UPDATE August 27, 2023: The trick R' = R - r_s does not reproduce the Einstein approximation metric accurately. If we switch to cartesian coordinates, there is a large cross term in the R' metric. It is not diagonal like the Einstein approximation metric. Generally, "bulging" coordinates will have cross terms in the metric because the coordinate lines are not orthogonal.
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UPDATE August 26, 2023: The most serious problem in the Einstein approximation formula is that it is not Lorentz covariant. Bulging coordinate lines will also cause problems.
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We may have found out what is the problem in traditional gravitoelectromagnetism, and other methods which use linearized Einstein equations to determine the effects that massive bodies have on a test mass.
Our blog builds our metric perturbation analysis on static Minkowski coordinates. We have treated the Schwarzschild coordinates around a massive body as "canonical coordinates" which have to be matched to the Minkowski coordinates.
If the mass is moving relative to our static coordinates, we perform a Lorentz transformation to map the effects on a test mass to our static Minkowski coordinates.
We have a good reason to believe that the Schwarzschild coordinates really are canonical. In our own Minkowski & newtonian gravity model, the slowdown of clocks and the stretching of the radial metric around a massive body are explained by changes in the inertia of a test mass. The changes in inertia do not indicate any stretching of the tangential metric. Thus, the Schwarzschild choice of keeping the tangential metric element as 1 is the right one. The Schwarzschild coordinates are in our model the "true", underlying Minkowski metric coordinates.
Einstein-Thirring coordinates are error-prone
"Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie", Phys. Zeit. 19 (1918), 33-39.
In the link we have the Hans Thirring paper from the year 1918, translated to English by D. H. Delphenich.
The metric in the paper is given in x, y, z, t coordinates, in this order. There is a uniform stretching of the spatial metric at the distance R from the gravitating body.
The radial and the tangential stretching in the Schwarzschild coordinates are
1 + 1/2 r_s / R
and
1,
where r_s is the Schwarzschild radius. We assume that r_s is very small.
Let us use a new radial coordinate
R' = R - 1/2 r_s.
If r_s is very small, then in terms of R', the radial stretching is essentially the same as for R:
1 + 1/2 r_s / R,
and the tangential stretching is approximately
R / R' = R / (R - 1/2 r_s)
= 1 + 1/2 r_s / R.
That is, by changing the radial coordinate to R' we were able to make the spatial metric locally uniformly stretched relative to the coordinates.
Our new coordinate R' looks innocuous, but it is error-prone for perturbation calculations.
Let us have two massive bodies M1 and M2. We sum the perturbations to the Minkowski metric to get an approximate solution.
● M1
--------------------------> photon
● M2
The configuration is symmetric. The photon will go along a straight line in the coordinates.
In the R' calculation, the tangential metric between M1 and M2 is stretched relative to a Schwarzschild calculation. The radial metric and the temporal metrics are almost exactly the same for R' and Schwarzschild.
A naive calculation in the R' coordinates gives a larger Shapiro delay from the Schwarzschild calculation, because there is more spatial stretching in the R' calculation.
The problem with R' is that the coordinate R' is "distorted" by a fixed amount 1/2 r_s throughout the space. A correct calculation should take into account this distortion.
If we have a two observers very far away from each other in space, and move the mass system M1, M2 between them, how to calculate the Shapiro delay correctly with the R' coordinates?
Naively, we would expect the observers to stay in the "same place" for the R' coordinates, but that cannot be the case. Their distance must shrink, to cut the Shapiro delay to be the same as in the Schwarzschild calculation.
The produre is clearer if we use the distorted R' coordinates in Minkowski space without the masses M1, M2. Then it is immediately clear that the distance of the observers shrinks in the R' coordinates, relative to the regular R polar coordinates.
If we use distorted coordinates in Minkowski space, then masses no longer move along straight lines. This makes any calculation error-prone.
Conclusions
We will look into the Einstein 1916 paper and the Thirring 1918 paper, and check if there are errors.
Albert Einstein (1916): Näherungsweise Integration der Feldgleichungen
der Gravitation.
One error in the Thirring paper (1918) is immediately obvious, above. If we set ω = 0, then the hollow sphere does not rotate. The metric above claims that the metric is stretched in the x and y directions (1 and 2), but not in the z direction g₃₃. That cannot be correct.
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