UPDATE January 19, 2022: Birkhoff's theorem requires strict spherical symmetry. The test mass has to be a spherical shell of matter. A shell does not affect the spatial metric inside it. Thus, pressure does not attract the test mass (= shell) if the test mass is outside the spherical mass. Birkhoff's theorem is saved.
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Birkhoff's theorem is derived from the Einstein field equations, and is extremely strict: the theorem claims that the metric in the vacuum area of a spherically symmetric system has to be the static Schwarzschild solution. It enforces conservation of energy.
If we allow energy non-conservation in the system, then obviously general relativity does not have a solution.
Let us analyze how having no solution is expressed in the action. The first thought is that since the action only looks for a local extreme value, there would be a solution for all systems.
However, if a singularity develops, then there is no solution.
Suppose that we want to remove energy from a spherically symmetric mass, while obeying conservation of energy. The mass may radiate, for example.
There exists a smooth solution since the outgoing radiation takes away the extra curvature of spacetime.
If we would just let the radiation disappear, then a discontinuity would appear in the attempted solution. A discontinuity is a kind of a singularity.
The problem of pressure changes - a rubber membrane model
One can temporarily change the pressure of a spherically symmetric system. For example, we may have springs which we temporarily release.
Pressure acts as a source of gravity in general relativity. The metric inside the spherical mass will change.
How can we match the change in the internal metric to the static external metric?
In a rubber membrane model, Birkhoff's theorem would mean that the membrane is frozen outside the spherical mass. If we increase the pressure inside the mass, we can make the depression in the membrane deeper. Is it possible to match the membrane outside the mass and inside?
Increased pressure starts to eject matter from the borders of the spherical mass. Could this help in matching?
Pressure probably increases the positive curvature inside the mass. If we want to keep the curvature outside static, we probably have to introduce negative curvature to the borders of the mass. What could produce such negative curvature? In general relativity, negative curvature is prohibited by a weak energy condition.
If we do not enforce Birkhoff's theorem to the rubber membrane, negative curvature does not appear, or does it? Yes it does. Faraway parts of the membrane do not have time to react when pressure pushes the depression deeper. A wave of negative curvature starts spreading from the mass.
Does general relativity prohibit negative curvature?
In a rubber membrane model there is nothing which prohibits negative curvature. What about general relativity? Can negative curvature exist in the vacuum?
No. If the stress-energy tensor is zero, then Ricci curvature has to be zero.
Could it be that the ejected matter at the border of the mass generates negative curvature?
Probably not. It is a common belief that moving matter only creates positive curvature.
We may imagine that the mass is enclosed into a rigid shell. When pressure tries to push matter out, it bumps into the shell. Negative pressure starts to build up in the shell. Could this negative pressure cancel the effect of a sudden pressure increase in the mass?
Probably not. It takes time for the negative pressure to build up, while the positive pressure increase is immediate.
Does general relativity have a solution for a change in pressure?
In the interior Schwarzschild solution pressure contributes to positive curvature along with mass-energy.
If we increase the mass-energy temporarily through energy non-conservation, then general relativity probably does not have a solution.
We can increase the pressure temporarily. Why would general relativity have a solution in that case?
If there is no solution, then the Einstein-Hilbert action is broken. The probable culprit is using the Ricci scalar R as the lagrangian of spacetime curvature. In our previous blog post we showed that the derived metric does not describe the orbit of an infinitesimal test mass correctly. The status of the metric and R as the "curvature of spacetime" is not clear.
If we increase the pressure symmetrically in a spherically symmetric mass, then the attraction from pressure probably does not grow outside the mass but the inertia of the test mass grows
In our Minkowski & newtonian model, stretching of the spatial metric is due to a potential gradient.
If we put an infinitesimal test mass dm close to the spherical mass, but not inside it, then the test mass makes the potential gradient steeper on one side of the spherical mass and less steep on the other side. The summed effect probably is zero.
The radial metric stretches on the far side, and pressure does work there. On the near side, pressure receives work when the radial metric contracts.
If we want to create an extra pulling force on the test mass, we should increase the pressure on the far side of the spherical mass, or we should lower the test mass inside the spherical mass.
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/ - - - \ negative energy
\ + + + / positive energy
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• test mass dm
Pressure does not increase the force on the test mass when it comes outside the spherical mass. But pressure does cause energy to move inside the spherical mass. This, in turn, increases the inertia of the test mass and changes the metric.
This is evidence against Birkhoff's theorem. By increasing pressure inside the spherical mass we can increase the inertia of the test mass outside it. Increased inertia means a slower speed of light, which can be explained by slower time or increased distances.
The effect on the rate of a mechanical clock seems to be ~ 1 / r², while for gravity it is ~ 1 / r. The effect is a tidal effect. This makes sense: Birkhoff's theorem is not aware of tidal effects on an infinitesimal test mass.
A slower speed of light means that rays of light are bent more close to the spherical mass. That implies that gravity feels stronger with increased pressure inside the spherical mass.
A higher pressure increases the interaction between masses. A strong interaction increases inertia, and slows down the speed of light. In our syrup model of gravity, pressure increases the viscosity of the syrup.
We must analyze the process in more detail. The effects of pressure can be surprising.
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