Leopold Infeld and Alfred Schild tackled the geodesic hypothesis in 1949 through studying a point particle which is a singularity in the gravity field. The particle moves in an external gravity field, and its neighborhood does not contain any other fields than the gravity field.
The point particle moves superluminally relative to a static coordinate system
We remarked in the blog post on March 17, 2023 that one cannot model the movement of a very heavy neutron star in a static coordinate system because the speed of light is very slow close to the surface of the neutron star, as seen by a faraway observer. If we would use a static coordinate system, then the neutron star would move superluminally relative to the coordinate system. One cannot define the stress-energy tensor T in such a case because matter can move superluminally relative to the coordinates.
The point mass of Infeld and Schild is a singularity, that is, a black hole. Any movement against a static coordinate system is superluminal.
An analysis of the proof
Infeld and Schild consider a timelike worldline L in a fixed background metric g. They choose coordinates such that at every point of L the metric g is equal to the flat Minkowski metric:
g = η.
Enrico Fermi proved the existence of such coordinates in 1922. The coordinates are "freely falling" on the line L relative to the background metric.
The authors then introduce comoving coordinates
t, z¹, z², z³
along L, in section 3 of their paper.
The authors let a point mass singularity move along the worldline L. They aim to show that L has to be a geodesic. They will prove that the approximate Schwarzschild metric around the point mass cannot be accelerating relative to a freely falling frame in the background metric.
There is an obvious problem: the singularity cannot truly move along the line L because the speed of light is zero close to the singularity. If we imagine a physical thread laid along the spatial path of L, the singularity will push the entire thread in front of it. However, the singularity can appear to move along L for outside observers.
In section 4 the authors assume that the metric with the particle is
g = η + a + m b,
where η + a is the background metric, m b is the correction from the existence of the particle, and m is the mass of the particle. Recall that a is zero on the line L - we chose the coordinates in such a way.
The argument of the authors is based on the asymptotic behavior of the metric close to the singularity. In (5.04) they write the metric in a form
β₋₁ + β₀ + ...
where β₋₁ varies roughly as 1 / r, β₀ is roughly a constant, and so on. Here r is the spatial distance from the singularity.
Here we see a problem in the proof: the β₋₁, β₀, ... are not uniquely determined. The authors claim that the terms of the same order in r must match. That would be true if the terms would be of the form
constant * 1 / rⁿ,
but they are not of that precise form.
An example: let
f(r) / r + h(r) = 0
for all r. That does not imply that f(r) and h(r) are zero for all r. If f(r) and h(r) were constants, then the implication would hold.
The authors continue to write equations believing that β₋₁, β₀, ... are uniquely determined. They proceed to show that the second time derivative of the position of the point mass ξ must be zero. That is, the point mass does not accelerate relative to a freely falling frame.
The proof may be erroneous.
Another problem in the proof: if the singularity moves on the worldline L, then close to the singularity, the static coordinate system x¹, x², x³ moves superluminally relative to the comoving coordinates z¹, z², z³. The authors base their argument on the asymptotic behavior close to the singularity. But one cannot use static coordinates there at all.
The singularity does not need to move along a geodesic of the background metric
A system should follow a path where the action has a local extreme value. That is, any small change to the metric or the position of particles increases the value of the integral if the extreme value is a minimum, and vice versa for a maximum.
Can we assume that there is a local extreme value of the Einstein-Hilbert action for the development of this system? It could happen that the value of the action explodes and there is no local extreme value at all. Let us assume that there is a local minimum.
Schwarzschild
metric
• ---------------------- #######
point mass m rigid object
Let us design the lagrangian L in such a way that a neighboring object repels the curved spatial Schwarzschild metric around a point mass. The object is very rigid and resists any deformation of its form away from the spatial metric which its own gravity field created.
If we remove the point mass, then the stress-energy tensor is not aware of the rigidity. That is, the rigidity does not affect the background metric created by the object. Let the background metric be
g.
We can tune the rigidity of the object without changing g.
Let us assume that the force resisting a deformation of the object is constant regardless of the compression or stretching ratio of the object.
The spatial stretching (strain) is approximately linearly dependent on m in the Schwarzschild solution. If we move the point mass m close to the object, we must do work worth
~ m.
Let us have a freely falling frame in the metric g, at the point mass m. The frame does not depend on the rigidity of the object.
We have a point mass accelerating relative to a freely falling frame of the background metric g.
The solution does satisfy the Einstein field equations, because the Einstein equations simply state that the path is a local extreme value of the Einstein-Hilbert action. Specifically, close to the point mass, the Ricci curvature tensor is zero. It is empty space.
In our example, the backreaction of the metric inside the object is very strong. Any small mass m causes a large pressures in the object. Can we find an example where the backreaction is small?
Yes. Our November 10, 2021 blog post discusses "tidal" effects on a test mass. The inertia of a test mass of a weight m grows near a pressurized sphere because it moves pressure energy around inside the sphere. The increase in the inertia is linearly proportional to m. Setting m very small sets the metric of the whole system very close to the metric of the sphere alone. Still, the inertia of m increases with some fixed ratio I > 1. That affects its path considerably.
Birkhoff's theorem states that the metric around a spherically symmetric system is the same regardless of what we do with the pressure inside the sphere. But the pressure affects how much pressure energy our test mass moves around, that is, the inertia of the test mass. Therefore, the background metric cannot determine the path of the test mass, even if m is very small.
We assume that the Einstein-Hilbert action understands inertia and the whole process. It can be used to determine the path of the system.
This is a counterexample to the claim of Infeld and Schild. We have a system where:
1. The path of a test mass differs considerably from a geodesic of the background metric.
2. The backreaction of the background metric is proportional to m and can be set arbitrarily small.
3. The system satisfies the Einstein equations for empty space around the test mass. This is because the path is an extreme value of the Einstein-Hilbert action, and the Einstein equations are derived through varying the metric around the path. The Einstein equations must be satisfied by the path.
Conclusions
Wikipedia states that proofs of the geodesic hypothesis are "controversial". We found two possible problems in the proof by Infeld and Schild. They do not address the superluminal movement problem in any way. Their decomposition of the β metric seems to be nonunique.
We also presented a counterexample to the claim of Infeld and Schild: a sphere where pressure modulates the inertia of a test mass close to the sphere. The metric of general relativity does not understand "tidal" processes, though the Einstein-Hilbert action presumably understands them.
We are not sure if the geodesic hypothesis holds for a set of pure point masses, either. There may be tidal forces also in that case, and the background metric does not understand them. Einstein, Infeld, and Hoffman in a 1938 paper claim that the geodesic hypothesis holds for slowly moving point masses. We should check their proof.
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