UPDATE August 21, 2023: We corrected the Lorentz transformation of the acceleration. It is (1 - v² / c²), not (1 - 1/2 v² / c²).
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https://en.wikipedia.org/wiki/Gravitoelectromagnetism
Let us show that the strange 4X rule breaks special relativity for gravitoelectromagnetism.
It, actually, is a direct consequence of the calculation in the section "In a linear motion, the magnetic force is really a Lorentz transformation of the electric force" of our August 4, 2023 blog text. If one multiplies the magnetic force by a factor of 4 there, the calculation breaks special relativity badly.
The Wikipedia article states that gravitoelectromagnetism does not conform to Lorentz transformations. We will present an example of this.
Gravitoelectromagnetism for a moving cylinder
Let us have a thin, long – but not infinitely long – cylinder, static in the laboratory frame.
m •
| acceleration a
v
r
==================
cylinder ρ
<--- v moving frame
The cylinder contains a small amount of mass ρ per unit length. The gravity field around the cylinder is almost the newtonian field.
We have a test mass m at the middle of the cylinder, initially static relative to the cylinder at a distance r. We let the test mass fall toward the cylinder.
Let the acceleration be
a.
Let us then switch to a frame which moves horizontally at a slow speed v to the left. The acceleration downward, measured in the moving frame, must be
a * (1 - v² / c²)
= a * (1 - v² / c²),
because of time dilation. We assume |v| small.
Let us then calculate in the moving frame using the formulae of gravitoelectromagnetism (GEM):
The gravity field in the laboratory frame at the test mass m at the distance r is
E_g = 1 / (2 π r) * 4 π G ρ
= 2 G ρ / r.
We have
a = E_g
= 2 G ρ / r.
In the moving frame, the cylinder is length contracted, and it possesses kinetic energy. Both involve a factor 1 / sqrt(1 - v² / c²). We conclude that
ρ' = ρ * 1 / (1 - v² / c²),
E_g' = E_g * 1 / (1 - v² / c²).
The magnetic field of an electric current J is
B = μ₀ J / (2 π r)
= 1 / (ε₀ c²) * J / (2 π r).
In the analogy, ε₀ corresponds to 1 / (4 π G). We get
B_g = 1 / c² * 4 π G J_g / (2 π r)
= 2 / c² * G J_g / r.
We have
J_g = v ρ' ≅ v ρ,
because |v| is small.
The gravitomagnetic field in the moving frame is
B_g = 2 / c² * G v ρ / r.
In the moving frame, the mass-energy of the test mass m has grown to
m' = m / sqrt(1 - v² / c²),
but that does not affect accelerations.
The gravity acceleration on the test mass in the moving frame is, downward in the diagram,
a_e = E_g'
= 2 G ρ / r * 1 / (1 - v² / c²)
= a (1 + v² / c²),
for |v| small.
The gravitomagnetic acceleration, upward in the diagram, is
a_g = -4 v B_g
= -4 v * 2 / c² * G v ρ / r
= -4 * 2 G ρ / r * v² / c²
= -a * 4 v² / c².
The total acceleration is
a * (1 - 3 v² / c²),
while it should be
a * (1 - v² / c²).
The electromagnetic analogue
If the cylinder is electrically charged, then the length contraction in the moving frame causes the acceleration due to the electric field to increase by a factor
a * (1 + 1/2 v² / c²),
while the magnetic field adds a factor
a * (1 - v² / c²).
The acceleration is then
a * (1 - 1/2 v² / c²).
The electromagnetic analogue does not calculate correctly, either.
In the case of gravity, if we would like to explain the acceleration with gravitomagnetism, then the magnetic force should be twice the analogous electromagnetic field. It does not look like that we can give a consistent definition of a gravitomagnetic field at all. The reason is that kinetic energy acts as a source of gravity and complicates calculations.
Gravitoelectromagnetism requires a static gravitomagnetic field
Athanasias Bakopoulos (2016) writes that for gravitoelectromagnetism to work, the gravitomagnetic field has to be static. That is not true above, when we switch to the moving frame.
Conclusions
We showed that the equations of gravitoelectromagnetism are not consistent with special relativity.
The gravitomagnetic effect of a rotating body in the Kerr metric and the Hartle-Thorne metric conforms to the gravitoelectromagnetic equations. The magnetic effect is four times the effect in the analogous electromagnetic system.
We have to figure out if gravitoelectromagnetism works correctly in those cases.
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