Chapter 8. Sources

8.1 Sources in general relativity

8.1.1 Point sources in a background-independent theory

Schrödinger equation and Maxwell's equations treat spacetime as a stage on which particles and fields act out their roles. General relativity, however, is essentially a theory of spacetime itself. The role played by atoms or rays of light is so peripheral that by the time Einstein had derived an approximate version of the Schwarzschild metric, and used it to find the precession of Mercury's perihelion, he still had only vague ideas of how light and matter would fit into the picture. In his calculation, Mercury played the role of a test particle: a lump of mass so tiny that it can be tossed into spacetime in order to measure spacetime's curvature, without worrying about its effect on the spacetime, which is assumed to be negligible. Likewise the sun was treated as in one of those orchestral pieces in which some of the brass play from off-stage, so as to produce the effect of a second band heard from a distance. Its mass appears simply as an adjustable parameter m in the metric, and if we had never heard of the Newtonian theory we would have had no way of knowing how to interpret m.

When Schwarzschild published his exact solution to the vacuum field equations, Einstein suffered from philosophical indigestion. His strong belief in Mach's principle led him to believe that there was a paradox implicit in an exact spacetime with only one mass in it. If Einstein's field equations were to mean anything, he believed that they had to be interpreted in terms of the motion of one body relative to another. In a universe with only one massive particle, there would be no relative motion, and so, it seemed to him, no motion of any kind, and no meaningful interpretation for the surrounding spacetime.

Not only that, but Schwarzschild's solution had a singularity at its center. When a classical field theory contains singularities, Einstein believed, it contains the seeds of its own destruction. As we've seen on page 210, this issue is still far from being resolved, a century later.

However much he might have liked to disown it, Einstein was now in possession of a solution to his field equations for a point source. In a linear, background-dependent theory like electromagnetism, knowledge of such a solution leads directly to the ability to write down the field equations with sources included. If Coulomb's law tells us the 1/r² variation of the electric field of a point charge, then we can infer Gauss's law. The situation in general relativity is not this simple. The field equations of general relativity, unlike the Gauss's law, are nonlinear, so we can't simply say that a planet or a star is a solution to be found by adding up a large number of point-source solutions. It's also not clear how one could represent a moving source, since the singularity is a point that isn't even part of the continuous structure of spacetime (and its location is also hidden behind an event horizon, so it can't be observed from the outside).

8.1.2 The Einstein field equation

The Einstein tensor

Given these difficulties, it's not surprising that Einstein's first attempt at incorporating sources into his field equation was a dead end. He postulated that the field equation would have the Ricci tensor on one side, and the stress-energy tensor T^ab (page 145) on the other,

where a factor of G/c⁴ on the right is suppressed by our choice of units, and the 8π is determined on the basis of consistency with Newtonian gravity in the limit of weak fields and low velocities. The problem with this version of the field equations can be demonstrated by counting variables. R and T are symmetric tensors, so the field equation contains 10 constraints on the metric: 4 from the diagonal elements and 6 from the off-diagonal ones. In addition, conservation of mass-energy requires the divergence-free property ∇_b T^ab=0, because otherwise, for example, we could have a mass-energy tensor that varied as T⁰⁰=kt, describing a region of space in which mass was uniformly appearing or disappearing at a constant rate. But this adds 4 more constraints on the metric, for a total of 14. The metric, however, is a symmetric rank-2 tensor itself, so it only has 10 independent components. This overdetermination of the metric suggests that the proposed field equation will not in general allow a solution to be evolved forward in time from a set of initial conditions given on a spacelike surface, and this turns out to be true. It can in fact be shown that the only possible solutions are those in which the traces and are constant throughout spacetime.

The solution is to replace R_ab in the field equations with the a different tensor G_ab, called the Einstein tensor, defined by G_ab=R_ab-(1/2)Rg_ab,

The Einstein tensor is constructed exactly so that it is divergence-free, ∇_b G^ab=0. (This is not obvious, but can be proved by direct computation.) Therefore any stress-energy tensor that satisfies the field equation is automatically divergenceless, and thus no additional constraints need to be applied in order to guarantee conservation of mass-energy.

Self-check: Does replacing R_ab with G_ab invalidate the Schwarzschild metric?

Interpretation of the stress-energy tensor

The stress-energy tensor was briefly introduced in section 5.2 on page 145. By applying the Newtonian limit of the field equation to the Schwarzschild metric, we find that T_tt is to be identified as the mass density ρ. The Schwarzschild metric describes a spacetime using coordinates in which the mass is at rest. In the cosmological applications we'll be considering shortly, it also makes sense to adopt a frame of reference in which the local mass-energy is, on average, at rest, so we can continue to think of T_tt as the (average) mass density. By symmetry, T must be diagonal in such a frame. For example, if we had T_tx≠ 0, then the positive x direction would be distinguished from the negative x direction, but there is nothing that would allow such a distinction.

Example 1: Dust in a different frame

As discussed in example 10 on page 118, it is convenient in cosmology to distinguish between radiation and “dust,” meaning noninteracting, nonrelativistic materials such as hydrogen gas or galaxies. Here “nonrelativistic” means that in the comoving frame, in which the average flow of dust vanishes, the dust particles all have |v| << 1. What is the stress-energy tensor associated with dust?

Since the dust is nonrelativistic, we can obtain the Newtonian limit by using units in which c ≠ 1, and letting c approach infinity. In Cartesian coordinates, the components of the stress-energy have units that cause them to scale like

In the limit of carrow∞, we can therefore take the only source of gravitational fields to be , which in Newtonian gravity must be the mass density ρ, so

Under a Lorentz boost by β in the x direction, the tensor transformation law gives

The over-all factor of γ² arises because of the combination of two effects: each dust particle's mass-energy is increased by a factor of γ, and length contraction also multiplies the density of dust particles by a factor of γ. In the limit of small boosts, the stress-energy tensor becomes

This motivates the interpretation of the time-space components of T as the flux of mass-energy along each axis. In the primed frame, mass-energy with density ρ flows in the x direction at velocity β, so that the rate at which mass-energy passes through a window of area A in the y-z plane is given by ρβ A.

In example 1, we found that had to be interpreted as the flux of (i.e., the flux of mass-energy) across the x axis. Lorentz invariance requires that we treat t, x, y, and z symmetrically, and this forces us to adopt the following interpretation: , where μ is spacelike, is the flux of the density of the mass-energy four-vector in the μ direction. In the comoving frame, in Cartesian coordinates, this means that , , and should be interpreted as pressures. For example, is the flux in the x direction of x-momentum. This is simply the pressure, P, that would be exerted on a surface with its normal in the x direction, so in the comoving frame we have . For a fluid that is not in equilibrium, the pressure need not be isotropic, and the stress exerted by the fluid need not be perpendicular to the surface on which it acts. The space-space components of T would then be the classical stress tensor, whose diagonal elements are the anisotropic pressure, and whose off-diagonal elements are the shear stress. This is the reason for calling T the stress-energy tensor.

The prediction of general relativity is then that pressure acts as a gravitational source with exactly the same strength as mass-energy density. This has important implications for cosmology, since the early universe was dominated by radiation, and a photon gas has P=ρ/3 (example 10, p. 118).

Experimental tests

But how do we know that this prediction is even correct? Can it be verified in the laboratory? The classic laboratory test of the strength of a gravitational source is the 1797 Cavendish experiment, in which a torsion balance was used to measure the very weak gravitational attractions between metal spheres. We could test this aspect of general relativity by doing a Cavendish experiment with boxes full of photons, so that the pressure is of the same order of magnitude as the mass-energy. This is unfortunately utterly impractical, since both P and ρ for a well-lit box are ridiculously small compared to ρ for a metal ball.

However, the repulsive electromagnetic pressure inside an atomic nucleus is quite large by ordinary standards --- about 10³³ Pa! To see how big this is compared to the nuclear mass density of ρ∼ 10¹⁸ kg/m³, we need to take into account the factor of c²≠1 in SI units, the result being that P/ρ is about 10^-2, which is not too small. Thus if we measure gravitational interactions of nuclei with different values of P/ρ, we should be able to test this prediction of general relativity. This was done in a Princeton PhD-thesis experiment by Kreuzer¹ in 1966.

Before we can properly describe and interpret the Kreuzer experiment, we need to distinguish the several different types of mass that could in principle be different from one another in a theory of gravity. We've already encountered the distinction between inertial and gravitational mass, which Eötvös experiments (p. 22) show to be equivalent to about one part in 10¹². But there is also a distinction between an object's active gravitational mass m_a, which measures its ability to create gravitational fields, and its passive gravitational mass m_p, which measures the force it feels when placed in an externally generated field. For experiments using laboratory-scale material objects at nonrelativistic velocities, the Newtonian limit applies, and we can think of active gravitational mass as a scalar, with a density T_tt=ρ.

To understand how this relates to pressure as a source of gravitational fields, it is helpful to consider a case where P is about the same as ρ, which occurs for light. Light is inherently relativistic, so the Newtonian concept of a scalar gravitational mass breaks down, but we can still use “mass” in quotes to talk qualitatively about an electromagnetic wave's active and passive participation in gravitational effects. Experiments show that general relativity correctly predicts the deflection of light by the sun to about one part in 10⁵ (p. 202). This is the electromagnetic equivalent of an Eötvös experiment; it shows that general relativity predicts the right thing about the proportion between a light wave's inertial and passive gravitational “masses.” Now suppose that general relativity was wrong, and pressure was not a source of gravitational fields. This would cause a drastic decrease in the active gravitational “mass” of an electromagnetic wave.

The Kreuzer experiment actually dealt with static electric fields inside nuclei, not electromagnetic waves, but it is still clear what we should expect in general: if pressure does not act as a gravitational source, then the ratio m_a/m_p should be different for different nuclei. Specifically, it should be lower for a nucleus with a higher atomic number Z, in which the electrostatic pressures are higher.

Kreuzer did a Cavendish experiment, figure b, using masses made of two different substances. The first substance was teflon. The second substance was a mixture of the liquids trichloroethylene and dibromoethane, with the proportions chosen so as to give a density as close as possible to that of teflon. Teflon is 76% fluorine by weight, and the liquid is 74% bromine. Fluorine has atomic number Z=9, bromine Z=35, and since the electromagnetic force has a long range, the pressure within a nucleus scales upward roughly like Z^1/3 (because any given proton is acted on by Z-1 other protons, and the size of a nucleus scales like Z^1/3, so P∝ Z/(Z^1/3)²). The solid mass was immersed in the liquid, and the combined gravitational field of the solid and the liquid was detected by a Cavendish balance.

Ideally, one would formulate the liquid mixture so that its passive-mass density was exactly equal to that of teflon, as determined by buoyancy. Any oscillation in the torque measured by the Cavendish balance would then indicate an inequivalence between active and passive gravitational mass.

c / The Kreuzer experiment. 1. There are two passive masses, P, and an active mass A consisting of a single 23-cm diameter teflon cylinder immersed in a fluid. The teflon cylinder is driven back and forth with a period of 400 s. The resulting deflection of the torsion beam is monitored by an optical lever and canceled actively by electrostatic forces from capacitor plates (not shown). The voltage required for this active cancellation is a measure of the torque exerted by A on the torsion beam. 2. Active mass as a function of temperature. 3. Passive mass as a function of temperature. In both 2 and 3, temperature is measured in units of ohms, i.e., the uncalibrated units of a thermistor that was immersed in the liquid.

In reality, the two substances involved had different coefficients of thermal expansion, so slight variations in temperature made their passive-mass densities unequal. Kreuzer therefore measured both the buoyant force and the gravitational torque as functions of temperature. He determined that these became zero at the same temperature, to within experimental errors, which verified the equivalence of active and passive gravitational mass to within a certain precision,

Kreuzer intended this exeriment only as a test of m_p∝ m_a, but it was reinterpreted in 1976 by Will² as a test of the coupling of sources to gravitational fields as predicted by general relativity and other theories of gravity. Crudely, we've already argued that m_p∝ m_a would be substance-dependent if pressure did not couple to gravitational fields. Will actually carried out a more careful calculation, of which I present a simplified summary. Suppose that pressure does not contribute as much to gravitational fields as is claimed by general relativity; its coupling is reduced by a factor 1-x, where x=0 in general relativity.³ Will considers a model consisting of pointlike particles interacting through static electrical forces, and shows that for such a system,

where U_e is the electrical energy. The Kreuzer experiment then requires |x| < 6× 10^-2, meaning that pressure does contribute to gravitational fields as predicted by general relativity, to within a precision of 6%.

One of the important ways in which Will's calculation goes beyond my previous crude argument is that it shows that when x=0, as it does for general relativity, the correction term x U_e/2 vanishes, and m_a=m_p exactly. This is interpreted as follows. Let a bromine nucleus be referred to with a capital M, fluorine with the lowercase m. Then when a bromine nucleus and a fluorine nucleus interact gravitationally at a distance r, the Newtonian approximation applies, and the total internal force acting on the pair of nuclei taken as a whole equals (m_pM_a-M_pm_a)/r² (in units where the Newtonian gravitational constant G equals 1). This vanishes only if m_pM_a-M_pm_a=0, which is equivalent to m_p/M_p=m_a/M_a. If this proportionality fails, then the system violates Newton's third law and conservation of momentum; its center of mass will accelerate along the line connecting the two nuclei, either in the direction of M or in the direction of m, depending on the sign of x.

Thus the vanishing of the correction term x U_e/2 tells us that general relativity predicts exact conservation of momentum in this interaction. This is comforting, but a little susprising on the face of it. Newtonian gravity treats active and passive massive perfectly symmetrically, so that there is a perfect guarantee of conservation of momentum. But relativity incorporates them in a completely asymmetric manner, so there is no obvious reason that we should have perfect conservation of momentum. In fact we don't have any general guarantee of conservation of momentum, since, as discussed in section 4.5.1 on page 133, the language of general relativity doesn't even give us the symbols we would need in order to state a global conservation law for a vector. General relativity does, however, allow local conservation laws. We will have local conservation of mass-energy and momentum provided that the stress-energy tensor's divergence ∇_b T^ab vanishes.

Bartlett and van Buren⁴ used this connection to conservation of momentum in 1986 to derive a tighter limit on x. Since the moon has an asymmetrical distribution of iron and aluminum, a nonzero x would cause it to have an anomalous acceleration along a certain line. Because lunar laser ranging gives extremely accurate data on the moon's orbit, the constraint is tightened to |x| < 1× 10^-8.

These are tests of general relativity's predictions about the gravitational fields generated by the pressure of a static electric field. In addition, there is indirect confirmation (p. 264) that general relativity is correct when it comes to electromagnetic waves.

Energy of gravitational fields not included in the stress-energy tensor

Summarizing the story of the Kreuzer and Bartlett-van Buren results, we find that observations verify to high precision one of the defining properties of general relativity, which is that all forms of energy are equivalent to mass. That is, Einstein's famous E=mc² can be extended to gravitational effects, with the proviso that the source of gravitational fields is not really a scalar m but the stress-energy tensor T.

But there is an exception to this even-handed treatment of all types of energy, which is that the energy of the gravitational field itself is not included in T, and is not even generally a well-defined concept. In Newtonian gravity, we can have conservation of energy if we attribute to the gravitational field a negative potential energy density .

Self-check: Convince yourself that the negative sign makes sense, by considering the case where two equal masses start out far apart and then fall together and combine to make a single body with twice the mass.

But the equivalence principle tells us that g is not a tensor, for we can always make g vanish locally by going into the frame of a free-falling observer, and yet the tensor transformation laws will never change a nonzero tensor to a zero tensor under a change of coordinates. Since the gravitational field is not a tensor, there is no way to add a term for it into the definition of the stress-energy, which is a tensor. The grammar and vocabulary of the tensor notation are specifically designed to prevent writing down such a thing, so that the language of general relativity is not even capable of expressing the idea that gravitational fields would themselves contribute to T.

As a concrete example, we observe that the Hulse-Taylor binary pulsar system (p. 202) is gradually losing orbital energy, and that the rate of energy loss exactly matches general relativity's prediction of the rate of gravitational radiation. There is a net decrease in the forms of energy, such as rest mass and kinetic energy, that are accounted for in the stress energy tensor T. We can account for the missing energy by attributing it to the outgoing gravitational waves, but that energy is not included in T, and we have to develop special techniques for evaluating that energy. Those techniques only turn out to apply to certain special types of spacetimes, such as asymptotically flat ones, and they do not allow a uniquely defined energy density to be attributed to a particular small region of space (for if they did, that would violate the equivalence principle).

8.1.3 Energy conditions

Physical theories are supposed to answer questions. For example:

1. Does a small enough physical object always have a world-line that is approximately a geodesic?
2. Do massive stars collapse to form black-hole singularities?
3. Did our universe originate in a Big Bang singularity?
4. If our universe doesn't currently have violations of causality such as the closed timelike curves exhibited by the Petrov metric (p. 235), can we be assured that it will never develop causality violation in the future?

We would like to “prove” whether the answers to questions like these are yes or no, but physical theories are not formal mathematical systems in which results can be “proved” absolutely. For example, the basic structure of general relativity isn't a set of axioms but a list of ingredients like the equivalence principle, which has evaded formal definition.⁵

Even the Einstein field equations, which appear to be completely well defined, are not mathematically formal predictions of the behavior of a physical system. The field equations are agnostic on the question of what kinds of matter fields contribute to the stress-energy tensor. In fact, any spacetime at all is a solution to the Einstein field equations, provided we're willing to admit the corresponding stress-energy tensor. We can never answer questions like the ones above without assuming something about the stress-energy tensor.

In example 10 on page 118, we saw that radiation has P=ρ/3 and dust has P=0. Both have . If the universe is made out of nothing but dust and radiation, then we can obtain the following four constraints on the energy-momentum tensor:

These are arranged roughly in order from strongest to weakest. They all have to do with the idea that negative mass-energy doesn't seem to exist in our universe, i.e., that gravity is always attractive rather than repulsive. With this motivation, it would seem that there should only be one way to state an energy condition: ρ > 0. But the symbols ρ and P refer to the form of the stress-energy tensor in a special frame of reference, interpreted as the one that is at rest relative to the average motion of the ambient matter. In this frame, the tensor is diagonal. Switching to some other frame of reference, the ρ and P parts of the tensor would mix, and it might be possible to end up with a negative energy density. The weak energy condition is the constraint we need in order to make sure that the energy density is never negative in any frame.

The dominant energy condition is like the weak energy condition, but it also guarantees that no observer will see a flux of energy flowing at speeds greater than c.

Geodesic motion of test particles

Question 1 on p. 247 was: “Does a small enough physical object always have a world-line that is approximately a geodesic?” In other words, do Eötvös experiments give null results when carried out in laboratories using real-world apparatus of small enough size? We would like something of this type to be true, since general relativity is based on the equivalence principle, and the equivalence principle is motivated by the null results of Eötvös experiments. Nevertheless, it is fairly easy to show that the answer to the question is no, unless we make some more specific assumption, such as an energy condition, about the system being modeled.

Before we worry about energy conditions, let's consider why the small size of the apparatus is relevant. Essentially this is because of gravitational radiation. In a gravitationally radiating system such as the Hulse-Taylor binary pulsar (p. 202), the material bodies lose energy, and as with any radiation process, the radiated power depends on the square of the strength of the source. The world-line of a such a body therefore depends on its mass, and this shows that its world-line cannot be an exact geodesic, since the initially tangent world-lines of two different masses diverge from one another, and these two world-lines can't both be geodesics.

Let's proceed to give a rough argument for geodesic motion and then try to poke holes in it. When we test geodesic motion, we do an Eötvös experiment that is restricted to a certain small region of spacetime S. Our test-body's world-line enters S with a certain energy-momentum vector p and exits with p'. If spacetime was flat, then Gauss's theorem would hold exactly, and the vanishing divergence ∇_b T^ab of the stress-energy tensor would require that the incoming flux represented by p be exactly canceled by the outgoing flux due to p'. In reality spacetime isn't flat, and it isn't even possible to compare p and p' except by parallel-transporting one into the same location as the other. Parallel transport is path-dependent, but if we make the reasonable restriction to paths that stay within S, we expect the ambiguity due to path-dependence to be proportional to the area enclosed by any two paths, so that if S is small enough, the ambiguity can be made small. Ignoring this small ambiguity, we can see that one way for the fluxes to cancel would be for the particle to travel along a geodesic, since both p and p' are tangent to the test-body's world-line, and a geodesic is a curve that parallel-transports its own tangent vector. Geodesic motion is therefore one solution, and we expect the solution to be nearly unique when S is small.

Although this argument is almost right, it has some problems. First we have to ask whether “geodesic” means a geodesic of the full spacetime including the object's own fields, or of the background spacetime B that would have existed without the object. The latter is the more sensible interpretation, since the question is basically asking whether a spacetime can really be defined geometrically, as the equivalence principle claims, based on the motion of test particles inserted into it. We also have to define words like “small enough” and “approximately;” to do this, we imagine a sequence of objects O_n that get smaller and smaller as n increases. We then form the following conjecture, which is meant to formulate question 1 more exactly: Given a vacuum background spacetime B, and a timelike world-line ℓ in B, consider a sequence of spacetimes S_n, formed by inserting the O_n into B, such that: (i) the metric of S_n is defined on the same points as the metric of B; (ii) O_n moves along ℓ, and for any r>0, there exists some n such that for , O_m is smaller than r;⁶ (iii) the metric of S_n approaches the metric of B as narrow∞. Then ℓ is a geodesic of B.

This is almost right but not quite, as shown by the following counterexample. Papapetrou⁷ has shown that a spinning body in a curved background spacetime deviates from a geodesic with an acceleration that is proportional to LR, where L is its angular momentum and R is the Riemann curvature. Let all the O_n have a fixed value of L, but let the spinning mass be concentrated into a smaller and smaller region as n increases, so as to satisfy (ii). As the radius r decreases, the motion of the particles composing an O_n eventually has to become ultrarelativistic, so that the main contribution to the gravitational field is from the particles' kinetic energy rather than their rest mass. We then have L∼ pr∼ Er, so that in order to keep L constant, we must have E∝ 1/r. This causes two problems. First, it makes the gravitational field blow up at small distances, violating (iii). Also, we expect that for any known form of matter, there will come a point (probably the Tolman-Oppenheimer-Volkoff limit) at which we get a black hole; the singularity is then not part of the spacetime S_n, violating (i). But our failed counterexample can be patched up. We obtain a supply of exotic matter, whose gravitational mass is negative, and we mix enough of this mysterious stuff into each O_n so that the gravitational field shrinks rather than growing as n increases, and no black hole is ever formed.

Ehlers and Geroch⁸ have proved that it suffices to require an additional condition: (iv) The O_n satisfy the dominant energy condition. This rules out our counterexample.

The Newtonian limit

In units with c ≠ 1, a quantity like ρ+P is expressed as ρ+P/c². The Newtonian limit is recovered as carrow∞, which makes the pressure term negligible, so that all the energy conditions reduce to . What would it mean if this was violated? Would ρ<0 describe an object with negative inertial mass, which would accelerate east when you pushed it to the west? Or would it describe something with negative gravitational mass, which would repel ordinary matter? We can imagine various possiblities, as shown in figure e. Anything that didn't lie on the main diagonal would violate the equivalence principle, and would therefore be impossible to accomodate within general relativity's geometrical description of gravity. If we had “upsidasium” matter such as that described by the second quadrant of the figure (example 2, p. 26), gravity would be like electricity, except that like masses would attract and opposites repel; we could have gravitational dielectrics and gravitational Faraday cages. The fourth quadrant leads to amusing possibilities like figure f.

Singularity theorems

An important example of the use of the energy conditions is that Hawking and Ellis have proved that under the assumption of the strong energy condition, any body that becomes sufficiently compact will end up forming a singularity. We might imagine that the formation of a black hole would be a delicate thing, requiring perfectly symmetric initial conditions in order to end up with the perfectly symmetric Schwarzschild metric. Many early relativists thought so, for good reasons. If we look around the universe at various scales, we find that collisions between astronomical bodies are extremely rare. This is partly because the distances are vast compared to the sizes of the objects, but also because conservation of angular momentum has a tendency to make objects swing past one another rather than colliding head-on. Starting with a cloud of objects, e.g., a globular cluster, Newton's laws make it extremely difficult, regardless of the attractive nature of gravity, to pick initial conditions that will make them all collide the future. For one thing, they would have to have exactly zero total angular momentum.

Most relativists now believe that this is not the case. General relativity describes gravity in terms of the tipping of light cones. When the field is strong enough, there is a tendency for the light cones to tip over so far that the entire future light-cone points at the source of the field. If this occurs on an entire surface surrounding the source, it is referred to as a trapped surface.

To make this notion of light cones “pointing at the source” more rigorous, we need to define the volume expansion Θ. Let the set of all points in a spacetime (or some open subset of it) be expressed as the union of geodesics. This is referred to as a foliation in geodesics, or a congruence. Let the velocity vector tangent to such a curve be u^a. Then we define Θ=∇_a u^a. This is exactly analogous to the classical notion of the divergence of the velocity field of a fluid, which is a measure of compression or expansion. Since Θ is a scalar, it is coordinate-independent. Negative values of Θ indicate that the geodesics are converging, so that volumes of space shrink. A trapped surface is one on which Θ is negative when we foliate with lightlike geodesics oriented outward along normals to the surface.

When a trapped surface forms, any lumpiness or rotation in the initial conditions becomes irrelevant, because every particle's entire future world-line lies inward rather than outward. A possible loophole in this argument is the question of whether the light cones will really tip over far enough. We could imagine that under extreme conditions of high density and temperature, matter might demonstrate unusual behavior, perhaps including a negative energy density, which would then give rise to a gravitational repulsion. Gravitational repulsion would tend to make the light cones tip outward rather than inward, possibly preventing the collapse to a singularity. We can close this loophole by assuming an appropriate energy condition. Penrose and Hawking have formalized the above argument in the form of a pair of theorems, known as the singularity theorems. One of these applies to the formation of black holes, and another one to cosmological singularities such as the Big Bang.

In a cosmological model, it is natural to foliate using world-lines that are at rest relative to the Hubble flow (or, equivalently, the world-lines of observers who see a vanishing dipole moment in the cosmic microwave background). The Θ we then obtain is positive, because the universe is expanding. The volume expansion is Θ=3H_o, where H_o≈ 2.3×10^-18 s^-1 is the Hubble constant (the fractional rate of change of the scale factor of cosmological distances). The factor of three occurs because volume is proportional to the cube of the linear dimensions.

Current status

The current status of the energy conditions is shaky. Although it is clear that all of them hold in a variety of situations, there are strong reasons to believe that they are violated at both microscopic and cosmological scales, for reasons both classical and quantum-mechanical.⁹ We will see such a violation in the following section.

8.1.4 The cosmological constant

Having included the source term in the Einstein field equations, our most important application will be to cosmology. Some of the relevant ideas originate long before Einstein. Once Newton had formulated a theory of gravity as a universal attractive force, he realized that there would be a tendency for the universe to collapse. He resolved this difficulty by assuming that the universe was infinite in spatial extent, so that it would have no center of symmetry, and therefore no preferred point to collapse toward. The trouble with this argument is that the equilibrium it describes is unstable. Any perturbation of the uniform density of matter breaks the symmetry, leading to the collapse of some pocket of the universe. If the radius of such a collapsing region is r, then its gravitational is proportional to r³, and its gravitational field is proportional to r³/r²=r. Since its acceleration is proportional to its own size, the time it takes to collapse is independent of its size. The prediction is that the universe will have a self-similar structure, in which the clumping on small scales behaves in the same way as clumping on large scales; zooming in or out in such a picture gives a landscape that appears the same. With modern hindsight, this is actually not in bad agreement with reality. We observe that the universe has a hierarchical structure consisting of solar systems, galaxies, clusters of galaxies, superclusters, and so on. Once such a structure starts to condense, the collapse tends to stop at some point because of conservation of angular momentum. This is what happened, for example, when our own solar system formed out of a cloud of gas and dust.

Einstein confronted similar issues, but in a more acute form. Newton's symmetry argument, which failed only because of its instability, fails even more badly in relativity: the entire spacetime can simply contract uniformly over time, without singling out any particular point as a center. Furthermore, it is not obvious that angular momentum prevents total collapse in relativity in the same way that it does classically, and even if it did, how would that apply to the universe as a whole? Einstein's Machian orientation would have led him to reject the idea that the universe as a whole could be in a state of rotation, and in any case it was sensible to start the study of relativistic cosmology with the simplest and most symmetric possible models, which would have no preferred axis of rotation.

Because of these issues, Einstein decided to try to patch up his field equation so that it would allow a static universe. Looking back over the considerations that led us to this form of the equation, we see that it is very nearly uniquely determined by the following criteria:

• It should be consistent with experimental evidence for local conservation of mass-energy and momentum.
• It should satisfy the equivalence principle.
• It should be coordinate-independent.
• It should be equivalent to Newtonian gravity in the appropriate limit.
• It should not be overdetermined.

This is not meant to be a rigorous proof, just a general observation that it's not easy to tinker with the theory without breaking it.

Example 3: A failed attempt at tinkering

As an example of the lack of “wiggle room” in the structure of the field equations, suppose we construct the scalar , the trace of the stress-energy tensor, and try to insert it into the field equations as a further source term. The first problem is that the field equation involves rank-2 tensors, so we can't just add a scalar. To get around this, suppose we multiply by the metric. We then have something like , where the two constants c₁ and c₂ would be constrained by the requirement that the theory agree with Newtonian gravity in the classical limit.

To see why this attempt fails, consider a beam of light directed along the x axis. Its momentum is equal to its energy (see page 114), so its contributions to the local energy density and pressure are equal. Thus its contribution to the stress-energy tensor is of the form . The trace vanishes, so the beam of light's coupling to gravity in the c₂ term is zero. As discussed on pp. 243-246, empirical tests of conservation of momentum would therefore constrain c₂ to be ≤sssim 10^-8.

One way in which we can change the field equation without violating any of these is to add a term Λ g_ab, giving

which is what we will refer to as the Einstein field equation.¹⁰ The universal constant Λ is called the cosmological constant. Einstein originally introduced a positive cosmological constant because he wanted relativity to be able to describe a static universe. To see why it would have this effect, compare its behavior with that of an ordinary fluid. When an ordinary fluid, such as the exploding air-gas mixture in a car's cylinder, expands, it does work on its environment, and therefore by conservation of energy its own internal energy is reduced. A positive cosmological constant, however, acts like a certain amount of mass-energy built into every cubic meter of vacuum. Thus when it expands, it releases energy. Its pressure is negative. Another way of verifying these statements is by observing that for a given cosmological constant, we can always observe the Λ g_ab term in the field equations into the 8π T_ab, as if the cosmological constant were some form of matter.

Now consider the following pseudo-classical argument. Although we've already seen (page 199) that there is no useful way to separate the roles of kinetic and potential energy in general relativity, suppose that there are some quantities analogous to them in the description of the universe as a whole. (We'll see below that the universe's contraction and expansion is indeed described by a set of differential equations that can be interpreted in essentially this way.) If the universe contracts, a cubic meter of space becomes less than a cubic meter. The cosmological-constant energy associated with that volume is reduced, so some energy has been consumed. The kinetic energy of the collapsing matter goes down, and the collapse is decelerated.

The addition of the Λ term constitutes a change to the vacuum field equations, and the good agreement between theory and experiment in the case of, e.g., Mercury's orbit puts an upper limit on Λ then implies that Λ must be small. For an order-of-magnitude estimate, consider that Λ has units of mass density, and the only parameters with units that appear in the description of Mercury's orbit are the mass of the sun, m, and the radius of Mercury's orbit, r. The relativistic corrections to Mercury's orbit are on the order of v², or about 10^-8, and they come out right. Therefore we can estimate that the cosmological constant could not have been greater than about (10^-8)m/r³ ∼ 10^-10 kg/m³, or it would have caused noticeable discrepancies. This is a very poor bound; if Λ was this big, we might even be able to detect its effects in laboratory experiments. Looking at the role played by r in the estimate, we see that the upper bound could have been made tighter by increasing r. Observations on galactic scales, for example, constrain it much more tightly. This justifies the description of Λ as cosmological: the larger the scale, the more significant the effect of a nonzero Λ would be.

Since the right-hand side of the field equation is 8π T_ab + Λ g_ab, it is possible to consider the cosmological constant as a type of matter contributing to the stress-energy tensor. Its negative pressure causes a violation of the strong energy condition. If the cosmological constant is a product of the quantum-mechanical structure of the vacuum, then this is not too surprising, because quantum fields are known to display negative energy. For example, the energy density between two parallel conducting plates is negative due to the Casimir effect.

8.2 Cosmological solutions

We are thus led to pose two interrelated questions. First, what can empirical observations about the universe tell us about the laws of physics, such as the zero or nonzero value of the cosmological constant? Second, what can the laws of physics, combined with observation, tell us about the large-scale structure of the universe, its origin, and its fate?

8.2.1 Evidence for the finite age of the universe

We have a variety of evidence that the universe's existence does not stretch for an unlimited time into the past.

When astronomers view light from the deep sky that has been traveling through space for billions of years, they observe a universe that looks different from today's. For example, quasars were common in the early universe but are uncommon today.

In the present-day universe, stars use up deuterium nuclei, but there are no known processes that could replenish their supply. We therefore expect that the abundance of deuterium in the universe should decrease over time. If the universe had existed for an infinite time, we would expect that all its deuterium would have been lost, and yet we observe that deuterium does exist in stars and in the interstellar medium.

The second law of thermodynamics predicts that any system should approach a state of thermodynamic equilibrium, and yet our universe is very far from thermal equilibrium, as evidenced by the fact that our sun is hotter than interstellar space, or by the existence of functioning heat engines such as your body or an automobile engine.

With hindsight, these observations suggest that we should not look for cosmological models that persist for an infinite time into the past.

8.2.2 Evidence for expansion of the universe

We don't only see time-variation in locally observable quantities such as quasar abundance, deuterium abundance, and entropy. In addition, we find empirical evidence for global changes in the universe. By 1929, Edwin Hubble at Mount Wilson had determined that the universe was expanding, and historically this was the first convincing evidence that Einstein's original goal of modeling a static cosmology had been a mistake. Einstein later referred to the cosmological constant as the “greatest blunder of my life,” and for the next 70 years it was commonly assumed that Λ was exactly zero.

Since we observe that the universe is expanding, the laws of thermodynamics require that it also be cooling, just as the exploding air-gas mixture in a car engine's cylinder cools as it expands. If the universe is currently expanding and cooling, it is natural to imagine that in the past it might have been very dense and very hot. This is confirmed directly by looking up in the sky and seeing radiation from the hot early universe. In 1964, Penzias and Wilson at Bell Laboratories in New Jersey detected a mysterious background of microwave radiation using a directional horn antenna. As with many accidental discoveries in science, the important thing was to pay attention to the surprising observation rather than giving up and moving on when it confounded attempts to understand it. They pointed the antenna at New York City, but the signal didn't increase. The radiation didn't show a 24-hour periodicity, so it couldn't be from a source in a certain direction in the sky. They even went so far as to sweep out the pigeon droppings inside. It was eventually established that the radiation was coming uniformly from all directions in the sky and had a black-body spectrum with a temperature of about 3 K.

This is now interpreted as follows. At one time, the universe was hot enough to ionize matter. An ionized gas is opaque to light, since the oscillating fields of an electromagnetic wave accelerate the charged particles, depositing kinetic energy into them. Once the universe became cool enough, however, matter became electrically neutral, and the universe became transparent. Light from this time is the most long-traveling light that we can detect now. The latest data show that transparency set in when the temperature was about 3000 K. The surface we see, dating back to this time, is known as the surface of last scattering. Since then, the universe has expanded by about a factor of 1000, causing the wavelengths of photons to be stretched by the same amount due to the expansion of the underlying space. This is equivalent to a Doppler shift due to the source's motion away from us; the two explanations are equivalent. We therefore see the 3000 K optical black-body radiation red-shifted to 3 K, in the microwave region.

It is logically possible to have a universe that is expanding but whose local properties are nevertheless static, as in the steady-state model of Fred Hoyle, in which some novel physical process spontaneously creates new hydrogen atoms, preventing the infinite dilution of matter over the universe's history, which in this model extends infinitely far into the past. But we have already seen strong empirical evidence that the universe's local properties (quasar abundance, etc.) are changing over time. The CMB is an even more extreme and direct example of this; the universe full of hot, dense gas that emitted the CMB is clearly nothing like today's universe.¹¹

8.2.3 Evidence for homogeneity and isotropy

These observations demonstrate that the universe is not homogeneous in time, i.e., that one can observe the present conditions of the universe (such as its temperature and density), and infer what epoch of the universe's evolution we inhabit. A different question is the Copernican one of whether the universe is homogeneous in space. Surveys of distant quasars show that the universe has very little structure at scales greater than a few times 10²⁵ m. (This can be seen on a remarkable logarithmic map constructed by Gott et al., astro.princeton.edu/universe.) This suggests that we can, to a good approximation, model the universe as being isotropic (the same in all spatial directions) and homogeneous (the same at all locations in space).

Further evidence comes from the extreme uniformity of the cosmic microwave background radiation, once one subtracts out the dipole anisotropy due to the Doppler shift arising from our galaxy's motion relative to the CMB. When the CMB was first discovered, there was doubt about whether it was cosmological in origin (rather than, say, being associated with our galaxy), and it was expected that its isotropy would be as large as 10%. As physicists began to be convinced that it really was a relic of the early universe, interest focused on measuring this anisotropy, and a series of measurements put tighter and tighter upper bounds on it.

Other than the dipole term, there are two ways in which one might naturally expect anisotropy to occur. There might have been some lumpiness in the early universe, which might have served as seeds for the condensation of galaxy clusters out of the cosmic medium. Furthermore, we might wonder whether the universe as a whole is rotating. The general-relativistic notion of rotation is very different from the Newtonian one, and in particular, it is possible to have a cosmology that is rotating without having any center of rotation (see problem 5, p. 238). In fact one of the first exact solutions discovered for the Einstein field equations was the Gödel metric, which described a bizarre rotating universe with closed timelike curves, i.e., one in which causality was violated. In a rotating universe, one expects that radiation received from great cosmological distances will have a transverse Doppler shift, i.e., a shift originating from the time dilation due to the motion of the distant matter across the sky. This shift would be greatest for sources lying in the plane of rotation relative to us, and would vanish for sources lying along the axis of rotation. The CMB would therefore show variation with the form of a quadrupole term, 3cos²θ-1. In 1977 a U-2 spyplane (the same type involved in the 1960 U.S.-Soviet incident) was used by Smoot et al.¹² to search for anisotropies in the CMB. This experiment was the first to definitively succeed in detecting the dipole anisotropy. After subtraction of the dipole component, the CMB was found to be uniform at the level of ∼ 3× 10^-4. This provided strong support for homogeneous cosmological models, and ruled out rotation of the universe with ω >rsim 10^-22 Hz.

8.2.4 The FRW cosmologies

The FRW metric and the standard coordinates

Motivated by Hubble's observation that the universe is expanding, we hypothesize the existence of solutions of the field equation in which the properties of space are homogeneous and isotropic, but the over-all scale of space is increasing as described by some scale function a(t). Because of coordinate invariance, the metric can still be written in a variety of forms. One such form is

where the spatial part is

To interpret the coordinates, we note that if an observer is able to determine the functions a and f for her universe, then she can always measure some scalar curvature such as the Ricci scalar or the Kretchmann invariant, and since these are proportional to a raised to some power, she can determine a and t. This shows that t is a “look-out-the-window” time, i.e., a time coordinate that we can determine by looking out the window and observing the present conditions in the universe. Because the quantity being measured directly is a scalar, the result is independent of the observer's state of motion. (In practice, these scalar curvatures are difficult to measure directly, so we measure something else, like the sky-wide average temperature of the cosmic microwave background.) Simultaneity is supposed to be ill-defined in relativity, but the look-out-the-window time defines a notion of simultaneity that is the most naturally interesting one in this spacetime. With this particular definition of simultaneity, we can also define a preferred state of rest at any location in spacetime, which is the one in which t changes as slowly as possible relative to one's own clock. This local rest frame, which is more easily determined in practice as the one in which the microwave background is most uniform across the sky, can also be interpreted as the one that is moving along with the Hubble flow, i.e., the average motion of the galaxies, photons, or whatever else inhabits the spacetime. The time t is interpreted as the proper time of a particle that has always been locally at rest. The spatial distance measured by is called the proper distance. It is the distance that would be measured by a chain of rulers, each of them “at rest” in the above sense.

These coordinates are referred as the “standard” cosmological coordinates; one will also encounter other choices, such as the comoving and conformal coordinates, which are more convenient for certain purposes. Historically, the solution for the functions a and f was found by de Sitter in 1917.

The spatial metric

The unknown function f(r) has to give a 3-space metric dℓ² with a constant Einstein curvature tensor. The following Maxima program computes the curvature.

load(ctensor);
dim:3;
ct_coords:[r,theta,phi];
depends(f,t);
lg:matrix([f,0,0],
          [0,r^2,0],
          [0,0,r^2*sin(theta)^2]);
cmetric();
einstein(true);

Line 2 tells Maxima that we're working in a space with three dimensions rather than its default of four. Line 4 tells it that f is a function of time. Line 9 uses its built-in function for computing the Einstein tensor . The result has only one nonvanishing component, . This has to be constant, and since scaling can be absorbed in the factor a(t) in the 3+1-dimensional metric, we can just set the value of G_tt more or less arbitrarily, except for its sign. The result is f=1/(1-kr²), where k=-1, 0, or 1.

The resulting metric, called the Robertson-Walker metric, is

The form of dℓ² shows us that k can be interpreted in terms of the sign of the spatial curvature. We recognize the k=0 metric as a flat spacetime described in spherical coordinates. To interpret the k≠ 0 cases, we note that a circle at coordinate r has proper circumference C=2π a r and proper radius . For k<0, we have f<1 and C>2π R, indicating negative spatial curvature. For k>0 there is positive curvature.

Let's examine the positive-curvature case more closely. Suppose we select a particular plane of simultaneity defined by t=constant and φ=π/2, and we start doing geometry in this plane. In two spatial dimensions, the Riemann tensor only has a single independent component, which can be identified with the Gaussian curvature (sec. 5.4, p. 150), and when this Gaussian curvature is positive and constant, it can be interpreted as the angular defect of a triangle per unit area (sec. 5.3, p. 146). Since the sum of the interior angles of a triangle can never be greater than 3π, we have an upper limit on the area of any triangle. This happens because the positive-curvature Robertson-Walker metric represents a cosmology that is spatially finite. At a given t, it is the three-dimensional analogue of a two-sphere. On a two-sphere, if we set up polar coordinates with a given point arbitrarily chosen as the origin, then we know that the r coordinate must “wrap around” when we get to the antipodes. That is, there is a coordinate singularity there. (We know it can only be a coordinate singularity, because if it wasn't, then the antipodes would have special physical characteristics, but the FRW model was constructed to be spatially homogeneous.) This “wrap-around” behavior is described by saying that the model is closed.

In the negative-curvature case, there is no limit on distances, b/3. Such a universe is called open. In the case of an open universe, it is particularly easy to demonstrate a fact that bothers many students, which is that proper distances can grow at rates exceeding c. Let particles A and B both be at rest relative to the Hubble flow. The proper distance between them is then given by L=aℓ, where is constant. Then differentiating L with respect to the look-out-the-window time t gives . In an open universe, there is no limit on the size of ℓ, so at any given time, we can make dL/dt as large as we like. This does not violate special relativity, since it is only locally that special relativity is a valid approximation to general relativity. Because GR only supplies us with frames of reference that are local, the velocity of two objects relative to one another is not even uniquely defined; our choice of dL/dt was just one of infinitely many possible definitions.

The distinction between closed and open universes is not just a matter of geometry, it's a matter of topology as well. Just as a two-sphere cannot be made into a Euclidean plane without cutting or tearing, a closed universe is not topologically equivalent to an open one. The correlation between local properties (curvature) and global ones (topology) is a general theme in differential geometry. A universe that is open is open forever, and similarly for a closed one.

The Friedmann equations

Having fixed f(r), we can now see what the field equation tells us about a(t). The next program computes the Einstein tensor for the full four-dimensional spacetime:

load(ctensor);
ct_coords:[t,r,theta,phi];
depends(a,t);
lg:matrix([1,0,0,0],
          [0,-a^2/(1-k*r^2),0,0],
          [0,0,-a^2*r^2,0],
          [0,0,0,-a^2*r^2*sin(theta)^2]);
cmetric();
einstein(true);

The result is

where dots indicate differentiation with respect to time.

Since we have with mixed upper and lower indices, we either have to convert it into G_ab, or write out the field equations in this mixed form. The latter turns out to be simpler. In terms of mixed indices, is always simply diag(1,1,1,1). Arbitrarily singling out r=0 for simplicity, we have g=diag(1,-a²,0,0). The stress-energy tensor is . Substituting into , we find

Rearranging a little, we have a set of differential equations known as the Friedmann equations,

The cosmology that results from a solution of these differential equations is known as the Friedmann-Robertson-Walker (FRW) or Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) cosmology.

8.2.5 A singularity at the Big Bang

The Friedmann equations only allow a constant a in the case where Λ is perfectly tuned relative to the other parameters, and even this artificially fine-tuned equilibrium turns out to be unstable. These considerations make a static cosmology implausible on theoretical grounds, and they are also consistent with the observed Hubble expansion (p. 257).

Since the universe is not static, what happens if we use general relativity to extrapolate farther and farther back in time?

If we extrapolate the Friedmann equations backward in time, we find that they always have a=0 at some point in the past, and this occurs regardless of the details of what we assume about the matter and radiation that fills the universe. To see this, note that, as discussed in example 10 on page 118, radiation is expected to dominate the early universe, for generic reasons that are not sensitive to the (substantial) observational uncertainties about the universe's present-day mixture of ingredients. Under radiation-dominated conditions, we can approximate Λ=0 and P=ρ/3 (example 10, p. 118) in the first Friedmann equation, finding

where ρ is the density of mass-energy due to radiation. Since is always negative, the graph of a(t) is always concave down, and since a is currently increasing, there must be some time in the past when a=0. One can readily verify that this is not just a coordinate singularity; the Ricci scalar curvature diverges, and the singularity occurs at a finite proper time in the past.

In section 6.3.1, we saw that a black hole contains a singularity, but it appears that such singularities are always hidden behind event horizons, so that we can never observe them from the outside. The FRW singularity, however, is not hidden behind an event horizon. It lies in our past light-cone, and our own world-lines emerged from it. The universe, it seems, originated in a Big Bang, a concept that originated with the Belgian Roman Catholic priest Georges Lema\^{i}tre.

Self-check: Why is it not correct to think of the Big Bang as an explosion that occurred at a specific point in space?

Does the FRW singularity represent something real about our universe?

One thing to worry about is the accuracy of our physical modeling of the radiation-dominated universe. The presence of an initial singularity in the FRW solutions does not depend sensitively on on assumptions like P=ρ/3, but it is still disquieting that no laboratory experiment has ever come close to attaining the conditions under which we could test whether a gas of photons produces gravitational fields as predicted by general relativity. We saw on p. 243 that static electric fields do produce gravitational fields as predicted, but this is not the same as an empirical confirmation that electromagnetic waves also act as gravitational sources in exactly the manner that general relativity claims. We do, however, have a consistency check in the form of the abundances of nuclei. Calculations of nuclear reactions in the early, radiation-dominated universe predict certain abundances of hydrogen, helium, and deuterium. In particular, the relative abundance of helium and deuterium is a sensitive test of the relationships among a, , and predicted by the FRW equations, and they confirm these relationships to a precision of about 5± 4%.¹³

An additional concern is whether the Big Bang singularity is just a product of the unrealistic assumption of perfect symmetry that went into the FRW cosmology. One of the Penrose-Hawking singularity theorems proves that it is not.¹⁴ This particular singularity theorm requires three conditions: (1) the strong energy condition holds; (2) there are no closed timelike curves; and (3) a trapped surface exists in the past timelike geodesics originating at some point. The requirement of a trapped surface can fail if the universe is inhomogeneous to >rsim 10^-4, but observations of the cosmic microwave background rule out any inhomogeneity this large (see p. 258). The other possible failure of the assumptions is that if the cosmological constant is large enough, it violates the strong energy equation, and we can have a Big Bounce rather than a Big Bang (see p. 278).

An exceptional case: the Milne universe

There is still a third loophole in our conclusion that the Big Bang singularity must have existed. Consider the special case of the FRW analysis, found by Milne in 1932 (long before FRW), in which the universe is completely empty, with ρ=0 and Λ=0. This is of course not consistent with the fact that the universe contains stars and galaxies, but we might wonder whether it could tell us anything interesting as a simplified approximation to a very dilute universe. The result is that the scale factor a varies linearly with time (problem 3, p. 290). If a is not constant, then there exists a time at which a=0, but this doesn't turn out to be a real singularity (which isn't surprising, since there is no matter to create gravitational fields). Let this universe have a scattering of test particles whose masses are too small to invalidate the approximation of ρ=0, and let the test particles be at rest in the (r,θ,φ) coordinates. The linear dependence of a on t means that these particles simply move inertially and without any gravitational interactions, spreading apart from one another at a constant rate like the raisins in a rising loaf of raisin bread. The Friedmann equations require k=-1, so the spatial geometry is one of constant negative curvature.

The Milne universe is in fact flat spacetime described in tricky coordinates. The connection can be made as follows. Let a spherically symmetric cloud of test particles be emitted by an explosion that occurs at some arbitrarily chosen event in flat spacetime. Make the cloud's density be nonuniform in a certain specific way, so that every observer moving along with a test particle (called a comoving observer) sees the same local conditions in his own frame; due to Lorentz contraction by a factor γ, this requires that the density be proportional to γ as described by the observer O who remained at the origin. This scenario turns out to be identical to the Milne universe under the change of coordinates from spatially flat coordinates (T,R) to FRW coordinates (t,r), where t=T/γ is the proper time and r=vγ.

The Milne universe may be useful as an innoculation against the common misconception that the Big Bang was an explosion of matter spreading out into a preexisting vacuum. Such a description seems obviously incompatible with homogeneity, since, for example, an observer at the edge of the cloud sees the cloud filling only half of the sky. But isn't this a logical contradiction, since the Milne universe does have an explosion into vacuum, and yet it was derived as a special case of the FRW analysis, which explicitly assumed homogeneity? It is not a contradiction, because a comoving observer never actually sees an edge. In the limit as we approach the edge, the density of the cloud (as seen by the observer who stayed at the origin) approaches infinity, and the Lorentz contraction also approaches infinity, so that O considers them to be like Hamlet saying, “I could be bounded in a nutshell, and count myself a king of infinite space.” This logic only works in the case of the Milne universe. The explosion-into-preexisting-vacuum interpretation fails in Big Bang cosmologies with ρ ≠ 0.

8.2.6 Observability of expansion

Brooklyn is not expanding!

The proper interpretation of the expansion of the universe, as described by the Friedmann equations, can be tricky. The example of the Milne universe encourages us to imagine that the expansion would be undetectable, since the Milne universe can be described as either expanding or not expanding, depending on the choice of coordinates. A more general consequence of coordinate-independence is that relativity does not pick out any preferred distance scale. That is, if all our meter-sticks expand, and the rest of the universe expands as well, we would have no way to detect the expansion. The flaw in this reasoning is that the Friedmann equations only describe the average behavior of spacetime. As dramatized in the classic Woody Allen movie “Annie Hall:” “Well, the universe is everything, and if it's expanding, someday it will break apart and that would be the end of everything!” “What has the universe got to do with it? You're here in Brooklyn! Brooklyn is not expanding!”

To organize our thoughts, let's consider the following hypotheses:

1. The distance between one galaxy and another increases at the rate given by a(t) (assuming the galaxies are sufficiently distant from one another that they are not gravitationally bound within the same galactic cluster, supercluster, etc.).
2. The wavelength of a photon increases according to a(t) as it travels cosmological distances.
3. The size of the solar system increases at this rate as well (i.e., gravitationally bound systems get bigger, including the earth and the Milky Way).
4. The size of Brooklyn increases at this rate (i.e., electromagnetically bound systems get bigger).
5. The size of a helium nucleus increases at this rate (i.e., systems bound by the strong nuclear force get bigger).

We can imagine that:

• All the above hypotheses are true.
• All the above hypotheses are false, and in fact none of these sizes increases at all.
• Some are true and some false.

If all five hypotheses were true, the expansion would be undetectable, because all available meter-sticks would be expanding together. Likewise if no sizes were increasing, there would be nothing to detect. These two possibilities are really the same cosmology, described in two different coordinate systems. But the Ricci and Einstein tensors were carefully constructed so as to be intrinsic. The fact that the expansion affects the Einstein tensor shows that it cannot interpreted as a mere coordinate expansion. Specifically, suppose someone tells you that the FRW metric can be made into a flat metric by a change of coordinates. (I have come across this claim on internet forums.) The linear structure of the tensor transformation equations guarantees that a nonzero tensor can never be made into a zero tensor by a change of coordinates. Since the Einstein tensor is nonzero for an FRW metric, and zero for a flat metric, the claim is false.

Self-check: The reasoning above implicitly assumed a non-empty universe. Convince yourself that it fails in the special case of the Milne universe.

We can now see some of the limitations of a common metaphor used to explain cosmic expansion, in which the universe is visualized as the surface of an expanding balloon. The metaphor correctly gets across several ideas: that the Big Bang is not an explosion that occurred at a preexisting point in empty space; that hypothesis 1 above holds; and that the rate of recession of one galaxy relative to another is proportional to the distance between them. Nevertheless the metaphor may be misleading, because if we take a laundry marker and draw any structure on the balloon, that structure will expand at the same rate. But this implies that hypotheses 1-5 all hold, which cannot be true.

Since some of the five hypotheses must be true and some false, and we would like to sort out which are which. It should also be clear by now that these are not five independent hypotheses. For example, we can test empirically whether the ratio of Brooklyn's size to the distances between galaxies changes like a(t), remains constant, or changes with some other time dependence, but it is only the ratio that is actually observable.

Empirically, we find that hypotheses 1 and 2 are true (i.e., the photon's wavelength maintains a constant ratio with the intergalactic distance scale), while 3, 4, and 5 are false. For example, the orbits of the planets in our solar system have been measured extremely accurately by radar reflection and by signal propagation times to space probes, and no expanding trend is detected.

General-relativistic predictions

Does general relativity correctly reproduce these observations? General relativity is mainly a theory of gravity, so it should be well within its domain to explain why the solar system does not expand while intergalactic distances do. It is impractical to solve the Einstein field equations exactly so as to describe the internal structure of all the bodies that occupy the universe: galaxies, superclusters, etc. We can, however, handle simple cases, as in example 8 on page 277, where we display an exact solution for the case of a universe containing only two things: an isolated black hole, and an energy density described by a cosmological constant. We find that the characteristic scale of the black hole, e.g., the radius of its event horizon, is still set by the constant mass m, so we can see that cosmological expansion does not affect the size of this gravitationally bound system. We can also imagine putting a test particle in a circular orbit around the black hole. Since the metric near the black hole is very nearly the same as an ordinary Schwarzschild metric, we find that the test particle's orbit does not expand by any significant amount.

Cooperstock et al. have carried out estimates for more realistic cosmologies and for actual systems of interest such as the solar system.¹⁵ For example, the predicted general-relativistic effect on the radius of the earth's orbit since the time of the dinosaurs is calculated to be about as big as the diameter of an atomic nucleus; if the earth's orbit had expanded according to a(t), the increase would have been millions of kilometers.

To see why the solar-system effect is so small, let's consider how it can depend on the function a(t). The Milne universe is just flat spacetime described in silly coordinates, and it has , i.e., a nonvanishing value of H_o. This shows that we should not expect any expansion of the solar system due to . The lowest-order effect requires . Since a rescaling like a(t)arrow 2a(t) has no physical meaning, we can guess that the effect is proportional to . This quantity should be on the order of the inverse square of the age of the universe, i.e., of order H_o²∼ 10^-35 s^-2. Based on units, we expect that multiplying this by the size of the solar system might give an estimate of the rate at which the solar system expands. The result is indeed in order-of-magnitude agreement with Cooperstock's rigorous result, but should not be taken too seriously since, e.g., it would also lead us to expect, erroneously, an expansion for atoms and nuclei. Cooperstock's calculation, which correctly incorporates the physics of a gravitationally bound system, does not say anything about atoms and nuclei.

It is more difficult to demonstrate by explicit calculation that atoms and nuclei do not expand, since we do not have a theory of quantum gravity at our disposal. It is, however, easy to see that such an expansion would violate either the equivalence principle or the basic properties of quantum mechanics. One way of stating the equivalence principle is that the local geometry of spacetime is always approximately Lorentzian, so that the the laws of physics do not depend on one's position or state of motion. Among these laws of physics are the principles of quantum mechanics, which imply that an atom or a nucleus has a well-defined ground state, with a certain size that depends only on fundamental constants such as Planck's constant and the masses of the particles involved.

This is different from the case of a photon traveling across the universe. The argument given above fails, because the photon does not have a ground state. The photon does expand, and this is required by the correspondence principle. If the photon did not expand, then its wavelength would remain constant, and this would be inconsistent with the classical theory of electromagnetism, which predicts a Doppler shift due to the relative motion of the source and the observer. One can choose to describe cosmological redshifts either as Doppler shifts or as expansions of wavelength due to cosmological expansion.

A nice way of discussing atoms, nuclei, photons, and solar systems all on the same footing is to note that in geometrized units, the units of mass and length are the same. Therefore the existence of any fundamental massive particle sets a universal length scale, one that will be known to any intelligent species anywhere in the universe. Since photons are massless, they can't be used to set a universal scale in this way; a photon has a certain mass-energy, but that mass-energy can take on any value. Similarly, a solar system sets a length scale, but not a universal one; the radius of a planet's orbit can take on any value. A universe without massive fundamental particles would be a universe without measurement. Geometrically, it would obey the laws of conformal geometry, in which angles and light-cones were the only geometrical measurements. This is the reason that atoms and nuclei, which are made of massive fundamental particles, do not expand.

More than one dimension required

Another good way of understanding why a photon expands, while an atom does not, is to recall that a one-dimensional space can never have any intrinsic curvature. If the expansion of atoms were to be detectable, we would need to detect it by comparing against some other meter-stick. Let's suppose that a hydrogen atom expands more, while a more tightly bound uranium atom expands less, so that over time, we can detect a change in the ratio of the two atoms' sizes. The world-lines of the two atoms are one-dimensional curves in spacetime. They are housed in a laboratory, and although the laboratory does have some spatial extent, the equivalence principle guarantees that to a good approximation, this small spatial extent doesn't matter. This implies an intrinsic curvature in a one-dimensional space, which is mathematically impossible, so we have a proof by contradiction that atoms do not expand.

Now why does this one-dimensionality argument fail for photons and galaxies? For a pair of galaxies, it fails because the galaxies are not sufficiently close together to allow them both to be covered by a single Lorentz frame, and therefore the set of world-lines comprising the observation cannot be approximated well as lying within a one-dimensional space. Similar reasoning applies for cosmological redshifts of photons received from distant galaxies. One could instead propose flying along in a spaceship next to an electromagnetic wave, and monitoring the change in its wavelength while it is in flight. All the world-lines involved in such an experiment would indeed be confined to a one-dimensional space. The experiment is impossible, however, because the measuring apparatus cannot be accelerated to the speed of light. In reality, the speed of the light wave relative to the measuring apparatus will always equal c, so the two world-lines involved in the experiment will diverge, and will not be confined to a one-dimensional region of spacetime.

Does space expand?

Finally, the balloon metaphor encourages us to interpret cosmological expansion as a phenomenon in which space itself expands, or perhaps one in which new space is produced. Does space really expand? Without posing the question in terms of more rigorously defined, empirically observable quantities, we can't say yes or no. It is merely a matter of which definitions one chooses and which conceptual framework one finds easier and more natural to work within. Bunn and Hogg have stated the minority view against expansion of space¹⁸, while the opposite opinion is given by Francis et al.¹⁹

As an example of a self-consistent set of definitions that lead to the conclusion that space does expand, Francis et al. give the following. Define eight observers positioned at the corners of a cube, at cosmological distances from one another. Let each observer be at rest relative to the local matter and radiation that were used as ingredients in the FRW cosmology. (For example, we know that our own solar system is not at rest in this sense, because we observe that the cosmic microwave background radiation is slightly Doppler shifted in our frame of reference.) Then these eight observers will observe that, over time, the volume of the cube grows as expected according to the cube of the function a(t) in the FRW model.

This establishes that expansion of space is a plausible interpretation. To see that it is not the only possible interpretation, consider the following example. A photon is observed after having traveled to earth from a distant galaxy G, and is found to be red-shifted. Alice, who likes expansion, will explain this by saying that while the photon was in flight, the space it occupied expanded, lengthening its wavelength. Betty, who dislikes expansion, wants to interpret it as a kinematic red shift, arising from the motion of galaxy G relative to the Milky Way Malaxy, M. If Alice and Betty's disagreement is to be decided as a matter of absolute truth, then we need some objective method for resolving an observed redshift into two terms, one kinematic and one gravitational. But we've seen in section 7.3 on page 226 that this is only possible for a stationary spacetime, and cosmological spacetimes are not stationary: regardless of an observer's state of motion, he sees a change over time in observables such as density of matter and curvature of spacetime. As an extreme example, suppose that Betty, in galaxy M, receives a photon without realizing that she lives in a closed universe, and the photon has made a circuit of the cosmos, having been emitted from her own galaxy in the distant past. If she insists on interpreting this as a kinematic red shift, the she must conclude that her galaxy M is moving at some extremely high velocity relative to itself. This is in fact not an impossible interpretation, if we say that M's high velocity is relative to itself in the past. An observer who sets up a frame of reference with its origin fixed at galaxy G will happily confirm that M has been accelerating over the eons. What this demonstrates is that we can split up a cosmological red shift into kinematic and gravitational parts in any way we like, depending on our choice of coordinate system (see also p. 232).

8.2.7 The vacuum-dominated solution

For 70 years after Hubble's discovery of cosmological expansion, the standard picture was one in which the universe expanded, but the expansion must be decelerating. The deceleration is predicted by the special cases of the FRW cosmology that were believed to be applicable, and even if we didn't know anything about general relativity, it would be reasonable to expect a deceleration due to the mutual Newtonian gravitational attraction of all the mass in the universe.

But observations of distant supernovae starting around 1998 introduced a further twist in the plot. In a binary star system consisting of a white dwarf and a non-degenerate star, as the non-degenerate star evolves into a red giant, its size increases, and it can begin dumping mass onto the white dwarf. This can cause the white dwarf to exceed the Chandrasekhar limit (page 129), resulting in an explosion known as a type Ia supernova. Because the Chandrasekhar limit provides a uniform set of initial conditions, the behavior of type Ia supernovae is fairly predictable, and in particular their luminosities are approximately equal. They therefore provide a kind of standard candle: since the intrinsic brightness is known, the distance can be inferred from the apparent brightness. Given the distance, we can infer the time that was spent in transit by the light on its way to us, i.e. the look-back time. From measurements of Doppler shifts of spectral lines, we can also find the velocity at which the supernova was receding from us. The result is that we can measure the universe's rate of expansion as a function of time. Observations show that this rate of expansion has been accelerating. The Friedmann equations show that this can only occur for Λ >rsim 4ρ. This picture has been independently verified by measurements of the cosmic microwave background (CMB) radiation. A more detailed discussion of the supernova and CMB data is given in section 8.2.10 on page 281.

With hindsight, we can see that in a quantum-mechanical context, it is natural to expect that fluctuations of the vacuum, required by the Heisenberg uncertainty principle, would contribute to the cosmological constant, and in fact models tend to overpredict Λ by a factor of about 10¹²⁰! From this point of view, the mystery is why these effects cancel out so precisely. A correct understanding of the cosmological constant presumably requires a full theory of quantum gravity, which is presently far out of our reach.

The latest data show that our universe, in the present epoch, is dominated by the cosmological constant, so as an approximation we can write the Friedmann equations as

This is referred to as a vacuum-dominated universe. The solution is

where observations show that Λ∼ 10^-26 kg/m³, giving years.

The implications for the fate of the universe are depressing. All parts of the universe will accelerate away from one another faster and faster as time goes on. The relative separation between two objects, say galaxy A and galaxy B, will eventually be increasing faster than the speed of light. (The Lorentzian character of spacetime is local, so relative motion faster than c is only forbidden between objects that are passing right by one another.) At this point, an observer in either galaxy will say that the other one has passed behind an event horizon. If intelligent observers do actually exist in the far future, they may have no way to tell that the cosmos even exists. They will perceive themselves as living in island universes, such as we believed our own galaxy to be a hundred years ago.

When I introduced the standard cosmological coordinates on page 260, I described them as coordinates in which events that are simultaneous according to this t are events at which the local properties of the universe are the same. In the case of a perfectly vacuum-dominated universe, however, this notion loses its meaning. The only observable local property of such a universe is the vacuum energy described by the cosmological constant, and its density is always the same, because it is built into the structure of the vacuum. Thus the vacuum-dominated cosmology is a special one that maximally symmetric, in the sense that it has not only the symmetries of homogeneity and isotropy that we've been assuming all along, but also a symmetry with respect to time: it is a cosmology without history, in which all times appear identical to a local observer. In the special case of this cosmology, the time variation of the scaling factor a(t) is unobservable, and may be thought of as the unfortunate result of choosing an inappropriate set of coordinates, which obscure the underlying symmetry. When I argued in section 8.2.6 for the observability of the universe's expansion, note that all my arguments assumed the presence of matter or radiation. These are completely absent in a perfectly vacuum-dominated cosmology.

For these reasons de Sitter originally proposed this solution as a static universe in 1927. But by 1920 it was realized that this was an oversimplification. The argument above only shows that the time variation of a(t) does not allow us to distinguish one epoch of the universe from another. That is, we can't look out the window and infer the date (e.g., from the temperature of the cosmic microwave background radiation). It does not, however, imply that the universe is static in the sense that had been assumed until Hubble's observations. The r-t part of the metric is

where a blows up exponentially with time, and the k-dependence has been neglected, as it was in the approximation to the Friedmann equations used to derive a(t).²⁰ Let a test particle travel in the radial direction, starting at event A=(0,0) and ending at B=(t',r'). In flat space, a world-line of the linear form r=vt would be a geodesic connecting A and B; it would maximize the particle's proper time. But in the this metric, it cannot be a geodesic. The curvature of geodesics relative to a line on an r-t plot is most easily understood in the limit where t' is fairly long compared to the time-scale of the exponential, so that a(t') is huge. The particle's best strategy for maximizing its proper time is to make sure that its dr is extremely small when a is extremely large. The geodesic must therefore have nearly constant r at the end. This makes it sound as though the particle was decelerating, but in fact the opposite is true. If r is constant, then the particle's spacelike distance from the origin is just r a(t), which blows up exponentially. The near-constancy of the coordinate r at large t actually means that the particle's motion at large t isn't really due to the particle's inertial memory of its original motion, as in Newton's first law. What happens instead is that the particle's initial motion allows it to move some distance away from the origin during a time on the order of T, but after that, the expansion of the universe has become so rapid that the particle's motion simply streams outward because of the expansion of space itself. Its initial motion only mattered because it determined how far out the particle got before being swept away by the exponential expansion.

Example 7: Geodesics in a vacuum-dominated universe

In this example we confirm the above interpretation in the special case where the particle, rather than being released in motion at the origin, is released at some nonzero radius r, with dr/dt=0 initially. First we recall the geodesic equation

from page 161. The nonvanishing Christoffel symbols for the 1+1-dimensional metric ds² = dt² - a² dr² are and . Setting T=1 for convenience, we have and .

We conjecture that the particle remains at the same value of r. Given this conjecture, the particle's proper time is simply the same as its time coordinate t, and we can therefore use t as an affine coordinate. Letting λ=t, we have

This confirms the self-consistency of the conjecture that r=constant is a geodesic.

Note that we never actually had to use the actual expressions for the Christoffel symbols; we only needed to know which of them vanished and which didn't. The conclusion depended only on the fact that the metric had the form ds² = dt² - a² dr² for some function a(t). This provides a rigorous justification for the interpretation of the cosmological scale factor a as giving a universal time-variation on all distance scales.

The calculation also confirms that there is nothing special about r=0. A particle released with r=0 and initially stays at r=0, but a particle released at any other value of r also stays at that r. This cosmology is homogeneous, so any point could have been chosen as r=0. If we sprinkle test particles, all at rest, across the surface of a sphere centered on this arbitrarily chosen point, then they will all accelerate outward relative to one another, and the volume of the sphere will increase. This is exactly what we expect. The Ricci curvature is interpreted as the second derivative of the volume of a region of space defined by test particles in this way. The fact that the second derivative is positive rather than negative tells us that we are observing the kind of repulsion provided by the cosmological constant, not the attraction that results from the existence of material sources.

Example 8: Schwarzschild-de Sitter space

The metric

is an exact solution to the Einstein field equations with cosmological constant Λ, and can be interpreted as a universe in which the only mass is a black hole of mass m located at r=0. Near the black hole, the Λ terms become negligible, and this is simply the Schwarzschild metric. As argued in section 8.2.6, page 266, this is a simple example of how cosmological expansion does not cause all structures in the universe to grow at the same rate.

The Big Bang singularity in a universe with a cosmological constant

On page 265 we discussed the possibility that the Big Bang singularity was an artifact of the unrealistically perfect symmetry assumed by our cosmological models, and we found that this was not the case: the Penrose-Hawking singularity theorems demonstrate that the singularity is real, provided that the cosmological constant is zero. The cosmological constant is not zero, however. Models with a very large positive cosmological constant can also display a Big Bounce rather than a Big Bang. If we imagine using the Friedmann equations to evolve the universe backward in time from its present state, the scaling arguments of example 10 on page 118 suggest that at early enough times, radiation and matter should dominate over the cosmological constant. For a large enough value of the cosmological constant, however, it can happen that this switch-over never happens. In such a model, the universe is and always has been dominated by the cosmological constant, and we get a Big Bounce in the past because of the cosmological constant's repulsion. In this book I will only develop simple cosmological models in which the universe is dominated by a single component; for a discussion of bouncing models with both matter and a cosmological constant, see Carroll, “The Cosmological Constant,” http://www.livingreviews.org/lrr-2001-1. By 2008, a variety of observational data had pinned down the cosmological constant well enough to rule out the possibility of a bounce caused by a very strong cosmological constant.

8.2.8 The matter-dominated solution

Our universe is not perfectly vacuum-dominated, and in the past it was even less so. Let us consider the matter-dominated epoch, in which the cosmological constant was negligible compared to the material sources. The equations of state for nonrelativistic matter (p. 118) are

so the Friedmann equations become

where for compactness ρ's dependence on a, with some constant of proportionality, is not shown explicitly. A static solution, with constant a, is impossible, and is negative, which we can interpret semiclassically in terms of the deceleration of the matter in the universe due to gravitational attraction. There are three cases to consider, according to the value of k.

The closed universe

We've seen that k=+1 describes a universe in which the spatial curvature is positive, i.e., the circumference of a circle is less than its Euclidean value. By analogy with a sphere, which is the two-dimensional surface of constant positive curvature, we expect that the total volume of this universe is finite.

The second Friedmann equation also shows us that at some value of a, we will have . The universe will expand, stop, and then recollapse, eventually coming back together in a “Big Crunch” which is the time-reversed version of the Big Bang.

Suppose we were to describe an initial-value problem in this cosmology, in which the initial conditions are given for all points in the universe on some spacelike surface, say t=constant. Since the universe is assumed to be homogeneous at all times, there are really only three numbers to specify, a, , and ρ: how big is the universe, how fast is it expanding, and how much matter is in it? But these three pieces of data may or may not be consistent with the second Friedmann equation. That is, the problem is overdetermined. In particular, we can see that for small enough values of ρ, we do not have a valid solution, since the square of would have to be negative. Thus a closed universe requires a certain amount of matter in it. The present observational evidence (from supernovae and the cosmic microwave background, as described above) is sufficient to show that our universe does not contain this much matter.

The flat universe

The case of k=0 describes a universe that is spatially flat. It represents a knife-edge case lying between the closed and open universes. In a semiclassical analogy, it represents the case in which the universe is moving exactly at escape velocity; as t approaches infinity, we have a arrow∞, ρarrow 0, and . This case, unlike the others, allows an easy closed-form solution to the motion. Let the constant of proportionality in the equation of state ρ ∝ a^-3 be fixed by setting -4πρ/3=-ca^-3. The Friedmann equations are

Looking for a solution of the form a∝ t^p, we find that by choosing p=2/3 we can simultaneously satisfy both equations. The constant c is also fixed, and we can investigate this most transparently by recognizing that is interpreted as the Hubble constant, H, which is the constant of proportionality relating a far-off galaxy's velocity to its distance. Note that H is a “constant” in the sense that it is the same for all galaxies, in this particular model with a vanishing cosmological constant; it does not stay constant with the passage of cosmological time. Plugging back into the original form of the Friedmann equations, we find that the flat universe can only exist if the density of matter satisfies ρ=ρ_crit=3H²/8π=3H²/8π G. The observed value of the Hubble constant is about 1/(14×10⁹ years), which is roughly interpreted as the age of the universe, i.e., the proper time experienced by a test particle since the Big Bang. This gives ρ_crit∼ 10^-26 kg/m³.

As discussed in subsection 8.2.10, our universe turns out to be almost exactly spatially flat. Although it is presently vacuum-dominated, the flat and matter-dominated FRW cosmology is a useful description of its matter-dominated era.

The open universe

The k=-1 case represents a universe that has negative spatial curvature, is spatially infinite, and is also infinite in time, i.e., even if the cosmological constant had been zero, the expansion of the universe would have had too little matter in it to cause it to recontract and end in a Big Crunch.

The time-reversal symmetry of general relativity was discussed on p. 193 in connection with the Schwarzschild metric.²¹ Because of this symmetry, we expect that solutions to the field equations will be symmetric under time reversal (unless asymmetric boundary conditions were imposed). The closed universe has exactly this type of time-reversal symmetry. But the open universe clearly breaks this symmetry, and this is why we speak of the Big Bang as lying in the past, not in the future. This is an example of spontaneous symmetry breaking. Spontaneous symmetry breaking happens when we try to balance a pencil on its tip, and it is also an important phenomenon in particle physics. The time-reversed version of the open universe is an equally valid solution of the field equations. Another example of spontaneous symmetry breaking in cosmological solutions is that the solutions have a preferred frame of reference, which is the one at rest relative to the cosmic microwave background and the average motion of the galaxies. This is referred to as the Hubble flow.

8.2.9 The radiation-dominated solution

For the reasons discussed in example 10 on page 118, the early universe was dominated by radiation. The solution of the Friedmann equations for this case is taken up in problem 11 on page 291.

8.2.10 Observation

Historically, it was very difficult to determine the universe's average density, even to within an order of magnitude. Most of the matter in the universe probably doesn't emit light, making it difficult to detect. Astronomical distance scales are also very poorly calibrated against absolute units such as the SI.

A strong constraint on the models comes from accurate measurements of the cosmic microwave background, especially by the 1989-1993 COBE probe, and its 2001-2009 successor, the Wilkinson Microwave Anisotropy Probe, positioned at the L2 Lagrange point of the earth-sun system, beyond the Earth on the line connecting sun and earth.²² The temperature of the cosmic microwave background radiation is not the same in all directions, and its can be measured at different angles. In a universe with negative spatial curvature, the sum of the interior angles of a triangle is less than the Euclidean value of 180 degrees. Therefore if we observe a variation in the CMB over some angle, the distance between two points on the surface of last scattering is actually greater than would have been inferred from Euclidean geometry. The distance scale of such variations is limited by the speed of sound in the early universe, so one can work backward and infer the universe's spatial curvature based on the angular scale of the anisotropies. The measurements of spatial curvature are usually stated in terms of the parameter Ω, defined as the total average density of all source terms in the Einstein field equations, divided by the critical density that results in a flat universe. Ω includes contributions from matter, Ω_M, the cosmological constant, Ω_\Lambda, and radiation (negligible in the present-day unverse). The results from WMAP, combined with other data from other methods, gives Ω=1.005± .006. In other words, the universe is very nearly spatially flat.

The supernova data described on page 274 complement the CMB data because they are mainly sensitive to the difference Ω_\Lambda-Ω_M, rather than their sum Ω=Ω_\Lambda+Ω_M. This is because these data measure the acceleration or deceleration of the universe's expansion. Matter produces deceleration, while the cosmological constant gives acceleration. Figure e shows some recent supernova data.²³ The horizontal axis gives the redshift factor z=(λ'-λ)/λ, where λ' is the wavelength observed on earth and λ the wavelength originally emitted. It measures how fast the supernova's galaxy is receding from us. The vertical axis is Δ(m-M)=(m-M)-(m-M)_empty, where m is the apparent magnitude, M is the absolute magnitude, and (m-M)_empty is the value expected in a model of an empty universe, with Ω=0. The difference m-M is a measure of distance, so essentially this is a graph of distance versus recessional velocity, of the same general type used by Hubble in his original discovery of the expansion of the universe. Subtracting (m-M)_empty on the vertical axis makes it easier to see small differences. Since the WMAP data require Ω=1, we need to fit the supernova data with values of Ω_M and Ω_\Lambda that add up to one. Attempting to do so with Ω_M=1 and Ω_\Lambda=0 is clearly inconsistent with the data, so we can conclude that the cosmological constant is definitely positive.

Figure f summarizes what we can conclude about our universe, parametrized in terms of a model with both Ω_M and Ω_\Lambda nonzero. ²⁴ We can tell that it originated in a Big Bang singularity, that it will go on expanding forever, and that it is very nearly flat. Note that in a cosmology with nonzero values for both Ω_M and Ω_\Lambda, there is no strict linkage between the spatial curvature and the question of recollapse, as there is in a model with only matter and no cosmological constant; therefore even though we know that the universe will not recollapse, we do not know whether its spatial curvature is slightly positive (closed) or negative (open).

Astrophysical considerations provide further constraints and consistency checks. In the era before the advent of high-precision cosmology, estimates of the age of the universe ranged from 10 billion to 20 billion years, and the low end was inconsistent with the age of the oldest globular clusters. This was believed to be a problem either for observational cosmology or for the astrophysical models used to estimate the age of the clusters: “You can't be older than your ma.” Current data have shown that the low estimates of the age were incorrect, so consistency is restored.

Another constraint comes from models of nucleosynthesis during the era shortly after the Big Bang (before the formation of the first stars). The observed relative abundances of hydrogen, helium, and deuterium cannot be reconciled with the density of “dust” (i.e., nonrelativistic matter) inferred from the observational data. If the inferred mass density were entirely due to normal “baryonic” matter (i.e., matter whose mass consisted mostly of protons and neutrons), then nuclear reactions in the dense early universe should have proceeded relatively efficiently, leading to a much higher ratio of helium to hydrogen, and a much lower abundance of deuterium. The conclusion is that most of the matter in the universe must be made of an unknown type of exotic non-baryonic matter, known generically as “dark matter.” A number of experiments are under way to detect dark matter directly, and the results so far are not definitive and sometimes contradict one another.²⁵ Even with the inclusion of dark matter, there is a problem with the abundance of lithium-7 relative to hydrogen, which models greatly overpredict.²⁶

8.3 Mach's principle revisited

8.3.1 The Brans-Dicke theory

Mach himself never succeeded in stating his ideas in the form of a precisely testable physical theory, and we've seen that to the extent that Einstein's hopes and intuition had been formed by Mach's ideas, he often felt that his own theory of gravity came up short. The reader has so far encountered Mach's principle in the context of certain thought experiments that are obviously impossible to realize, involving a hypothetical universe that is empty except for certain apparatus (e.g., section 3.5.2, p. 104). It would be easy, then, to get an impression of Mach's principle as one of those theories that is “not even wrong,” i.e., so ill-defined that it cannot even be falsified by experiment, any more than Christianity can be.

But in 1961, Robert Dicke and his student Carl Brans came up with a theory of gravity that made testable predictions, and that was specifically designed to be more Machian than general relativity. Their paper²⁷ is extremely readable, even for the non-specialist. In this theory, the seemingly foolproof operational definition of a Lorentz frame given on p. 26 fails. On the first page, Brans and Dicke propose one of those seemingly foolish thought experiments about a nearly empty universe:

\begin{quotation} The imperfect expression of [Mach's ideas] in general relativity can be seen by considering the case of a space empty except for a lone experimenter in his laboratory. [...] The observer would, according to general relativity, observe normal behavior of his apparatus in accordance with the usual laws of physics. However, also according to general relativity, the experimenter could set his laboratory rotating by leaning out a window and firing his 22-caliber rifle tangentially. Thereafter the delicate gyroscope in the laboratory would continue to point in a direction nearly fixed relative to the direction of motion of the rapidly receding bullet. The gyroscope would rotate relative to the walls of the laboratory. Thus, from the point of view of Mach, the tiny, almost massless, very distant bullet seems to be more important than the massive, nearby walls of the laboratory in determining inertial coordinate frames and the orientation of the gyroscope. \end{quotation}

They then proceed to construct a mathematical and more Machian theory of gravity. From the Machian point of view, the correct local definition of an inertial frame must be determined relative to the bulk of the matter in the universe. We want to retain the Lorentzian local character of spacetime, so this influence can't be transmitted via instantaneous action at a distance. It must propagate via some physical field, at a speed less than or equal to c. It is implausible that this field would be the gravitational field as described by general relativity. Suppose we divide the cosmos up into a series of concentric spherical shells centered on our galaxy. In Newtonian mechanics, the gravitational field obeys Gauss's law, so the field of such a shell vanishes identically on the interior. In relativity, the corresponding statement is Birkhoff's theorem, which states that the Schwarzschild metric is the unique spherically symmetric solution to the vacuum field equations. Given this solution in the exterior universe, we can set a boundary condition at the outside surface of the shell, use the Einstein field equations to extend the solution through it, and find a unique solution on the interior, which is simply a flat space.

Since the Machian effect can't be carried by the gravitational field, Brans and Dicke took up an idea earlier proposed by Pascual Jordan²⁸ of hypothesizing an auxiliary field φ. The fact that such a field has never been detected directly suggests that it has no mass or charge. If it is massless, it must propagate at exactly c, and this also makes sense because if it were to propagate at speeds less than c, there would be no obvious physical parameter that would determine that speed. How many tensor indices should it have? Since Mach's principle tries to give an account of inertia, and inertial mass is a scalar,²⁹ φ should presumably be a scalar (quantized by a spin-zero particle). Theories of this type are called tensor-scalar theories, because they use a scalar field in addition to the metric tensor.

The wave equation for a massless scalar field, in the absence of sources, is simply ∇_i ∇ⁱ φ=0. The solutions of this wave equation fall off as φ∼ 1/r. This is gentler than the 1/r² variation of the gravitational field, so results like Newton's shell theorem and Birkhoff's theorem no longer apply. If a spherical shell of mass acts as a source of φ, then φ can be nonzero and varying inside the shell. The φ that you experience right now as you read this book should be a sum of wavelets originating from all the masses whose world-lines intersected the surface of your past light-cone. In a static universe, this sum would diverge linearly, so a self-consistency requirement for Brans-Dicke gravity is that it should produce cosmological solutions that avoid such a divergence, e.g., ones that begin with Big Bangs.

Masses are the sources of the field φ. How should they couple to it? Since φ is a scalar, we need to construct a scalar as its source, and the only reasonable scalar that can play this role is the trace of the stress-energy tensor, . As discussed in example 3 on page 254, this vanishes for light, so the only sources of φ are material particles.³⁰ Even so, the Brans-Dicke theory retains a form of the equivalence principle. As discussed on pp. 39 and 33, the equivalence principle is a statement about the results of local experiments, and φ at any given location in the universe is dominated by contributions from matter lying at cosmological distances. Objects of different composition will have differing fractions of their mass that arise from internal electromagnetic fields. Two such objects will still follow identical geodesics, since their own effect on the local value of φ is negligible. This is unlike the behavior of electrically charged objects, which experience significant back-reaction effects in curved space (p. 39). However, the strongest form of the equivalence principle requires that all experiments in free-falling laboratories produce identical results, no matter where and when they are carried out. Brans-Dicke gravity violates this, because such experiments could detect differences between the value of φ at different locations --- but of course this is part and parcel of the purpose of the theory.

We now need to see how to connect φ to the local notion of inertia so as to produce an effect of the kind that would tend to fulfill Mach's principle. In Mach's original formulation, this would entail some kind of local rescaling of all inertial masses, but Brans and Dicke point out that in a theory of gravity, this is equivalent to scaling the Newtonian gravitational constant G down by the same factor. The latter turns out to be a better approach. For one thing, it has a natural interpretation in terms of units. Since φ's amplitude falls off as 1/r, we can write φ∼ Σ m_i/r, where the sum is over the past light cone. If we then make the identification of φ with 1/G (or c²/G in a system wher c ≠ 1), the units work out properly, and the coupling constant between matter and φ can be unitless. If this coupling constant, notated 1/ω, were not unitless, then the theory's predictive value would be weakened, because there would be no way to know what value to pick for it. For a unitless constant, however, there is a reasonable way to guess what it should be: “in any sensible theory,” Brans and Dicke write, “ω must be of the general order of magnitude of unity.” This is, of course, assuming that the Brans-Dicke theory was correct. In general, there are other reasonable values to pick for a unitless number, including zero and infinity. The limit of ωarrow∞ recovers the special case of general relativity. Thus Mach's principle, which once seemed too vague to be empirically falsifiable, comes down to measuring a specific number, ω, which quantifies how non-Machian our universe is.³¹

8.3.2 Predictions of the Brans-Dicke theory

Returning to the example of the spherical shell of mass, we can see based on considerations of units that the value of φ inside should be ∼ m/r, where m is the total mass of the shell and r is its radius. There may be a unitless factor out in front, which will depend on ω, but for ω∼ 1 we expect this constant to be of order 1. Solving the nasty set of field equations that result from their Lagrangian, Brans and Dicke indeed found , where the constant in square brackets is of order unity if ω is of order unity. In the limit of ωarrow∞, φ=0, and the shell has no physical effect on its interior, as predicted by general relativity.

Brans and Dicke were also able to calculate cosmological models, and in a typical model with a nearly spatially flat universe, they found φ would vary according to

where ρ_o is the density of matter in the universe at time t=t_o. When the density of matter is small, G is large, which has the same observational consequences as the disappearance of inertia; this is exactly what one expects according to Mach's principle. For ωarrow∞, the gravitational “constant” G= 1/φ really is constant.

Returning to the thought experiment involving the 22-caliber rifle fired out the window, we find that in this imaginary universe, with a very small density of matter, G should be very large. This causes a frame-dragging effect from the laboratory on the gyroscope, one much stronger than we would see in our universe. Brans and Dicke calculated this effect for a laboratory consisting of a spherical shell, and although technical difficulties prevented the reliable extrapolation of their result to ρ_oarrow 0, the trend was that as ρ_o became small, the frame-dragging effect would get stronger and stronger, presumably eventually forcing the gyroscope to precess in lock-step with the laboratory. There would thus be no way to determine, once the bullet was far away, that the laboratory was rotating at all --- in perfect agreement with Mach's principle.

8.3.3 Hints of empirical support

Only six years after the publication of the Brans-Dicke theory, Dicke himself, along with H.M. Goldenberg³² carried out a measurement that seemed to support the theory empirically. Fifty years before, one of the first empirical tests of general relativity, which it had seemed to pass with flying colors, was the anomalous perihelion precession of Mercury. The word “anomalous,” which is often left out in descriptions of this test, is required because there are many nonrelativistic reasons why Mercury's orbit precesses, including interactions with the other planets and the sun's oblate shape. It is only when these other effects are subtracted out that one sees the general-relativistic effect calculated on page 197. The sun's oblateness is difficult to measure optically, so the original analysis of the data had proceeded by determining the sun's rotational period by observing sunspots, and then assuming that the sun's bulge was the one found for a rotating fluid in static equilibrium. The result was an assumed oblateness of about 1×10^-5. But we know that the sun's dynamics are more complicated than this, since it has convection currents and magnetic fields. Dicke, who was already a renowned experimentalist, set out to determine the oblateness by direct optical measurements, and the result was (5.0± 0.7)× 10^-5, which, although still very small, was enough to put the observed perihelion precession out of agreement with general relativity by about 8%. The perihelion precession predicted by Brans-Dicke gravity differs from the general relativistic result by a factor of (4+3ω)/(6+3ω). The data therefore appeared to require ω≈ 6± 1, which would be inconsistent with general relativity.

8.3.4 Mach's principle is false.

The trouble with the solar oblateness measurements was that they were subject to a large number of possible systematic errors, and for this reason it was desirable to find a more reliable test of Brans-Dicke gravity. Not until about 1990 did a consensus arise, based on measurements of oscillations of the solar surface, that the pre-Dicke value was correct. In the interim, the confusion had the salutary effect of stimulating a renaissance of theoretical and experimental work in general relativity. Often if one doesn't have an alternative theory, one has no reasonable basis on which to design and interpret experiments to test the original theory.

Currently, the best bound on ω is based on measurements³³ of the propagation of radio signals between earth and the Cassini-Huygens space probe in 2003, which require ω>4×10⁴. This is so much greater than unity that it is reasonable to take Brans and Dicke at their word that “in any sensible theory, ω must be of the general order of magnitude of unity.” Brans-Dicke fails this test, and is no longer a “sensible” candidate for a theory of gravity. We can now see that Mach's principle, far from being a fuzzy piece of philosophical navel-gazing, is a testable hypothesis. It has been tested and found to be false, in the following sense. Brans-Dicke gravity is about as natural a formal implementation of Mach's principle as could be hoped for, and it gives us a number ω that parametrizes how Machian the universe is. The empirical value of ω is so large that it shows our universe to be essentially as non-Machian as general relativity.

Homework Problems

6. The Einstein field equations are

G_ab = 8π T_ab + Λ g_ab ,

and when it is possible to adopt a frame of reference in which the local mass-energy is at rest on average, we can interpret the stress-energy tensor as

where ρ is the mass-energy density and P is the pressure. Fix some point as the origin of a local Lorentzian coordinate system. Analyze the properties of these relations under a reflection such as x arrow -x or t arrow -t. (solution in the pdf version of the book)