____
/ \
| • | • = test mass m
\______/
pressurized vessel
We slowly move a test mass m from far away into the vessel. The test mass might be a small massless box filled with light. That is, the mass-energy is in the form of light.
How much energy is released when we move the test mass?
1. the test mass m extends the radial metric but keeps the tangential metric as is: the volume of the vessel grows and energy from the pressure is released;
2. each bouncing photon gains inertia because the mechanism of item 1 makes pressure energy to move around: more inertia means less kinetic energy, and energy is released from the photons.
Item 2 might explain the temporal metric of general relativity. The pressure in the Schwarzschild interior solution slows down clocks at the center of the the ball of incompressible liquid.
Our earlier calculations in this blog suggested that also item 1 would explain the metric. Do we have double gravity here?
The Einstein-Hilbert action knows the metric and is certainly also aware of item 1. When we move the test mass m to the center, the slow flow of time effectively releases some of the mass-energy m. But the action also calculates the energy released by the volume expansion of the vessel.
How quickly does the extra inertia affect a movement of m?
Let us move m abruptly a short distance inside the vessel. That will make pressure energy to move around in the vessel, and the inertia of m feels larger. But since changes in the gravity field only propagate at the speed of light, the extra inertia should first be less and then grow?
General relativity believes that it is the metric which "implements" the extra inertia. In general relativity, the inertia stays the same throughout the movement.
However, in general relativity, gravitational waves exist, which shows that general relativity is aware of the speed limit on the gravity field. Is there a contradiction here?
Does Einstein-Hilbert action count the extra inertia twice? First through the slowdown of the time in the metric, and after that through the flow of pressure energy in the vessel? Is the action smart enough to understand that pressure energy flows inside the vessel and adds to the inertia of m?
What takes care of action/reaction? Noether's theorem
The Einstein-Hilbert action describes the direct gravitational attraction between masses M1 and M2 . For them, the metric may implement the action/reaction law of newtonian mechanics. M1 falls in the field of M2, and vice versa.
What about pressure and even more indirect interactions of the matter lagrangian L and gravity? What ensures that momentum is conserved, i.e., there is an action/reaction? We have to check if the ADM formalism, which is a hamiltonian formulation of general relativity, proves conservation of momentum.
Generally, Noether's theorem states that a lagrangian, which only has conservative forces, preserves linear momentum if the lagrangian is invariant with respect to spatial translations. The Einstein-Hilbert action seems to conserve energy, and it is invariant under spatial translations. Thus, it most probably also conserves linear momentum.
On the other hand, if we try to analyze the system using a metric, then it is not clear what would alter the metric around M to cause M to move. This might be yet another instance where the geodesic hypothesis fails.
Pressure in M and a test mass m: a hamiltonian seems to conserve momentum
In the case of pressure inside a large mass M, and a test mass m within, momentum conservation comes from the following process.
Let us have a hamiltonian which is equivalent to the Einstein-Hilbert action. If we move m and M closer to each other, then energy is freed from potential energy to kinetic energy. That is, the lagrangian T - V increases. The increase is the fastest if we move both m and M closer to each other. That is because if we free some potential energy ΔV and change it to kinetic energy of m and M, then the distance between m and M shrinks the fastest if both approach each other. The lagrangian favors the fastest conversion of potential energy to kinetic energy.
This is the mechanism of action/reaction.
Conclusions
In our own Minkowski & newtonian gravity model, pressure frees energy through the two mechanisms described in items 1 and 2.
In general relativity, it may be that the extra inertia of the test mass is double counted. We have to study the Einstein-Hilbert action in detail to figure out if that is the case.
The speed of the extra inertia remains an open problem. Does the test mass m possess the extra inertia immediately (like it does in a metric), or does it only acquire the extra inertia when pressure energy starts to flow?
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