Chapter 7. Sources

7.1 Sources in general relativity

7.1.1 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 177, 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).

7.1.2 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 energy-momentum tensor T^ab (page 125) 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 energy-momentum 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?

Further interpretation of the energy-momentum tensor

The energy-momentum tensor was briefly introduced in section 5.2 on page 125. 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. 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. The spacelike components are associated with the pressure, P. The form of the tensor with mixed upper and lower indices has the simple form .

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 rotations.

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:

• The equivalence principle is satisfied.
• 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 1: 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 energy-momentum 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.

This particular attempt fails, because it violates the equivalence principle. Consider a beam of light directed along the x axis. Its momentum is equal to its energy (see page 102), so its contributions to the local energy density and pressure are equal. Thus its contribution to the energy-momentum tensor is of the form . The trace vanishes, so its coupling to gravity in the c₂ term is zero. But this violates the equivalence principle, which requires that all forms of mass-energy contribute equally to gravitational mass.

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.¹ 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.

Now consider the following pseudo-classical argument. Although we've already seen (page 169) 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.

7.2 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 tell us about the large-scale structure of the universe, its origin, and its fate?

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).

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

In these coordinates, the time t is interpreted as the proper time of a particle that has always been at rest. Events that are simultaneous according to this t are events at which the local properties of the universe --- i.e., its curvature --- are the same. 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 unknown function f(r) has to make the 3-space metric have 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 form of shows us that k can be interpreted in terms of the sign of the spatial curvature. The k=0 case gives a flat space. For negative k, a circle of radius r centered on the origin has a circumference 2π r f(r) that is less than its Euclidean value of 2π r. The opposite occurs for k>0. The resulting metric, called the Robertson-Walker metric, is

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 energy-momentum 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.

7.2.1 Evidence for expansion of the universe

By 1929, Edwin Hubble at Mount Wilson had determined that the universe was expanding rather than static, so that Einstein's original goal of allowing a static cosmology became pointless. The universe, it seemed, had originated in a Big Bang (a concept that originated with the Belgian Roman Catholic priest Georges Lema\^{i}tre). This now appears natural, since the Friedmann equations would only allow a constant a in the case where Λ was perfectly tuned relative to the other parameters. 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.

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

The existence of the Big Bang is confirmed directly by looking up in the sky and seeing it. 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. Soon after the Big Bang, 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 around 4× 10⁵ years after the big bang, 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.

7.2.2 A singularity at the Big Bang

In section 6.3.1, we saw that a black hole contains a singularity. However, it appears that black hole singularities are always hidden behind event horizons, so that we can never observe them from the outside. Now 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 9 on page 105, 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=0 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. If this singularity in the model corresponds to a singularity in the real universe, then it is not a singularity that is hidden behind an event horizon. It lies in our past light-cone, and our own world-lines emerged from it.

We may ask, however, whether this singularity is merely an unrealistic artifact of the model. Early relativists suspected, with good reason, that it was. 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. Since Newton's laws have time-reversal symmetry, the same is true for the past: if we choose a random set of present positions and velocities for a set of particles interacting gravitationally, there is zero probability that extrapolation into the past will show them having come out of an explosion from a single point. It is therefore natural to be skeptical about the Big Bang singularity implied by the Friedmann equations, since the Friedmann equations were derived under the unrealistic assumption that the universe was perfectly symmetric. For many decades, there was thus a suspicion that the Big Bang might not have been a singularity at all, but rather a moment at which all the matter and radiation in the universe came very close together before flying apart again.

We now know that this is not the case, and the reasons are very similar to the reasons why we know that black hole singularities occur generically, not just when the star's collapse is calculated in some unrealistically symmetric model. 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 either the entire future light-cone or the entire past light-cone converges at the source of the field. This statement is formalized by a series of theorems by Penrose and Hawking, known as the singularity theorems. The detailed statement of these theorems is technical and beyond the scope of this book, but they prove the existence of a Big Bang singularity, provided that the cosmological constant is not too large; but regarding this point, see page 200.

7.2.3 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. It might seem as though the expansion would be undetectable, in the sense that general relativity is coordinate-independent, and therefore 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-space 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-space metric, the claim is false.

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 4 on page 199, 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. Estimates have also been carried out for more realistic cosmologies and for actual systems of interest such as the solar system.² 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.

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.

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 space⁵, while the opposite opinion is given by Francis et al.⁶ 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 according to the function a(t) in the FRW model.

7.2.4 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 116), 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 7.2.6 on page 202.

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

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 186, 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 7.2.3 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).⁷ 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 d r 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 3: 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 d r/d t=0 initially. First we recall the geodesic equation

from page 140. The nonvanishing Christoffel symbols for the 1+1-dimensional metric d s² = d t² - a² d r² 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 d s² = d t² - a² d r² 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 4: 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 7.2.3, page 190, 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 190 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 9 on page 105 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.

7.2.5 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. 105) 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 →∞, ρ→ 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=2H²/8π=2H²/8π 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.

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.

a / The angular scale of fluctuations in the cosmic microwave background can be used to infer the curvature of the universe.

b / A Hubble plot for distant supernovae. Each data point represents an average over several different supernovae with nearly the same z.

c / The cosmological parameters of our universe, after Perlmutter et al., arxiv.org/abs/astro-ph/9812133.

7.2.6 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. 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 195 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 b shows some recent supernova data.⁸ 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 c summarizes what we can conclude about our universe, parametrized in terms of a model with both Ω_M and Ω_\Lambda nonzero. ⁹ 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.”

7.3 Mach's principle revisited

7.3.1 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.~96). 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 paper¹⁰ is extremely readable, even for the non-specialist. On the first page, they 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 Jordan¹¹ 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, 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 1 on page 184, this vanishes for light, so only material particles are sources of φ. Even so, the Brans-Dicke theory retains a form of the equivalence principle. As discussed on pp.~34 and 28, 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.~34). 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 ω→∞ 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.¹²

7.3.2 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 ω→∞, 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 ω→∞, 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_o→ 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.

a / The apparatus used by Dicke and Goldenberg to measure the oblateness of the sun was essentially a telescope with a disk inserted in order to black out most of the light from the sun.

7.3.3 Hints of empirical support

Only six years after the publication of the Brans-Dicke theory, Dicke himself, along with H.M.~Goldenberg¹³ 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 167. 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.

7.3.4 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 measurements¹⁴ 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.