UPDATE March 15, 2024: The Einstein approximation formula (see the post on August 15, 2023) produces a much more reasonable metric around the spherical shell, if we add perturbations. However, the "bulging coordinates" problem, which we studied in August 2023, may make the produced metric somewhat wrong.
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The Einstein equations say that in vacuum, the Ricci curvature is zero.
Let us have metric perturbations h₁ and h₂, such that the Ricci curvature of
η + hi
is zero. Here η is the Minkowski metric.
The derivatives in the Christoffel symbols Γ are assumed to have a very small absolute value. In the formula of the Ricci tensor, the cross terms Γ * Γ' then have extremely small absolute values. We are tempted to drop off the cross terms altogether. Then the Ricci tensor would be linear on perturbations.
Is the sum
η + h₁ + h₂
a good guess for a metric which would have the Ricci tensor zero?
No. Let us construct a metric around a spherical shell of matter by adding the metric perturbations from each small element of the shell. We will show that the linearly summed metric is totally wrong. The correct metric, of course, is the Schwarzschild metric.
A small element of the shell, dm, when alone in space, carries a Schwarzschild metric around it.
We assume that the radius of the shell is and mass / area of the shell is 1.
Let us then estimate the radial and the tangential metric around the shell, by adding the spatial metric perturbations. Let us denote the angle between OB and OC by α.
Let the radius of the sphere be 1. The element dm is at B in the diagram. We
The element dm stretches the spatial metric at C by
dm / (2 sin(α / 2)).
The radial stretch component is
dm / (2 sin(α / 2)) * sin(α / 2)
= dm / 2,
and the tangential stretch component is
~ dm / (2 sin(α /2)) * cos(α / 2).
The area of a C-centric narrow circular strip through B on the shell is
2 π sin(α) dα.
The contribution to the stretching of the tangential metric is
π
t₁ = ∫ 1 / (2 sin(α / 2)) * cos(α / 2)
0
* 2 π sin(α) dα
= 9.9,
where we used the numerical integrator at:
The contribution to the radial metric is
π
r₁ = ∫ 1 / 2 * 2 π sin(α) dα
0
= 2 π
= 6.3.
We see that the sum of metric perturbations stretches the tangential metric at C more than the radial metric. Let us draw two circles around the sphere, such that the radius of one is 1, and the other is at a proper distance dr farther from 1. The length of the larger circle would be > 2 π dr larger than the smaller circle. The spatial metric would have a negative Ricci curvature around the sphere.
In the correct, Schwarzschild, metric, the spatial metric has a positive Ricci curvature around the sphere.
Conclusions
We showed that one cannot get the correct metric around a spherical shell by summing perturbations of the metric for each mass element in the shell. The sum of the perturbations is grossly wrong.
This implies that one cannot rely on sums of perturbations when working with general relativity. We can say that the Einstein equations are "horribly nonlinear".
In the next blog posts we will analyze this further. Suppose that we are working on some mass distribution and have found a metric g which is "almost" right: the Ricci tensor in the vacuum area only differs from zero by a tiny amount. But the correct metric may still be very different from g. Let us then calculate the Einstein tensor T for g. We are able to satisfy the Einstein equations if we place a small amount of matter at certain locations, so that the stress-energy tensor becomes T.
Is this physically reasonable? Placing a small amount of matter switches the metric radically. Nature does not seem to function like that.
Andrew Strominger et al. (2011) show that Navier-Stokes equations in p + 1 dimensions are equivalent to Einstein equations in a certain setup in p + 2 dimensions. We know that the Navier-Stokes equations behave horribly badly. There is turbulence, and the existence of solutions is a one of the Millennium Prize problems in mathematics:
The existence of molecules and atoms saves us from the potentially infinite complexity of the Navier-Stokes equations in the real world.
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