|__________/\___________|
wall wrinkle wall
Let
S = the volume between the carpet
and the floor.
We can determine the "action" S by integrating the difference of the elevation of the carpet versus the floor over the entire room.
We want to find a minimum of the action to determine how the carpet should settle itself. But the minimum is not unique. We can put the wrinkle at any location of in the room, and S has the same value.
A "locally" stationary point for the carpet is one where the carpet touches the floor. A globally stationary point is such that S is at the minimum. The carpet has no globally stationary point which would be locally stationary everywhere.
Magically adding mass to the Einstein-Hilbert action
Suppose that we have a spherically symmetric mass M and the metric around it is Schwarzschild. We use a magic trick to increase the mass M to M'. What does the Einstein-Hilbert action say about the time development of the system after that?
-------___●___-------- metric
M
------_ _------ metric
--- ● ---
M'
After adding the extra mass, the action is not in a locally stationary point. There is a wrinkle in the metric around M'. The wrinkle corresponds to a negative mass M - M'. We cannot get rid of the wrinkle instantaneously because the speed of the light is the limit. Actually, we can never get rid of the wrinkle because we cannot expel it past the infinity.
The system is not locally stationary because the required negative mass does not exist at the wrinkle. It could be globally stationary, though. A globally stationary point is not necessarily locally stationary everywhere.
We do not know if a supposed globally stationary point determines the time development of the system in the Einstein-Hilbert action.
Changing pressure inside M
Our example may describe what happens in the Einstein-Hilbert action if we change the pressure inside M. After that, the system may still be in a globally stationary point, but it is not locally stationary any more. It is like adding a wrinkle to the wall-to-wall carpet by sewing a new patch into the carpet.
This shows that the Einstein-Hilbert action might be able to handle changes in pressure, but then the Einstein field equations would not hold in the entire space.
"Negative mass" propagating from M when the internal pressure of M is increased
• • •
• • • test masses m
• • • |
v acceleration
● M
Suppose that we are able to increase the attraction of M from zero to a non-zero value suddenly. Let us have a cube of test masses floating in space. Does the volume of the cube increase?
The lowest surface of the cube starts accelerating toward M. The volume of the cube increases. This shows that a sudden increase of the attraction causes a "defocusing" effect. It is like negative mass-energy would be propagating from M.
The variation which is used to derive the Einstein field equations
David Hilbert (1862 - 1943) (photo Wikipedia)
The Einstein field equations assume that the system is everywhere locally stationary. That is, the local metric matches the local stress-energy tensor. Is this assumption incorrect for dynamically changing systems?
Let us check the derivation of the Einstein field equations in Wikipedia.
The derivation does not specify what variations of the metric,
δg^μν
are allowed. We remarked in an earlier blog post that if we add the Schwarzschild metric of a small mass dm, the volume integral of the change in the metric over the entire 3D space is infinite.
Also, the time development of the system must be such that faster-than-light signals are not allowed. Even if a certain development would be a stationary point of the action, it is not allowed if it involves superluminal signals.
"By Stokes' theorem, this only yields a boundary term when integrated. The boundary term is in general non-zero, because the integrand depends not only on δg^μν, but also on its partial derivatives
...
However when the variation of the metric δg^μν vanishes in a neighbourhood of the boundary or when there is no boundary, this term does not contribute to the variation of the action. Thus, we can forget about this term"
Wikipedia claims that we can forget about the behavior of the variation δg^μν far away: it does not affect the action integral.
Suppose that we magically add some more mass to a spherically symmetric M. Does the metric afterwards obey the Einstein field equations at every event?
Suppose that we are able to find a stationary point of the action, such that it extends the history after the magic trick.
If there would exist a variation which only adjusts the value of the Ricci tensor R at the event we are looking at, then we could argue that the Ricci tensor R must match the stress-energy tensor T at that event. But there probably exists no such variation. If the variation does not extend over the entire 3D space at the time of the event, then the variation presumably modifies R at the event, as well as in many other locations. There is no guarantee that we can make R to match T at all events. It is like the carpet which we can make to touch the floor at any one location, but cannot make it to touch the floor everywhere.
Another way to explain the problem in the Einstein field equations: optimizing the action requires global information of the system because the metric and its curvature have global dependencies. If we increase the curvature in a certain volume of spacetime, we have to reduce it elsewhere. This because we cannot change the metric simultaneously everywhere in the 3D space. The Einstein field equations are strictly local. They cannot understand the global optimization problem.
Pressure and dynamic systems
Adding pressure obviously requires a change in the metric inside M. There is no guarantee that we can make R and T match everywhere.
The same may be true for almost every dynamic system where there are accelerating masses. There is no proof that a stationary point of the Einstein-Hilbert action is locally stationary, that is, that R and T match at every event.
We know a few solutions where we are able to make R and T to match everywhere. The Schwarzschild exterior and interior solutions (1916) are the best known ones. The Oppenheimer-Snyder collapse (1939) is a known dynamic solution.
Conclusions
The derivation of the Einstein field equations from the Einstein-Hilbert action is incomplete, and probably erroneous. This explains why Birkhoff's theorem seems to clash with Tolman's paradox.
It looks like that most changes in the pressure of a system cause the system to enter a state where a stationary point of the action is not locally stationary everywhere. That is, the Einstein fields equations are not satisfied by a stationary point of the Einstein-Hilbert action. In practice, this means the Einstein field equations do not have a solution for any realistic physical system, because such pressure changes will always happen.
Birkhoff's theorem is probably false for gravity, as gravity occurs in nature. The attractive force of a spherically symmetric mass M does change when we manipulate the pressure inside it.
The Einstein-Hilbert action itself is probably incorrect, too. It is based on a very optimistic hypothesis that a "metric" can capture all the complicated phenomena associated with gravity.
No comments:
Post a Comment