by the fourfold gravitomagnetic effect of the Kerr metric when compared to the analogous rotating electric charge. Maybe it is a result of the nonlinearity of gravity close to a black hole? The strong field may "screen" the far side of a rotating black hole, making the magnetic effect of the near side more prominent.
Let us study magnetic effects observed by a test charge q or a test mass m close to a moving spherical charge Q or mass M. We may as well assume that Q or M is static, and let q or m move.
We can use the Schwarzschild metric to determine the behavior of a test mass m.
A test charge approaches a moving charge from the side
Q ●
^ V
|
|
• -> v
q
This is an example of "frame dragging" in electromagnetism. A large charge Q is static. The test charge has a velocity vector V + v. Let us assume that |v| is very small. V points directly to Q. The test charge q has the opposite sign to Q.
Our starting point is a classic Coulomb force calculation with no relativistic corrections. We calculate two corrections to the classic orbit.
Calculation A.
As q descends a distance toward Q, q gains the energy W from the common field of Q and q. Let us assume that the inertia of q grows by the mass-energy W gained.
The inertia of q also grows because it acquires W as additional kinetic energy.
The total gain of inertia is 2 W / c². The velocity vector v of q is corrected by
Δ = -v * 2 W / (m c²),
where m is the mass of the test charge q.
Calculation B.
Let us then calculate the process in a frame where q does not move sideways. Again, the starting point is a classic calculation where we add corrections.
v <- ● Q
^ V
|
|
•
q
y
^
|
--------> x
Let E be the electric field of Q at q (E_y). The magnetic field B_z of Q is then
B_z = v E / c².
The sideways force on q is
F = -q V v E / c².
Let q move for a short time t. It gains the energy
W = q E V t.
It acquires a momentum p = F t, and a velocity
Δ' = -F t / m
= - (q V v E t / c²) / m
= -v W / (m c²).
That is the correction from the magnetic field. From the table above we see that there is no correction to E_x.
We have Δ' = 1/2 Δ. The corrections were only a half of Calculation A!
We believe that Calculation A gives the correct value. Calculation B with just the magnetic field does not capture correctly the changes in the inertia of q.
It is hard to measure the correct inertia experimentally because the electric force greatly dominates the magnetic force.
If Q and q would have the same sign, then the first calculation would give Δ = 0, while the second would have the sign of Δ' switched.
A test mass approaches a moving mass
M ●
^ V
|
|
• -> v
m
Let us repeat the first calculation, this time with masses, and using the Schwarzschild metric.
Our starting point is a classical newtonian orbit with no relativistic corrections. We check what corrections the Schwarzschild metric adds to it.
For the test mass m, the specific angular momentum h = L / m stays constant:
where we can assume that the reduced mass μ is equal to m if the test mass m is infinitesimal.
We are interested in the angular velocity of the test mass in Schwarzschild coordinates. That is how a distant observer sees the process: dφ / dt.
For a classic newtonian orbit,
L / m = r² dφ / dt
is constant. Let us assume that the test mass descends from a larger radius R to r, to a lower potential, such that the test mass m gains the energy W from gravity.
The proper time in the Schwarzschild metric is
dτ = sqrt(1 - r_s / r) * sqrt(1 - V'² / c²) dt,
where V' is the velocity of the test mass measured by a static local observer.
Let us assume that the gravity field is weak and V' is not very large. Then
sqrt(1 - r_s / r) * sqrt(1 - V'² / c²)
= (1 - P / (m c²)) * (1 - 1/2 m V'² / (m c²)),
where P is the (negative) gravity potential of the test mass and 1/2 m V'² is the locally measured kinetic energy of the test mass. Both factors are only slightly less than 1.
In the newtonian calculation, let us use the local time at the original position R as the global time.
When the mass descends down, the above factor must be multiplied by
(1 - W / (m c²)) * (1 - W / (m c²))
= 1 - 2 W / (m c²).
To keep r² dφ / dτ constant, we must reduce dφ.
The tangential velocity r dφ / dt is then corrected by
Δ = -v * 2 W / (m c²).
In this case, the frame dragging, or the gravitomagnetic effect, agrees with electromagnetism, if we believe Calculation A in the preceding section.
Why is frame dragging surprisingly large in the Kerr metric and the Hartle-Thorne metric?
The Hartle-Thorne metric describes a rotating body which is not a black hole. Frame dragging in it is similar to the Kerr metric, in terms of J / M.
The Kerr frame dragging is fourfold compared to the purely magnetic correction in an analogous electromagnetic system.
If we also include the inertia of kinetic energy, like in Calculation A above, then the Kerr frame dragging is double.
We have to find out from where does the extra frame dragging originate.
A rotating body is a much more complex example. Let us try to determine frame dragging in that case.
For a rotating sphere of electric charge, the Poynting vector shows energy rotating in the field along with the charge. If we lower a test charge on the surface of the sphere, that may affect the Poynting vector quite a lot. Maybe the test charge through this mechanism acquires surprisingly lot of angular momentum?
The Poynting vector B × E is linear on the electric field E of the test charge, and we know empirically that test charges under a strong external magnetic field B (and no strong external electric field) obey the magnetic force, nothing else. Thus, the Poynting vector hardly can explain the strong frame dragging.
A test mass stretches the spatial metric of a rotating central body. This could create a "tidal force" on the test mass. However, general relativity usually is not aware of tidal forces.
Conclusions
For a simple case of a test charge, or a test mass, approaching a moving object from a side, frame dragging is perfectly analogous for gravity and electromagnetism, if we use Calculation A above.
For a rotating mass, general relativity gives a 4-fold frame dragging relative to electromagnetism. We have to find out why.
According to Wikipedia, there is a perfect analogy between electromagnetism and gravitoelectromagnetism for weak gravity fields, but the Lorentz force is 4-fold for a gravitomagnetic field:
A mystery: we calculated above that for the Schwarzschild orbit, electromagnetism and general relativity agree perfectly for Calculation A. They would NOT agree if we make the effect of the magnetic field 4-fold in the electromagnetic calculation! Could it be that the perturbation technique which is used to derive the gravitoelectromagnetic equations miscalculates Schwarzschild orbits? But then we face the mystery why it would get Kerr orbits right.
Since much of magnetism can be derived from the Coulomb force and special relativistic corrections, it is strange if gravitoelectromagnetism contains a superfluous factor 4.
General relativity seems to predict closed timelike curves for certain rotating objects. Such solutions are nonsensical. This suggests that general relativity handles rotating objects incorrectly. But it would be strange if there would be an error in the Kerr solution.
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