UPDATE October 12, 2023: We forgot the v_x² "pressure" element T_xx from the stress-energy tensor T. The tensor is aware of the movement of dm and dm'. We have to check if this changes anything.
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UPDATE August 20, 2023: Linearized general relativity does not "lose crucial information" in the configuration, because the x momentum term (g_tx) changes when we move to the y direction.
The Christoffel symbol Γ^y_tx is not zero because the metric g_tx changes in the y direction as we come closer to dm and recede from dm'. Linearized general relativity does predict an acceleration in the y direction:
d²y / dτ²
is not zero. The value seems to be 4/3 of what we calculate below. Linearized general relativity agrees with gravitoelectromagnetism.
The Thomas precession is caused by Lorentz transformations when an electron orbits the nucleus. It might be associated with this.
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We may have found the problem in gravitoelectromagnetism. The Einstein approximation formula loses crucial information about the movement of masses.
^
| v
• dm' • dm
| v
v
r = distance (m, dm)
^
| V
|
• m
^ x
|
--------> y
Einstein, Albert, Näherungsweise Integration der Feldgleichungen der Gravitation, June 22, 1916.
To calculate the metric at the test mass m, we sum the "contributions" of dm' and dm and divide by r, which is the distance (m, dm). The distance (m, dm') is the same and the masses dm and dm' are equal.
The contribution of dm in the x momentum part of the stress-energy tensor T is
dm v,
while it is
-dm v
for dm'. These cancel each other out. The approximate metric at the test mass is not aware of the fact that dm' is approaching and dm is moving away!
Our own analysis shows that this relative movement causes y acceleration. Let us use the notation of August 10, 2023. There, α = π / 2 and β is a small angle for which sin(β) is approximately R / R_m. Also, r is approximately R_m.
1. The radial correction is:
G / c² * dm / r² * v sin(α) V * sin(β)
= ω V G / c² * 1 / R_m³ * R² dm.
2. The term "dm' approaches faster" is:
-2 ω V G / c² * 1 / R_m³ * R² dm.
3. The term "dm moves to the right" is:
ω V * 2 G / c² * dm R / r² * cos²(β) cos(α + β)
= -2 ω V G / c² * 1 / R_m³ * R² dm,
where we used the fact that for a small angle β, cos(β) is approximately 1, and that cos(α + β) = sin(β).
The total effect is
a_y = -3 ω V G / c² * 1 / R_m³ * R² dm
= V G / c² * 1 / R_m³ * -3 R v dm
= V G / c² * 1 / R_m³ * -3 J.
The acceleration is as if the system would have a gravitomagnetic moment -3 J.
People have been using the Einstein approximation formula since 1916, but never checked if it takes into account all relevant factors.
The 1916 paper
The original printed article is freely readable in the link above. Equation (8) calculates the "trace-reversed" version of the tensor γ' by subtracting the sum of the elements on its diagonal.
We have an observer who is static in space and we want to calculate an approximate metric at his location.
Equation (9) sums the contributions to the metric at the observer, by adding the stress-energy tensor for each volume element V₀, divided by its distance r from the observer. Equation (9) is like the retarded potential formula in electromagnetism.
In our previous blog posts we repeatedly saw that the magnetic field of a moving electric charge does not correctly give the frame dragging effect.
We have to analyze what does the Einstein approximation formula actually calculate. It does give an approximation of the metric but does not predict right the y acceleration in our example configuration.
In Section 3 of the paper, Albert Einstein proceeds to calculate the energy loss of "material systems by emissions of gravitational waves". We know that linearized Einstein equations do predict the power of gravitational waves correctly. It may be that the Einstein approximation formula in that case is adequate.
Our example shows that general relativity clashes with special relativity?
There are good reasons to believe that general relativity approaches linearized gravity if the masses dm' and dm go to zero.
However, our example configuration shows that linearized gravity does not handle Lorentz transformations correctly.
This may imply that general relativity is not Lorentz covariant. Since we firmly trust special relativity, this would prove that general relativity is an incorrect theory.
General relativity attempts to capture Lorentz transformations into a metric. It is a difficult task. Maybe general relativity failed?
A brief Internet search does not retrieve any proof of Lorentz covariance for general relativity. The basic idea of general relativity is that the metric alone determines orbits of test masses. If general relativity gives incorrect predictions, it is an incorrect theory.
The Schwarzschild metric can be derived from general relativity; the metric is empirically verified for the weak field around Earth.
The Hulse-Taylor binary pulsar proves that general relativity predicts the power of gravitational waves correctly, within an uncertainty margin of 0.3%.
Measurements of gravitomagnetic effects are controversial at this stage. Subtler phenomena associated with Lorentz transformations are difficult to measure. Anyway, a failure to obey Lorentz covariance leads to many mathematical contradictions. We cannot call a non-Lorentz-covariant theory correct.
Our own Minkowski & newtonian gravity model does not have problems with Lorentz covariance. We do not claim that there exists a "metric" which would determine orbits of test masses. We calculate orbits usually in a comoving frame of a large mass M, and then Lorentz transform the orbit into the laboratory frame.
Electromagnetism does not claim either that some global field would determine orbits of charges perfectly. In electromagnetism we choose a convenient frame and then Lorentz transform if needed.
If general relativity, after all, handles Lorentz transformations correctly, that has to be due to some nonlinear mechanism. But it would be a small miracle if nonlinearity in the Einstein equations would reproduce Lorentz transformations.
Evidence for the claim that general relativity breaks Lorentz covariance
1. There seems to be no proof that general relativity respects Lorentz covariance, not even for weak fields.
2. Gravitoelectromagnetism is derived from linearized general relativity, and Wikipedia states that gravitoelectromagnetism breaks Lorentz covariance.
3. Our own calculations suggest that linearized general relativity breaches Lorentz covariance.
4. It is hard to build Lorentz transformations into a metric. Albert Einstein himself probably did not check Lorentz covariance. We are not aware of such a paper.
Conclusions
It looks like that it is not the Einstein approximation formula, but general relativity itself which is to blame for the discrepancy in the calculations of frame dragging around rotating masses.
In this blog we have repeatedly argued that it is not possible to describe a complex physical lagrangian through a "metric". Our example in this blog post is further evidence for that. Our view in this blog is that there is no such thing as "curved spacetime". Phenomena attributed to a "spacetime geometry" are really ordinary interactions through the force of gravity.
An accurate measurement of frame dragging around a rotating mass may be possible within a few decades. Then we would know if Lorentz covariance gives the correct value, or if general relativity does. Gravity Probe B was designed to measure the precession of a top in the gravitomagnetic field of Earth. Unfortunately, static electric charges inside the rotating spheres spoiled the measurement. The team claims that they still were able to infer frame dragging at a precision of 20%, and it agreed with general relativity.
If general relativity breaches Lorentz covariance, then we are virtually certain that special relativity is the correct theory and general relativity the incorrect one. Without Lorentz covariance we end up in many paradoxes.
If general relativity breaches Lorentz covariance, then it might be that an asymptotic Kerr solution is incorrect for a rotating mass. We will look into this.
We have the goal of proving that overspinning a black hole will disassemble it. We will study that, too.
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