The Levi-Civita metric is not physically realistic though, because it assumes an infinitely long cylinder.
Is there any static solution for a finite cylinder? Or for two spherical masses held apart by a very lightweight rod?
Let us investigate the case of two lightweight masses. We assume that a "magic" mechanism keeps them static in space, so that they do not crash together.
● ●
M₁ M₂
Each M alone would have the Schwarzschikd metric around it. The field is weak. We may write the metric
η + h,
where η is the Minkowski metric, and h is a perturbation.
Our first guess for two masses M is, of course, the sum of the perturbations
η + h₁+ h₂.
Each h has Ricci curvature zero in vacuum.
The Cristoffel symbols Γ are linear in the metric g. If the directional Ricci curvatures Rⱼₖ were linear in the Christoffel symbols, then the sum of the perturbations h₁ + h₂ would have Ricci curvature zero. But we have the annoying cross terms:
Since the fields are weak, the Christoffel symbols have small absolute values, and the cross terms have very small absolute values. Since the equations are nonlinear, this does not guarantee that η + h₁+ h₂ is close to a solution. Maybe the solution does not exist at all?
This eerily reminds us of our paper bending experiments on March 8, 2024. If we were allowed to stretch the paper a little bit, then we could find a solution for many types of "masses". But we are not allowed to do that, and the cone is the only beautiful solution that we know.
On March 11, 2024 we calculated Ricci curvatures for various components of the Schwarzschild metric. Let us try to determine what the cross terms look like if we sum two Schwarzschild metrics.
M₁ M₂
----> x ● ●
| • test mass
v y
In R₀₀, we have, for example, the product
Γ¹₁₁ Γ¹₀₀.
Let us take the first Γ from h₁ and the second Γ from h₂. The cross term has an approximate value
-1/2 * 1 / x₁² * 1/2 * 1 / x₂²,
where x₁ is the x distance from M₁ and x₂ is the x distance from M₂. Recall that we set
2 G M / c² = 1
on March 11, 2024. Thus, they are actually very strong fields.
Iterative methods do not work?
Let us have the lightweight masses M₁ and M₂ above. Our first try to solve the Einstein field equations is the metric
g = η + h₁+ h₂
above.
The metric does not precisely satisfy the Einstein field equations, because of the tiny cross terms between the perturbations h₁ and h₂:
The right side of the Einstein equations obtains some residual value T', where T' is a tensor close to zero.
Suppose further that only the 00 component of T' differs from zero. It is like a mass distribution M(r) in space, where r is the position vector. The mass M(r) is very small compared to M₁ and M₂.
Let us assume that we have a Schwarzschild-like solution
η + hM
for the mass distribution M(r) in otherwise empty space. The perturbation hM is much closer to zero than h₁ and h₂ are. We obtain a much better approximate solution for the original equation in the metric:
g - hM.
"Better" here means that the new residual value on the right, T'', is much closer to zero than T'.
If all the above assumptions were true, then we might be able to prove that the iteration quickly converges to a solution. We would have a powerful existence theorem. But such an existence theorem is not known. If our simple method would work, someone would have spotted it soon after the year 1915.
A problem is that the residual tensor T' has also other non-zero components besides the 00 component. T' does not describe a pure mass distribution.
Is there a "Schwarzschild" solution for an arbitrary stress-energy tensor component? For example, if we assume an isolated pressure component T₁₁ which occupies a spherical volume V in space?
For a moment, we do not worry if such isolated pressure can exist in the real world.
A Schwarzschild-like solution for pressure?
There is probably no Schwarzschild-like solution for a component of pressure. A central feature in the Schwarzschild solution is that it is spherically symmetric. Pressure is directed.
P pressure
------- T₁₁ non-zero
-----------
-------
F' anomalous
F ^ ^
\ |
• m test mass
^ y
|
------> x
The test mass m stretches the radial metric around it. We expect the potential of m to be lower if m is to the x direction from the pressure volume P. The force F does not pull m directly toward P, but there is also an "anomalous" component F' which pulls m up so that m would be located to the x direction from P.
We have to check from the Einstein-Hilbert action if our reasoning is correct.
The anomalous force F' probably has the divergence
∇² F'
non-zero in vacuum. But is the anomalous force known to the metric g around P, or is F' a more complicated consequence of the Einstein-Hilbert action?
The metric in general relativity certainly understands some effects of pressure, but does the metric understand the force F'?
Conclusions
If a simple iterative method would work, then we would have a wealth of exact solutions in general relativity. We would probably have existence theorems, too.
We conclude that no one has found a good iterative method for solving the Einstein equations.
We will next investigate what general relativity understands about the pressure area P.
The Einstein approximation formula (1916) below suggests a very simple metric around the pressure area P. Does the formula give a good approximate metric?
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