and so will still give a Lorentz gauge. Thus, the Lorentz gauge is really a class of gauges. In this gauge, Eq. (8.32) becomes (see Exer. 10, § 8.6)

Then the weak-field Einstein equations are

These are called the field equations of ‘linearized theory’, since they result from keeping terms linear in hαβ .

8.4 N e w t o n i a n g ra v i t a t i o n a l fi e l d s

Newtonian limit

Newtonian gravity is known to be valid when gravitational fields are too weak to produce velocities near the speed of light: |φ| 1, |v| 1. In such situations, GR must make the same predictions as Newtonian gravity. The fact that velocities are small means that the components Tαβ typically obey the inequalities |T00| |T0i| |Tij|. We can only say ‘typically’ because in special cases T0i might vanish, say for a spherical star, and the second inequality would not hold. But in a strongly rotating Newtonian star, T0i would greatly exceed any of the components of Tij. Now, these inequalities should be expected to trans-

about making too broad a statement: we can add in any solution to the homogeneous form of Eq. (8.42), where the right-hand-side is set to zero. In such a solution, the sizes of the components would not be controlled by the sizes of the components of Tαβ . These homogeneous solutions are what we call gravitational waves, as we shall see in the next chapter.

¯

So the ordering given here on the components of hαβ holds only in the absence of significant gravitational radiation. Newtonian gravity, of course, has no gravitational waves, so the ordering is just what we need if we want to reproduce Newtonian gravity in general relativity. Thus, we can expect that the dominant ‘Newtonian’ gravitational field comes

where we use the fact that T00 = ρ + 0(ρv2). For fields that change only because the sources move with velocity v, we have that ∂/∂t is of the same order as v ∂/∂x, so that

This is identical to the metric given in Eq. (7.8). We saw there that this metric gives the correct Newtonian laws of motion, so the demonstration here that it follows from Einstein’s equations completes the proof that Newtonian gravity is a limiting case of GR. Importantly, it also confirms that the constant 8π in Einstein’s equations is the correct value of k.

Most astronomical systems are well-described by Newtonian gravity as a first approximation. But there are many systems for which it is important to compute the corrections beyond Newtonian theory. These are called post-Newtonian effects, and in Exers. 19 and 20, § 8.6, we encounter two of them. Post-Newtonian effects in the Solar System include the famous fundamental tests of general relativity, such as the precession of the perihelion of Mercury and the bending of light by the Sun; both of which we will meet in Ch. 11. Outside the Solar System the most important post-Newtonian effect is the shrinking of the orbit of the Binary Pulsar, which confirms general relativity’s predictions concerning gravitational radiation (see Ch. 9). Post-Newtonian effects therefore lead to important highprecision tests of general relativity, and the theory of these effects is very well developed. The approximation has been carried to very high orders (Blanchet 2006, Futamase and Itoh 2007).

The far field of stationary relativistic sources

For any source of the full Einstein equations, which is confined within a limited region of space (a ‘localized’ source), we can always go far enough away from it that its gravitational field becomes weak enough that linearized theory applies in that region. We say that such a spacetime is asymptotically flat: spacetime becomes flat asymptotically at large distances from the source. We might be tempted, then, to carry the discussion we have just gone through over to this case and say that Eq. (8.50) describes the far field of the source, with φ the Newtonian potential. This method would be wrong, for two related reasons. First, the derivation of Eq. (8.50) assumed that gravity was weak everywhere, including inside the source, because a crucial step was the identification of Eq. (8.45) with Eq. (8.1) inside

the source. In the present discussion we wish to make no assumptions about the weakness of gravity in the source. The second reason the method would be wrong is that we do not know how to define the Newtonian potential φ of a highly relativistic source anyway, so Eq. (8.50) would not make sense.

So we shall work from the linearized field equations directly. Since at first we assume the source of the field Tμν is stationary (i.e. independent of time), we can assume that far away from it hμν is independent of time. (Later we will relax this assumption.) Then Eq. (8.42) becomes

far from the source. This has the solution

where Aμν is constant. In addition, we must demand that the gauge condition, Eq (8.33), be satisfied:

¯μν

where the sum on ν reduces to a sum on the spatial index j because h is time independent, and where nj is the unit radial normal,

¯00

for all μ and j. This means that only h survives or, in other words, that, far from the source

These conditions guarantee that the gravitational field does indeed behave like a Newtonian field out there, so we can reverse the identification that led to Eq. (8.46) and define the ‘Newtonian potential’ for the far field of any stationary source to be

With this identification, Eq. (8.50) now does make sense for our problem, and it describes the far field of our source.

Definition of the mass of a relativistic body

Now, far from a Newtonian source, the potential is

where M is the mass of the source (with G = 1). Thus, if in Eq. (8.52) we rename the constant A00 to be 4M, the identification, Eq. (8.57), says that