UPDATE October 12, 2023: We forgot the γ M v² / r term T_xx from the stress-energy tensor T. We have to check if that changes anything.
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UPDATE 2 August 27, 2023: The metric seems to understand that it moves with M, after all.
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UPDATE August 27, 2023: The geodesic equation thinks that stretching of the spatial metric is "attached" to the coordinates, while it is attached to M. This may be a fundamental problem in general relativity.
We also corrected the error: the factor in the reverse Lorentz transformation is γ, not 1 / γ.
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Let us analyze in detail why the Einstein approximation formula leads to a breach of Lorentz covariance when we switch to a moving frame. This is the simple configuration which we studied in our August 21, 2023 blog post. If the coordinates were used correctly, then Lorentz covariance would be preserved.
We showed in our August 15, 2023 post Summing metric perturbations is error-prone for wrong choices of coordinates that the Einstein-Thirring coordinates are obtained from Schwarzschild coordinates by defining a new radial coordinate R':
R' = R - 1/2 r_s,
where r_s is the Schwarzschild radius for the central mass M. If we draw coordinate lines for R' = 1, R' = 2, ... , the lines are r_s farther from M than the corresponding lines for the Schwarzschild coordinates. In this sense, the Einstein-Thirring coordinates are "bulging" relative to the Schwarzschild coordinates. The pseudo-cartesian coordinates in the image above are bulging relative to standard cartesian coordinates.
Throughout this blog post we set the speed of light c = 1. We use the (- + +) sign convention.
The metric in the laboratory frame where the mass M and the test mass m are static
● M
/
/ r
/
R = distance (m, M)
^ d²y / dt² acceleration
|
• m
^ y
|
-------> x
Let us start from a static mass M and a test mass m. The test mass is initially static but accelerates toward M. The coordinate distance (m, M) is R. We use (almost) cartesian coordinates. We want to calculate the y acceleration of m.
The Einstein approximation formula gives as the mapped stress-energy tensor T_μν / r at an arbitrary location at a distance r from M:
M / r 0 0
0 0 0
0 0 0
Trace reversing gives us a metric perturbation of the Minkowski metric η:
h₀ =
4 G *
1/2 M / r 0 0
0 1/2 M / r 0
0 0 1/2 M / r
The full metric is:
η + h₀ =
-1 + 2 G M / r 0 0
0 1 + 2 G M / r 0
0 0 1 + 2 G M / r
Let us calculate the y acceleration of the test mass m. Since m is initially static, only dt / dτ differs from zero in the geodesic equation:
Γ_ytt = -1/2 dg_tt / dy
= -1/2 d(4 G * 1/2 M / R ) / dy
= -G M / R².
Then
Γ^y_tt = -1 / g_yy * G M / R²,
where
g_yy = 1 - 2 G M / R.
We get:
d²y / dt² * dt² / dτ²
= dt² / dτ² * 1 / g_yy * G M / R²,
or
d²y / dt² = 1 / g_yy * G M / R².
If M is small, then we can assume that g_yy = 1. The acceleration is the familiar newtonian one.
Switch to moving coordinates: the Einstein approximation formula
M ● ----> v
/
/ r
/
R = distance (m, M)
^ d²y' / dt'² acceleration
|
m • ----> v
^ y'
|
-------> x'
The new coordinates move to the left at a velocity v:
|v| << 1.
We denote the new coordinates with the prime'. The Einstein approximation gives as the mapped stress-energy tensor T_μν / r at an arbitrary location:
γ M / r γ M v / r 0
γ M v / r 0 0
0 0 0
where
γ = 1 / sqrt(1 - v²).
Trace reversing gives us the perturbation to the Minkowski metric:
h₁ =
2 G M / r *
γ -2 γ v 0
-2 γ v γ 0
0 0 γ
=
2 G M / r *
1 + 1/2 v² -2 v 0
-2 v 1 + 1/2 v² 0
0 0 1 + 1/2 v²
where we used the fact that |v| is small and discarded ~ v³ terms.
The perturbation h₁ has to be summed to the Minkowski metric η to obtain the full metric:
η =
-1 0 0
0 1 0
0 0 1
Lorentz transformation Λ of the laboratory metric to moving coordinates
Let us compare the above result for h₁ to the Lorentz transformation of the metric η + h₀ in the laboratory frame into the moving frame. Let M move to the right so that its coordinates at a time t' are
(x', y') = (v t', R).
Then
r = sqrt( (x' - v t')² + (y' - R)² ).
The coordinates of m are
(x', y') = (v t', 0).
The transformation Λ to the coordinates (marked with the prime ') in the moving frame is
t' = γ (t + v x),
x' = γ (x + v t),
y' = y.
The inverse transformation is:
t = γ (t' - v x'),
x = γ (x' - v t'),
y = y'.
Let us first look at the transformation of the Minkowski metric η. If we write
dt = γ * (dt' - v dx'),
dx = γ * (dx' - v dt'),
dy = dy',
we obtain
Λ(η) = η =
-1 0 0
0 1 0
0 0 1
as the metric in the moving frame. We have
γ² * v² = (1 + v²) * v² = v²,
if we assume |v| << 1.
The transformation of the perturbation h₀ is
Λ(h₀) = h₂ =
2 G M / r * γ² *
1 + v² -2 v 0
-2 v 1 + v² 0
0 0 1 * 1 / γ²
We write
γ² = 1 + v².
We get:
h₂ =
2 G M / r *
1 + 2 v² -2 v 0
-2 v 1 + 2 v² 0
0 0 1
where r = sqrt( (x' - v t')² + (y' - R)² ).
We dropped ~ v³ terms. We see that h₁ and h₂ differ in the g_t't' and g_x'x' elements.
We once again use the geodesic equation to determine the y' acceleration of the test mass m. Let us multiply the geodesic equation (the first equation above) by dτ² / dt'². Then we can assume that ds in the equation is dt'.
In the geodesic equation, μ = y', and αβ can be any of t't', t'x', x't', x'x'.
For the location of the test mass m:
dg_t't' / dy' = 2 G M (1 + 2 v²) * 1 / R²
Γ^y'_t't' = 1 / g_y'y' * -G M (1 + 2 v²) / R².
Also,
dg_t'x' / dy' = -4 G M v / R²,
Γ^y'_t'x' = 1 / g_y'y' * 2 G M v / R²,
Γ^y'_x't' = 1 / g_y'y' * 2 G M v / R²,
dg_x'x' / dy' = 2 G M (1 + 2 v²) * 1 / R²,
Γ^y'_x'x' = 1 / g_y'y' * -G M (1 + 2 v²) / R².
Multiplying with v or v² where appropriate, and summing, we obtain the y' acceleration:
d²y' / dt'² = 1 / g_y'y' * (
G M (1 + 2 v²) / R²
- 2 G M v² / R²
- 2 G M v² / R²
+ G M v² / R²
)
= 1 / g_y'y' * (1 - v²) * G M / R²
= (1 - v²) * G M / R²,
if M is small. We used the fact that g_y'y' goes to 1 when M goes to zero. We dropped terms ~ v⁴.
In the moving frame, the system (M, m) moves at the velocity v to the right. In the formula d²y / dt², the time dt is dilated by a factor 1 + 1/2 v² when looked at in the moving frame. Thus, the acceleration measured in the moving frame should be slower by a factor 1 - v². Our result matches the expectation!
What if we use the Schwarzschild coordinates? Does the geodesic equation work?
The above calculation uses the bulging Einstein-Thirring coordinates. If we instead use the Schwarzschild coordinates, then in the laboratory metric perturbation h₀, we have the metric x,x component zero.
Removing the contribution of the perturbation in the x,x metric causes the following changes to the formula of d²y' / dt'²:
1. G M (1 + 2 v²) / R² becomes G M (1 + v²) / R²,
2. both -2 G M v² / R² terms become -G M v² / R²,
3. + G M v² / R² drops off.
The y' acceleration stays the same, and is the correct one.
Analysis of the Einstein approximation error
The error was not in the bulging coordinates, after all, in this case. The test mass m moves directly below M, and the bulge in the y' coordinate lines does not generate a spurious y' acceleration to the movement of m.
The error simply is that the Einstein formula does not calculate the Lorentz transformation of the static metric which we had around M in the laboratory frame.
The Einstein formula does a Lorentz transformation of the stress-energy tensor, but the transformation for the (Schwarzschild) metric is more complicated, as we calculated above.
Conclusions
General relativity seems to be Lorentz covariant, though we have not yet proved that in detail.
The Einstein approximation formula does not calculate the Lorentz transformation of the metric correctly.
For other orbits of the test mass m, the bulging of the coordinate lines would produce spurious coordinate accelerations
d²y' / dt'².
Also, the bulging coordinate lines may cause serious errors in the summing of perturbations for several masses M1, M2, etc. The bulging metric has "unnatural" stretching of tangential distances. For example, the metric for a long cylinder may be incorrect using the Einstein approximation. We will study this and other questions in subsequent blog posts.
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