The speed of gravitational waves in general relativity

Question

Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein field equations? The approach via the linearization of the EFE's, which is used in many books on relativity, does not seem to provide an exact proof that disturbances in spacetime propagate at the speed $ c $.

Answer 1

What is the speed of a wave in a non-linear theory? Answering before considering your question is important, because that answer will tell you where to look for your answer.

A useful notion is that of domain of dependence (see for example a decent book on GR for a detailed discussion, e.g., Wald or Hawking & Ellis). If $S\subseteq \Sigma$ is a subset of a Cauchy surface, then the domain of dependence $D(S)$ is the region of the spacetime where the solution is completely determined by the initial data on $S$, irrespective of what initial data is specified on the complement $\Sigma \setminus S$. Thus, the "slope" of the boundaries of $D(S)$ when represented in a spacetime diagram can be interpreted as the rate at which the influence of the initial data from $\Sigma \setminus S$ is encroaching on the spacetime region where the solution is determined by the initial data on $S$ alone. In other words, the "slope" of the boundary of $D(S)$ determines the speed of the propagation of disturbances in the solutions of the PDE you are solving. This speed also agrees with the speed of traveling wave solutions in the case of a linear PDE with constant coefficients.

Note that this "speed" of propagation could be independent of the underlying solution itself (the case in electromagnetism) or may actually depend on the solution (the case in GR). It happens to be a fact of life that for both electromagnetism and GR that the boundary of $D(S)$ is always traced out by the null rays of the spacetime metric (which is dynamical in the latter case, but need not be in the former). This fact then implies that the speed of electromagnetic and gravitational waves is the same, independent of any linearization.

The above result on the boundary of the domain of dependence is standard in the hyperbolic PDE literature and can be found, for instance, in the books of John (Partial differential equations), Lax (Hyperbolic partial differential equations) and Hoermander (Lectures on nonlinear hyperbolic differential equations). Briefly, the boundary is always "characteristic", where for PDEs of Lorentzian wave equation type the "characteristics" coincide with the null directions of the metric.

Answer 2

The question asks whether it is possible to mathematically prove the speed of propagation of gravity waves without linearization.

It does not yet appear possible. Please let me explain.

Einstein's GR postulates the equality of a curvature tensor $R_{ij}-R g_{ij}$ with a mass-energy quantity of the form $\kappa T_{ij}$ where $T$ is a term representing mass-energy. That is $R_{ij}-Rg_{ij}=\kappa T_{ij}$

The equality demands:

Hypothesis (i) that the LHS term of Einstein's GR equation $R_{ij}-Rg_{ij}$ is an invariant tensor, and has vanishing tensor divergence $div(R_{ij}-Rg_{ij})=0$. (And this is true, as verified by calculations).

Hypothesis (ii): that the RHS mass-energy quantity $T_{ij}$ be a tensor quantity, and that $T_{ij}$ have well-defined tensor divergence which vanishes identically.

The GR equation appears to be well formed only if (ii) is satisfied.

In Einstein's formulation of GR (1916) an expression for the mass energy $T$ was given, but which failed to satisfy (ii). The error is an improper index contraction. The master of the absolute differential calculus, Levi-Civita corrected Einstein's original expression $t^\alpha_\sigma$ for the so-called "energy components" $t^\alpha_\sigma$ of the gravitational fields. Levi-Civita's form of GR equations yields proper tensor divergences $div T$ if $T$ is tensorial. See Ch.XI, SS 20-25 in Levi-Civita's "Absolute Differential Calculus". However Levi-Civita's appears to confuse mass with matter, invoking the continuity equation of incompressible fluid flow as parallel for the energy density using $e=mc^2=\rho c^2$. But we see no reason why mass needs satisfy a continuity equation. For we acknowledge that matter is neither created nor destroyed, but mass is mutable. This is controversial, relating to Newton's definition of mass as a measure of matter.

The tensoriality of $T$ is not readily established, being dependant on the physical model used.

As far as Einstein's original expression $t=t^\alpha_\sigma$, P.A.M. Dirac said ``in general, gravitational energy cannot be localized. The best we can do is use a pseudotensor...which gives us approximate information about gravitational energy, which in some special cases can be accurate." (See Dirac's book "General Theory of Relativity").

A.S. Eddington similarly concluded the nonpossibility, writing: ``If coordinates are chosen so as to satisfy a certain condition which has no very clear geometrical importance, the speed [of gravity waves] is that of light; if the coordinates are slightly different the speed is altogether different from that of light. The result stands or falls by the choice of coordinates, ...". (See Eddington, The Mathematical Theory of Relativity, S 57).

The key point -- as realized by Einstein, Eddington, Dirac, Hoyle, Abrams, even Crothers -- is that Einstein's so-called "gravitational energy tensor" is not a tensor at all! To quote Einstein: "The quantities $t^\alpha_\sigma$ we call the 'energy components' of the gravitational field,..., it is to be noted that $t^\alpha_\sigma$ is not a tensor". (See Einstein's ``The Foundation of the General Relativity, 1916, S.15).

Einstein noted well that $t$ is not a tensor, but is invariant under linear unimodular change of coordinates. This is elaborated in many excellent articles by E.Norton (see his articles on General Covariance and Einstein's Point-Coincidence Argument, and the long documented struggles which Einstein had in developing satisfactory covariant equations.

A further difficult is that Einstein apparently discovers the conservation of gravitational energy by evaluating the coordinate divergence(!) of $t$ and finding $\partial t^\alpha_\sigma / \partial x_\alpha=0$. He says "This equation expresses the law of conservation of momentum and of energy for the gravitational field." (Ibid) However, the vanishing of a coordinate divergence of a nontensor object is not a covariant object except (in this case) for observers who share the same volume form, i.e. unimodular linear change of coordinates change of coordinates.

Dirac says further, "Let us consider the energy of these waves. Oweing to the pseudo-tensor not being a real tensor, we do not get, in general, a clear result independant of the coordinate system. But there is one special case in which we do get a clear result; namely when the waves are all moving in the same direction", (Ibid).

So as of yet, the answer to the OP's question appears Negative.