Finite or infinite?
News •
An issue that arises when one contemplates the universe at large is whether space and time are infinite or finite. After many centuries of thought by some of the best minds, humanity has still not arrived at conclusive answers to these questions. Aristotle’s answer was that the material universe must be spatially finite, for if stars extended to infinity, they could not perform a complete rotation around Earth in 24 hours. Space must then itself also be finite because it is merely a receptacle for material bodies. On the other hand, the heavens must be temporally infinite, without beginning or end, since they are imperishable and cannot be created or destroyed.
Except for the infinity of time, these views came to be accepted religious teachings in Europe before the period of modern science. The most notable person to publicly express doubts about restricted space was the Italian philosopher-mathematician Giordano Bruno, who asked the obvious question that, if there is a boundary or edge to space, what is on the other side? For his advocacy of an infinity of suns and earths, he was burned at the stake in 1600.
In 1610 the German astronomer Johannes Kepler provided a profound reason for believing that the number of stars in the universe had to be finite. If there were an infinity of stars, he argued, then the sky would be completely filled with them and night would not be dark! This point was rediscussed by the astronomers Edmond Halley of England and Jean-Philippe-Loys de Chéseaux of Switzerland in the 18th century, but it was not popularized as a paradox until Wilhelm Olbers of Germany took up the problem in the 19th century. The difficulty became potentially very real with American astronomer Edwin Hubble’s measurement of the enormous extent of the universe of galaxies with its large-scale homogeneity and isotropy. His discovery of the systematic recession of the galaxies provided an escape, however. At first people thought that the redshift effect alone would suffice to explain why the sky is dark at night—namely, that the light from the stars in distant galaxies would be redshifted to long wavelengths beyond the visible regime. The modern consensus is, however, that a finite age for the universe is a far more important effect. Even if the universe is spatially infinite, photons from very distant galaxies simply do not have the time to travel to Earth because of the finite speed of light. There is a spherical surface, the cosmic event horizon (13.8 billion light-years in radial distance from Earth at the current epoch), beyond which nothing can be seen even in principle; and the number (roughly 1010) of galaxies within this cosmic horizon, the observable universe, are too few to make the night sky bright.
When one looks to great distances, one is seeing things as they were a long time ago, again because light takes a finite time to travel to Earth. Over such great spans, do the classical notions of Euclid concerning the properties of space necessarily continue to hold? The answer given by Einstein was: No, the gravitation of the mass contained in cosmologically large regions may warp one’s usual perceptions of space and time; in particular, the Euclidean postulate that parallel lines never cross need not be a correct description of the geometry of the actual universe. And in 1917 Einstein presented a mathematical model of the universe in which the total volume of space was finite yet had no boundary or edge. The model was based on his theory of general relativity that utilized a more generalized approach to geometry devised in the 19th century by the German mathematician Bernhard Riemann.
Gravitation and the geometry of space-time
The physical foundation of Einstein’s view of gravitation, general relativity, lies on two empirical findings that he elevated to the status of basic postulates. The first postulate is the relativity principle: local physics is governed by the theory of special relativity. The second postulate is the equivalence principle: there is no way for an observer to distinguish locally between gravity and acceleration. The motivation for the second postulate comes from Galileo’s observation that all objects—independent of mass, shape, colour, or any other property—accelerate at the same rate in a (uniform) gravitational field.
Einstein’s theory of special relativity, which he developed in 1905, had as its basic premises (1) the notion (also dating back to Galileo) that the laws of physics are the same for all inertial observers and (2) the constancy of the speed of light in a vacuum—namely, that the speed of light has the same value (3 × 1010 centimetres per second [cm/sec], or 2 × 105 miles per second [miles/sec]) for all inertial observers independent of their motion relative to the source of the light. Clearly, this second premise is incompatible with Euclidean and Newtonian precepts of absolute space and absolute time, resulting in a program that merged space and time into a single structure, with well-known consequences. The space-time structure of special relativity is often called “flat” because, among other things, the propagation of photons is easily represented on a flat sheet of graph paper with equal-sized squares. Let each tick on the vertical axis represent one light-year (9.46 × 1017 cm [5.88 × 1012 miles]) of distance in the direction of the flight of the photon, and each tick on the horizontal axis represent the passage of one year (3.16 × 107 seconds) of time. The propagation path of the photon is then a 45° line because it flies one light-year in one year (with respect to the space and time measurements of all inertial observers no matter how fast they move relative to the photon).
The principle of equivalence in general relativity allows the locally flat space-time structure of special relativity to be warped by gravitation, so that (in the cosmological case) the propagation of the photon over thousands of millions of light-years can no longer be plotted on a globally flat sheet of paper. To be sure, the curvature of the paper may not be apparent when only a small piece is examined, thereby giving the local impression that space-time is flat (i.e., satisfies special relativity). It is only when the graph paper is examined globally that one realizes it is curved (i.e., satisfies general relativity).
In Einstein’s 1917 model of the universe, the curvature occurs only in space, with the graph paper being rolled up into a cylinder on its side, a loop around the cylinder at constant time having a circumference of 2πR—the total spatial extent of the universe. Notice that the “radius of the universe” is measured in a “direction” perpendicular to the space-time surface of the graph paper. Since the ringed space axis corresponds to one of three dimensions of the actual world (any will do since all directions are equivalent in an isotropic model), the radius of the universe exists in a fourth spatial dimension (not time) which is not part of the real world. This fourth spatial dimension is a mathematical artifice introduced to represent diagrammatically the solution (in this case) of equations for curved three-dimensional space that need not refer to any dimensions other than the three physical ones. Photons traveling in a straight line in any physical direction have trajectories that go diagonally (at 45° angles to the space and time axes) from corner to corner of each little square cell of the space-time grid; thus, they describe helical paths on the cylindrical surface of the graph paper, making one turn after traveling a spatial distance 2πR. In other words, always flying dead ahead, photons would return to where they started from after going a finite distance without ever coming to an edge or boundary. The distance to the “other side” of the universe is therefore πR, and it would lie in any and every direction; space would be closed on itself.
Now, except by analogy with the closed two-dimensional surface of a sphere that is uniformly curved toward a centre in a third dimension lying nowhere on the two-dimensional surface, no three-dimensional creature can visualize a closed three-dimensional volume that is uniformly curved toward a centre in a fourth dimension lying nowhere in the three-dimensional volume. Nevertheless, three-dimensional creatures could discover the curvature of their three-dimensional world by performing surveying experiments of sufficient spatial scope. They could draw circles, for example, by tacking down one end of a string and tracing along a single plane the locus described by the other end when the string is always kept taut in between (a straight line) and walked around by a surveyor. In Einstein’s universe, if the string were short compared to the quantity R, the circumference of the circle divided by the length of the string (the circle’s radius) would nearly equal 2π = 6.2837853…, thereby fooling the three-dimensional creatures into thinking that Euclidean geometry gives a correct description of their world. However, the ratio of circumference to length of string would become less than 2π when the length of string became comparable to R. Indeed, if a string of length πR could be pulled taut to the antipode of a positively curved universe, the ratio would go to zero. In short, at the tacked-down end the string could be seen to sweep out a great arc in the sky from horizon to horizon and back again; yet, to make the string do this, the surveyor at the other end need only walk around a circle of vanishingly small circumference.
To understand why gravitation can curve space (or more generally, space-time) in such startling ways, consider the following thought experiment that was originally conceived by Einstein. Imagine an elevator in free space accelerating upward, from the viewpoint of a woman in inertial space, at a rate numerically equal to g, the gravitational field at the surface of Earth. Let this elevator have parallel windows on two sides, and let the woman shine a brief pulse of light toward the windows. She will see the photons enter close to the top of the near window and exit near the bottom of the far window because the elevator has accelerated upward in the interval it takes light to travel across the elevator. For her, photons travel in a straight line, and it is merely the acceleration of the elevator that has caused the windows and floor of the elevator to curve up to the flight path of the photons.
Let there now be a man standing inside the elevator. Because the floor of the elevator accelerates him upward at a rate g, he may—if he chooses to regard himself as stationary—think that he is standing still on the surface of Earth and is being pulled to the ground by its gravitational field g. Indeed, in accordance with the equivalence principle, without looking out the windows (the outside is not part of his local environment), he cannot perform any local experiment that would inform him otherwise. Let the woman shine her pulse of light. The man sees, just like the woman, that the photons enter near the top edge of one window and exit near the bottom of the other. And just like the woman, he knows that photons propagate in straight lines in free space. (By the relativity principle, they must agree on the laws of physics if they are both inertial observers.) However, since he actually sees the photons follow a curved path relative to himself, he concludes that they must be bent by the force of gravity. The woman tries to tell him there is no such force at work; he is not an inertial observer. Nonetheless, he has the solidity of Earth beneath him, so he insists on attributing his acceleration to the force of gravity. According to Einstein, they are both right. There is no need to distinguish locally between acceleration and gravity—the two are in some sense equivalent. But if that is the case, then it must be true that gravity—“real” gravity—can actually bend light. And indeed it can, as many experiments have shown since Einstein’s first discussion of the phenomenon.
It was the genius of Einstein to go even further. Rather than speak of the force of gravitation having bent the photons into a curved path, might it not be more fruitful to think of photons as always flying in straight lines—in the sense that a straight line is the shortest distance between two points—and that what really happens is that gravitation bends space-time? In other words, perhaps gravitation is curved space-time, and photons fly along the shortest paths possible in this curved space-time, thus giving the appearance of being bent by a “force” when one insists on thinking that space-time is flat. The utility of taking this approach is that it becomes automatic that all test bodies fall at the same rate under the “force” of gravitation, for they are merely producing their natural trajectories in a background space-time that is curved in a certain fashion independent of the test bodies. What was a minor miracle for Galileo and Newton becomes the most natural thing in the world for Einstein.
To complete the program and to conform with Newton’s theory of gravitation in the limit of weak curvature (weak field), the source of space-time curvature would have to be ascribed to mass (and energy). The mathematical expression of these ideas constitutes Einstein’s theory of general relativity, one of the most beautiful artifacts of pure thought ever produced. The American physicist John Archibald Wheeler and his colleagues summarized Einstein’s view of the universe in these terms:
Curved spacetime tells mass-energy how to move;
mass-energy tells spacetime how to curve.
Contrast this with Newton’s view of the mechanics of the heavens:
Force tells mass how to accelerate;
mass tells gravity how to exert force.
Notice therefore that Einstein’s worldview is not merely a quantitative modification of Newton’s picture (which is also possible via an equivalent route using the methods of quantum field theory) but represents a qualitative change of perspective. And modern experiments have amply justified the fruitfulness of Einstein’s alternative interpretation of gravitation as geometry rather than as force. His theory would have undoubtedly delighted the Greeks.
Relativistic cosmologies
Einstein’s model
To derive his 1917 cosmological model, Einstein made three assumptions that lay outside the scope of his equations. The first was to suppose that the universe is homogeneous and isotropic in the large (i.e., the same everywhere on average at any instant in time), an assumption that the English astrophysicist Edward A. Milne later elevated to an entire philosophical outlook by naming it the cosmological principle. Given the success of the Copernican revolution, this outlook is a natural one. Newton himself had it implicitly in mind when he took the initial state of the universe to be everywhere the same before it developed “ye Sun and Fixt stars.”
The second assumption was to suppose that this homogeneous and isotropic universe had a closed spatial geometry. As described above, the total volume of a three-dimensional space with uniform positive curvature would be finite but possess no edges or boundaries (to be consistent with the first assumption).
The third assumption made by Einstein was that the universe as a whole is static—i.e., its large-scale properties do not vary with time. This assumption, made before Hubble’s observational discovery of the expansion of the universe, was also natural; it was the simplest approach, as Aristotle had discovered, if one wishes to avoid a discussion of a creation event. Indeed, the philosophical attraction of the notion that the universe on average is not only homogeneous and isotropic in space but also constant in time was so appealing that a school of English cosmologists—Hermann Bondi, Fred Hoyle, and Thomas Gold—would call it the perfect cosmological principle and carry its implications in the 1950s to the ultimate refinement in the so-called steady-state theory.
To his great chagrin Einstein found in 1917 that with his three adopted assumptions, his equations of general relativity—as originally written down—had no meaningful solutions. To obtain a solution, Einstein realized that he had to add to his equations an extra term, which came to be called the cosmological constant. If one speaks in Newtonian terms, the cosmological constant could be interpreted as a repulsive force of unknown origin that could exactly balance the attraction of gravitation of all the matter in Einstein’s closed universe and keep it from moving. The inclusion of such a term in a more general context, however, meant that the universe in the absence of any mass-energy (i.e., consisting of a vacuum) would not have a space-time structure that was flat (i.e., would not have satisfied the dictates of special relativity exactly). Einstein was prepared to make such a sacrifice only very reluctantly, and, when he later learned of Hubble’s discovery of the expansion of the universe and realized that he could have predicted it had he only had more faith in the original form of his equations, he regretted the introduction of the cosmological constant as the “biggest blunder” of his life. Ironically, observations of distant supernovas have shown the existence of dark energy, a repulsive force that is the dominant component of the universe.
De Sitter’s model
It was also in 1917 that the Dutch astronomer Willem de Sitter recognized that he could obtain a static cosmological model differing from Einstein’s simply by removing all matter. The solution remains stationary essentially because there is no matter to move about. If some test particles are reintroduced into the model, the cosmological term would propel them away from each other. Astronomers now began to wonder if this effect might not underlie the recession of the spiral galaxies.
Friedmann-Lemaître models
In 1922 Aleksandr A. Friedmann, a Russian meteorologist and mathematician, and in 1927 Georges Lemaître, a Belgian cleric, independently discovered solutions to Einstein’s equations that contained realistic amounts of matter. These evolutionary models correspond to big bang cosmologies. Friedmann and Lemaître adopted Einstein’s assumption of spatial homogeneity and isotropy (the cosmological principle). They rejected, however, his assumption of time independence and considered both positively curved spaces (“closed” universes) as well as negatively curved spaces (“open” universes). The difference between the approaches of Friedmann and Lemaître is that the former set the cosmological constant equal to zero, whereas the latter retained the possibility that it might have a nonzero value. To simplify the discussion, only the Friedmann models are considered here.
The decision to abandon a static model meant that the Friedmann models evolve with time. As such, neighbouring pieces of matter have recessional (or contractional) phases when they separate from (or approach) one another with an apparent velocity that increases linearly with increasing distance. Friedmann’s models thus anticipated Hubble’s law before it had been formulated on an observational basis. It was Lemaître, however, who had the good fortune of deriving the results at the time when the recession of the galaxies was being recognized as a fundamental cosmological observation, and it was he who clarified the theoretical basis for the phenomenon.
The geometry of space in Friedmann’s closed models is similar to that of Einstein’s original model; however, there is a curvature to time as well as one to space. Unlike Einstein’s model, where time runs eternally at each spatial point on an uninterrupted horizontal line that extends infinitely into the past and future, there is a beginning and end to time in Friedmann’s version of a closed universe when material expands from or is recompressed to infinite densities. These instants are called the instants of the “big bang” and the “big squeeze,” respectively. The global space-time diagram for the middle half of the expansion-compression phases can be depicted as a barrel lying on its side. The space axis corresponds again to any one direction in the universe, and it wraps around the barrel. Through each spatial point runs a time axis that extends along the length of the barrel on its (space-time) surface. Because the barrel is curved in both space and time, the little squares in the grid of the curved sheet of graph paper marking the space-time surface are of nonuniform size, stretching to become bigger when the barrel broadens (universe expands) and shrinking to become smaller when the barrel narrows (universe contracts).
It should be remembered that only the surface of the barrel has physical significance; the dimension off the surface toward the axle of the barrel represents the fourth spatial dimension, which is not part of the real three-dimensional world. The space axis circles the barrel and closes upon itself after traversing a circumference equal to 2πR, where R, the radius of the universe (in the fourth dimension), is now a function of the time t. In a closed Friedmann model, R starts equal to zero at time t = 0 (not shown in barrel diagram), expands to a maximum value at time t = tm (the middle of the barrel), and recontracts to zero (not shown) at time t = 2tm, with the value of tm dependent on the total amount of mass that exists in the universe.
Imagine now that galaxies reside on equally spaced tick marks along the space axis. Each galaxy on average does not move spatially with respect to its tick mark in the spatial (ringed) direction but is carried forward horizontally by the march of time. The total number of galaxies on the spatial ring is conserved as time changes, and therefore their average spacing increases or decreases as the total circumference 2πR on the ring increases or decreases (during the expansion or contraction phases). Thus, without in a sense actually moving in the spatial direction, galaxies can be carried apart by the expansion of space itself. From this point of view, the recession of galaxies is not a “velocity” in the usual sense of the word. For example, in a closed Friedmann model, there could be galaxies that started, when R was small, very close to the Milky Way system on the opposite side of the universe. Now, 1010 years later, they are still on the opposite side of the universe but at a distance much greater than 1010 light-years away. They reached those distances without ever having had to move (relative to any local observer) at speeds faster than light—indeed, in a sense without having had to move at all. The separation rate of nearby galaxies can be thought of as a velocity without confusion in the sense of Hubble’s law, if one wants, but only if the inferred velocity is much less than the speed of light.
On the other hand, if the recession of the galaxies is not viewed in terms of a velocity, then the cosmological redshift cannot be viewed as a Doppler shift. How, then, does it arise? The answer is contained in the barrel diagram when one notices that, as the universe expands, each small cell in the space-time grid also expands. Consider the propagation of electromagnetic radiation whose wavelength initially spans exactly one cell length (for simplicity of discussion), so that its head lies at a vertex and its tail at one vertex back. Suppose an elliptical galaxy emits such a wave at some time t1. The head of the wave propagates from corner to corner on the little square grids that look locally flat, and the tail propagates from corner to corner one vertex back. At a later time t2, a spiral galaxy begins to intercept the head of the wave. At time t2, the tail is still one vertex back, and therefore the wave train, still containing one wavelength, now spans one current spatial grid spacing. In other words, the wavelength has grown in direct proportion to the linear expansion factor of the universe. Since the same conclusion would have held if n wavelengths had been involved instead of one, all electromagnetic radiation from a given object will show the same cosmological redshift if the universe (or, equivalently, the average spacing between galaxies) was smaller at the epoch of transmission than at the epoch of reception. Each wavelength will have been stretched in direct proportion to the expansion of the universe in between.
A nonzero peculiar velocity for an emitting galaxy with respect to its local cosmological frame can be taken into account by Doppler-shifting the emitted photons before applying the cosmological redshift factor; i.e., the observed redshift would be a product of two factors. When the observed redshift is large, one usually assumes that the dominant contribution is of cosmological origin. When this assumption is valid, the redshift is a monotonic function of both distance and time during the expansional phase of any cosmological model. Thus, astronomers often use the redshift z as a shorthand indicator of both distance and elapsed time. Following from this, the statement “object X lies at z = a” means that “object X lies at a distance associated with redshift a”; the statement “event Y occurred at redshift z = b” means that “event Y occurred a time ago associated with redshift b.”
The open Friedmann models differ from the closed models in both spatial and temporal behaviour. In an open universe the total volume of space and the number of galaxies contained in it are infinite. The three-dimensional spatial geometry is one of uniform negative curvature in the sense that, if circles are drawn with very large lengths of string, the ratio of circumferences to lengths of string are greater than 2π. The temporal history begins again with expansion from a big bang of infinite density, but now the expansion continues indefinitely, and the average density of matter and radiation in the universe would eventually become vanishingly small. Time in such a model has a beginning but no end.
