"Conservation of pressure" in general relativity?

Our previous blog post brought up a possibility that general relativity actually has a "conservation of pressure" principle. Any volume of positive pressure has to be enclosed into a shell of negative pressure. Then the metric changed by an internal pressure cannot spread out of a spherically symmetric mass M. Birkhoff's theorem is saved.

The stress-energy tensor T

Let us check how T is precisely defined. Could it be that an accelerating mass at the surface of M actually must be counted as negative pressure?

https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor

The component T¹¹ is defined in Wikipedia as the flux of x₁ momentum across a surface of a constant x₁ coordinate.

  

                     

          pressure P        mass M

              ---------------> ■■■■■■■

                                       ---> 

                                       |   |             |

                                       X   Y            Z    

      ----> x

Let us analyze the flux in the above configuration. What is the flux at X, Y, and Z?

Let us consider the volume element between X and Y. The pressure at X is higher than at Y. The element "eats" pressure.

"Eating" pressure would be defined as a process where a mass dm accelerates inside the volume element.

But this will not work as a definition of negative pressure for M. If P is negative, then M still "eats" pressure.

And the volume containing a positive pressure P may be arbitrarily large. Assigning the right amount to compensate it with a negative pressure within M is hard.

Conservation of pressure in a large block of an elastic material

Suppose that we have a large block of material with constant Young's modulus. If we stretch the material at one position, we have to squeeze it at another position. The sum of pressures is conserved.

We have to analyze this from the perspective of Noether's theorem. Could it be that the Einstein-Hilbert action implies conservation of pressure?

In a rubber sheet model, masses put on the sheet produce negative pressure (stretching). But otherwise, pressure is conserved.

Is the proof of Birkhoff's theorem correct?

https://www.roma1.infn.it/teongrav/VALERIA/TEACHING/RELATIVITA_GENERALE/AA2015inpoi/birkhoff_theorem.pdf

The proof is based on defining new coordinates based on the usual coordinates t and r, and showing that the metric stays as the Schwarschild metric in the new coordinates.

   t

   ^

   |                |        |          • m

   |         ----------------------

   |               /            \

   |              |            | 

   |         ----------------------

   |              |            |

    ------------------------------------>  x

                          ●

                          M

But is it ok to switch the coordinates in an arbitrary way? Let us have test masses which are initially static in the old coordinates. If we define new coordinates which move at an accelerating speed relative the old coordinates, are such coordinates allowed?

If the metric would be Minkowski, reasonable coordinate systems are inertial.

In the diagram we have a case where the spatial metric around M suddenly expands. The x coordinate lines make a turn. If we have test masses m floating around M, their proper distance from M suddenly grows.

But if we define new spatial coordinates, we can make the spatial metric to appear constant in the new coordinates. As if nothing would have happened. In the diagram, the new coordinates would "accelerate" relative to the old coordinates.

Could it be that in the proof of Birkhoff's theorem new coordinates hide the fact that the metric has changed relatively to freely floating test masses?

Conclusions

There seems to be no simple and intuitive way of defining a "negative pressure" of accelerating mass, such that the negative pressure would compensate a positive pressure.

It looks like Birkhoff's theorem is broken by pressure.