We assume that we start from the static Schwarzschild interior and exterior metrics for a uniform sphere of incompressible fluid. The sphere does not collapse under gravity because it has an internal pressure.
A change in the pressure makes the system dynamic: hence, no solution for the Einstein equations
Suppose that there would exist a solution g for the Einstein field equations. Like in the previous blog post, we can apply an infinitesimal rogue metric variation δg, which does not change the action integral S of the Ricci scalar R, but raises a mass m to a higher gravity potential. Such a variation decreases the action integral S of LM.
Then g is not a stationary point of the Einstein-Hilbert action, and cannot be a solution of the Einstein field equations. We have a contradiction.
One could try to remedy the problem by introducing canonical coordinates and a canonical kinetic energy to LM.
But that might not help in the pathological problem of a non-conserved "pressure charge" in general relativity.
Raising the pressure suddenly inside a spherical uniform mass M
Let the radius of a uniform spherical mass M be r₀, its proper density ρ₀, and its proper pressure p₀. Initially, the system is static. The pressure exactly cancels the gravity force.
The system has the Schwarzschild interior and exterior metric g. We assume weak fields. The metric g is only slightly perturbed from the flat metric η. We use the signature (- + + +).
The Ricci tensor inside the mass M must be approximately
R ≈
κ * 1/2 *
ρ + 3 p 0 0 0
0 ρ - p 0 0
0 0 ρ - p 0
0 0 0 ρ - p.
The Ricci curvature R₀₀ can be determined with a cube of test masses m which are initially static and then fall freely. Let U₀ be the initial proper volume of the cube.
The proper volume U(t) of the cube decreases according to R₀₀:
U(t) = U₀ * (1 - C R₀₀ t²),
where C is a constant and t is the time that the test masses have fallen freely.
Let us put test masses m everywhere inside M, and also outside it, up to a radial coordinate r₁ > r₀. We then have a spherical constellation of test masses. The volume of the constellation decreases with time because infinitesimal cubes in the constellation shrink inside M (where R₀₀ > 0). The cubes outside M do not shrink (there R₀₀ = 0).
Since
R₀₀(ρ, p) = 1/2 κ (ρ + 3 p),
we can modify the value of R₀₀ as we wish by changing the pressure p inside M.
• •
• • • •
• • • • m test masses
• •
<---- 2 r --->
If the sphere is static and we release the test masses, then the outer radius r of the test masses must decrease at a speed which is consistent with the shrinking of the cubes U inside the mass M.
Birkhoff's theorem says that the metric outside M cannot change. The outermost test masses fall the same regardless of p.
Let us suddenly raise the pressure p by Δp.
The shrink rate of the cubes U inside M increases. As the test masses outside M still fall at the same rate, does this lead to a contradiction? The proper volume of the test mass constellation would shrink faster, even though its surface moves as before.
Because of the increased pressure, the average density of the sphere M starts to decline:
ρ(t) = (1 - C₂ t²) ρ₀,
where C₂ is a constant.
Let us update the formula:
U(t) = U₀ * ( 1 - C * R₀₀((1 - C₂ t²) ρ₀, p) * t² ).
If t is small, we can ignore the change in the density ρ, because the effect on U(t) is only ~ t⁴.
Is it possible that the pressure increase causes the radial metric g₁₁ to shrink, so that it could explain away the shrinking of the cubes U?
In the interior Schwarzschild solution, the spatial metric of a slice of the sphere M in the x, y plane is the surface of a sphere. The radius ℛ of this surface only depends on the density ρ, not on the pressure p at all.
Thus, we believe that the change in the pressure does not affect the spatial metric, and we have a contradiction. But we need an exact proof.
Let us raise the pressure by a fixed amount Δp at radii
r < r₂ < r₀.
At radii
r₂ < r < r₀,
we let the extra pressure Δp to fall off, so that the pressure is zero at the surface of M.
It is a good guess that the new metric g will imitate the metric inside a static larger sphere whose density is ρ. The added outer layers in a larger sphere cause a uniform extra pressure.
But the pressure falls off rapidly near the surface of M. A rogue variation prevents the Einstein field equations from having a solution there, because we can reduce the potential energy of the extra pressure by moving particles to a larger radius r.
Let us assume that we are able to fix the lagrangian LM by introducing a canonical kinetic energy into it. We might pretend that the Δp is 0 at
r₂ < r < r₀,
and ignore the sharp jump of the pressure at r₂.
Then the Schwarzschild interior solution says that the spatial metric is unchanged at r₂ < r < r₀. The spatial metric is unchanged in the entire sphere M. This leads to a contradiction because after raising the pressure, the proper volume of the test mass constellation shrinks faster. The radius of the outermost test masses should shrink faster, but it does not. The outermost test masses fall as fast as before.
Raising the pressure in M and Gauss's law for gravity
The Ricci curvature component R₀₀ describes the "focusing power" of gravity.
Outside the spherical mass M, we have R₀₀ = 0, and the lines of force of gravity must be continuous: Gauss's law holds there.
Let us have a cube of test masses inside M. The component R₀₀ tells us how much "source" of gravity is inside the cube. The Komar mass formula gives the total source of gravity inside a static M.
But we can increase the pressure p inside M. Then R₀₀ grows, and suddenly we have more source of gravity inside M. More lines of force start there.
This necessarily breaks Gauss's law, because Birkhoff's theorem says that outside M, we cannot increase the number of lines of force. General relativity has a fundamental problem here.
Pressure generates gravity also in our Minkowski & newtonian gravity model. We do not have Birkhoff's theorem. Therefore, this is not a problem for us.
The Einstein equations do not have a solution for a rotating mass M – the Kerr metric
If M is held static by a pressure, then a rogue metric variation does not change the value of the Einstein-Hilbert action S.
But if the mass is kept "stationary" through rotation, then we obviously can change the value of S by raising some mass m to a higher gravity potential, or lowering m to a lower gravity potential.
It has been an open problem if any rotating mass can generate the Kerr metric. We here solved the problem, in the negative.
It may be possible to remedy general relativity by switching to canonical coordinates and adding the kinetic energy to LM.
In August and September 2023 we noted that general relativity "has problems handling accelerating mass flows". Here we proved a major problem: the Einstein equations have no solution for them.
Is the Kerr metric itself reasonable? We were not able to solve the question in the Fall of 2023, using our own Minkowski & newtonian gravity model.
Conclusions
Pressure does generate gravity – also in our own Minkowski & newtonian gravity model. Since one can change pressure, there is no conservation of the "pressure charge". We have to abandon Gauss's law for the gravity generated by pressure.
General relativity implies Birkhoff's theorem, which, in turn, implies that Gauss's law must hold for weak gravity fields.
If we want to repair general relativity, radical changes are needed in it. On the other hand, our own gravity model has no problems handling pressure changes.
We will next look at cosmological FLRW models. Our hypothesis is that only a universe with a totally uniform mass density ρ satisfies the Einstein equations. This means that general relativity does not have a solution for any realistic cosmological model.
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