We wrote about this problem also in our November 12, 2021 blog post.
https://edition-open-sources.org/media/sources/10/17/sources10chap15.pdf
Later, Leopold Infeld and Alfred Schild (1949) wrote another paper on the problem:
Wikipedia says that the papers are "controversial".
David B. Malament (2012) writes that various proofs require energy conditions. They do not follow from the Einstein equations alone.
Steven Weinberg's derivation of the geodesic equation in Wikipedia
The Wikipedia page presents a derivation of the geodesic equation by Steven Weinberg (1972).
Let us assume that we have coordinates and a metric for spacetime.
Weinberg assumes the following equivalence principle for a freely falling observer: the observer can define local coordinates such that the local metric then looks cartesian.
If we have a stationary observer in the Schwarzschild solution, then the proper time in his neighborhood runs at a rate which depends linearly on the radial distance r. He cannot define cartesian coordinates in his immediate neighborhood.
But a freely falling observer can define locally cartesian coordinates. Weinberg shows that the geodesic equation holds for the path of the freely falling observer.
Weinberg's argument proves the following. Let us assume:
1. the Einstein equations determine a metric which correctly describes time intervals and distances as they are measured by observers;
2. there is a "natural" way to match the clock times and measuring rods in any local coordinates of a moving observer to the metric;
3. a freely falling observer can define locally cartesian co-moving coordinates.
Then the path of the freely falling observer must obey the geodesic equation.
But is the derivation circular? If rays of light would not obey the geodesic equation, could it be that the stationary observer in the Schwarzschild metric could see clocks running at the same rate regardless of the radial distance r of the observer? We must assume that rays of light cannot propagate to the past. Let us add:
4. rays of light cannot go to the past in the metric. This is included in the word "natural" in item 2.
The dominant energy condition fails? Probably not
Let us have two people pulling on a very rigid, tense, almost massless rope.
tense rope
------------------
o/ \o
| |
/\ /\
There are also other people holding on the rope, placed at every meter of the rope. At some time t, the people let the rope to move to the right some distance. The person at the left end can harvest some energy E from the tension of the rope, and the person on the right loses that same energy E.
Did the energy E travel superluminally from the right end to the left end? If the rope is almost massless, then that is the only conclusion that we can make. The dominant energy condition fails. But we cannot construct almost massless ropes. This saves the condition.
Erik Lentz (2020) has constructed a superluminal soliton. The basic idea in his soliton might be similar to our rope example.
If there is interference of waves, constructive interference may appear to move faster than light. That may be yet another violation of the dominant energy condition.
Let us assume that we have two circularly polarized waves A and B moving to the same direction. The rotation of the waves is clockwise. B has a somewhat lower frequency than A. Let both waves move at the speed of light. If A and B have a full constructive interference at some point P, then P moves at the speed of light to the same direction as A and B. Let P be at a location x at t = 0 s, and at y at t = 1 s.
Let us now make B a little slower than light. We assume a constructive interference at x at 0 s.
B lagging
\ | A not lagging
\ |
\ |
location y at t = 1 s
Then at y, after 1 second, the phase of B is "lagging behind" a little, compared to what it would have been were it moving at the speed of light. The wave A already had a constructive interference with B before t = 1 s. We conclude that the point of constructive interference moves at a superluminal speed. Energy appears to be transported superluminally.
For B to propagate slower than A, they have to be in a medium which is not a vacuum. This may save the dominant energy condition: the apparent superluminal speed may be a result of a small part of the total system energy concentrating at the location of a constructive interference. The total energy pool moves much slower than light.
The 2012 proof by David Malament
Let us check the proof of the geodesic hypothesis by David Malament in his 2012 paper. It is proposition 3.1 in his paper.
The proposition assumes that an arbitrary small "body" moves along a path I, and its stress-energy tensor T satisfies the conservation condition, such that the energy and the momentum of the body is conserved. That is supposed to mean that there are no external forces on the body.
Malament also assumes that the effect of the small body on the "background metric" is "negligible".
Our counterexamples to the geodesic hypothesis versus Malament's and Weinberg's proofs
Our counterexamples in this blog are such that the test mass does not affect the "background metric" at all.
In order to capture what people usually understand with the geodesic hypothesis, Malament's proposition should contain the following part: for an arbitrary (neutral) body there exists a path where the conservation condition for T holds.
Our counterexamples do not refute Malament's proof because in our examples, complex interactions, which act through the gravity field of the body, change T along its path.
Our counterexamples show the following:
Theorem. If we have a test mass m which does not affect the background metric at all, and if m has no other charges than the gravity charge, then it can still happen that m does not move along a geodesic path.
Our Theorem shows that the metric of general relativity does not really capture the movement of test masses. However, the Einstein-Hilbert action might capture it.
Weinberg's proof assumes that the metric correctly describes time intervals and distances measured by an observer. Customarily, it is assumed that the observer uses rays of light and clocks to make measurements. In our counterexamples, rays of light do not obey the metric. Weinberg's assumption is not correct in the counterexamples.
Conclusions
The proofs of the geodesic principle seem to ignore complex interactions which are mediated through the gravity field of a test mass.
General relativity, if we define it through the Einstein-Hilbert action, probably captures these complex interactions. Proofs of the geodesic hypothesis assume things which are not true when general relativity is defined this way.
The proofs are, in a sense, circular. They make assumptions which are quite close to assuming the geodesic hypothesis itself.
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