a supermassive black hole. King, A. (2003). “Black Holes, Galaxy Formation, and the MBH-σ Relation.” The Astrophysical Journal, Letters 596 (1): 27–29. Chalk up another one for gut physics.


References
Susskind, Leonard. The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. New York: Little, Brown and Company, 2005. Paperback edition, New York: Back Bay Books / Little, Brown and Company, 2006.

—. The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. New York: Little, Brown and Company, 2008.
March 20, 2013

Attaining the Unattainable

How to “defeat” relativity and travel at the speed of light in two easy lessons. These methods are not, however, without their consequences, which are mostly of the rather unpleasant variety.


Method One: Build yourself a photon rocket
Consider the ordinary rocket. The efficiency of a rocket is a function of the speed of the exhaust. You cannot get a better exhaust speed than the speed of light. So we have here another thought experiment.

Thought experiments are great aren’t they? Einstein constructed both his theories of special relativity and general relativity on the results of a couple of thought experiments. Experiments so simple that almost anyone could have thought of them—only nobody ever had before.

So let us assume that we have built a magic photon rocket. I call it a photon rocket because its exhaust consists of photons. This rocket has two magic devices built into it. The first is a perfect parabolic reflector that will reflect any photons no matter how energetic from the focus backward in a perfectly collimated beam. The second magic device directs a beam at the focus of the magic reflector, a beam that will instantly transform any matter located at the focus into the energy which it comprises: mc2. This energy, in the form of photons, is then reflected off the magic reflector and directed backwards—propelling the rocket forward.

We climb aboard and start the rocket engine, a simple (and realizable) device that spits tiny amounts of matter roughly the size of molecules through a tiny aperture and into the focus of the magic mirror. There it is converted into energy and begins to move the rocket. Slowly at first, but then faster and faster. Eventually relativistic effects begin to assert themselves.

When I first devised this little thought experiment, I made a mistake. I realized that as the speed of the ship increased, so would its mass. Its total effective (or inertial) mass would be equal to

m = m0 / (1 – v2/c2)½

where m
0 is the initial mass of the rocket before takeoff. But the mass of the particles emitted to be turned into energy would be increased in the same ratio. It sounded to me like a Mexican standoff. I put this conundrum to several graduate physics students and some physics professors, but none of them could explain the apparent paradox. Then one day as I was crossing a street in Newark, New Jersey, on my way home from Newark College of Engineering, it hit me like a thunderbolt: Of course! Time dilation: I had completely failed to take it into account. Although the energy of the rocket exhaust would keep up with the relativistic mass of the rocket, it would be expelled at an ever slower rate (as noted by an observer on earth). So my imaginary rocket ship could not just blast past the speed of light unrestrained after all. Chalk up another one for the big man. Say what you will, Albert Einstein was a genius of the first rank.

But the big question remains: Will our little rocket ship ever attain the speed of light? The answer is “yes,” but with some caveats. One day in math class at the University of Colorado in Boulder, my mind began to wander while the good professor was explaining some intricacy of differential equations. (Although I didn’t know it at the time, I had attention deficit disorder, so my mind was apt to wander often when it shouldn’t have.) While the professor was blatting away about some partial differential equation, I was quietly working out the integral equation for my photon rocket. It turned out to be quite simple, and the answer was that the ship would attain the speed of light at precisely the instant when the last microgram of fuel was used up.

In this instance the “last microgram of fuel” includes the ship itself and any passengers thereon. Working out the integral equation involved is left as an exercise for the reader. Suffice it to say, however, that there is a much simpler way of discovering what I did. The energy required to accelerate a mass m to the speed of light is exactly equal to mc2—which is, of course, the result I obtained with the integral equation.

As I remarked before, caveats. You’re gone, used up. Call it a Pyrrhic victory if you will.


Method Two: Fall into a black hole
The second method of attaining the speed of light is to fall into a black hole. To explain this method, we first start out with some examples based on ordinary experience. For these preliminary thought experiments, we suppose that our earth does not have an atmosphere. Air resistance muddies up the equations of ballistics and makes them difficult to solve, so we simply ignore it.

Remember the old saying, what goes up must come down? In the earth’s gravitational field, this saying is true unless the object is going straight up at a speed equal to or greater than the escape velocity (in this case it really is a velocity, because it is a vector with direction perpendicular to the surface of the earth), which is about 11.6 km/sec. A rocket that is propelled upward at a speed greater than this will never come down. It will keep going forever—unless it runs into something. To take the inverse of this, what will be the speed of an object coming from very far away (infinity) if it falls squarely on the surface of the earth? The answer is, of course, 11.6 km/sec. For a body large enough to have a strong gravitation field, any object approaching from a great distance and falling on the surface of that body will hit with a speed equal to the escape velocity of the body.

What is the escape velocity for a black hole? By definition, it is the speed of light. The event horizon of a black hole defines a surface from which nothing can escape, not even light itself. An object falling into a black hole from a great distance will attain the speed of light at precisely the instant at which it crosses the event horizon of the black hole.

But this is not all that happens to such an object.


Several things happen as an object approaches the speed of light—as seen from a frame of reference not moving at that speed. First, the length of the object shortens in the direction of its movement. Second, the inertial mass of the object increases. And thirdly, the rate at which time passes slows down. All three of these vary by the same proportion. At a speed of 86.6 percent of the speed of light, the object’s length will be half of what it was originally; its mass will be twice what it was before; and time will pass only half as fast as it did before. At 99.5 percent of the speed of light, the object’s length will be one-tenth what it was, its mass ten times as great, and time will pass at a rate of only one second for every ten (on earth).

Imagine that you are an astronaut who has had the misfortune of falling into a black hole. And imagine that another astronaut is watching you fall into it from the vantage point of a space ship that is at a safe distance from the black hole. To him, it will seem as though you are falling ever more slowly toward the event horizon of the black hole; that you have become immensely heavier than you were; and that you have flattened out like some cartoon character (think Wily Coyote after he hits the bottom of a deep canyon).

You see a lot of nonsense printed about things like this. Nigel Calder, in his otherwise excellent book Einstein’s Universe, writes:

An astronaut falling into a black hole would be stretched and squeezed so strongly that he would become like a long length of spaghetti, even before he reached the surface of the black hole. The … spaghettification of the astronaut give[s] tangible expression to the local curvature of space.


Einstein’s Universe, page 121 (Penguin Edition)


Nonsense! The lengthwise shrinking dictated by relativity would cancel out the curvature of space, so there would not be enough of the astronaut to stretch into spaghetti.

Note, however, one very important fact that emerges from these considerations. It would, in fact, take forever for you to finish your fall into the black hole as seen by your non-falling astronaut companion. This has very important consequences for the physics of black holes in general.


As for you, the falling astronaut, you would be aware of none of these effects. For you the universe would seem to flatten out—including the event horizon of the black hole—and you would simply fall through it. To … where? That my friend belongs in the company of the big questions of philosophy, and I shall make no attempt to answer it. I deal from here on only with what can be seen by observers who are not falling into the black hole.

A black hole allegedly has a singularity at the center of it, a point where all the matter (mass) that went into the black hole collapsed on itself. In point of fact, nobody knows that it looks like inside a black hole. It is literally terra incognita. Nobody has been there. All is conjecture. Actually, there are three different ways that an object with the external properties of a black hole can be formed. I assume here an object of mass M (the capital M to distinguish the mass of the black hole from that of objects with which it may interact), with a Schwarzschild radius of R. Here are the three possibilities:
  • All of the mass M is concentrated at the center in a singularity.
  • The mass M is distributed uniformly in a very small central core.
  • The mass M is contained in a very thin layer at the radius R and the interior is empty.
The first of these is the conventional viewpoint. But Albert Einstein hated singularities. Nature hates singularities. I hate singularities. Singularities Suck!

(I may be crazy; I may be wrong; but I’m not afraid to say what I think!)

The second choice is intriguing, and I shall deal with it in a later blog.

But what about the third one? According to Susskind (The Black Hole War), anything that impacts the event horizon of a black hole is absorbed by it, spreading over the entire extent of the event horizon the way a drop of detergent spreads rapidly over the entire surface of a basin of water. What if the event horizon itself comprises all of the mass contained in the black hole, held in a layer perhaps one Planck size in thickness (admittedly, that’s a SWAG—a scientific wild-assed guess—on my part)? From the outside, it would still behave like a black hole should. All the differences would be on the inside.

In this scenario the material of a collapsing star would, as soon as it has compacted enough to comprise a black hole, begin to migrate to the event horizon, like iron filings attracted to a magnet. The only place where the gravity of the material comprising the event horizon layer was neutral would be the exact, precise center of the black hole. But even so small a particle as a hadron would, sooner or later, wander off center—if for no other reason, because of the Heisenberg uncertainty principle. It would then be instantly attracted to the event horizon and would stick there like a fly on fly paper. Eventually the entire inside of the black hole would be empty. The layer comprising the event horizon layer may be extremely thin, but it is most definitely not a singularity, a mere mathematical point.

If all of the material that formed a black hole condensed to a mathematical point, it would have no existence in this universe. Where would it be? And why would it continue to create an immense gravitational field? Since when do we get something out of nothing? I like my scenarios much better.

Just don’t try to go there!


More relativistic nonsense
Here is yet another example of disinformation concerning Einstein’s theory of special relativity. This is from Nigel Calder’s book Einstein’s Universe:

Many accounts of relativity say, quite incorrectly, that a passing spaceship appears unnaturally squashed or contracted along its length. It does appear foreshortened but only in accordance with the entirely natural perspective of an object seen from an angle.


Einstein’s Universe, page 161 [emphasis in the original]


Poppycock! We need look no further than Einstein’s own book on relativity. There on pages 36-37 of the new edition we find the following discussion regarding a meter-stick located along the x-axis of a frame of reference K that is moving with speed v with respect to another reference frame, K:

But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity v is (1 - v2/c2)½ of a metre. The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod.


Relativity: The Special and the General Theories, pp, 36-37


Now, unless there is some fundamental difference between a meter-stick and a spaceship that allows the latter to avoid the consequences of the special theory of relativity, Calder’s statement is grievously in error. Why he should have made such a blunder escapes me completely. It is inexplicable.

I do not address here the question of foreshortening caused by viewing the passing spaceship at an angle. That is an entirely different matter from the shortening attributable to the spaceship’s speed relative to the observer’s frame of reference, which is presumably somewhere on the earth.

For an illustration of what this foreshortening would look like, see the illustration below. In conformance with the special theory of relativity, an observer on a passing spaceship would see the earth as suffering the contraction of length (in the direction in which the spaceship is moving). This is shown in the bottom picture below.

I define here a new term: “Max,” which is analogous to “Mach” for speeds relative to the speed of sound. Mach 1 is the speed of sound; Mach 2, twice that. Max 1 is the speed of light. Max 0.5 is half the speed of light, etc. So all values of speed expressed as v/c can be called “Max speeds.”

And fools continue to rush in where angels fear to tread.


Rocket--Blog+Composite
Three Spaceships Passing by the Earth


The spaceship on the bottom is traveling at Max 0.1 (v/c = 0.1) so it appears almost normal. Relativity shortens its apparent length by only one-half of one percent. The one in the middle is moving at Max 0.866, and its apparent length is only half of what it would be at rest on the earth. The one at the top is speeding by at Max 0.995—99.5 percent of the speed of light—and its apparent length has shrunk to a mere 10 percent of what it would be at rest.

To observers on the top spaceship, the earth would look something like a discus, with its width only ten percent of its height. See below.

Earth+at+10+Percent.JPG
The Earth as Seen From Spaceship Three


Spaceship three is traveling at Max 0.995 (v/c = 0.995)

March 20, 2013

The Light of Our World

This is an outline paper—a sort of “back of the envelope” development—of what may the most important idea I’ve ever thought of. I had the key to it in 1956 when I was an undergraduate at the University of Colorado in Boulder, taking an honors course under George Gamow, the famous physicist (an early advocate and developer of LeMaître’s Big Bang theory, among other things).

One day I was thinking about the equation in special relativity for the addition of speeds, which is given by:
W = (w + v) / (1 + wv/c2) (1) (Einstein, page 40)1
where W is the combined (forward) speed obtained by adding the two speeds w and v, and c is the speed of light. Something about this equation seemed oddly familiar to me. I soon decided that the speeds involved should be normalized to the speed of light. In other words they should be expressed in some way as fractions of the speed of light; for example, as light-seconds per second, rather than, say, meters per second. With this slight redefinition of speeds, which should have no effect on anything except for the units involved in their measurement, (1) becomes:
Wc = (wc+ vc / (1 + wcvc) (2)
where the subscript c indicates normalized speeds. Since the normalization involves dividing the speeds by the speed of light, c, the term c2 in (1) disappears, although it is still inherent in the resulting equation.

If we now drop the subscript c from (2) we have the simple expression:
W = (w + v) / (1 + wv) (3)
It was this simplified equation that rang a bell in my memory. Compare (3) with the formula for the hyperbolic tangent, tanh, of the sum of x and y in terms of tanh(x) and tanh(y):
tanh (x + y) = ( tanh(x) + tanh(y) ) / (1 + tanh(x)tanh(y) ) (4)
from a book of mathematical tables, formulas, and curves (New York: Rinehart and Company, 1948, 1953, ed. of 1956). It is immediately obvious that the form of (4) is identical to that of (3), which implied to me that there is some connection between hyperbolic functions and the Lorentz Transformation that is the basis of the special theory of relativity.

But what did this mean? I showed a short development of this idea to Professor Gamow, who read it through quickly and then asked me, “So?” Like an idiot, I did not have a ready answer for him. I should have anticipated his question and prepared an answer for it, but I had not done so. I hesitated to pursue the matter with such a distinguished man as Gamow, so I dropped it. I felt that if there were something to the idea, he would have instantly realized it. Perhaps I was wrong about that.

Later, I showed my little “paper” to a physics professor named Wesley Britton, who read it over and nodded, saying, “What you have discovered is the inherently hyperbolic nature of the theory of relativity.”

For many years this idea haunted my thoughts from time to time. I instinctively felt the answer was very simple yet too bizarre to entertain, so I did not pursue it further.

Recently (January 16, 2013) I happened to pick up Einstein’s book on relativity and began to read it over once again. (It has been a favorite of mine since about 1949; an old copy, yellowed with age, dated 1931 by the Henry Regnery Company in Chicago, is still on my bookshelf.) I came across this text on page 34 of Einstein’s book (new edition), where he is discussing the old method of simply adding two speeds to each other to get the result:

This system of equations is often termed the “Galilei transformation.” The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the [speed]2 of light in the latter transformation.


But what if this were literally true, and we simply do not know it?


The speed of light
Nearly everyone with a basic working knowledge of physics knows that the speed of light is about 300,000 kilometers per second. That is what we measure it to be. And all frames of reference, whether moving with respect to other frames of reference or not, yield the same value for the speed of light. Everywhere that we measure it, we obtain the same value.

The question is: what are we really measuring? Is this so-called universal value actually the true speed of light?

If we investigate this question in some depth, we can quickly come to another conclusion. That is, the speed of light is itself dependent in some way on the frame of reference. How can this be so? Consider the matter from the point of view of a photon of light. A photon travels at the speed of light by definition. Like the bus in the movie Speed, the photon cannot slow down or it will cease to exist (if stopped by some quantum interaction, it may become a so-called particle of some kind; it has, in effect, been “imprisoned”—not for speeding but for going too slow!)

Consider travel from a photon’s point of view. Let us suppose that we have a special photon with a tiny clock inside it. Of course that is an impossibility and this is only a thought experiment, but physics is full of such ridiculous things as frictionless pulleys, which are used to simplify problems so they can be solved in simple terms (to demonstrate principles of mechanics, etc.). What does this photon-clock do as the photon travels on its way through the universe? Nothing. Absolutely nothing. The equations of special relativity, which have been proved to be correct in dozens of critical experiments, state that at the speed of light time stops from the reference point of the system of coordinates that is moving at that speed. Our clocks on the earth will keep on ticking for the millions of years we think the photon takes to arrive at some distant point in the universe.

Consider a photon that was emitted from the “surface of last scattering,” at which time, about 370,000 years after the original “big bang,” the decoupling of matter and radiation occurred. This photon was emitted approximately 13.8 billion years ago and arrives at our experimental location on the earth 13.8 billion years later after travelling (by definition) 13.8 billion light-years to get here. The tiny, imaginary sub-atomic clock that it carried “piggy back” all that way will not have ticked off even a single nanosecond in that entire length of (our) time. Should this photon now impact an electron or some other particle and be annihilated, the tiny clock would be thrown off willy-nilly and begin ticking off earth-seconds as any ordinary clock would.

(The light generated by this photon and others like it is the “background” radiation, which represents the “afterglow” of the Big Bang. It was detected by Penzias and Wilson of the Bell Labs in 1964.)

So we have here two clocks measuring what should be the same thing. The one on board our photon, which measured no time at all, and the other here on earth (that earth did not even exist 13.5 billion years ago is a mere detail not worth our bother), which ticked off the seconds in a period of 13.5 billion years—and that comprises a whale of a lot of seconds! (Roughly, 4 x 101717 seconds.)

But which clock is correct? The answer is: both. They are simply measuring the passage (or non-passage) of time in different frames of reference that are moving with respect to each other at the speed of light. The answer to the conundrum lies in that simple fact.

A photon can go anywhere it wants to in no time at all. That statement would seem to satisfy the definition of an infinite speed. And that, in fact, is what I believe to be true. The true speed of light is infinite, but all we can see of it is the projection of that infinite speed on our four-dimensional frame of reference here on the earth.

This is where the hyperbolic functions come in.


Hypothesis
The true speed of light is infinite, and what we measure within the known universe is merely the projection of that infinite speed on our finite space-time continuum.


Development
First, methods of measurement. We start with the proposal that all speeds (or velocity magnitudes) be measured in terms of the speed of light (as observed). In other words, instead of using meters or kilometers, we use light-seconds or light-years. This proposal was first made by the author and a senior colleague, Kenneth A. Norton, at the National Bureau of Standards in Boulder, Colorado, in the 1960s. It had the great advantage for us (at the time) of removing the uncertainty in the measured value of the speed of light from all equations involving light or radio wave propagation. It now becomes useful in working with relativistic effects.

Systems of rods and measures, even those proposed by the good professor in his book on relativity, are really quite useless. Measuring rods change as soon as they begin to move. They shrink. Imperceptibly at first, but then more and more as the speed of the frame of reference containing them increases. In the limit, as the speed of the reference frame reaches the speed of light, even the so-called standard meter will vanish, its length reduced to nothing by the effects so well predicted by the special theory of relativity. A vanishing standard meter: not of much value, that.

There is only one quantity that does not vary: the speed of light itself. It forms an invariant limit to the speed that any particle or material object can attain.


The world we really live in
We are all accustomed to thinking that we live in a three-dimensional world, with up and down, left and right, forward and backward as directions in which we can move, and with time inexorably ticking off the seconds everywhere—the great leveler of human endeavors. We set our watch to the clock in the tower, and when we return later—if both our watch and the clock in the tower are faithful timekeepers—the time on our watch agrees with the time on the tower clock.

But all is not what it seems. We live in a four-dimensional world, and though many of us give lip service to this concept, few of us realize that it is literally and inescapably true. We speak of “now,” but now never exists except perhaps at one point—within our own minds. Everything around us we see through the lens of time. This fourth dimension controls everything we can see, everything we can experience.

We gaze at distant objects and see them as if in a strange looking-glass. We can see nothing that is not in the past. Allow me to demonstrate.

Back in the 1960s when I was working with a group that was measuring the propagation of radio waves, we developed a convenient yardstick: Light travels one foot per nanosecond (one-billionth of a second). When you look at a building that is 1,000 feet (about 300 meters) away, you are seeing it not as it is “now” but as it was one microsecond ago. If you observe a tall mountain that is 100 miles away (you need a clear day for that!) you are seeing it as it was some 530 microseconds in the past. If you see the sun rising above the horizon you are seeing it as it was 8 minutes ago. It could have flared into a supernova 4 minutes ago and you would not know it for another 4 minutes (not a likely event, please be assured). If you are lucky enough to spot the galaxy of Andromeda (visible to the naked eye under good sky conditions), you are seeing it as it was some 2.5 million years ago. Everything that you can see around you is seen through the lens of time: the fourth dimension. Nothing is as it seems to be.

Let us conduct yet another thought experiment. This time we shall use a couple of clocks, very accurate, so-called atomic clocks, clocks that can measure time to the nanosecond.

Imagine that you have two such atomic clocks that measure time to the nanosecond, and that these two clocks can communicate with each other (they are very sophisticated). You can synchronize these clocks by pushing a button on one of them, causing it to synchronize with the time given by the second clock. You do that. Then you carry the second clock to a bell tower about 300 meters (1,000 feet) away and leave it there.


Returning to your original position, you check your clock and discover to your annoyance that it is no longer in sync with the second clock, which is now sending signals from the bell tower. You press the synchronize button, and all is well once again. But your curiosity has been aroused by this apparent shift in one or the other clock, so you decide to carry your clock to the bell tower and check things again.


But as you begin to walk toward the bell tower, you notice something very strange. The second clock seems to be gaining time as you walk toward it. This distresses you, because these clocks are supposed to be extremely stable and should not gain or lose even one nanosecond in hundreds of years. Yet the closer you approach the bell tower, the more the second clock gains on the one you are carrying. In fact, when you reach the second clock in the bell tower you find that it is fully 1,000 nanoseconds fast as compared with the other one.

You watch the two clocks for about five minutes, but neither one gains or loses time with respect to the other. It seems impossible to you that the mere act of walking could have affected one of the clocks. You are correct. That is not what happened. When you were at your original position, the signals from the second clock in the bell tower were delayed by 1,000 nanoseconds (one foot per nanosecond), so when you received the other clock’s time signal of t = 0, it was already showing t = 1,000 (nanoseconds). You synchronized your clock to the wrong time, because you did not take into account that the clock in the bell tower had now retreated to a time that was 1,000 nanoseconds in your past.

Such are the vicissitudes of life in a four-dimensional world.


Relativistic speeds
The escape velocity from the earth is approximately 11.6 km (kilometers) per second. This forms an appropriate upper value for the speeds currently attained by man-made devices, such as rockets or missiles. Taking 12 km/sec as a good round value to represent such speeds, what does this amount to in light-seconds per second? Dividing 12 by 300,000 we get 4 x 10-5 or 4 parts in 100,000.

Assuming for the sake of argument that the projection of “absolute” (the meaning of this term will be made clear later) speed, s, in our frame of reference is given by v in the equation:
v = tanh (s)      (5)
What does this imply about speeds attained by man-made devices? The Taylor expansion for tanh is:
tanh (s) =  s - 1/3 (s3) + 2/15 (s5)  - …          (6)
or: tanh (s) =  s [1 - 1/3 (s2) + 2/15 (s4) - … ]      (6a)

It should be immediately obvious that the second term in this series is almost entirely negligible for speeds in the neighborhood of 12 km/sec, since (in 6a) it becomes 16/3 x 10
-10, or about 5 parts in one billion. For all practical purposes tanh(s) = s for s = 4 x 10-5. In other words, these relativistic effects disappear for the speeds attained by man-made objects, because they become vanishingly small. (In the same manner, Einstein concluded that ordinary Newtonian mechanics is accurate for most of the movements of heavenly bodies, such as the moon and planets. It forms a first approximation to the more accurate relativistic equations that apply.)

But what about speeds attained by very high speed particles, or of light itself? There a very different situation obtains. For photons, which travel at the speed of light, the normalized speed is one, because v = c and by definition the normalized speed is v/c. This implies from equation (5) that:
tanh (s) =  1 (7)
Where s is the “absolute” speed and 1 represents the normalized speed of light. Solving for the variable s we find that:
s = arctanh (1) (8)
The arc-hyperbolic tangent of one can only be approached in the limit as the argument approaches 1, since the expression cannot be evaluated at x = 1. The answer is, of course, infinity. The arc hyperbolic tangent of one is infinite. This is what we set out to show, Q.E.D. The true, “absolute” speed of light is infinite.


Life in the fast lane
A recent trend in physics has been to eliminate the use of the term “rest mass” for the mass of a particle or other object that is at rest with respect to the frame of reference of the observer. The term “mass” is now used as a synonym for “rest mass” and any other kind of mass is evidently dismissed out of hand. I believe this is a big mistake. Nowadays we read things like this:

Photons are particles of light, and as Einstein explained, light always moves at the speed of light. A photon can never be brought to rest; instead of slowing down, it would just disappear. Thus the mass of a photon is zero. Any particle that travels at the speed of light is said to be massless.


—Susskind, Leonard, The Cosmic Landscape, page 103


Of course, the “mass” that Susskind is referring to here is what we used to call “rest mass,” and the word “rest” in Susskind’s quotation above applies to the observer’s frame of reference (otherwise the term “rest” has no meaning). If a photon had any rest mass, it would have infinite mass while moving at the speed of light. So it obviously has none. I designate so-called rest mass as m0.

But a photon is definitely not massless. It has an amount of energy associated with it that is given by
E = hν (9)
where h is Planck’s constant (about 6.6 x 10-27 erg-seconds) and ν is the frequency of the radiation in hertz (a photon is a quantum of electro-magnetic radiation).

Einstein’s famous equation E = mc2 still holds true, even for photons, so we have:
hν = mc2        (10)
Solving for the mass, m, we get
m = hν / c2        (11)

True, this is an infinitesimal amount of mass for any reasonable value of the frequency, ν, but it is also most definitely greater than zero. A photon can be said to have mass only by virtue of its energy, which is—at least in part—owing to its speed of movement. As Dr. Susskind correctly points out, a photon cannot slow down without losing its identity.


If it could attain the speed of light, a particle of mass m
0 would have infinitely more mass than it did when it was at “rest” (whatever that really means). But if that “rest mass” is zero, then we have the situation where the mass at the speed of light is zero multiplied by infinity. This is an indeterminate value, a value that in this instance is given by (11) for photons.

The name “rest mass” is itself a contradiction in terms, since “rest” must refer to some known frame of reference. I suggest that a better name for the rest mass is “proper mass”; although the reader may find some other name preferable. Perhaps “essential mass” would be even better. The name “mass” should be reserved for that mass which is conserved by the principle of the conservation of mass-energy, the mass that is implied by the equation E = mc2. Nothing exists that does not have this property of mass.

The ultimate speed of light as derived in equation (8) is infinite. But the proper mass of photons is zero, so there is still no contradiction. Zero times infinity is always an indeterminate finite quantity:  0 v   ∞.


Room at the top and at the bottom
So how much mass does a photon possess? For run-of-the-mill photons, the answer is: not very much. In quantum electrodynamics, the electron volt, or eV, is a convenient unit of energy. One eV is about 1.6 x 10-19 joules, or 1.6 x 10-12 ergs. Photons of visible light have energies in the range 1.6 eV (red light) to 3.4 eV (violet light).  When substituted for hν in (11) we find that the mass of 2.5 eV photons is on the order of 4 x 10-30 mg (milligrams). An ordinary aspirin tablet has a mass of 325 mg, so 2.5 eV represents a tiny amount of mass.

If we want to find some fairly “heavy” photons, we have to look at the cosmic rays. These are very energetic photons. When I was taking physics at the University of Colorado in Boulder, in 1955-56, we did an experiment to detect cosmic rays. This involved some apparatus called a “coincidence detector” mounted amongst a pile of bricks in a high tower of the physics building. It was supposed to detect only particles that went straight down from the top to the bottom of the apparatus. But during the two weeks or so that we operated this equipment, we noticed that a few times it recorded a particle that apparently came straight up from below and passed through our coincidence detector backwards. These were rare cosmic rays that were so energetic that they passed completely through the earth without being stopped by anything. So much for “solid” objects (Einstein’s “ponderable bodies”).

The most energetic cosmic ray that has ever been detected is the so called “Oh-My-God particle,” a cosmic ray photon that measured 300 EeV (exa-electron volts), or 3 x 1020 eV. This “heavyweight” came in at 2 x 10-11 milligrams. That’s still not very much, but it is certainly a lot more than zero. Something like that makes you feel as though you’re walking around with a target painted on the top of your head. I certainly wouldn’t want one of those 300 EeV cosmic ray photons shooting through my poor old head. It might knock the last little bit of sense out of me, and then where would I be?

But the real point of this discussion is that all photons are scooting around in their own private frame of reference, which moves at the speed of light with respect to any other frame of reference (bar none), and they move freely at infinite speed in that cozy environment. In fact, since photons have no concept of time, nor are they affected by time, they could be said—in a certain sense—to be everywhere at once. Eternal, without beginning or end. Sort of like God. (Oops! Did I really say that? Scratch that and replace “God” with … what?)

No, I’m not trying to say that God exists (I may believe it, but that’s my point of view, in my own private frame of reference), nor am I arguing for anything like “intelligent design.” Leonard Susskind did a better job of destroying that myth than I ever could do, so let’s just leave it at that.

What I am saying here is that there is a frame of reference somewhere that is moving at the speed of light with respect to our pitiful little human frames of reference (such as galaxies with a hundred or so billion stars), and that everything that happens there is projected onto our frames of reference by the hyperbolic functions that are inherent in the theory of relativity (at least the special theory, and probably the general theory as well).

We have here the ultimate frame of reference. But instead of it being a reference frame “at rest” (a meaningless term under relativity theory), it is a frame of reference that moves at the speed of light with respect to any reference frame within the universe itself. We have been looking in the wrong place for over a century, looking in vain for a reference frame within which to define everything else. Now, at last, we have one that cannot be denied—a reference frame for all ages.


The ultimate, absolute frame of reference
Throughout the history of physics and astronomy humans have sought to find some benchmark, some ground, from which to measure and evaluate things. From the earth as the center of the universe, this search branched out—almost desperately—until, around the time of Einstein’s theories, we settled on something we called “the fixed stars.” What nonsense! There is no such thing as a “fixed star.” Fixed with respect to what? Even so astute and far-ranging an intellect as Albert Einstein felt compelled to embrace this concept as a group of stellar bodies so arranged that they somehow formed a “base” from which to evaluate other objects. With his intuitive knowledge of the relativity of all frames of reference, he should have known better.

We all should have known better. But it turns out that all this time we have been working things backwards. In fact, we have been looking at the universe upside-down, as it were. The fixed frame of reference that we have needed to anchor everything was there all along but we did not see it: a frame of reference that moves at the speed of light. With respect to that frame of reference, everything else is rock solid. Unmoving. It was not a zero-based reference system that we needed to explain our theories, it was one that moved faster than anything else in the universe. The “fixed stars” were a myth. The “ether” was a myth. Space itself was a myth—and so the good Professor Einstein stated in his Appendix V: Relativity and the Problem of Space. He wrote:

There is no such thing as an empty space, i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field. … Thus Descartes was not so far from the truth when he believed he must exclude the existence of empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. It requires the idea of the field as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes’ idea; there exists no space “empty of field.”


—Einstein, Albert. Relativity, pp. 157-158


Everything we see, everything we measure, every theory we propose, all are seen as projections from the infinity of the speed-of-light reference frame on our own space-time fields and constructs.

This is relativity with a vengeance.

Thank you, Albert. We owe you more than you possibly ever were aware of. It may seem presumptuous for me, a lowly physicist who could not solve a tensor equation to save his life, to address you as an equal, yet I do so nevertheless. I only wish I could have known you in person. I suppose I should be content to have known Richard Feynman and George Gamow during my brief life. You were all giants in the field of physics.


Notes
1 All references to relativity theory in this outline come directly from Albert Einstein himself, in his book Relativity: The Special and the General Theory (see below).

2 The book has the word “velocity” at this point, but velocity is a vector quantity and Professor Einstein is here speaking of a scalar. The word used may have been a mistake in translation by someone who did not appreciate the difference between vectors and scalars.


Bibliography
Einstein, Albert. Relativity: The Special and the General Theory. Authorized translation by Robert W. Lawson; New York: Tess Press, date not given in the book, but probably 2005 since it is identified as a 100th anniversary edition.

Susskind, Leonard. The Cosmic Landscape. New York: Back Bay Books, 2006



Note: The further development of this theory is left as an exercise for the reader. Now doesn’t that sound like a physics professor?
March 11, 2013

Strangers in a Strange Land


What kind of world do we live in? This is not a casual question, and the answer is neither simple nor easy to comprehend. Let’s begin by describing what our world—our universe—looks like to us and go on from there. Scientists discovered in 1964 that we are surrounded by the afterglow of the Big Bang that started our universe some 14 billion years ago. This afterglow is called the cosmic microwave background, or CMB. It comes to us from almost the dawn of time, probably some 370,000 years after the Big Bang (in its original meaning). It glows with the heat of a black body at a temperature of 2.725 degrees Kelvin, which is very cold indeed. It started out with a temperature roughly the same as that of the surface of our sun—around 3,000 degrees Kelvin. But it has been red-shifted nearly out of existence by the expansion of our universe since that long ago time.

And it is here that the paradox confronts us. We are seeing the CMB as it was almost 14 billion years ago, when it was a spherical surface with a circumference of some 2.3 million light-years. But it appears to us that we are inside this sphere of radiation and that it has a circumference of about 88 billion light-years: 38,000 times as large as it actually was all those 14 billion years ago. How can this be?

The answer to this lies in the nature of the space-time continuum implied by the theory of general relativity developed by Albert Einstein almost 100 years ago. Our “mentor,” Leonard Susskind, in his book The Cosmic Landscape, discusses this very subject on pages 310-312. He writes of our universe as a “bubble” in the megaverse, a bubble that expands at the speed of light at its outer boundary (he writes “almost” the speed of light, but I believe it is actually at the speed of light). Susskind adds:

One would expect that an observer inside the bubble would experience a finite world that at every instant is bounded by a growing wall. But that’s not at all what he sees. The view from inside the bubble is very surprising. [Earlier] we encountered the three basic kinds of expanding universes: the closed-and-bounded universe of Alexander Friedman, the flat universe, and the infinite open universe with negative curvature. All the standard universes are homogeneous and none of them has an edge or wall. One might think that an inhabitant in the interior of a bubble would observe the expanding domain wall and conclude that he didn’t live in any of the standard universes. Surprisingly, this is incorrect: that inhabitant of the bubble would observe an infinite open universe with negatively curved space! How a finite expanding bubble can look like an infinite universe from the inside is one of those mysterious paradoxes of non-Euclidian Einsteinian geometry. [Emphasis in the original.]


—Susskind, Leonard. The Cosmic Landscape, pages 310-311 (paperback)


To understand this paradox, Susskind invokes the example of the Mercator projection of the surface of the earth in which Greenland appears to be larger than South America. I believe, however, that the world I call “Flatland” is an even better example, because some of the features of our actual universe can be demonstrated in that model.

sphere
We start with a view of the spherical universe inhabited by the Flatlanders. They live inside the surface of this spherical universe, and for them there are only two dimensions: right or left, and forward or backward. There is no “up or down” as there is for us in our three-dimensional space continuum. A model of this spherical universe looks like this wire-frame model. The lines going from the top to the bottom of this spherical model are important. They are called “geodesics.” A geodesic is the shortest distance between two points, anywhere. We are familiar with them from the term “great circle route,” which describes the path taken by long-distance aircraft in order to get from one place on the earth to another. Plot one of these routes on a Mercator projection and it looks anything but straight. But on a spherical surface—and the earth is almost a perfect sphere—the shortest distance between two points opposite each other on the sphere is any line connecting them. Yet remember that the Flatlanders cannot see any of this spherical geometry because to them their world looks as flat as a billiard table. They see things along geodesic lines, as must any being looking at things that are shown to him by rays or beams of light—in our universe as well as in Flatland.

flatlanders
So what, exactly, does the Flatlander see when he looks at his universe? He sees something like the drawing on the left, which is a projection of his world on a flat piece of paper. In this admittedly crude drawing, the geodesics along which the Flatlander must view his universe are the straight lines radiating out from the center. The length of these lines should equal the length of the geodesic lines on the wire model of the spherical universe shown above. Assuming the age of the Flatlander’s universe to be the same as ours, those lines are about 14 billion light-years long. The circumference of the circle established by these radiating lines is 88 billion light years, just as it is in our universe. Notice also that no matter what kind of experiment the Flatlander devises, he will conclude that his universe is a flat, Euclidean space. He will not see, and he cannot detect, the curvature of these geodesic lines that is imposed by the shape of his universe. That shape is impossible for the Flatlander to even imagine, because he has no way of envisioning a three-dimensional world any more that we have a way of imagining a four-dimensional world. Our minds are simply not “wired” that way, as Leonard Susskind remarks in several places. Not only this, but the Flatlander sees the CMB of his universe near the outer boundary of this circle, which is “running away” from him at the speed of light—and the CMB, nearly as fast as that.

Susskind goes on to describe negatively curved space—as opposed to our Flatland, which is a positively curved space.

The same is true if we try to flatten a negatively curved surface so that it can be drawn on a plane. It’s not easy to draw such a space, but fortunately, a famous artist has already done the work. The artist M.C. Escher’s famous woodcut “Limit Circle IV” is nothing but a uniformly negatively curved space drawn on a flat piece of paper. All the angels are the same size, and so are all the devils. They can roughly be thought of as galaxies. But to flatten the space, the center has to be stretched, and the distant parts must be compressed.


—Susskind, Leonard. The Cosmic Landscape, page 311.



escher
Really, the central part is actual size, and only the components away from the center are compressed. See how this looks? And guess what kind of curve describes the compression of objects as they approach the outer limit of the circle? Right you are: it is our old friend the hyperbolic curve. Also, notice that if you start in the middle where the feet of the three angels meet, geodesic lines radiate out to the limits of the circle as very straight lines, just as they did in the case of the spherical universe projected onto a plane circle. The difference here is that the number of angels and devils increases without limit as the outer edge of the circle is approached. Susskind writes that “the distance from the center of the space to the boundary is infinite. An infinite number of devils (or angels) must be crossed in order to get to the edge. Because each devil is the same size as all the others, the distance is also infinite. Nevertheless, the whole infinite space appears as the interior of a circle when it is flattened onto a plane. With this in mind it is not so difficult to picture the infinite geometry fitting into a finite bubble.” (The Cosmic Landscape, page 312.)

Nevertheless, we are limited as to what we can observe by the CMB, which mocks us in a cosmic version of “you can’t catch me.” No matter where we go, the CMB will always appear to be some 14 billion light-years away from us. And no matter if our universe is infinite: anything that is more than that same 14 billion light-years away from us cannot be part of our universe because it would be older than our universe. Time began at the
instant of the (original) Big Bang, some 370,000 years before the CMB was created by the surface of last scattering. Nothing in our universe can be older than that.

Getting back to Escher’s woodcut, I scaled the distortion of the images from the center of the picture to its outer edges. I did this from the full size reproduction in a coffee-table edition of Escher art, in which Limit Circle IV had a diameter of 216 mm. (roughly 8.5 inches).

Graph

The results are shown in the figure above. In this hand-drawn graph the measurements taken from the Escher drawing are shown as round dots. You can see from the smooth curve drawn through these points that even so great an artist as Escher was not perfect. But then, who is? The curve does show an unmistakable resemblance to a hyperbolic curve as I suggested. But it is not at all clear exactly what hyperbolic function (or some combination of functions) would fit this curve. The answer is likely simple, but often the easiest things will elude us. So far, the best function that fits the points taken from the Escher woodcut is a rather simple function of the proportion between the distance from the center and the radius of the circle:

y = 0.82 / (1  -  x )

where x = distance from the center divided by 108. This function is plotted in red on the left-hand side of the graph above. It matches the data plotted from the Escher woodcut except for the outlier mentioned earlier, for which it gives a value of 11.1 instead of the plotted value of 14.9 (although 11.1 actually matches the curve as drawn rather well), and the area around 50 mm, which was probably plotted too low because no distortion data were available within the inner angels. The top of the curve is also off a bit because of the uncertainty in reading values from the woodcut near the outer limit of the circle; for this reason I do not believe that this discrepancy is significant.

Leonard Susskind continues from the sentence quoted earlier and adds the following observations, which seem especially cogent:

What is especially strange is that if the astronomer wanted to study the expanding domain wall, he would always find it infinitely far away. Interior to the bubble, the geometry of space is unbounded, despite … that at any moment … an exterior observer sees the bubble as a bounded sphere. It’s not that an astronomer inside the bubble can’t detect light coming from the domain wall. But that light does not seem to be coming from a boundary of space; rather, it seems to come from a boundary of time—from what appears to be a Big Bang taking place in the past. This is a most paradoxical situation, an infinite expanding universe inside a finite expanding bubble.


Knowing that we live in an open, negatively curved universe would be a strong reason to believe that our pocket universe evolved from some point in history during which it was a bubble in an exponentially expanding space. That seems like a clear prediction, but it may be impossible to confirm. The observable universe is just too big, and so far we have seen only a tiny part of it. We simply don’t see enough to know if it is curved or flat.


—Susskind, Leonard. The Cosmic Landscape, pages 212-213 (paperback)


A few comments about this most intriguing discussion of Susskind’s: Some evidence for our universe (Susskind’s “pocket” universe) being open and negatively curved exists in the inherent hyperbolic nature of Einstein’s relativity theories. And can it be that Dr. Susskind has in this passage described a sort of complementarity principle for “bubble” universes? The paradoxical nature of such a universe as seen from within and without closely parallels the similar nature of black holes as seen from within and without. And on page 371 of The Cosmic Landscape Susskind adds these comments: “If we are very, very lucky, the largest lumps in the CMB might date to a time just before the usual Inflation got started…. Indeed, there is some evidence that the very largest lumps are weaker than the others. It is a long shot, but those large-scale density contrasts could have information about the formation of our bubble from a previous epoch with a large cosmological constant. … If we are that lucky, then the Inflation did not go on long enough to wipe out evidence for the curvature of space. Here again bubble nucleation has a distinct signature. If our pocket universe was born in a bubble-nucleation event, the universe must be negatively curved.” (Emphasis in the original.)

The space depicted by M.C. Escher’s “Limit Circle IV” is a negatively curved space. Personally, I would be inclined to bet a goodly sum that our universe will eventually be found to be open, infinite, and negatively curved. But then I’m not a betting man.

Of course, I’m not at all convinced that this “Inflation” actually took place—at least, within our own “pocket” universe. That it occurred in the case of the megaverse is all but certain. If not, where the deuce did it come from? Nevertheless, Susskind’s comments are applicable. If some of these “large lumps” are artifacts from the time prior to the surface of last scattering they could contain information pertaining to the curvature of our universe.


All of us see strangers every day, others who move among and around us, whom we do not know and have no contact with: they are strangers to us. But none of them is as strange as the world we all inhabit. Our world nearly defies description and certainly is beyond the capability of our minds to grasp and understand. Yet it seems perfectly ordinary and normal as we move about it in our daily actions. How many of us ever take the time to try to comprehend the true nature of our universe?

Strangers in a strange land, indeed!

A black hole seen from the inside
As paradoxical as it seems, living in our universe is something like living inside a black hole. Except this black hole is “inside out.” The spherical boundary of our “bubble” as described by Susskind is running away from us at the speed of light. In my version of cosmology, this boundary has been expanding at the speed of light ever since the Big Bang some 14 billion years ago. This makes the boundary an event horizon, just like the one enclosing a black hole. But this event horizon is expanding without limit. We would see it from the inside if we could see it at all. We cannot see it, however, because our view of it is cut off by the cosmic microwave background—the CMB—which forms an impenetrable wall of light at almost the distance of the event horizon. The CMB was formed when our universe was only about 372,000 years old, which means that it is closer to us than the event horizon that forms the outer boundary of our universe by only about one part in 38,000 of the distance to that surface. This is a mere 0.0026 percent of the 14 billion light-years. Almost not worth mentioning. For all practical purposes, when we observe the CMB, we are seeing our universe at the time of its birth.

The apparent temperature of the CMB has been reduced by the red shift at the incredible distance of some 14 billion light-years from roughly 5,000 degrees Kelvin to a tiny 2.725 degrees Kelvin. Something that should look as bright as our sun—spread over all of the space around us—has been reduced by the red shift to a wavelength so long that it isn’t even in the infra-red any more; it is so far from visible light wavelengths that it takes very sensitive radio receivers even to detect its presence. In fact, that is just how Penzias and Wilson first discovered the CMB. They were testing a sensitive radio horn receiver at Holmdel, New Jersey, and were annoyed by a constant noise they were picking up on their equipment. It turned out to be the CMB that was making all the “noise.” If anyone ever picked up a Nobel prize by accident, it was these two. That does not diminish the enormity of their discovery. It was confirmation of the accuracy of the Big Bang theory of the origin of our universe, something that had been a bone of contention amongst cosmologists for a couple of decades before the discovery by Penzias and Wilson settled the matter for once and for all.

When I say that the CMB should look as bright as our sun, you might think that surely it would have cooled off a great deal during the past 14 billion years. Of course it would have. But remember—we are seeing it as it was 14 billion years ago, when it did have a temperature of some 5,000 degrees Kelvin. It is only the relativistic red shift that saves us from a blazing hot night sky. Think about that one the next time you are outside at night and look up at the stars (however many of them you can see).

Is this a strange universe we inhabit, or what?

A frame of reference that is at rest?
In his theory of special relativity, Albert Einstein shows that no frame of reference can be presumed to be at rest with respect to other frames of reference. In other words, there is no meaning in saying that such and such a body is “at rest.” The good professor would immediately respond with, “At rest with respect to what?”

Nevertheless, it appears that we may now actually have a frame of reference that is at rest with respect to the objects in our universe. The work of NASA projects COBE (Cosmic Background Explorer, also known as Explorer 66, a satellite dedicated to cosmology and measurements of the CMB) and WMAP (Wilkinson Microwave Anisotropy Probe) has revealed that the CMB radiation is subject to a shift in frequency caused by the motion of the solar system through space.

The data that show this motion of the solar system (the speed of the earth as it goes around the sun is small compared with the combined motion of the Milky Way galaxy and the sun as it orbits the center of our galaxy1) are revealed in this series of pictures made by the COBE and WMAP projects. A discussion of these images by NASA personnel follows.


cobe
“In the comparison of the images [at the right], images on the left produced by the COBE science team show three false color images of the sky as seen at microwave frequencies. The images on the right show one of our computer simulations of what the WMAP experiment detects. Note that WMAP detects much finer features than are visible in the COBE maps of the sky. This additional angular resolution allows scientists to infer a great deal of additional information, beyond that supplied by COBE, about conditions in the early universe.

“The orientation of the maps is such that the plane of the Milky Way runs horizontally across the center of each image. The top pair of figures show the temperature of the microwave sky in a scale in which blue is zero [degrees] Kelvin (absolute zero) and red is 4 [degrees] Kelvin. Note that the temperature appears completely uniform on this scale. The actual temperature of the microwave background is 2.725 [degrees] Kelvin. The middle image pair show the same map display in a scale such that the blue corresponds to 2.721 [degrees] Kelvin and red is 2.729 [degrees] Kelvin. The “yin-yang” pattern is the dipole anisotropy that results from the motion of the Sun relative to the rest frame of the cosmic microwave background. The bottom figure pair shows the microwave sky after the dipole anisotropy has been subtracted from the map. This removal eliminates most of the fluctuations in the map: the ones that remain are thirty times smaller. On this map, the hot regions, shown in red, are 0.0002 [degrees] Kelvin hotter than the cold regions, shown in blue.


“There are two main sources for the fluctuations seen in the [bottom maps].
  • Emission from the Milky Way dominates the equator of the maps but is quite small away from the equator.
  • Fluctuating emission from the edge of the visible universe dominates the regions away from the equator.
  • There is also residual noise from the instruments themselves, but this noise is quite small compared with the signals in these maps.

“These cosmic microwave temperature fluctuations are believed to trace fluctuations in the density of matter in the early universe, as they were imprinted shortly after the Big Bang. This being the case, they reveal a great deal about the early universe and the origin of galaxies and large scale structure in the universe.” [I added the term “degrees” to make the text readable by ordinary mortals.]

Aside from the extreme sensitivity of the measurements—the difference between the red and blue areas on the bottom two maps is only two ten-thousandths of a degree Kelvin—the most interesting part of this discussion is about the “yin-yang” pattern of red and blue in the middle two maps, which is caused by the motion of the Sun (i.e., the solar system of which we are a part) relative to the CMB. The CMB evidently comprises a reference frame that is “at rest” compared with the other objects in the observable universe. This is precisely the sort of at-rest reference frame that the theory of relativity says should not exist. Yet, there it is.

The cosmic microwave background is not only a paradox—an inside out universe, so to speak—but is also the at-rest frame of reference that physicists had long ago abandoned hope of obtaining. Any object—whether a planet, a star, or a spaceship—that measures no dipole anisotropy when looking at the CMB in fine detail is necessarily at rest in the universe. This is, of course, aside from the expansion of our universe that is continually taking place, although one would be hard put to design an experiment that could measure any changes owing to this expansion. In terms of the length of a human life, the expansion of the universe is extremely slow—even though the outer limits are expanding at the speed of light. I doubt that Penzias and Wilson had any idea of what a Pandora’s Box they were opening when they first discovered the source of the noise in their microwave radio receiver.

Is there a Nobel prize for confusing other human beings?

Kelvin, Kelvin, wherefore art thy degrees?
There is a large number of stupid things that have been decided in recent years either by organizations or by groups of individuals with a common occupation. One of these was depriving Dr. John Langdon Down of his discovery of the syndrome formerly known as “Down’s syndrome.” The National Institutes of Health decreed in 1975 that “eponyms” should be abandoned. Presumably they did this because “Down’s syndrome” was slightly harder to say than, e.g., Alzheimer’s disease.

In the field of physics two equally stupid things have been decided, more or less by common consent, in the past fifty years or so. One of these was to abandon the term “rest mass” and replace it with “mass,” which then deprived everything that did not have a rest mass of any mass at all. This is, of course, sheer nonsense. Anything that exists has some energy, and so by Einstein’s famous equation E= mc2, it has mass. A photon of light has no rest mass, but it has a definite mass and so “falls” when passing through a gravitational field, just like anything else. Of course, it goes by so fast that it does not drop very much as it passes by even a massive object like our sun. General relativity also predicts that photons will be deflected approximately twice as much as the Newtonian equations of gravity would require. But this deflection can be and has been measured quite accurately in recent years.

The second stupid idea was to drop the term “degree” from “degrees Kelvin.” Never mind that the entire public does not understand the idea of a temperature without some kind of “degrees” relating to it. What does “2.725 Kelvin” mean to you? To me, it means nothing at all without a qualifier. So I have added the word “degree” or “degrees” to every mention of a Kelvin temperature in my articles. If an object has a temperature of 100 degrees Celsius, then it has an absolute temperature of 373.2 degrees Kelvin (zero on the Celsius scale being 273.2 degrees Kelvin above absolute zero).

By the way, I use the term “gravitational field”—as did Einstein—to represent the distortion of space-time caused by gravity and not in the sense of a Newtonian gravitational field.

End of rant.

Notes
1. The earth moves around the sun at an orbital speed of about 30 km/sec, whereas the sun orbits our galactic center at about 220 km/sec, and the Milky Way galaxy itself moves through space at about 600 km/sec. The motion of the sun in its orbit around the galactic center is almost at right angles to the motion of the galaxy through space, so when added as vectors the total for the average speed of the earth through space is about 640 km/sec—more than 20 times the earth’s orbital speed around the sun.

March 4, 2013

The Cosmology of Our Universe


First off, why do I say “our universe”? Because I want to differentiate it from all of the other universes that may inhabit the megaverse—Leonard Susskind’s term for it. He prefers that name to multiverse, likely because the term “multiverses” can just as easily refer to the universes that are contained in the megaverse. I agree with him. So I shall use megaverse throughout this article wherever it is necessary to mention it.

Ground rules
If we are to discuss the origins and growth of our universe (i.e., the cosmology of it) intelligently, we need to set down a few ground rules. There are not many of them, and they are not complicated.

1) The speed of light cannot be exceeded between any two frames of reference in our universe, no matter how far apart they may be;

2) Our universe began with a “Big Bang” roughly 14.1 billion years ago and has been expanding ever since then.


3) Occam’s razor shall be our guiding principle whenever difficult or obscure choices have to be made. Briefly put, Occam’s razor says that the solution with the fewest assumptions, or which is the simplest among competing theories, is the best one.


The first rule conforms to Albert Einstein’s theories of relativity (both the special and general theories). No one has ever demonstrated that anything can exceed the speed of light in our universe. I shall not even attempt to argue with such a great man as Einstein, and I see no reason to. The second rule has been established as the most likely origin of our universe, as proposed by physicists such as George Gamow and Georges LeMaître (who also first described the expanding universe prior to Hubble). The third rule is just common sense, or so it appears to me. The first two rules may also be described as hypotheses, the first one confirmed by experimental evidence many times now, and the second one still theoretical—since nobody was around to observe it and we can never see any evidence directly pertaining to it (see the cosmic microwave background, later). The 14.1 billion years comes from the latest data released by NASA on the results of the Wilkinson Microwave Anisotropy Probe, which yield a value of 69.32 ± 0.8 km/sec/Mpc (Mpc = megaparsec), giving the age of our universe as 14.1 billion years;  the value of H0 in light-years is 21.24 ± 0.5 km/sec/LY, which divided into c gives the age of our universe.

Taking it from the top
Most physicists agree that our universe began from a singularity (if so, why doesn’t the singularity at the center of a black hole explode?), or possibly a Planck-sized quantum of immense density, which exploded and began to expand rapidly. At first our infant universe comprised only a very hot mixture of photons, electrons,  and baryons. No atoms or molecules yet existed. Our infant universe was essentially an extremely hot plasma. The outer limits of this little universe were expanding at the speed of light, and everything inside that barrier was being stretched—and cooling off in the process.

After some 372,000 years our baby universe, now some 744,000 light-years in diameter, had cooled off enough that electrons could hook up with protons and form hydrogen atoms. This made things transparent to photons, and the light began to scatter in all directions. Much has been made of the isotropy of the radiation given off by the “surface of last scattering” as the physicists like to call it. They all seem to believe that there would be no good reason for the light to be almost uniform in all directions. So Alan Guth, a physicist at the Stanford Linear Accelerator Center, invented a theory called “inflation” that supposedly explained this. I do not, however, see any reason why inflation should be necessary. I assume our universe began as a perfectly spherical “bomb” and spread in all directions uniformly, simply because there was no good reason for it not to do so. (Occam’s razor, here.)

How do we know that this radiation was almost isotropic? Because we can see it today. It is called the cosmic microwave background (CMB). It was detected in 1964 by two scientists from the Bell Telephone Laboratories, Arno Penzias and Robert Wilson, for which they received the Nobel Prize in physics for 1978. It had a temperature of about 3,000-5,000 degrees Kelvin when it was first released (roughly the temperature of our sun) but now has a temperature of only 2.725 degrees Kelvin, barely above absolute zero. This relict of the Big Bang is all around us, in every direction we look. It has a perfect black body radiation curve, exactly as predicted by the Big Bang theory.

CMB+Black+Body+Curve--Adj
Black Body Spectrum of CMB

The CMB has become the most precisely measured spectrum in nature. It is, however, not perfectly isotropic. Small irregularities exist that comprise about one part in 100,000 of the average value of the CMB. These were discovered by two experiments, “Boomerang,” and later, WMAP. The latter showed precisely how large these anisotropies were. They are exactly the right size to have caused the later formation of stars and galaxies—not to mention a few massive black holes, which went on to gather large collections of stars that orbit around them in massive patterns of great beauty. We call these “galaxies.” They are the most majestic objects in our universe.

The CMB is the most distant object—if that’s the right word—we can see today. It comes from a distance of about 14.1 billion light-years and has been so red-shifted that instead of wave lengths on the order of 500 micro-meters as it originally had, it now has wave lengths of about 1.5 cm—in the microwave region at frequencies of about 200 GHz.

In an earlier blog I theorized that the slight anisotropies in the CMB were caused by quantum fluctuations in the cosmic soup that preceded the surface of last scattering (which, by the way, lasted for some 115,000 years before it was finished scattering things about). Then I discovered this passage in Leonard Susskind’s Cosmic Landscape:

Quantum mechanics and its jittery consequences are normally thought to apply to the world of the very small, not galaxies and other cosmic-scale phenomena. But it now appears all but certain that galaxies and other large-scale structures are remnants of original minute quantum fluctuations that were expanded and enhanced by the unrelenting effect of gravity.


—Susskind, Leonard. The Cosmic Landscape. Page 166 (paperback edition)


So, count another small victory for gut physics. That’s all it was; I just had a gut feeling that quantum fluctuations were the only things that could account for these anisotropies of one part in 100,000.

What does all of this mean? Any object that is farther away than 14.1 billion light-years—regardless of whether or not light from that object could have reached the earth—must have existed more than 14.1 billion years ago. This means it would have to be older than our universe, which is obviously impossible, unless such an object was outside our universe and not a part of it. At a distance of 14.1 billion light-years (at least, approximately) there is an event horizon, similar to the one that defines the outer limits of a black hole. Except that we are inside this event horizon, not outside it. This horizon (the existence of which is correctly pointed out by Susskind, both in The Cosmic Landscape and The Black Hole War) comprises, as it were, the “edge” of the universe. Beyond it nothing can exist—at least, nothing that is a part of this universe. This is so for the simple reason that anything beyond that limit would have had to come into existence before the Big Bang, before the universe itself. Time itself is part and parcel of our universe, and it came into existence at the precise instant1 that the universe was “jump started” by the primeval Big Bang. Asking what came before the Big Bang is something like asking what is north of the north pole. The question is simply meaningless. Asking what lies beyond the event horizon of our universe is equally meaningless. Guth’s “inflation” probably applies to the megaverse, not to our universe. He and his followers even talk about “bubbles” forming in the megaverse, and Susskind (ibid.) says that cosmologists now believe that there may be more than 10500 universes (“bubbles”) in the megaverse.

I used the term “Big Bang” above in its original meaning as the very beginning of the universe. Its usage has been shifted somewhat in recent years to mean those events that occurred when the decoupling of matter and radiation occurred, at which time photons were first released. This is referred to as the surface of last scattering. Susskind (Cosmic Landscape, p. 163) refers to the Big Bang as the “conventional Big Bang” and refers it to the time of the release of photons from the surface of last scattering. This redefined Big Bang occurred roughly 372,000 years after the creation of the universe, prior to which time the proposed “Inflation” occurred.

Inflation is supposed to have caused the nascent universe to expand exponentially to a size far larger than today’s observable universe. I suppose this is possible—but only if the laws of physics that hold today were not valid during that period of time. The speed of light—which is invariant and impossible to exceed according to Einstein’s theory of relativity—would seem to limit anything of this sort. This “cosmic speed limit” applies to the universe itself, at least insofar as it comprises anything that possesses mass-energy.2 If a universe contains no mass-energy, then what is it composed of? Nothing?

To the best of my knowledge, no one has ever proposed a solution to this conundrum. How could a newly born universe expand like some enraged banshee in violation of all known laws of physics for a few hundred thousand years, and then suddenly settle down to a peaceful existence in accord with those same laws? This theory of Inflation may explain something, but it is entirely too ad hoc and speculative to sit well with me.

I firmly believe that the universe began expanding at the speed of light from time zero right up until the present, and that any “inflation” is purely subjective and simply an ad hoc construct to explain something that could not easily be explained by existing theories. The correct theory has yet to be discovered.

Inflation is a theory first proposed by Alan Guth of the Stanford Linear Accelerator Center, who based his theory on earlier work done by a Soviet physicist named Alexei Starobinsky (ibid., p. 161). Guth’s theory was proposed to explain why the cosmic background radiation is so nearly isotropic. As Susskind explains it, one assumes that the universe began “in a deflated state, badly shriveled up, with lots of wrinkles like a dried prune.” (ibid., p. 162) He poses the question as to how the universe became so homogeneous that the CMB [cosmic microwave background, or cosmic background radiation] looks exactly the same in every direction. (Actually, it doesn’t, but the variations are less than one part in 100,000.) He goes on to show that this cannot be explained without something like Guth’s inflation theory.

But who is to say that the universe began wrinkled like a dried prune? Why not assume that it began in a perfectly symmetrical state? If it did, then we have only to explain how the slight anisotropy that we observe now came about, not how the almost-perfect isotropy came to be. Since nobody was there to observe the origin of the universe, it seems to me that one theory is as good as another. The principle of “Occam’s razor”—the best theory is the one with the fewest assumptions—points to the “smooth” beginning as the best theory here, not one in which an ad hoc inflationary theory must be invoked in order to produce the observed result.

Albert Einstein maintained—for good reasons—that thought experiments are the best way to derive new and better theories with which to explain the world. All conjecture about the way the universe first came into existence must perforce be the results of thought experiments, since we have no instruments with which to make any definitive measurements on the subject. And of course we never will have any such instruments, for the very good reason that the cosmic background radiation forms an impenetrable barrier to anyone who seeks to know anything about the earlier history of our universe. Just beyond that barrier lies the event horizon of the universe, roughly 372,000 light-years farther away—a mere 0.0026 percent of the 14.1 billion light-years that lie between the outer background radiation barrier and ourselves.

From the beginning….
Since we are talking cosmology here, what is the picture we can get from the assumptions I have made above? And how does it compare to Guth’s theory of inflation? We start with the usual assumption: a singularity. That is, the universe began as a point source of incredibly dense material, which—no matter of what it was made—contained an enormous amount of mass-energy. Of course, it instantly exploded.

When I worked with radio measuring equipment back in the 1960s, we used a convenient yardstick: light travels one foot per nanosecond. A nanosecond is a mere one-billionth of a second. If the universe expanded at the speed of light, as I have proposed, then it had a radius of one foot one billionth of a second after the Big Bang (in its original meaning). After two nanoseconds, the radius increased to two feet. After one microsecond (1,000 nanoseconds) it would have been roughly 2,000 feet across (twice the radius). One millisecond after it was created, it would have attained a radius of about 983,600 feet (299.8 kilometers). After a full second it would have grown to a radius of 299,792.5 kilometers. After some 372,000 years, at the time of matter-radiation separation, or the surface of last scattering, it would have had a radius of 372,000 light-years.

The universe continued expanding with its outer “shell” receding at the speed of light, until it attained a radius of some 14.1 billion light-years—as it is today. The actual event horizon would be some 372,000 light-years beyond the distance to the cosmic background radiation, but this is a negligible fraction of 14.1 billion.

The Hubble constant
The currently accepted value of the Hubble constant is 69.32 ± 0.80 km / sec / megaparsec, or 21.24 ± 0.25 km / sec / million light-years (mly). Since my cosmology predicts that the radius of the universe should be equal to the speed of light divided by the Hubble constant, this yields a radius of 299,792.5 / 21.24 = 14,110 million light-years. The age of the universe would therefore be 14.1 billion years. This value agrees well with the accepted age of 13.8 billion years. I consider this confirmation that my cosmology is essentially correct: the universe has been expanding at the speed of light ever since its creation. In addition, if this theory is correct, then the exact value of the Hubble constant is 299,792.5 / 13770 = 21.77 km / sec / mly (millions of light-years)—assuming the actual age of the universe is 13.77 billion years, as currently accepted. The value of 21.24 is well within the interval of uncertainty for current measurements of the Hubble constant: 21.0 to 21.5, in the same units as above. In the usual units of kilometers per second per megaparsec, the current value of the Hubble constant is 69.3. (Data from NASA’s WMAP project gave a value of 70.0 ± 2.2 for the Hubble constant.)3

Why do I say “current” value? Because the Hubble constant is not really a constant at all. It is more properly a variable, a ratio. Its value depends on the size, and hence on the age, of the universe. The equation that defines the Hubble ratio is R = c / H, where R is the radius of the universe—which if given in light-years is also its age—c is the speed of light, and H is the Hubble “constant.” Thus H = c / R. The current value of the Hubble ratio, usually denoted by Ho, is defined by

H
o = c / Ro, 

where R
o is the current radius (or age) of the universe. The Hubble ratio was much larger when the universe was young, and will grow gradually smaller as the universe ages—and thus, grows. (Susskind also noted this in The Cosmic Landscape in a footnote on page 136.) How fast is the Hubble ratio changing? One thousand years from today its value will have shrunk by only about one part in fifteen million, which is essentially not worth mentioning. For all practical purposes, the Hubble ratio really is a constant.

Imagine! A simple three-term equation defining the age and size of the universe from the rate of cosmic expansion and the speed of light. Whether this is correct or incorrect, I can’t help feeling that Einstein would have loved this equation. It probably deserves second place right after E = mc2.

So how and why are we here?
We come now to the so-called Anthropic Principle. I recall many years ago reading an article in American Scientist (the in-house publication of Sigma Xi—The Scientific Research Society of America) titled “The Universe as a Home for Man.” In this article I first encountered the Anthropic Principle. It seemed right at the time. Here everything was arranged just perfectly to support life as we know it. Back then I did not know it, but the operative words were “life as we know it.” The hypothetical Anthropic Principle holds that the world was so designed that we could live in it, observe it, and you could be here to read these words. Nowadays the Anthropic Principle is a hot topic, with many scientists supporting it, and many others bitterly opposed to it. They don’t realize it, but they are all wasting their time. You see, the Anthropic Principle is stated exactly backwards. Instead of the universe having been designed to support life as we know it, life as we know it was designed to fit into this universe.

As is so often the case with scientists, who seem more often than not to have their noses buried in their work, they have once again put the cart before the horse. Believe it or not, it was Michael Crichton, the author of Jurassic Park who supplied the answer. In the film version of Jurassic Park, the character Dr. Ian Malcolm (played by Jeff Goldblum) says at one point, “Life is infinitely adaptable.” Exactly. Even here on earth we have discovered life forms flourishing in the most unlikely places, such as Antarctica, where temperatures drop to more than 100 °F below zero, and in the effluent from volcanic vents at the bottom of the ocean, where temperatures exceed the boiling point of water, places where life should not be able to exist at all. But hey—life is infinitely adaptable. Why should we not extend this reasoning to include the basic parameters upon which a universe is based?

In other words, life adapts itself to the environment in which it finds itself. True, some of the multiple universes in the so-called megaverse could be hostile to any form of life, or even to the basics such as stars and planets that might be required to support some kind of life. But out of 10500 possibilities, surely there must be a large number where infinitely adaptable life could find a niche. So we’ve no right to assume that our “little” universe is the only one of this huge number of possible universes where life exists. Besides that, there is only the possibility of 10500 different types of universes in the megaverse. Who is to say that there actually are that many? It is also possible, since we cannot investigate any of them (besides our own), that all of them are like ours. Perhaps the megaverse is generating multiple universes using a single copy of some unknown “genetic code,” a code that will generate clones of our universe. Like a giant salamander that lays 1,000 eggs, each of which hatches into a little replica of its mother. So long as we are guessing here—and that’s exactly what we are doing—we may as well guess anything we wish. Nobody will ever be able to prove us right or wrong.

But mark my words here and mark them well. Somebody, someday, is going to figure out what the “genetic code” was that laid out our universe exactly the way we find it today. And that somebody is going to become as famous as Albert Einstein. His name shall be enshrined in history along with other pioneers such as Darwin, Einstein, and Newton—people who figured out something really big.

This “genetic code” will certainly not be a double-helix like DNA. No, it will be something much simpler and smaller than DNA.4 But it will prove to have been just as effective in describing the way our universe would evolve as DNA is in describing what a life-form is going to become as an adult.

So the reason we are here is because life is infinitely adaptable. It adapted to the unique aspects of our universe and unerringly went on to develop more and more sophisticated forms, until it came up with the thinking beings we call humans.

But the “why” of all of this is a bit beyond the scope of this work, as authors are fond of saying. The why is left as an exercise for the reader. If you should come up with a really good reason as to why we are here, please let me know. I’m dying of curiosity. But I suspect old age will catch up to me before anyone comes up with a satisfactory answer to that question.

The Rules of Life
I state here for the record the rules of life. There are only three of them, but they are of the utmost importance if we are to understand how the Anthropic Principle has been “stood on its head” by almost all the physicists, astronomers, and cosmologists in the world.

The first rule of life is as inviolate as the rule that the speed of light cannot be exceeded in relativity theory. The second rule reverses the so-called Anthropic Principle and puts the horse back in front of the cart. In these rules, a sustainable environment means a universe that will not collapse upon itself or explode. There are likely to be many universes in the Cosmic Landscape (megaverse) that will be sustainable in this respect.
Who knows? Perhaps all of these multiverses are sustainable. With probability theory, anything is possible, and one choice may be as appropriate as any other.

Our almost certainly closed universe
Now we come to some rather heavy stuff. I must admit—along with countless other physicists—that I cannot wrap my head around a three-dimensional universe that is finite yet unbounded. Our minds are not wired to visualize that sort of thing. But it is easy enough to come up with a two-dimensional analogy, in which our third dimension allows for the curvature of space that causes this effect.

Imagine that we live in a two-dimensional world, a world in which there is both forward / back and left / right motion, but nothing in the vertical. That dimension does not exist for us. We live upon the surface of this spherical universe. When we look at things that are far away from us, our universe looks perfectly flat to us. We do not see the spherical curvature of this imaginary universe. It looks something like the picture, seen as a wire frame model.

Spherical+Universe--Gray+Scale
Wire Frame Spherical Universe

Now imagine that we live at the very top of this spherical universe, where all the lines diverge. And imagine that at the bottom of the sphere is the CMB (the remains of this little universe’s big bang). The distance from the top of the sphere to the bottom is 14.1 billion light-years. As half the circumference of the sphere, this distance is equal to πR, where R is the radius of curvature of this spherical universe. This value of R would be about 4.5 billion light-years.

But when we beings in this imaginary universe look at things that are very distant, we do not see the curvature of space. It would be as if we were living in a flat circular space with a radius of 14.1 billion light-years. The CMB, which in this instance we assume to have the same radius as its counterpart in our universe, would appear stretched along the circumference of this doubly-imaginary circle. No matter in which direction we inhabitants of sphere-land looked, we would see this stretched out CMB at a distance of some 14 billion light-years.

Divided+Circle+2
Sphere Land as Seen by Us

The tiny center of this spherical universe at its origin has been spread around the circumference of this circular flatland by the expansion of that universe since its own “big bang” some 14 billion years in the past. Notice also that the geometry of this flatland is perfectly Euclidean. No matter what measurements we inhabitants might undertake, they would not reveal the true curvature of this imaginary spherical universe. They would prove to be Euclidean measurements in every experiment we make. The circumference of this flatland universe would be 2πR, where R is about 14 billion light-years, or some 88 billion light-years. A bit too far for your typical Sunday drive.

Now imagine that we, as inhabitants of sphere-land, are very intelligent and have discovered a way to travel through the space of our spherical universe—which to us looks perfectly flat. As soon as we begin to move through space, say along one of the diverging lines shown in the picture, the position of the CMB would shift, so that it was always exactly opposite us on the sphere. If we traveled completely around the sphere, we would come back to the place we started—after a journey of 88 billion light-years. A lot of distance to cover to accomplish essentially nothing. This spherical universe is expanding just as our real universe is. The CMB recedes from our view in this strange world at very nearly the speed of light, and the actual antipodes of the spherical universe are expanding at the speed of light. So the entire sphere grows, just as our real universe is expanding. And the things on the sphere—galaxies, quasars, and the like—move apart, just as those in our real universe move apart. Galaxies remain the same size because gravity holds them together against the expansion of their universe. We can even derive a Hubble constant for our spherical world.

Sector+5
What the Flatlanders See

Our four-dimensional space-time continuum probably works somewhat like this imaginary spherical universe I have described here. But our minds are not wired to conceive of something that complex. So we must be content with two-dimensional analogies like this one. The picture on the right shows the difference between what the people of Flatland should see when they look at the CMB of their spherical universe and what they actually see. They are looking from the right-hand apex of the gray sphere sector. But what they see is shown by the magenta colored lines,5 which end at the cosmic microwave background (CMB) of their spherical universe. The CMB is shown by the red colored arc—which is way too thick, but you get the idea.

But note this, and note it well: Just as in the apparent flatland of this hypothetical universe no matter in which direction the inhabitants look they see the CMB in the far distance, the same thing applies to our real universe: Every way we look we see the CMB in the distance. So it is more than just likely that our three-dimensional plus one (time) universe works somehow like the hypothetical spherical universe that appears to be flat to its inhabitants. This is a fact that cannot be avoided. It must be taken into account when describing our universe. It cannot be dismissed out of hand. It may be true that our best efforts at measuring any curvature to our three-dimensional universe have come to naught. But these measurements are somehow not definitive. They do not explain why we see the CMB surrounding us. Yet there it is; implacable and incapable of being explained by a truly flat (Euclidean) universe. What’s more, relativity which holds that all frames of reference are equal, tells us that like the flatlanders on their sphere, the CMB will appear to be at the same distance—about 14 billion light-years—no matter where we go within our universe.

In other words, we are not in the middle of our universe. We are just someplace inside it. If we move a sizeable distance (not likely in the foreseeable future), the CMB will always look as if it is about 14 billion light-years away. In the words of Alice of Wonderland fame, we have to run as fast as we can just to stay in the same place. Wicked game, that.

Just as our counterparts in sphere-land, we can never get to the “edge” of our universe. It has no edge in that sense. Yet just as real, it has a boundary. But we can never get there—another Catch 22.

Do we, like Alice, have to think of seven impossible things before breakfast?

Notes
1 Note the logical quandary here: If time did not yet exist, how could there be an “instant” at which time began?

2 Arguments that the speed of light limit can be exceeded when space itself is expanding are just verbalisms. They smack of resurrection of the discredited theory of the “ether” (demolished once and for all by the famous Michelson-Morley experiment). Space has no existence of its own; it is merely the distance between objects. (See Einstein’s Relativity, page 157.) To say that two objects can have a relative velocity greater than c simply because the “space” between them is expanding contradicts both the special and general theories of relativity. Can those who make these arguments also feel the ether blowing through their hair?

3 The value of 69.32 ± 0.80 km/sec/Mpc for the Hubble constant was published by NASA on December, 20, 2012, derived using data from the Wilkinson Microwave Anisotropy Probe (WMAP). This value puts the age of the universe at 14.1 billion years.

4 The “mother of all strings”?

5 The magenta-colored lines follow the line of sight of the sphere sector at the origin, where the Flatlanders are looking toward the boundary. They are completely unaware that the sector lines curve back on themselves and converge on the CMB at the other end of the sphere sector. Instead, they see the triangular sector that ends with an expanded portion of the CMB at the outer boundary, which is some 14 billion light-years away from the Flatlanders. The same sort of thing applies to our universe, except that it happens in three dimensions instead of only two. The direction of curvature is complex and imaginary (as in complex variables) and may involve two more dimensions that we are not aware of in any way. (After all, string theory invokes no less than nine dimensions plus time.)

Remember, when we look at far distant objects, we are looking backward in time. When we see the CMB, we are really seeing it as it was 14 billion years ago, at which time it had a diameter of some 744 thousand light-years. So we are seeing it through a sort of magnifying glass, just as the Flatlanders see their CMB magnified by the curvature of their space-time continuum. Seen through the magnifying glass of time and curved space, our CMB appears to have a diameter of some 28 billion light-years.

References
Einstein, Albert. Relativity: The Special and the General Theory. 100th Anniversary Edition. New York: Tess Press, undated, but probably 2005. English edition originally published in 1931.

Susskind, Leonard. The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. New York: Little, Brown and Company, 2005. Paperback edition, New York: Back Bay Books / Little, Brown and Company, 2006.

—. The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. New York: Little, Brown and Company, 2008.