Willie Wong (2012) in the link explains the current status of the problem. The claim that the test mass follows the metric is usually taken as an axiom of general relativity.
Several authors have tried to prove the axiom from the Einstein field equations.
The Ehlers and Geroch theorem (2003)
Jürgen Ehlers and Robert Geroch (2003) proved a theorem whose content is roughly the following:
Assume that we have a body of continuous matter (no pointlike particles). Let us have an orbit γ in spacetime. We assume that the body follows γ as we let the mass of the body tend to zero. We assume that we have a solution to the metric in an environment of the orbit γ, and that solution converges strongly everywhere toward the background metric. Then the orbit γ is a geodesic of the background metric.
"Converges strongly" means that the metric and its first derivatives only differ from the background metric by at most ε if we choose a suitable environment.
Let us analyze the theorem. For a pointlike particle, strong convergence cannot hold.
We cannot prove the existence of a metric which is a solution. Therefore, we do not know if the metric, if any, converges strongly toward the background metric. However, it is a reasonable conjecture to assume that such metrics exist.
Let us put a spherical mass with pressure close to the test body. Then the gravity field of the test body is coupled to the pressure. We believe that the gravity field of the test body imposes a self-force on the test body. We do not think that the self-force can be expressed with a metric which is visible to all observers. Ehlers and Geroch do not consider this mechanism. A self-force makes the body to deviate from a geodesic.
The Gralla and Wald model (2008)
Samuel E. Gralla and Robert M. Wald (2008) in their paper "A rigorous derivation of gravitational self-force" start from an orbit γ of a test body, just like Ehlers and Geroch. They study the near field and the far field of the test body. They allow the body to be even a mini black hole.
Gralla and Wald, too, assume a family of well-behaved metrics g(λ), where λ is the parameter, and that the family converges nicely toward a background metric as we reduce the mass of the test body.
We can object, just like in the case of Ehlers and Geroch, that we do not know if such well-behaved families exist. But we can make a conjecture that they do exist.
In the last half of the paper the authors seem to calculate some kind of a "vertex function" for gravity. When the test body is accelerated, its own field may impose a self-force on the test body.
We in this blog believe that gravity is almost totally analogous with electromagnetism. Therefore, a vertex function for gravity must exist.
The authors do not consider a self-force which would come from pressure or some complex interaction of the gravity field of the test body with matter fields.
The authors mention that there is no legitimate derivation of the self-force on a charged particle in electromagnetism, despite more than a century of work. We in this blog have tried to determine the electromagnetic self-force. Our October 1, 2021 blog post is a step to that direction.
Conclusions
In general relativity, the claim that an infinitesimal test mass follows a geodesic, is taken as an axiom.
There exist various attempted proofs of the axiom from the Einstein field equations, but the consensus is that the proofs are not general enough. The proofs have to assume the existence of well-behaved metrics.
Our blog posts from the past few days suggest that pressure, or other complex interaction, imposes a self-force on the test mass. By a self-force we mean that the field of the test mass is like steel wires attached to it, and these wires impose substantial forces on the test mass. It does not follow a geodesic, then.
The paper by Gralla and Wald raised the possibility that there exists a vertex function, just like in electromagnetism, where the field of a particle interacts with the particle itself and steers it away from the geodesic. We believe that a vertex function exists for gravity.
The vertex function, too, can be understood with the steel wire model of the field of the particle. If the particle is accelerated, the wires resist the change in the velocity vector. Our "rubber plate" model of electromagnetism is equivalent to the steel wire model.
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