UPDATE August 26, 2023: The Einstein field equations seem to be Lorentz covariant. The error is in the Einstein approximation formula which is not Lorentz covariant.
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UPDATE August 19, 2023: The Einstein equations are not Lorentz covariant. No such proof exists, and our most recent blog posts suggest that it is not Lorentz covariant.
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But gravitoelectromagnetism requires a static gravitomagnetic field. That is not true for a moving cylinder.
Gravitoelectromagnetism seems to fail for a closed loop
Let us check if gravitoelectromagnetism works correctly for a rotating loop whose gravitomagnetic field is static.
Consider the following loop:
v <-- mass flow
________
/ \
| __________ | M
2 r
• m
mass flow --> v
The length of the straight cylindrical part is 2 r, and there is a half-circle whose radius is r.
We keep the mass of the loop M and the velocity of the mass flow v constant, and let r grow.
We can always move the test mass m closer to the cylinder, so that the gravity attraction toward the cylinder is some "moderate" force F.
The forces from the far parts of the system, besides the moving straight cylinder, go to zero.
Gravitoelectromagnetism calculates that the field at m is essentially the field of the cylinder. We showed in the previous blog post that it produces results which conflict with special relativity.
We proved that general relativity clashes with special relativity? If that is the case, special relativity is almost certainly the correct theory of the two. General relativity miscalculates the metric of a moving mass. The Einstein equations are wrong.
The fact that general relativity produces closed timelike loops, already shows that the equations probably are incorrect. Our example, however, is much more mundane than examples of closed timelike loops.
Our counterexample requires the following assumptions:
1. Gravitoelectromagnetism gives a correct approximate solution for the Einstein equations for the loop above.
2. One can sum moderately weak fields, and gets a correct approximation.
3. The field of the "far" parts of the loop really approaches zero as we let r grow.
Our own, though sketchy, Minkowski & newtonian gravity model does not have this problem. It calculates the field of a moving cylinder correctly.
The Einstein equations and the geodesic equation are Lorentz covariant
In our counterexample, gravitoelectromagnetism breaks Lorentz covariance for the moving cylinder.
But it has been shown that the Einstein equations and the geodesic equation are Lorentz covariant. Thus, the problem probably is in gravitoelectromagnetism, not in the Einstein equations.
What does the parameter a actually mean in the Kerr metric?
According to Wikipedia, there is (probably) no known solution for the interior of a rotating mass, such that it can be matched to the Kerr metric. Could it be that an asymptotic Kerr metric does not describe a rotating body, after all?
The parameter
a = J / (M c)
in the Kerr metric is assumed to mean the angular momentum of the system. This is in contrast to electromagnetism: the magnetic moment
m = 1/2 J_e,
where J_e is the "angular momentum"
J_e = Σ qⱼ v × r
j
of the electric charges qⱼ.
Could it be that a in the Kerr metric actually should be 1/4 of J / (M c)? How do we know what value of J does the parameter a actually correspond to?
In the Schwarzschild metric, we match the newtonian gravity to the metric far away from the central mass, and guess that M is the total mass-energy of the system. In the case of the Kerr metric, there is no similar method to guess what the parameter a means.
The error is in the linearization, which is used to derive gravitoelectromagnetism?
In our August 5, 2023 post we showed that an orbit in the Schwarzschild metric agrees with the analogous electromagnetic problem. Since in gravitoelectromagnetism the gravitomagnetic field exerts a 4-fold force on a test mass, gravitoelectromagnetism would give a wrong orbit. If gravitoelectromagnetism fails to calculate such a simple problem correctly, why would it work for any system?
Could it be that the linearization of Einstein equations, which is used to derive gravitoelectromagnetism, is not a suitable approximation technique? The Schwarzschild solution is derived from the full Einstein equations and does not suffer from problems in linearization.
Valery V. Vasiliev and Leonid V. Fedorov (2020) discuss the derivation of newtonian gravity from linearized Einstein equations. It turns out to be very complicated.
Conclusions
We are suspecting that the linearized Einstein equations miscalculate gravitomagnetic effects.
For the Kerr metric, the role of the parameter a is not clear from the solution itself. It might be that people have used gravitomagnetic calculations to determine that
a = J / (M c)
in the Kerr metric. The error in gravitomagnetism would have propagated to the Kerr metric.
We have to find out if linearized equations are unsuitable for calculating gravitomagnetic effects.
According to our own calculation on August 5, 2023 about the Schwarzschild orbits, the correct interpretation might be
a = 1/2 J / (M c).
A half of this would be a true "magnetic" effect. The other half would come from the increase of the kinetic energy, which reduces the velocity of the test mass relative to the large gravitating mass, but not the momentum of the test mass.
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