In the preceding chapter, it was pointed out that according to the singularity theorem, the Universe must have evolved from or into a very dense state.
And just exactly what is a singularity? It is of the utmost importance to clarify this concept before we proceed any further, for herein lie the very roots of our beginnings, herein is where we come face to face with infinity itself, herein is where mathematics and logic as we know them lose all meaning.
Mathematically speaking, the concept of a singularity can be easily understood with a simple example. Take, for instance, the following formula:
Let us assume in the above formula that we can give Q1 and Q2 any numerical value we so desire, and for our discussion we will assume that k is a numerical constant that will be given the value of one. If we assume that r takes also the value of one, then it is not difficult to verify that if Q1 has a value of 2 and Q2 has a value of 3, then F will be computed to have a value of 6. Let us now assume that both Q1 and Q2 have a value of one and that r takes on a value of 0.5, in which case F can be computed to take on a value of 4. If we give r a smaller value such as 0.1, then F will take on a value of 100. If we give r an even smaller value such as 0.01, then F will have an even larger value of 10,000. If we give r an even smaller value such as 0.00001, then F will have an even greater value, which is:
F = 1,000,00,00,000
It is not hard to see that as we keep on giving r even smaller values, F becomes larger and larger and begins to grow without bounds. Since the smallest value we could possibly give to r, besides very tiny values which are close to zero, would be zero itself, it becomes obvious that the value
r = 0
cannot be used for an actual numerical computation, since the result for F could only be denoted by a number so large that the number would be none other than infinity itself. This is what we mean by a singularity.
Imagine now for a moment that you had the strength of a mythical Titan, and that you could be able to go to you car, grasp it with your hand, and applying enormous pressure you were able to compress that car into a tiny sphere the size of a marble. Now imagine you could grasp your entire city and applying enormous pressure you could compress everything to fit in the palm of your hand. As a next step, imagine yourself with enough strength to grasp and compress the entire planet Earth, with all its mountains, all its rivers, all its seas, all its continents, into a tiny sphere the size of marble. This marble, even if small in size, would be an enormously massive object, capable of exerting a tremendous gravitational pull on any other objects nearby. Having compressed planet Earth into a small marble, you would then reach for the Moon, compress it to the size of a marble, and compress both marbles into one, at which point you could go to grab another planet and repeat the process, crunching everything into the size of a small marble. But at some point, as you kept on adding more mass to the marble, you would be amazed at what would happen next. First, there would be a tremendous explosion, because you would have just created a nova by the sheer act of compressing so much matter into such a small volume, with an outer shell of debris being thrown out into space and a very massive inner shell remaining. After this, because of the enormous gravitational pull concentrated in such a small volume of space, the remaining accumulated mass would collapse into something known as a neutron star. But if you are unlucky, the gravitational pull of the concentrated mass might just be enough to enable it to compress itself into something even smaller than the marble you had in your hand, even smaller than the head of a pin. The accumulated mass would still be there, somewhere, but its gravitational pull would be so strong that even you, the mighty Titan, would be sucked into such strange object. Not even light itself can escape from such object. It goes without saying that this object would have to be something way beyond belief. This object, in fact, was predicted nearly a century ago by Einstein’s theory of general relativity, and is commonly known as a black hole. A black hole, brought about by this process of gravitational collapse, is not just a theoretical curiosity to explain a possible solution to the equations of general relativity. As a matter of fact, its existence in several parts of the Universe has been confirmed by recent astronomical data, and there is strong reason to believe that at the very center of our own galaxy, the Milky Way, there is a massive black hole powering a galactic engine almost beyond our capabilities of comprehension [since 1971, Cygnus X-1 in the Milky Way galaxy has now been accepted as a black hole. Other black holes in our galaxy are V404 Cygni, GS 2000+25, H1705-250, GRO J1655-40 and A 06020-00. At the nucleus of nearby spiral galaxy NGC 4258 there is another black hole. By June 1999, according to an article published in Scientific American entitled Revisiting the Black Hole, the authors Roger Blanford and Neil Gehrels mention that we had about 15 mass estimates for black holes in the nuclei of nearby galaxies that are quite secure, adding "No longer is there any serious debate as to whether black holes exist. We know that they must be quite common, and we can now study them in increasing detail"].
It was the German astronomer Karl Schwarzschild who in the winter of 1915, in an accomplishment that impressed Einstein, found a simple yet exact solution to the equations of general relativity. Among the conclusions drawn from the solution was that for a sufficiently compact mass there was a finite radius at which emitted light waves would have an infinitely long wavelength, which in simple terms implies that light cannot escape from such object.
In the book Gravitation by Charles Misner, Kip Thorne and John Archibald Wheeler, we find a more accurate description of what would happen to an unlucky traveler who might happen to pass near a black hole and get trapped by its gravitational pull:
“Consider the plight of an experimental astrophysicist who stands on the surface of a free falling star as it collapses to R = 0. As the collapse proceeds toward R = 0, the various parts of the astrophysicist’s body experience different gravitational forces. His feet, which are on the surface of the star, are attracted toward the star’s center by an infinitely mounting gravitational force; while his head, which is farther away, is accelerated downward by a somewhat smaller, though ever rising force. The difference between the two accelerations (tidal force) mounts higher and higher as the collapse proceeds, finally becoming infinite as R reaches zero. The astrophysicist’s body, which cannot withstand such extreme forces, suffers unlimited stretching between head and foot as R drops to zero …But this is not all. Simultaneous with this head-to-foot stretching, the astrophysicist is pulled by the gravitational field into regions of spacetime with ever decreasing circumferential area, 4πr². In order to accomplish this, tidal gravitational forces must compress the astrophysicist on all sides as they stretch him from head to foot. The circumferential compression is actually more extreme than the longitudinal stretching; so the astrophysicist, in the limit as R approaches zero, is crushed to zero volume and infinitely extended length.”
As William Poundstone more aptly puts it in his book Labyrinths of Reason, “A human body that enters a black hole is transformed into Euclid’s ideal line”.
The description given above can be found under the heading “THE FATE OF A MAN WHO FALLS INTO THE SINGULARITY AT R = 0”. This is because at the center of the black hole there is a singularity, a point of infinite compression and infinite curvature of space-time.
Let us now go back to our original line of thought, and imagine that once we have created a black hole with all of the combined masses of the solar system, from the Sun all the way out to Pluto, we bring into such black hole millions upon millions of other stars and planets from the galaxy, to form an incredibly massive black hole. For an outside observer, it would appear as if first the entire solar system had disappeared out of existence (well, not quite so, for in the process of falling into a black hole the accelerating matter emits a lot of very intense radiation, including X-rays, which can be used by the outside observer to infer that were there once was a system of stars there is now a massive singularity in the space-time continuum), and after that the entire galaxy had also seemingly disappeared into nowhere. After this, we go to other galaxies, with their millions upon millions of stars and planets and any black holes they might contain, and toss those entire galaxies into our super-massive black hole [from the equations of general relativity it can be formally proven that two black holes can be merged into one; the proof is contained in the work by Stephen Hawking entitled "Black Holes in General Relativity" published in 1972 on "Communications in Mathematical Physics"]. Continuing this process, we go even further and toss not just galaxies but entire clusters of galaxies with their millions upon millions of galaxies, stars, planets and black holes into our colossal black hole. We go even further, and toss super clusters of galaxies, with their millions upon millions of clusters of galaxies into the black hole. As a final push, we reach out towards the entire Universe, and bring all of its contents into our super-colossal black hole. It is possible at this point that such an incredibly massive object would have enough gravitational pull to suck into it not just the entire matter contained in the Universe but even the fabric itself upon which the Universe is built, the space-time continuum. In effect, everything in the entire Universe, and by this we mean absolutely everything, including space-time itself, would fold unto itself and vanish into a singularity that we cannot even possibly try to comprehend, much less analyze with our current knowledge.
Yet, it is precisely from a singularity such as this one where the Universe is supposed to have begun some ten to fifteen billion years ago. The singularity from which the Universe was born would indeed be the ultimate black hole, the mother of all black holes.
But if nothing can escape from a black hole, how then could the Universe have begun, with everything bursting out of the singularity in the midst of a tremendous explosion? This is one of the major puzzles being confronted by cosmologists nowadays, and several explanations have been advanced, among them the observation that if we go back in time the Universe will contract into a point for which quantum effects such as Wolfgang Pauli’s exclusion principle will become important. Also, recent astronomical evidence points to a growing possibility that the rate of expansion of the Universe is not slowing down nor is it keeping a steady rate but rather it is accelerating, and such effect can only be due to another unexplained force that is counteracting the pull of gravity on a cosmic scale.
One thing appears to be certain. If such a massive concentration of mass and energy and space and time and everything else there may be into a small point like object could be duplicated “somewhere else”, the concoction would be highly unstable. Our own Universe is “living proof” of that. It has been estimated that when our Universe was only a tenth of a nanosecond old (one-tenth of one-billionth of a second) the temperature was in the neighborhood of about 1015 degrees Kelvin (to get the equivalent in degrees Centigrade we add 273 to whatever reading we may have in the scale of our Kelvin degrees thermometer). Written out in full, this is a temperature of about
1,000,000,000,000,000 degrees
(Just for comparison purposes, if the outside temperature on a hot summer day was about 100 degrees Fahrenheit, then its equivalent would be a “cool” 310 degrees Kelvin.)
This enormous temperature has been decreasing ever since the explosion took place; down to the 2.7 degrees Kelvin it is today (the heat left over from the Big Bang) permeating all of outer space as the cosmic microwave background radiation (in the Centigrade scale, this temperature is close to 270 degrees Centigrade below freezing).
Besides, we now know that a black hole such as the ones we would expect to find in our own Universe at this very moment will not last forever, thanks to a discovery that took the scientific community by storm. Stephen Hawking proved it by resorting to quantum mechanical arguments that black holes are “evaporating” continuously by sending out a radiation that is now known as the Hawking radiation. In relation to his discovery, Hawking tells us in his article “The Quantum Mechanics of Black Holes” published in the January 1977 issue of Scientific American:
“To my great surprise I found (in 1974) that the black hole seemed to emit particles at a steady rate. Like everyone else at that time, I accepted the dictum that a black hole could not emit anything … What finally convinced me it was a real physical process was that outgoing particles have a spectrum that is precisely thermal: the black hole creates and emits particles and radiation just as if it were an ordinary hot body with a temperature that is proportional to the surface gravity and inversely proportional to the mass …Since that time the mathematical evidence that black holes can emit thermally has been confirmed by a number of other people with various different approaches … As a black hole emits particles its mass and size steadily decrease … In the long run every black hole in the universe will evaporate in this way. For large black holes, however, the time it will take is very long indeed: a black hole with the mass of the sun will last about 1066 years … The final stage of the evaporation of a black hole would proceed so rapidly that it would end in a tremendous explosion.”
There is, however, one important difference between an ordinary black hole and the singularity from which our Universe was born. The black holes that inhabit our universe and which are constantly evaporating by sending out their radiation do so into a vast preexisting space, whereas the primeval singularity created its own space, since before the singularity exploded there was no space in which to explode! And since all of the space that fills our cosmos came from within the confines of that singularity, the singularity vanished into every point of our expanding Universe. In effect, any region of all the known space of the Universe can now be considered to be the “center” of the primeval explosion, and no observer in any part of the Universe can take himself to be a “privileged observer” in the sense that while he standing still is at the very center of the Universe all of the other stars and galaxies are receding from him at a very fast pace. This agrees completely with the precepts of both the special and the general theories of relativity, in the sense that there are no privileged observers capable of detecting “absolute” motion with respect to some hypothetical point of reference, since no absolute point of reference exists in the Universe. While it is true that the more massive the black hole is the longer it will take to evaporate into outer space, we cannot infer from this dictum that the primeval singularity –being the most massive object in existence with an infinite density of matter- should have been the most stable object of them all, since whereas a black hole radiates particles into the space-time continuum which the singularity threw out, the singularity itself became the space-time continuum before any black holes could be born. The enormously high temperature at the moment of creation and the enormous speed at which the galaxies are now receding from each other confirms that a singularity like the one that gave birth to our Universe is inherently unstable.
Reviewing the singularity theorem which states that the Universe must have evolved from or into a very dense state, we can see that if the Universe will keep on expanding forever, without any possibility of evolving in the future into a singularity because of its never ending expansion, then it must have evolved from a singularity in the past. There can be no other conclusion at present.
It is ironical that in order to explore the Universe itself which in all likelihood began as a point like particle, scientists are now resorting to the tools provided by relatively new subjects such as “quantum field theory” and their offspring “unified field theories”, which deal with almost point like subatomic particles, in order to explain the first stages of the newly born Universe. But these models only allow us to go back so far, and at present the farthest back we can go with our current knowledge is about 10-35 second after the Big Bang took place, which written in full reads out as follows:
0.000,000,000,000,000,000,000,000,000,000,000,01 second
This is a very small time lapse indeed. But as remarkable as this achievement sounds, it stills leaves many doubts as to what events could have taken place between the actual moment of creation (time = 0) and the 10-35 second after that.
On final analysis, whatever characteristics the singularity from which the Universe came from could have had at the precise moment of creation (time = 0), it is those characteristics the ones which ultimately set the stage for the creation of stars, galaxies, comets, and planets capable of sustaining intelligent life. At the very moment of creation the singularity had all of the right initial conditions required for these things to happen. Once the explosion has taken place, the singularity cannot go back and reconstitute itself in order to alter those initial conditions, at least not within the framework of any of the workable theoretical models we have nowadays. The primeval explosion appears to have been such a fiery and overwhelming event that it just doesn’t seem possible for it to be undone on its own once it has started, no matter how small the time span after the actual moment of creation, be it 10-35 second, 10-100 second, or even 10-1,000,000,000 second (try to write this number down; it is a one preceded by one million zeroes!)
We have assumed throughout that general relativity is still valid at the start of the early Universe. Of course, we could all be wrong, and there is a possibility that all known physical laws including relativity itself may break down completely once we have reached the singularity, and our known physical laws would have to be replaced by other yet-unknown physical laws to describe the point of origin. Nevertheless, we need to carry on with what we already have and which we know is true and has been confirmed by experimental data here on Earth, for the alternative is to just sit back and do nothing while something else comes to mind. It has been said that “good things come to those who wait”, but in this case this old adage would be out of place, and a more proper one would be “good things come to those who work.”
All around us, we are surrounded with what appear to be infinities, from the infinitely small (unless we take the subatomic particles from which everything is built to put an absolute limit to the extension of how small the very small can be) to the infinitely big (unless we consider the possibility that our Universe is a “closed” Universe with a finite though very large volume, a possibility allowed by the equations of general relativity, which would put an absolute limit to how big the very big can be). But we know ourselves all too well to be finite creatures, confined to live in a finite planet which can be measured with our finite yardsticks, inside a finite solar system which can also be measured with finite yardsticks, inside a finite galaxy which can also be measured with finite yardsticks, and within us are the finite discrete parts we call our internal organs, made up of a finite amount of differentiated cells, with each finite cell and everything else such as the water molecules which flow through them made up of a finite amount of atoms. The dilemma many might have asked themselves at one time or another is this: How can things that are the purest embodiment of infinity itself give rise to things that are very definitely finite?
It turns out that, at least from the perspective of mathematics, this is not only possible but indeed it should be expected. To see how this could take place, imagine you have a cube in your hand containing a known finite volume V. Assume each side of the cube measures one centimeter, in which case the cube will have a volume of one cubic centimeter. If we double the height h of the original cube, then in order to contain the same volume V of one cubic centimeter as in the original cube we must reduce each side l of its square base to:
By the same token, if we increase the height h of the original cube tenfold, in order to still contain the same volume V of one cubic centimeter the cross sectional area of its base must now be a square with an even smaller length l of about 0.306 centimeter. That the volume remains the same can readily be verified the reader as follows:
V = (0.316 centimeter)(0.316 centimeter)(10 centimeters) = 1 cubic centimeter
This procedure can be repeated indefinitely with no end in sight, and we can have an ever-thinner rod with an ever-growing length still containing the same finite volume V provided that as we increase the height we adjust the sides of the base accordingly:
| Side of square base | Height | Volume = 1 cubic centimeter |
1
|
1
|
(1)(1)(1)
|
0.707
|
2
|
(0.707)(0.707)(2)
|
0.316
|
10
|
(0.316)(0.316)(10)
|
:
:
:
|
:
:
:
|
:
:
:
|
0.1
|
100
|
(0.1)(0.1)(100)
|
0.001
|
1,000,000 |
(0.001)(0.001)(1,000,000)
|
:
:
:
|
:
:
:
|
:
:
:
|
0
|
∞
|
1 cm3
|
In the last row, we have used on the first column the symbol zero to represent a quantity so vanishingly small (though not exactly zero which stands for “nothing”) that to us it would appear to be the smallest quantity we can think of (by this we mean a quantity so small that whatever small number we may write down we may assume that this quantity is even smaller), and we have used the symbol “∞” to represent a quantity so large that it cannot even be written down. Yet, the product of these disparate quantities is known to yield a very finite result. Taking this procedure to the very extreme, we can see how the product of a quantity that is infinitely small times a quantity that is infinitely big will produce a result which is neither infinitely small nor infinitely big but, to the contrary, will be quite finite. Indeed, this is what happens to our astrophysicist who fell into a black hole becoming Euclid’s “ideal” line with infinite length and zero thickness but still managing to retain a finite volume as he falls deeper and deeper into the singularity. This remarkable fact can be expressed symbolically in the following manner:
In other words, the product of something that can be taken to be infinitely small times something that can be taken to be infinitely big can yield any finite number we can think of, whether it is ten, one million, one thousandth, or whatever. This is one way in which infinities can meet together to yield finite results. This agrees somehow with our common everyday experience, as we ourselves having finite bodies are fully aware that towards the outside of our bodies we peek at the infinitely big and towards the inside we can go deeper and deeper into the infinitely small, and thus the surface of our skins acts like an “interface” between these two extremes, a place where the two meet together.
In the physical world, it is quite possible that infinity may just be an illusion. The modern science of Quantum Mechanics owes its very existence to the fact that matter cannot be split down forever with a knife without encountering sooner or later those discrete finite units we call “atoms” and “molecules”. [It was precisely an abhorrence of the concept of infinity which lead Democritus nearly 2,500 years ago to postulate that matter could not be split down forever without encountering that it was made up of those small discrete finite units for which he himself coined the Greek word atomos which means that it cannot be "cut down" any further]. Likewise, the intensity of a beam of light cannot be dimmed indefinitely without finding out that at some point the beam of light is made up of small yet finite discrete lumps we call “photons” which cannot be split any further (whether these tiny lumps are themselves made up of something else even smaller is at present just a matter of speculation). Even Albert Einstein believed in the possibility of a very big but finite “closed” Universe, a possibility that is allowed by the equations of general relativity and mathematics itself within the framework of what we call “non-Euclidean” geometries. Nevertheless, many infinite quantities can be rigorously defined and manipulated. The taming of infinities becomes crucial in order to have workable theories that can predict many physical phenomena we observe in our laboratories, especially if those theories are to be extended for the exploration of the Universe and its origin. In the book Perfect Symmetry: The Search for the Beginning of Time, we read the following from Professor Heinz Pagels:
“Although the ideas of relativistic quantum-field theory successfully predicted the existence of antimatter, theoretical physicists in the 1930s and 1940s found lots of mathematical difficulties and problems with these new ideas. If they calculated quantum interaction processes using these new ideas they obtained infinite numbers, so clearly something was going wrong. Nature does not have physical quantities that are infinite … Yet others continued to struggle with this problem and eventually managed to tame these infinities by a mathematical trick called the ‘renormalization procedure’. They showed that the infinite numbers appeared only in calculations of a few quantities like the mass or electric charge of the quantum particles involved, and that if these quantities were redefined, or ‘renormalized,’ by subtracting an infinitely large number, they would then yield finite predictions for all experimentally measurable quantities. Subtracting these infinities seemed like a mathematical trick, but it worked … When the renormalization procedure was carefully carried out, the calculational results of quantum electrodynamics could be compared with precision experiments. To many people’s amazement, the theory, in spite of its abstract mathematical tricks, agreed decimal place for decimal place with experiments. Not since the time of Newton’s predictions of planetary motions had theory and observation accorded with each other so completely. Even physicists were astonished by the experimental success of quantum electrodynamics … Not every relativistic quantum-field theory is renormalizable –the mathematics of renormalization works for only a few kinds of quantum-particle interactions out of a possible infinite number. Remarkably, the renormalizable interactions are precisely the ones we observe. Is Nature trying to tell us something by using only renormalizable interactions? Some physicists, struck by this fact, think that renormalizability is a fundamental imposition by nature, just like the principle of special relativity.”
