Tag Archives: general relativity

Astronomy, Physics

Decay of the False Vacuum

The Decay of the False Vacuum

Written by Sten Odenwald. Copyright (C) 1983 Kalmbach Publishing. Reprinted by permission

In the recently developed theory by Steven Weinberg and Abdus Salam, that unifies the electromagnetic and weak forces, the vacuum is not empty. This peculiar situation comes about because of the existence of a new type of field, called the Higgs field. The Higgs field has an important physical consequence since its interaction with the W, W and Z particles (the carriers of the weak force) causes them to gain mass at energies below 100 billion electron volts (100 Gev). Above this energy they are quite massless just like the photon and it is this characteristic that makes the weak and electromagnetic forces so similar at high energy.

On a somewhat more abstract level, consider Figures 1 and 2 representing the average energy of the vacuum state. If the universe were based on the vacuum state in Figure 1, it is predicted that the symmetry between the electromagnetic and weak interactions would be quite obvious. The particles mediating the forces would all be massless and behave in the same way. The corresponding forces would be indistinguishable. This would be the situation if the universe had an average temperature of 1 trillion degrees so that the existing particles collided at energies of 100 Gev. In Figure 2, representing the vacuum state energy for collision energies below 100 Gev, the vacuum state now contains the Higgs field and the symmetry between the forces is suddenly lost or ‘broken’. Although at low energy the way in which the forces behave is asymmetric, the fundamental laws governing the electromagnetic and weak interactions remain inherently symmetric. This is a very remarkable and profound prediction since it implies that certain symmetries in Nature can be hidden from us but are there nonetheless.

During the last 10 years physicists have developed even more powerful theories that attempt to unify not only the electromagnetic and weak forces but the strong nuclear force as well. These are called the Grand Unification Theories (GUTs) and the simplist one known was developed by Howard Georgi, Helen Quinn,and Steven Weinberg and is called SU(5), (pronounced ‘ess you five’). This theory predicts that the nuclear and ‘electroweak’ forces will eventually have the same strength but only when particles collide at energies above 1 thousand trillion GeV corresponding to the unimaginable temperature of 10 thousand trillion trillion degrees! SU(5) requires exactly 24 particles to mediate forces of which the 8 massless gluons of the nuclear force, the 3 massless intermediate vector bosons of the weak force and the single massless photon of the electromagnetic force are 12. The remaining 12 represent a totally new class of particles called Leptoquark bosons that have the remarkable property that they can transform quarks into electrons. SU(5) therefore predicts the existence of a ‘hyperweak’ interaction; a new fifth force in the universe! Currently, this force is 10 thousand trillion trillion times weaker than the weak force but is nevertheless 100 million times stronger than gravity. What would this new force do? Since protons are constructed from 3 quarks and since quarks can now decay into electrons, through the Hyperweak interaction, SU(5) predicts that protons are no longer the stable particles we have always imagined them to be. Crude calculations suggest that they may have half-lives between 10^29 to 10^33 years. An immediate consequence of this is that even if the universe were destined to expand for all eternity, after ‘only’ 10^32 years or so, all of the matter present would catastrophically decay into electrons, neutrinos and photons. The Era of Matter, with its living organisms, stars and galaxies, would be swept away forever, having represented but a fleeting episode in the history of the universe. In addition to proton decay, SU(5) predicts that at the energy characteristic of the GUT transition, we will see the affects of a new family of particles called supermassive Higgs bosons whose masses are expected to be approximately 1 thousand trillion GeV! These particles interact with the 12 Leptoquarks and make them massive just as the Higgs bosons at 100 GeV made the W, W and Z particles heavy. Armed with this knowledge, let’s explore some of the remarkable cosmological consequences of these exciting theories.

The GUT Era

To see how these theories relate to the history of the universe, imagine if you can a time when the average temperature of the universe was not the frigid 3 K that it is today but an incredable 10 thousand trillion trillion degrees (10^15 GeV). The ‘Standard Model’ of the Big Bang, tells us this happened about 10^-37 seconds after Creation. The protons and neutrons that we are familiar with today hadn’t yet formed since their constituent quarks interacted much too weakly to permit them to bind together into ‘packages’ like neutrons and protons. The remaining constituents of matter, electrons, muons and tau leptons, were also massless and traveled about at essentially light-speed; They were literally a new form of radiation, much like light is today! The 12 supermassive Leptoquarks as well as the supermassivs Higgs bosons existed side-by-side with their anti-particles. Every particle-anti particle pair that was annihilated was balanced by the resurrection of a new pair somewhere else in the universe. During this period, the particles that mediated the strong, weak and electromagnetic forces were completely massless so that these forces were no longer distinguishable. An inhabitant of that age would not have had to theorize about the existence of a symmetry between the strong, weak and electromagnetic interactions, this symmetry would have been directly observable and furthermore, fewer types of particles would exist for the inhabitants to keep track of. The universe would actually have beed much simpler then!

As the universe continued to expand, the temperature continued to plummet. It has been suggested by Demetres Nanopoulis and Steven Weinberg in 1979 that one of the supermassive Higgs particles may have decayed in such a way that slightly more matter was produced than anti-matter. The remaining evenly matched pairs of particles and anti-particles then annihilated to produce the radiation that we now see as the ‘cosmic fireball’.

Exactly what happened to the universe as it underwent the transitions at 10^15 and 100 GeV when the forces of Nature suddenly became distinguishable is still under investigation, but certain tantalizing descriptions have recently been offered by various groups of theoriticians working on this problem. According to studies by Alan Guth, Steven Weinberg and Frank Wilczyk between 1979 and 1981, when the GUT transition occured, it occured in a way not unlike the formation of vapor bubbles in a pot of boiling water. In this analogy, the interior of the bubbles represent the vacuum state in the new phase, where the forces are distinguishable, embedded in the old symmetric phase where the nuclear, weak and electromagnetic forces are indistinguishable. Inside these bubbles, the vacuum energy is of the type illustrated by Figure 2 while outside it is represented by Figure 1. Since we are living within the new phase with its four distinguishable forces, this has been called the ‘true’ vacuum state. In the false vacuum state, the forces remain indistinguishable which is certainly not the situation that we find ourselves in today!

Cosmic Inflation

An exciting prediction of Guth’s model is that the universe may have gone through at least one period in its history when the expansion was far more rapid than predicted by the ‘standard’ Big Bang model. The reason for this is that the vacuum itself also contributes to the energy content of the universe just as matter and radiation do however, the contribution is in the opposite sense. Although gravity is an attractive force, the vacuum of space produces a force that is repulsive. As Figures 1 and 2 show, the minimum energy state of the false vacuum at ‘A’ before the GUT transition is at a higher energy than in the true vacuum state in ‘B’ after the transition. This energy difference is what contributes to the vacuum energy. During the GUT transition period, the positive pressure due to the vacuum energy would have been enormously greater than the restraining pressure produced by the gravitational influence of matter and radiation. The universe would have inflated at a tremendous rate, the inflation driven by the pressure of the vacuum! In this picture of the universe, Einstein’s cosmological constant takes on a whole new meaning since it now represents a definite physical concept ; It is simply a measure of the energy difference between the true and false vacuum states (‘B’ and ‘A’ in Figures 1 and 2.) at a particular time in the history of the universe. It also tells us that, just as in de Sitter’s model, a universe where the vacuum contributes in this way must expand exponentially in time and not linearly as predicted by the Big Bang model. Guth’s scenario for the expansion of the universe is generally called the ‘inflationary universe’ due to the rapidity of the expansion and represents a phase that will end only after the true vacuum has supplanted the false vacuum of the old, symmetric phase.

A major problem with Guth’s original model was that the inflationary phase would have lasted for a very long time because the false vacuum state is such a stable one. The universe becomes trapped in the cul-de-sac of the false vacuum state and the exponential expansion never ceases. This would be somewhat analogous to water refusing to freeze even though its temperature has dropped well below 0 Centigrade. Recent modifications to the original ‘inflationary universe’ model have resulted in what is now called the ‘new’ inflationary universe model. In this model, the universe does manage to escape from the false vacuum state and evolves in a short time to the familiar true vacuum state.

We don’t really know how exactly long the inflationary phase may have lasted but the time required for the universe to double its size may have been only 10^-34 seconds. Conceivably, this inflationary period could have continued for as ‘long’ as 10^-24 seconds during which time the universe would have undergone 10 billion doublings of its size! This is a number that is truely beyond comprehension. As a comparison, only 120 doublings are required to inflate a hydrogen atom to the size of the entire visible universe! According to the inflationary model, the bubbles of the true vacuum phase expanded at the speed of light. Many of these had to collide when the universe was very young in order that the visible universe appear so uniform today. A single bubble would not have grown large enough to encompass our entire visible universe at this time; A radius of some 15-20 billion light years. On the other hand, the new inflationary model states that even the bubbles expanded in size exponentially just as their separations did. The bubbles themselves grew to enormous sizes much greater than the size of our observable universe. According to Albrecht and Steinhardt of the University of Pennsylvania, each bubble may now be 10^3000 cm in size. We should not be too concerned about these bubbles expanding at many times the speed of light since their boundaries do not represent a physical entity. There are no electrons or quarks riding some expandind shock wave. Instead, it is the non-material vacuum of space that is expanding. The expansion velocity of the bubbles is not limited by any physical speed limit like the velocity of light.

GUMs in GUTs

A potential problem for cosmologies that have phase transitions during the GUT Era is that a curious zoo of objects could be spawned if frequent bubble mergers occured as required by Guth’s inflationary model. First of all, each bubble of the true vacuum phase contains its own Higgs field having a unique orientation in space. It seems likely that no two bubbles will have their Higgs fields oriented in quite the same way so that when bubbles merge, knots will form. According to Gerhard t’Hooft and Alexander Polyakov, these knots in the Higgs field are the magnetic monopoles originally proposed 40 years ago by Paul Dirac and there ought to be about as many of these as there were bubble mergers during the transition period. Upper limits to their abundance can be set by requiring that they do not contribute to ‘closing’ the universe which means that for particles of their predicted mass (about 10^16 GeV), they must be 1 trillion trillion times less abundant than the photons in the 3 K cosmic background. Calculations based on the old inflationary model suggest that the these GUMs (Grand Unification Monopoles) may easily have been as much as 100 trillion times more abundant than the upper limit! Such a universe would definitly be ‘closed’ and moreover would have run through its entire history between expansion and recollapse within a few thousand years. The new inflationary universe model solves this ‘GUM’ overproduction problem since we are living within only one of these bubbles, now almost infinitly larger than our visible universe. Since bubble collisions are no longer required to homogenize the matter and radiation in the universe, very few, if any, monopoles would exist within our visible universe.

Horizons

A prolonged period of inflation would have had an important influence on the cosmic fireball radiation. One long-standing problem in modern cosmology has been that all directions in the sky have the same temperature to an astonishing 1 part in 10,000. When we consider that regions separated by only a few degrees in the sky have only recently been in communication with one another, it is hard to understand how regions farther apart than this could be so similar in temperature. The radiation from one of these regions, traveling at the velocity of light, has not yet made it across the intervening distance to the other, even though the radiation may have started on its way since the universe first came into existence. This ‘communication gap’ would prevent these regions from ironing-out their temperature differences.

With the standard, Big Bang model, as we look back to earlier epochs from the present time, the separations between particles decrease more slowly than their horizons are shrinking. Neighboring regions of space at the present time, become disconnected so temperature differences are free to develope. Eventually, as we look back to very ancient times, the horizons are so small that every particle existing then literally fills the entire volume of its own, observable universe. Imagine a universe where you occupy all of the available space! Prior to the development of the inflationary models, cosmologists were forced to imagine an incredably well-ordered initial state where each of these disconnected domains (some 10^86 in number) had nearly identical properties such as temperature. Any departure from this situation at that time would have grown to sizable temperature differences in widely separated parts of the sky at the present time. Unfortunately, some agency would have to set-up these finely-tuned initial conditions by violating causality. The contradiction is that no force may operate by transmitting its influence faster than the speed of light. In the inflationary models, this contradiction is eliminated because the separation between widely scattered points in space becomes almost infinitly small compared to the size of the horizons as we look back to the epoc of inflation. Since these points are now within each others light horizons, any temperature difference would have been eliminated immediatly since hotter regions would now be in radiative contact with colder ones. With this exponentially-growing, de Sitter phase in the universe’s early history we now have a means for resolving the horizon problem.

Instant Flat Space

Because of the exponential growth of the universe during the GUT Era, its size may well be essentially infinite for all ‘practical’ purposes . Estimates by Albrecht and Steinhardt suggest that each bubble region may have grown to a size of 10^3000 cm by the end of the inflationary period. Consequently, the new inflationary model predicts that the content of the universe must be almost exactly the ‘critical mass’ since the sizes of each of these bubble regions are almost infinite in extent. The universe is, for all conceivable observations, exactly Euclidean (infinite and flat in geometry) and destined to expand for all eternity to come. Since we have only detected at most 10 percent of the critical mass in the form of luminous matter, this suggests that 10 times as much matter exists in our universe than is currently detectable. Of course, if the universe is essentially infinite this raises the ghastly spectre of the eventual annihilation of all organic and inorganic matter some 10^32 years from now because of proton decay.

In spite of its many apparent successes, even the new inflationary universe model is not without its problems. Although it does seem to provide explainations for several cosmological enigmas, it does not provide a convincing way to create galaxies. Those fluctuations in the density of matter that do survive the inflationary period are so dense that they eventually collapse into galaxy-sized blackholes! Neither the precise way in which the transition to ordinary Hubbel expansion occurs nor the duration of the inflationary period are well determined.

If the inflationary cosmologies can be made to answer each of these issues satisfactorily we may have, as J. Richard Gott III has suggested, a most remarkable model of the universe where an almost infinite number of ‘bubble universes’ each having nearly infinite size, coexist in the same 4-dimensional spacetime; all of these bubble universes having been brought into existence at the same instant of creation. This is less troublesome than one might suspect since, if our universe is actually infinite as the available data suggests, so too was it infinite even at its moment of birth! It is even conceivable that the universe is ‘percolating’ with new bubble universes continually coming into existence. Our entire visible universe, out to the most distant quasar, would be but one infinitessimal patch within one of these bubble regions. Do these other universes have galaxies, stars, planets and living creatures statistically similar to those in our universe? We may never know. These other universes, born of the same paroxicism of Creation as our own, are forever beyond our scrutiny but obviously not our imaginations!

Beyond The Beginning…

Finally, what of the period before Grand Unification? We may surmise that at higher temperatures than the GUT Era, even the supermassive Higgs and Leptoquark bosons become massless and at long last we arrive at a time when the gravitational interaction is united with the weak, electromagnetic and strong forces. Yet, our quest for an understanding of the origins of the universe remains incomplete since gravity has yet to be brought into unity with the remaining forces on a theoretical basis. This last step promises to be not only the most difficult one to take on the long road to unification but also appears to hold the greatest promise for shedding light on some of the most profound mysteries of the physical world. Even now, a handful of theorists around the world are hard at work on a theory called Supergravity which unites the force carriers (photons, gluons, gravitons and the weak interaction bosons) with the particles that they act on (quarks, electrons etc). Supergravity theory also predicts the existence of new particles called photinos and gravitinos. There is even some speculation that the photinos may fill the entire universe and account for the unseen ‘missing’ matter that is necessary to give the universe the critical mass required to make it exactly Euclidean. The gravitinos, on the other hand, prevent calculations involving the exchange of gravitons from giving infinite answers for problems where the answers are known to be perfectly finite. Hitherto, these calculations did not include the affects of the gravitinos.

Perhaps during the next decade, more of the details of the last stage of Unification will be hammered out at which time the entire story of the birth of our universe can be told. This is, indeed, an exciting time to be living through in human history. Will future generations forever envy us our good fortune, to have witnessed in our lifetimes the unfolding of the first comprehensive theory of Existence?

Astronomy, Physics

What is Space? Part I

Does Space Have More Than 3 Dimensions?
Written by Sten Odenwald
Copyright (C) 1984 Kalmbach Publishing. Reprinted by permission

The intuitive notion that the universe has three dimensions seems to be an irrefutable fact. After all, we can only move up or down, left or right, in or out. But are these three dimensions all we need to describe nature? What if there aree, more dimensions ? Would they necessarily affect us? And if they didn’t, how could we possibly know about them? Some physicists and mathematicians investigating the beginning of the universe think they have some of the answers to these questions. The universe, they argue, has far more than three, four, or five dimensions. They believe it has eleven! But let’s step back a moment. How do we know that our universe consists of only three spatial dimensions? Let’s take a look at some “proofs.”

On a 2-dimensional piece of paper you can draw an infinite number of polygons.  But when you try this same trick in 3-dimensions you run up against a problem.There are five and only five regular polyhedra. A regular polyhedron is defined as a solid figure whose faces are identical polygons – triangles, squares, and pentagons – and which is constructed so that only two faces meet at each edge. If you were to move from one face to another, you would cross over only one edge. Shortcuts through the inside of the polyhedron that could get you from one face to another are forbidden. Long ago, the mathematician Leonhard Euler demonstrated an important relation between the number of faces (F), edges (E), and corners (C) for every regular polyhedron: C – E + F = 2. For example, a cube has 6 faces, 12 edges, and 8 corners while a dodecahedron has 12 faces, 30 edges, and 20 corners. Run these numbers through Euler’s equation and the resulting answer is always two, the same as with the remaining three polyhedra. Only five solids satisfy this relationship – no more, no less.

Not content to restrict themselves to only three dimensions, mathematicians have generalized Euler’s relationship to higher dimensional spaces and, as you might expect, they’ve come up with some interesting results. In a world with four spatial dimensions, for example, we can construct only six regular solids. One of them – the “hypercube” – is a solid figure in 4-D space bounded by eight cubes, just as a cube is bounded by six square faces. What happens if we add yet another dimension to space? Even the most ambitious geometer living in a 5-D world would only be able to assemble thee regular solids. This means that two of the regular solids we know of – the icosahedron and the dodecahedron – have no partners in a 5-D universe.
For those of you who successfully mastered visualizing a hypercube, try imagining what an “ultracube” looks like. It’s the five- dimensional analog of the cube, but this time it is bounded by one hypercube on each of its 10 faces! In the end, if our familiar world were not three-dimensional, geometers would not have found only five regular polyhedra after 2,500 years of searching. They would have found six (with four spatial dimension,) or perhaps only three (if we lived in a 5-D universe). Instead, we know of only five regular solids. And this suggests that we live in a universe with, at most, three spatial dimensions.

All right, let’s suppose our universe actually consists of four spatial dimensions. What happens? Since relativity tells us that we must also consider time as a dimension, we now have a space-time consisting of five dimensions. A consequence of 5-D space-time is that gravity has freedom to act in ways we may not want it to.

To the best available measurements, gravity follows an inverse square law; that is, the gravitational attraction between two objects rapidly diminishes with increasing distance. For example, if we double the distance between two objects, the force of gravity between them becomes 1/4 as strong; if we triple the distance, the force becomes 1/9 as strong, and so on. A five- dimensional theory of gravity introduces additional mathematical terms to specify how gravity behaves. These terms can have a variety of values, including zero. If they were zero, however, this would be the same as saying that gravity requires only three space dimensions and one time dimension to “give it life.” The fact that the Voyager space- craft could cross billions of miles of space over several years and arrive vithin a few seconds of their predicted times is a beautiful demonstration that we do not need extra-spatial dimensions to describe motions in the Sun’s gravitational field.

From the above geometric and physical arguments, we can conclude (not surprisingly) that space is three-dimensional – on scales ranging from that of everyday objects to at least that of the solar system. If this were not the case, then geometers would have found more than five regular polyhedra and gravity would function very differently than it does – Voyager would not have arrived on time. Okay, so we’ve determined that our physical laws require no more than the three spatial dimensions to describe how the universe works. Or do they? Is there perhaps some other arena in the physical world where multidimensional space would be an asset rather than a liability?

Since the 1920s, physicists have tried numerous approaches to unifying the principal natural interactions: gravity, electromagnetism, and the strong and weak forces in atomic nuclei. Unfortunately, physicists soon realized that general relativity in a four-dimensional space-time does not have enough mathematical “handles” on which to hang the frameworks for the other three forces. Between 1921 and 1927, Theodor Kaluza and Oskar Klein developed the first promising theory combining gravity and electromagnetism. They did this by extending general relativity to five dimensions. For most of us, general relativity is mysterious enough in ordinary four-dimensional space-time. What wonders could lie in store for us with this extended universe?

General relativity in five dimensions gave theoreticians five additional quantities to manipulate beyond the 10 needed to adequately define the gravitational field. Kaluza and Klein noticed that four of the five extra quantities could be identified with the four components needed to define the electromagnetic field. In fact, to the delight of Kaluza and Klein, these four quantities obeyed the same types of equations as those derived by Maxwell in the late 1800s for electromagnetic radiationl Although this was a promising start, the approach never really caught on and was soon buried by the onrush of theoretical work on the quantum theory of electromagnetic force. It was not until work on supergravity theory began in 1975 that Kaluza and Klein’s method drew renewed interest. Its time had finally come.

What do theoreticians hope to gain by stretching general relativity beyond the normal four dimensions of space-time? Perhaps by studying general relativity in a higher-dimensional formulation, we can explain some of the constants needed to describe the natural forces. For instance, why is the proton 1836 times more massive than the electron? Why are there only six types of quarks and leptons? Why are neutrinos massless? Maybe such a theory can give us new rules for calculating the masses of fundamental particles and the ways in which they affect one another. These higher-dimensional relativity theories may also tell us something about the numbers and properties of a mysterious new family of particles – the Higgs bosons – whose existence is predicted by various cosmic unification schemes. (See “The Decay of the False Vacuum,” ASTRONOMY, November 1983.)

These expectations are not just the pipedreams of physicists – they actually seem to develop as natural consequences of certain types of theories studied over the last few years. In 1979, John Taylor at Kings College in London found that some higher- dimensional formalisms can give predictions for the maximum mass of the Higgs bosons (around 76 times that of the proton.) As they now stand, unification theories can do no more than predict the existence of these particles – they cannot provide specific details about their physical characteristics. But theoreticians may be able to pin down some of these details by using extended theories of general relativity. Experimentally, we know of six leptons: the electron, the muon, the tauon, and their three associated neutrinos. The most remarkable prediction of these extended relativity schemes, however, holds that the number of leptons able to exist in a universe is related to the number of dimensions of space-time. In a 6-D space-time, for example, only one lepton – presumably the electron – can exist. In a 10-D space-time, four leptons can exist – still not enough to accommodate the six we observe. In a 12-D space- time, we can account for all six known leptons – but we also acquire two additional leptons that have not yet been detected. Clearly, we would gain much on a fundamental level if we could increase the number of dimensions in our theories just a little bit.

How many additional dimensions do we need to consider in order to account for the elementary particles and forces that we know of today? Apparently we require at least one additional spatial dimension for every distinct “charge” that characterizes how each force couples to matter. For the electromagnetic force, we need two electric charges: positive and negative. For the strong force that binds quarks together to form, among other things, protons and neutrons, we need three “color” charges – red, blue, and green. Finally, we need two “weak” charges to account for the weak nuclear force. if we add a spatial dimension for each of these charges, we end up with a total of seven extra dimensions. The properly extended theory of general relativity we seek is one with an 11 -dimensional space-time, at the very least. Think of it – space alone must have at least 10 dimensions to accomodate all the fields known today.

Of course, these additional dimensions don’t have to be anything like those we already know about. In the context of modern unified field theory, these extra dimensions are, in a sense, internal to the particles themselves – a “private secret,” shared only by particles and the fields that act on them! These dimensions are not physically observable in the same sense as the three spatial dimensions we experience; they’stand in relation to the normal three dimensions of space much like space stands in relation to time.

With today’s veritable renaissance in finding unity among the forces and particles that compose the cosmos, some by methods other than those we have discussed, these new approaches lead us to remarkably similar conclusions. It appears that a four-dimensional space-time is simply not complex enough for physics to operate as it does.

We know that particles called bosons mediate the natural forces. We also know that particles called fermions are affected by these forces. Members of the fermion family go by the familiar names of electron, muon, neutrino, and quark; bosons are the less well known graviton, photon, gluon, and intermediate vector bosons. Grand unification theories developed since 1975 now show these particles to be “flavors” of a more abstract family of superparticies – just as the muon is another type of electron. This is an expression of a new kind of cosmic symmetry – dubbed supersymmetry, because it is all-encompassing. Not only does it include the force-carrying bosons, but it also includes the particles on which these forces act. There also exists a corresponding force to help nature maintain supersymmetry during the various interactions. It’s called supergravity. Supersymmetry theory introduces two new types of fundamental particles – gravitinos and photinos. The gravitino has the remarkable property of mathematically moderating the strength, of various kinds of interactions involving the exchange of gravitons. The photino, cousin of the photon, may help account for the “missing mass” in the universe.

Supersymmetry theory is actually a complex of eight different theories, stacked atop one another like the rungs of a ladder. The higher the rung, the larger is its complement of allowed fermion and boson particle states. The “roomiest” theory of all seems to be SO(8), (pronounced ess-oh-eight), which can hold 99 different kinds of bosons and 64 different kinds of fermions. But SO(8) outdoes its subordinate, SO(7), by only one extra dimension and one additional particle state. Since SO(8) is identical to SO(7) in all its essential features, we’ll discuss SO(7) instead. However, we know of far more than the 162 types of particles that SO(7) can accommodate, and many of the predicted types have never been observed (like the massless gravitino). SO(7) requires seven internal dimensions in addition to the four we recognize – time and the three “every day” spatial dimensions. If SO(7) at all mirrors reality, then our universe must have at least 11 dimensions! Unfortunately, it has been demonstrated by W. Nahm at the European Center for Nuclear Research in Geneva, Switzerland that supersymmetry theories for space-times with more than 11 dimensions are theoretically impossible. SO(7) evidently has the largest number of spatial dimensions possible, but it still doesn’t have enough room to accommodate all known types of particles.

It is unclear where these various avenues of research lead. Perhaps nowhere. There is certainly ample historical precedent for ideas that were later abandoned because they turned out to be conceptual dead-ends. Yet what if they turn out to be correct at some level? Did our universe begin its life as some kind of 11-dimensional “object” which then crystallized into our four- dimensional cosmos?

Although these internal dimensions may not have much to do with the real world at the present time, this may not always have been the case. E. Cremmer and J. Scherk of I’Ecole Normale Superieure in Paris have shown that just as the universe went through phase transitions in its early history when the forces of nature became distinguishable, the universe may also have gone through a phase transition when mensionality changed. Presumably matter has something like four external dimensions (the ones we encounter every day) and something like seven internal dimensions. Fortunately for us, these seven extra dimensions don’t reach out into the larger 4-D realm where we live. If they did, a simple walk through the park might become a veritable obstacle course, littered with wormholes in space and who knows what else!

Alan Chocos and Steven Detweiler of Yale University have considered the evolution of a universe that starts out being five- dimensional. They discovered that while the universe eventually does evolve to a state where three of the four spatial dimensions expand to become our world at large, the extra fourth spatial dimension shrinks to a size of 10^-31 centimeter by the present time. The fifth dimension to the universe has all but vanished and is 20 powers of 10 – 100 billion billion times – smaller than the size of a proton. Although the universe appears four- dimensional in space-time, this perception is accidental due to our large size compared to the scale of the other dimensions. Most of us think of a dimension as extending all the way to infinity, but this isn’t the full story. For example, if our universe is really destined to re-collapse in the distant future, the three- dimensional space we know today is actually limited itself – it will eventually possess a maximum, finite size. It just so happens that the physical size of human beings forces us to view these three spatial dimensions as infinitely large.

It is not too hard to reconcile ourselves to the notion that the fifth (or sixth, or eleventh) dimension could be smaller than an atomic nucleus – indeed, we can probably be thankful that this is the case.

Astronomy, Physics

The Cosmological Redshift

Galaxy Redshifts Reconsidered

Written by Sten Odenwald 
Copyright (C) 1993 Sky Publishing Corporation. Reprinted by permission. See February 1993 issue

Since its discovery nearly 65 years ago, the cosmological redshift has endured as one of the most persuasive ‘proofs’ that our universe is expanding. The steps leading to its discovery are well known. Soon after Christian Doppler discovered that motion produces frequency shifts in 1842, astronomers began an aggressive spectroscopic program to measure the velocities of stars and planets using their Doppler shifts. This continued through the first few decades of the 20th century ‘culminating’ in the work by Vesto Slipher, Edwin Hubble and Milton Humason on the so-called spiral nebulae — distinctly non- stellar objects that also seemed to display star-like Doppler shifts. So long as velocities of only a few hundred kilometers per second were measured, no one questioned that the frequency shifts for the spiral nebulae indicated relative motion just as they had for stars and planets.
But, during the 1920’s and 30’s spiral nebulae with Doppler shifts of over 34,000 kilometers per second were discovered. In a letter by Hubble to the Dutch cosmologist Willem De Sitter in 1931, he stated his concerns about these velocities by saying “… we use the term ‘apparent velocities’ in order to emphasize the empirical feature of the correlation. The interpretation, we feel, should be left to you and the very few others who are competent to discuss the matter with authority.” Dispite this cautionary note, the fact of the matter was that the redshifts measured for the distant galaxies LOOKED like Doppler shifts. The terms ‘recession velocity’ and ‘expansion velocity’ were quickly brought into service by astronomers at the telescope, and by popularizers, to describe the physical basis for the redshift.

As astronomers explored the universe to greater depths, galaxies and quasars appeared to be rushing away at faster and faster speeds. It seems to be a completely natural consequence of the outrushing of matter from the big bang. Like a sparkling display of fireworks on a warm summer evening, we imagine ourselves standing on one of those galactic ‘cinders’, watching the others rush past us into the dark void of infinite space. Upon closer examination, however, this intuitively-compelling and seductive mental image is both inadequate and misleading.

The Mysteries of Relativity

Big bang cosmology is based on Einstein’s general theory of relativity. It is a theory transcending both Newton’s mechanics and Einstein’s special theory of relativity, introducing us to concepts that do not exist within the older theories. Nor are these concepts easily comprehensible by our common sense which has been honed by organic evolution to see the world only through a narrow set of glasses.

For example, special relativity is based on the difficult-to-fathom postulate that the speed of light is absolutely constant when measured in reference frames moving at a constant speed. From this emerges the concept of ‘spacetime’ which then becomes the arena for all phenomena involving time dilation, length contraction and the Twin Paradox. Beyond special relativity lies the incomparably more alien landscape of general relativity. Gravitational fields now become geometric curvatures of spacetime. This has no analog in special relativity based as it is on a perfectly flat spacetime that remains aloof from any influence on it by matter or energy.

Just as the constancy of the speed of light led to the Twin Paradox, the curvature of spacetime leads to its own menageri of peculiar phenomena. One of these involves the slowing-down of clocks in the presence of a strong gravitational field. Related to this is the “gravitational redshift” which occurs when the frequency of light sent from the surface of a body is shifted to lower frequencies during the journey to the observer. This redshift is not related to the famous Doppler shift since the observer is not in motion relative to the body emitting the light signal!

A second phenomenon predicted by general relativity that also has no analog in special relativity is the cosmological redshift. Simply stated, the cosmological redshift occurs because the curvature of spacetime was smaller in the past when the universe was younger than it is now. Light waves become stretched en route between the time they were emitted long ago, and the time they are detected by us today.

The Doppler shift and cosmology

It is tempting to refer to cosmological redshifts as Doppler shifts. This choice of interpretation has in the years since Hubble’s work led to an unfortunate misunderstanding of big bang cosmology, obscurring one of its most mysterious beauties. As noted with a hint of frustration by cosmologists such as Steven Weinberg and Jaylant Narlikar and John Wheeler, “The frequency of light is also affected by the gravitational field of the universe, and it is neither useful nor strictly correct to interpret the frequency shifts of light…in terms of the special relativistic Doppler effect.”.

By refering to cosmological redshifts as Doppler shifts, we are insisting that our Newtonian intuition about motion still applies without significant change to the cosmological arena. A result of this thinking is that quasars now being detected at redshifts of Z = 4.0 would have to be interpreted as traveling a speeds of more than V = Z x c or 4 times the speed of light. This is, of course, quite absurd, because we all know that no physical object may travel faster than the speed of light.

To avoid such apparently nonsensical speeds, many popularizers use the special relativistic Doppler formula to show that quasars are really not moving faster than light. The argument being that for large velocities, special relativity replaces Newtonian physics as the correct framework for interpreting the world. By using a special relativistic velocity addition formula the quasar we just discussed has a velocity of 92 percent the speed of light. Although we now have a feeling that Reason has returned to our description of the universe, in fact, we have only replaced one incomplete explanation for another. The calculation of the quasar’s speed now presupposes that special relativity ( a theory of flat spacetime) is applicable even at cosmological scales where general relativity predicts that spacetime curvature becomes important. This is equivalent to a surveyor making a map of the state of California, and not allowing for the curvature of the earth!

The adoption of the special relativistic Doppler formula by many educators has led to a peculiar ‘hybrid’ cosmology which attempts to describe big bang cosmology using general relativity, but which is still firmly mired in the ruberik of special relativity. For instance, under the entry ‘redshift’ in the Cambridge Encyclopedia of Astronomy it is explicitly acknowledged that the redshift is not a Doppler shift, but less than two paragraphs later, the special relativistic Doppler formula is introduced to show how quasars are moving slower than the speed of light! It is also common for popularizers of cosmology to describe how ‘space itself stretches’ yet continue to describe the expansion of the universe as motion governed by the restrictions of special relativity. What’s going on here?

General relativity to the rescue

By adopting general relativity as the proper guide, such contradictions are eliminated. General relativity leads us to several powerful conclusions about our cosmos: 1) special relativity is inapplicable for describing the larger universe; 2) the concepts of distance and motion are not absolutely defined and 3) Preexisting spacetime is undefined. Each of these conclusions are as counter-intuitive as the Twin Paradox or as the particle/wave dualism of quantum mechanics. As Nobel Physicist John Wheeler once put it “If you are not completely confused by quantum mechanics, you do not understand it” The same may be said for general relativity.

The first conclusion means that we cannot trust even the insights hard won from special relativity to accurately represent the ‘big picture’ of the universe. General relativity must replace special relativity in cosmology because it denies a special role to observers moving at constant velocity, extending special relativity into the arena of accelerated observers. It also denies a special significance to special relativity’s flat spacetime by relegating it to only a microscopic domain within a larger geometric possibility. Just as Newtonian physics gave way to special relativity for describing high speed motion, so too does special relativity give way to general relativity. This means that the special relativistic Doppler formula should not, in fact cannot, be used to quantify the velocity of distant quasars. We have no choice in this matter if we want to maintain the logical integrity of both theories.

Distance and motion

The second conclusion is particularly upsetting because if we cannot define what we mean by distance, how then can we discuss in meaningful terms the ‘motion’ of distant quasars, or a Hubble Law interpreted as a distance versus velocity relation? In a small region of spacetime, we can certainly define motion as we always have because space has a static, flat geometry. When a body moves from point x to point y in a time interval, T, we say it is moving with a speed of S = (x – y)/T. There are also specific experimental ways of measuring x, y and T to form the quotent S by using clocks and rulers. The crucil feature behind these measurements is that nothing happens to the geometry of space during the experiment to change the results of the measuring process.

In the cosmological setting which we believe is accurately described by general relativity, we have none of these luxuries! Astronomers cannot wait millions of years to measure quasar proper motions. They cannot, like Highway Patrol officers, bounce radar beams off distant galaxies to establish their relative distances or speeds. Unlike all other forms of motion that have been previously observed, cosmological ‘motion’ cannot be directly observed. It can only be INFERRED from observations of the cosmological redshift, which general relativity then TELLS US means that the universe is expanding.

In big bang cosmology, galaxies are located at fixed positions in space. They may perform small dances about these positions in accordance with special relativity and local gravitational fields, but the real ‘motion’ is in the literal expansion of space between them! This is not a form of movement that any human has ever experienced. It is, therefore, not surprising that our intuition reels at its implication and seeks other less radical interpretations for it including special relativity. But even the exotic language and conundrums of special relativity cannot help us. Instead we are forced to interrogate the mathematics of general relativity itself for whatever landmarks it can provide. In doing so, we are left, however, with a riddle as profound as that of the Twin Paradox, and equally challenging to explain.

Two galaxies permanently located at positions (x1 , y1 , z1 ) and ( x2 , y2 , z2 ) at one time find themselves one billion light years apart. Then a few billion years later while located at the same coordinates, they find themselves 3 billion light years apart. The galaxies have not ‘moved’, nevertheless, their separations have increased. In fact, when the universe was only one year old, the separations between these galaxies were increasing at 300 times the speed of light! Space can expand faster than the speed of light in general relativity because space does not represent matter or energy. The displacements that arise from its dilation produce an entirely new kind of motion for which even our special relativistically-trained intuitions remain profoundly silent. Like that gentleman from Main once said “You can’t get there [to general relativity] from here [special relativity]”. To the extent that general relativity has been tested and found correct, we have no choice but to accept its consequences at face value.

Space, time and matter

The last conclusion drawn from general relativistic cosmology is that, unlike special relativity, it is not physically meaningful to speak of spacetime existing independently of matter and energy. In big bang cosmology, both space and time came into existence along side matter and energy at ‘time zero’. If our universe contains more than a critical density of matter and energy, its spacetime is forever finite and bounded, in a shape analogous to a sphere. Beyond this boundary, space and time simply do not exist. In fact, general relativity allows the Conservation of Energy to be suspended so that matter and energy may be created quite literally from the nothingness of curved spacetime. General relativity provides a means for ‘jump-starting’ Creation!

Big bang cosmology is both a profoundly beautiful, and disturbing, model for our universe, its shape and its destiny. It contains many surprises which have yet to be completely worked-out. But one feature of the evolving universe seems absolutly clear, the big bang was not some grand fireworks display, but an event of a completely different order. It resembled more an expanding soap bubble film upon which galactic dust motes are carried along for the ride. This film represents the totality of all the space and matter in our universe, and it expands into a mysterious primordial void which is itself empty of space, dimension, time or matter.

In the future it is hoped that a death knell will finally have sounded for the last vestage of the older thinking. With the Doppler interpretation of the cosmological redshift at last reconsidered, and rejected, we will finally be able to embrace the essential beauty and mystery of cosmic expansion as it was originally envisioned by its discoverers.