system
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human
Where in Einstein's field equations (or other equations if needed) does gravity propagate at the speed of light?

Obviously c is in the equation but it's in the denominator of a term, and it's to the fourth power. How does this translate into gravity propagating at the speed of light/causation? (Relatedly, is it significant that that term has 4 radial rotations about the circle in the numerator and c^4 in the denominator?)

Is gravity's speed just assumed? Is it part of the stress-energy tensor and I just need to study that more closely? Am I looking at the wrong equation entirely?
gpt
Let's first answer your question about the significance of c^(4) in the equations. The Einstein equations can be written as

> R*_μν_*-(R/2)g*_μν_*+Λg*_μν_* = (8πG/c^(4))T*_μν_*

The terms are R*_μν_* = Ricci tensor, R = scalar curvature, g*_μν_* = metric tensor, Λ = cosmological constant, T*_μν_* = stress-energy tensor.

The metric tensor is, strictly speaking, a dimensionless quantity. The components of the Ricci tensor are obtained via second derivatives of the metric, which means that the Ricci tensor has units of (length)^(-2). Hence *R* and Λ also have units of (length)^(-2) if the left side of the field equations is to make sense. The units of the stress-energy tensor is that of an energy density, or energy per unit volume, or (energy)(length)^(-3). The left and right sides of the equations must have the same units. Hence the constant multiplying T*_μν_* on the right side must have units of length per energy, or (length)(energy)^(-1). The unique combination of the gravitational constant *G* and *c* that gives those units is G/c^(4). So there really is no significance to the constant other than that of units.

(The factor of 8π, however, cannot be deduced by dimensional analysis, and it does have some significance. If the field equations are to reduce to Poisson's equation in the limit of Newtonian gravity, then we need the factor of 8π.)

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Now for your question about gravitational waves. The standard way of describing weak g-waves is to write the metric tensor as some fixed background metric plus a small perturbation. For the type of wave that was discovered by LIGO, the simplest choice of background metric would be the flat Minkowski metric. Symbolically,

> g*_μν_* = η*_μν_*+h*_μν_*

with |h*_μν_*| << 1. This has the effect of ignoring all terms that are at least quadratic in *h* when we calculate the components of the curvature tensors, Christoffel symbols, etc.

There is a bunch of math at this point where you substitute this form of g*_μν_* into the field equations to get some set of equations for the perturbation h*_μν_*. There are some mathematical subtleties here. For one, the decomposition of g*_μν_* is not unique: there may be many coordinates systems in which g*_μν_* takes the form of a flat background plus a small perturbation. This freedom is related to a so-called *gauge freedom*, and there are several choices one can make. A particular convenient choice is the *Lorenz gauge* (similar to the one used in classical electrodynamics). If the wave is far from any sources, then we can instead impose a so-called *transverse traceless* gauge. But, whatever, those details are not important for the purpose of answering your question.

You stick this form of the metric into the field equations after choosing an appropriate gauge. The equations that pop out take the form

> (-∂*_tt_*/c^(2)+Δ)h*_μν_* = -(16πG/c^(4))T*_μν_*

(Strictly speaking it's not *h* that appears here but rather the trace-reverse perturbation, but that's not important.) Hence each component of the perturbation satisfies an inhomogeneous wave equation. You can read off the source term as the corresponding component of the stress-energy tensor. You can also read off the speed of the wave, which is just the number that multiplies *t* in the time derivative part of the wave operator. In this case, that number is just *c*. It is well known that solutions to this wave equation may be interpreted as signals that propagate at the finite speed of *c*.

**In summary, (1) write metric as FLAT+SMALL, (2) substitute into the field equations, (3) obtain wave equations for the components of SMALL, (4) note that the speed of the wave is just *c*.** So it's not completely obvious from the general form of the field equations that gravity waves travel at *c*. For one, there is quite a bit of machinery that has to be developed just to define what we mean by BACKGROUND+SMALL. But, to answer your question, no, the speed of gravity waves is not defined or put in by fiat. The fact that gravity waves even exist (i.e., are predicted by the theory) is something to be discovered, as is their speed.