The summed spatial metric perturbation is quite different from the one solved from the Einstein field equations
What about general relativity? Let us first calculate the metric perturbation from an initially static lightweight shell.
M
___
/ • dm
\____/
R = distance (center of M, m)
r = distance (dm, m)
• m
Each part dm causes a slowdown of time at the test mass m by a factor
1 - 1/2 rₛ / r
where
rₛ = 2 G dm / c²,
and r is the distance of dm and m.
The Newton shell theorem states that the sum of the metric perturbations for gₜₜ is
1 - 2 G M / (c² R).
Also, dm stretches the radial metric, parallel to r, at m by a factor
1 + 1/2 rₛ / r.
The stretching of the spatial metric for each dm is to the direction of their individual vector (dm, m). The stretching is not parallel for all dm. If we sum the stretchings in the direction (center of M, m), the sum is not
2 G M / (c² R),
but less. There is also stretching in the tangential metric.
The stretching of the spatial volume is similar to how much time "shrinks". But the division of the stretching into the polar coordinates r, φ, θ is not at all the same as in the Schwarzschild metric around an equivalent small mass M!
What does this mean? Are the Einstein field equations "magically nonlinear", after all, so that even tiny faraway masses can substantially influence the metric perturbation created by a mass dm?
Or is there a switch of coordinates which brings the perturbation caused by the linear sum close to the metric around M?
Bending of light close to the Sun
We know that the bending of light close to the Sun conforms to the Schwarzschild metric. Do we get the same bending by summing the bending effect for all mass elements dm of the Sun?
Yes. For each element dm, exactly a half of the bending comes from the perturbation of the metric of time, and the other half from the perturbation of the spatial metric.
If we sum the perturbations of time by each dm, we obtain almost exactly the metric of time of the Schwarzschild solution around the Sun. Let α be the deflection angle due to time. Then
2 α
is the total deflection angle for the sum of all dm, and the total deflection angle is the same for the Schwarzschild metric around the Sun.
Since the deflection angle is the same, it may be difficult to measure which is the correct spatial metric: the one obtained by summing or the one obtained by solving the Einstein field equations.
If we have an initially static mass m close to the shell, its acceleration is solely from the metric of time. Tests with a static mass m will not reveal which is the correct metric.
Can we find coordinates where the summed metric is close to the Schwarzschild metric?
___ ___ shell
-- • --_________------
dm
β = angle (m, dm) vs. y axis
\ |
\ |
\|
• m
^ y
|
------> x
Close to the shell, the summed spatial metric perturbation differs substantially from the Schwarzschild metric of the equivalent M. The angle β above is quite large for most elements dm.
Suppose that the Schwarzschild metric close to m has r stretched by a factor
1 + b.
The summed metric has r stretched by
1 + b',
and the tangential metric by
1 + c'.
The stretching of the volume tells us that
b = b' + 2 c.
If m is close to the shell, the value of b' may be substantially smaller than b. The difference might be 10%.
Can we rescale the r coordinate to make the metric look like the Schwarzschild metric?
The radial metric, of course, can be made identical through rescaling. A possible rescale is
r' = r * (1 + b - b').
We may be able to make the metric to look rather similar to the Schwarzschild metric. But if we define the radial coordinate
r = proper circumference / (2 π),
then the metric does differ significantly from the Schwarzschild metric. No coordinate transformation will change that.
Does there exist a solution in general relativity for the shell?
The Einstein field equations have exact solutions for a ball of incompressible fluid (Schwarzschild 1916) and for the collapse of a ball of uniform dust (Oppenheimer and Snyder 1939).
We have to check the literature if there is a solution for a shell of dust. Since the acceleration of the inner surface of the shell is zero, the collapse becomes quite complicated. The outer surface will press against the inner surface.
If there exists a solution, then Birkhoff's theorem dictates that the outer metric has to be Schwarzschild – and we showed that it would differ considerably from the summed metric.
Is it plausible that gravity is nonlinear in such a drastic way?
It is a natural assumption that weak interactions are almost linear. But here we seem to have a case where an extremely lightweight shell would fundamentally modify the field of its parts dm. Does this make sense?
Are there other natural phenomena which would be so extremely nonlinear? So that even tiny effects would coordinate between themselves, and modify each other substantially?
The solution to the problem: the metric perturbation comes from the collective perturbation of the time? Or a "rotation" is easy?
The perturbation of time is the same in Schwarzschild and in the sum of perturbations.
We have claimed that the stretching of the radial metric in Schwarzschild comes from shipping of energy around in the field. When the test mass m approaches M, energy comes from, on the average, the distance between M and m and is absorbed by m. The shipping of energy increases the inertia of m, which simulates stretched radial metric.
We have also claimed that the Schwarzschild field allows field energy to rotate around M "easily". If the tangential spatial metric were stretched, that would indicate that field energy would not rotate easily.
● M • m ● M'
^ y
|
-------> x
What about the configuration above? We believe that we must sum the radial metric perturbations by M and M' at m. In this configuration, field energy does not "rotate" around, when we move m to the x direction. There is extra inertia, which causes stretching of the x metric at m.
Conclusions
The correct metric around the spherical shell probably is the Schwarzschild metric. The field energy for each pair (m, dm) is not "private" enough to cause the tangential metric to stretch.
We have to investigate if and how this affects frame dragging of a rotating disk.
Also, we have to find a precise definition: when does energy shipping affect inertia of m, and how much?
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