UPDATE September 3, 2023: Since the mass in the hollow sphere is accelerating, the radial acceleration term J below may be incorrect. See our post today.
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Gravitomagnetic moment at the center of a rotating lightweight circle
Let the test mass m be at the center of a rotating lightweight circle. The circle rotates clockwise at the speed ω R. The mass of the circle per length is ρ.
---> ω
dm' dm --> v
____•------------•____ circle
V ^ β = angle V vs. (m, dm)
| /
|/
• m
R = radius
-----•_________•----- circle
dm''' dm''
ω <---
^ x
|
---------> y
Let us check how much the symmetry cancels various acceleration terms. We let
0 < β < π / 2,
and check what the symmetry does. We copy the framework of this analysis from our August 10, 2023 post.
The radial acceleration term a' toward dm
Since m approaches dm as fast as dm', the Schwarzschild correction terms cancel each other out perfectly, because, using the August 10 notation, α + β = π / 2.
The Lorentz correction from the first term is
-a' (2 v • V / c²),
and the third term
-V (a' • v / c²).
We have a' • v = 0.
The projection of the correction on the y axis is
2 G / c² * dm / r² * v sin(β) V * sin(β).
The correction points to the right, in contrast to the August 10 post. That holds also for dm'.
For dm''' both a and v are the same as for dm, but the sign flipped. We see that all the mirror images of dm contribute acceleration to the same direction.
The mass element is
dm = ρ dβ.
The integral tells the y acceleration:
π / 2
a_y = 4 * ∫ 2 G / c² * 1 / R² * v V ρ sin²(β) dβ
0
= 2 G / c² * 1 / R² * v V ρ * π
= G / c² * 1 / R² * V ω R ρ * 2 π
= V G / c² * 1 / R³ * R ω R M
= V G / c² * 1 / R³ * J.
where J is the angular momentum of the circle. We can interpret the formula as the circle having a gravitomagnetic moment J with respect to an observer at the center. However, other terms will contribute more.
The term "dm' approaches faster"
This term is zero because the test mass m approaches dm' as fast as it approaches dm.
The term "dm moves to the right"
The test mass approaches both dm' and dm at the same speed, and both move to the right at the same speed. They contribute equally to a_y. The mirror images dm'' and dm''' move to the left, but from their point of view, V has its sign flipped: they contribute to a_y as much as dm' and dm. The term is:
(V cos(β) v cos(β) * r_s / r) * cos(β),
where the last cos(β) comes from the fact that we project the tangential acceleration relative to (m, dm) to the y axis. We have v = ω R:
π / 2
a_y' = 4 * ∫ V * 2 G / c² * 1 / R² * ω R cos³(β) ρ dβ
0
= V * 8 G / c² * 1 / R * ω * ρ * 2/3
= 2/3 V * 8 G / c² * 1 / (2 π R³)
* R ω R * ρ * 2 π
= V * G / c² * 1 / R³ * 8 / (3 π) * J.
This term corresponds to a gravitomagnetic moment 8 / (3 π) * J.
The total gravitomagnetic moment is
(1 + 8 / (3 π)) J = 1.85 J.
Comparison to magnetism
The magnetic field of a current loop at a height z from the center of the loop is (ρ_e = charge per length, Q = total charge):
B = μ₀ / (4 π) * R / R³ * 2 π R ρ_e v
= μ₀ / (4 π) * 1 / R³ * Q v R
= μ₀ / (4 π) * 1 / R³ * J_e,
where J_e is the charge angular momentum which we defined on August 10, 2023. The magnetic field B at the center is as if there would be a magnetic moment J_e generating it.
Note that electromagnetic frame dragging at is analogous to the gravitomagnetic frame dragging, and corresponds to a magnetic moment of 1.85 J_e. The magnetic field creates only a part of the frame dragging.
In the formula above, σ is charge per surface area. The total charge is
Q = σ * 4 π R².
The charge angular momentum is
J_e = 2/3 Q R² ω.
Then
B(0) / J_e = 2/3 μ₀ R σ ω
/ (2/3 * σ * 4 π R² * R² ω)
= μ₀ / (4 π) * 1 / R³,
or
B(0) = μ₀ / (4 π) * 1 / R³ * J_e.
The formula is the same as for a current loop.
The Thirring formula for the center of a rotating hollow sphere
Since we are lazy to calculate, let us guess that the formula for frame dragging at the center of a hollow sphere is the same as at the center of a circle. Then we can compare our result to what Hans Thirring calculated in 1918.
Hans Thirring obtained the equations of motion which are listed above. There a denotes the radius of the sphere and k is G. Thirring has set c = 1.
The angular momentum of a hollow sphere is
J = 2/3 M R² ω.
The gravitomagnetic acceleration on a test mass which is at (x, y) = (0, 0) and moves at a velocity V along the x axis is
a_y = V * 8/3 G M / R * ω
= V * G / c² * 1 / R³ * 4 * 2/3 M R² ω
= V * G / c² * 1 / R³ * 4 J.
That corresponds to a gravitomagnetic moment of 4 J. In the preceding section we saw that the magnetic moment of the analogous electric charge system is J_e. The Thirring result thus is consistent with what people studying gravitoelectromagnetism have found: the force on a test mass is four times of what it is for the analogous electric charge system.
Our own value is 1.85 J. The order of magnitude is the same.
Conclusions
Hans Thirring (1918) obtained the same value for the gravitomagnetic field as later researchers of gravitoelectromagnetism have calculated.
The Albert Einstein 1916 paper contains the approximation method which Thirring uses to sum the fields of individual masses. Einstein in that paper does not calculate the field of a rotating sphere.
Bahram Mashhoon (2003) uses the Einstein approximation method in his review of gravitoelectromagnetism.
We will look at research on gravitoelectromagnetism and try to pinpoint the error, if any. The error might be the unsuitable coordinates in the Einstein approximation formula, which is shown above. This would explain why gravitoelectromagnetism does not satisfy Lorentz covariance.
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