Where does the extra momentum go?
We can use, for example, a very long spring to accelerate the mass. The other end of the spring is attached to another mass far away. We assume that we can ignore the field of that other mass, at least for a moment.
Note that if we rotate the mass, then waves which transport energy at the speed of light, can transport a considerable amount of angular momentum if the waves "start their journey" at a great distance r away from the rotating mass. We can keep pumping angular momentum to the field and light-speed waves can take that angular momentum away.
For the electric field, we have suggested that the extra momentum goes to pulling the mass-energy E^2 of the outer parts of the electric field up to the pace of the acceleration. This is completely analogous to pushing a flexible solid object from the middle.
In the case of gravity, there is no concept of an energy density of the field. There can be positive energy in the field, as shown by gravitational waves. A binary neutron star keeps losing its angular momentum to gravitational waves. This means that the waves are carrying mass-energy out of the system.
If we use a newtonian approximation, then the field energy of a gravitational field appears to have a negative energy. Or does it? Suppose that we assign the whole rest mass of an object to its field energy, to lift the field energy positive everywhere. The combined field would contain the newtonian negative energy field plus the rest mass field. The extra momentum could be put to bringing the combined field up to the pace of the acceleration.
We have not found a way to dump the extra momentum to a field of negative energy.
Suppose that we have a static shell of mass, such that it is very close to its Schwarzschild radius. In that case, the negative energy of the newtonian field almost offsets the positive energy of the rest mass field.
General relativity does not have a concept of gravitational field energy. The sources of the stress energy tensor are rest mass, momentum, pressure, and shear stress. None of these is connected to the gravitational field.
The talk about momentum conservation and field energy may make sense in a newtonian approximation of general relativity.
When a binary neutron star loses its angular momentum and energy to gravitational waves, maybe we can interpret that the waves are in the combined field of rest mass and the negative energy newtonian field?
We can think that the whole mass-energy of an electron resides in its electric field. Similarly, we might think that the whole mass-energy of a 1 kg weight resides in its combined field.
Spacetime as a flexible plane
https://arxiv.org/abs/1603.07655
The Mechanics of Spacetime - a Solid Mechanics Perspective on the Theory of General Relativity (2018)
T. G. Tenev and M. F. Horstemeyer have an analogous model of general relativity, where spacetime is a flexible surface which is a flat plane if it is the Minkowski space. Let us check what their model says about the field energy of the gravitational field.
The flexible plane has an elastic modulus Y and its thickness is L.
Gravitational waves are transverse waves of the plane.
The Landau-Lifshitz pseudotensor
The Landau-Lifshitz pseudotensor seems to satisfy the properties which we would expect from the "energy" of a gravitational field.
For a static, spherically symmetric mass, the L-L pseudotensor must be negative, since the total energy of the system is the stress-energy tensor T plus the L-L pseudotensor. The total energy is less than T.
This does not help us. We have the problem where to put the extra momentum when we start pushing the matter, and we have not found a way to put it to a field of negative energy.
Pushing with a rod from far away
For an electric charge, its inertial mass feels less at the start of the push. How could we make the inertial mass of arbitrary matter feel less at the start of the push? Suppose that we are far away, and start pushing with a long rod. We may assume that the rod is attached to the mass. It is kind of a handle, by which a remote observer can move the mass.
The push create pressure in the rod, which in turn starts to attract the pushed mass. Could this attractive force absorb the surplus momentum from the pushed mass?
Suppose that after a while, the global observer is pushing with a force F. The mass feels a force F' from the rod. If the global observer pushes 1 meter, the observer on the surface of the mass observes the rod to move the same 1 meter. The force F' has to be greater than F, because if the surface observer would send the work done by F' as light to the global observer, there would be a redshift.
Suppose that the outside observer applies a 1 newton force to the rod for 1 second. Since gravitational waves can take a negligible amount of momentum away, the 1 newton second momentum must be absorbed by the system mass & rod & their gravitational field.
Exactly as in the case of an electric charge, the extra energy which goes to gravitational waves can be explained if the push takes a longer distance than it would with a totally rigid object.
Transforming momentum in a low gravitational potential to a high gravitational potential
How do we compare a momentum in a low gravitational potential to a momentum in the surrounding Minkowski space? Suppose that we have a weight inside a spherical mass shell and the weight is attached to a long rod which extends far away to the Minkowski space.
The far-away observer uses the rod to move (slowly) the weight.
Let the gravitational potential be V, where V = 1 corresponds to the Minkowski space, and V = 0 would correspond to a black hole horizon. The redshift + 1 is 1/V.
Suppose that the far-away observer applies a force of 1 newton to the rod for 1 second. The rod moves 1 meter, which corresponds to 1 joule of work. He may interpret that he was pushing a weight of 0.5 kg. The final speed of the weight is 2 m/s. Its momentum is 1 newton second and the kinetic energy is p^2/(2m) = 1 joule.
The local observer inside the shell measures that the weight moved 1 meter and it took V seconds. Its final speed is 1/V * 2 m/s. If he measures the weight as 0.5 kg, then its momentum is 1/V newton second and its kinetic energy is 1/V^2 joules. This does not match the fact that if the local observer sends the energy back to the far-away observer, he will receive 1/V joules. We have a perpetuum mobile.
The solution for the discrepancy is that the weight which the local observer measures is only V * 0.5 kg. Then
1/2 mv^2 = 1/2 * V * 0.5 * 2^2 * 1/V^2 J
= 1/V J.
The two observers agree on the length of 1 meter and they agree on the momentum of the weight in newton seconds.
The far-away observer thinks that the inertia of the weight is 1/V times of what the local observer measures. He also measures a time which is 1/V times the time measured by the local observer.
The far-away observer thinks that the energy and the force are V times the energy, and the force which the local observer measures.
Note that if the far-away observer pushes the whole gravitational system, then he will see the inertia of a weight less than the local observer.
Pressure inside a vessel makes the 3D space to stretch
Suppose that we have a spherical vessel with a very strong wall. We put there liquid and apply a very large pressure. Then there is positive pressure inside the vessel and a negative pressure in its wall. The pressure inside the vessel acts as a source in the stress-energy tensor. The geometry inside the vessel probably is not flat.
It would be logical if the space inside the vessel, that is, the 3 spatial dimensions, would stretch slightly to ease the pressure.
Birkhoff's theorem says that the geometry outside a closed spherically symmetric system is a fixed Schwarzschild geometry and cannot change through any process inside the closed system.
If the liquid is incompressible, then we can increase the pressure greatly with just a little bit of locally stored energy.
It has to be that the negative pressure in the vessel wall exactly cancels the geometry change inside the vessel, so that an outside observer sees no change in the geometry.
This analysis suggests that we can consider the 3-dimensional space, at least in slowly changing setups, as flexible, such that it stretches to reduce an increase in pressure.
Consider the following thought experiment. We build a spherical mass by lowering down thin spherical shells of matter. We collect the potential energy E which we gain in the process.
After completion, we enclose this mass with an infinitely rigid weightless spherical shell and fill the shell with incompressible weightless fluid. The metric allows us to put there more fluid than we could in a Minkowski geometry.
After completion, we enclose this mass with an infinitely rigid weightless spherical shell and fill the shell with incompressible weightless fluid. The metric allows us to put there more fluid than we could in a Minkowski geometry.
We then remove the mass gradually, by lifting spherical shells of mass up. We assume that we have some magic equipment which can move the mass through the infinitely rigid wall. We have to use an energy E'.
By Birkhoff's theorem, the metric outside the wall is a Schwarzschild metric. The circumference of the wall measured in global Schwarzschild coordinates is the same as in local coordinates. The circumference stays the same, as the material of the wall is infinitely rigid.
Once we have moved all mass away, we have a vessel which contains more incompressible liquid than it could in a Minkowski space. There has to be pressure inside which modifies the metric, so that the volume can fit in. The pressure can do work E'' if we let the fluid come out of the vessel.
Anyway, we proved that pressure can make more incompressible liquid to fit inside an infinitely rigid vessel. Pressure makes the 3D space to stretch. An incompressible liquid is not incompressible in the global view of an observer just looking at the walls of the vessel, even though it is incompressible for a local observer. This is because the 3D space itself can stretch.
When pushing with a rod, the energy for the gravitational wave comes from the stretching of the 3D space?
Suppose that we have an infinitely rigid rod. That is, the local observer will always measure the distance of atoms in the rod exactly the same, even if the rod is under heavy stress. When a far-away observer pushes a mass with the rod, there is pressure inside the rod. If the 3D space stretches to ease the pressure, the far-away pusher will interpret that the rod is elastic, even though local observers think it is perfectly rigid.
If the pusher pushes for 1 second with a force of 1 newton, he notices that the force will do work over a longer distance than if he would be pushing a totally rigid system. He has to do more work than he would need in the case of a totally rigid system. This extra work does not go to the kinetic energy of the mass but in the deformation of the rod, which itself is caused by the deformation of space. The pusher has pushed some energy into the deformation of space. This energy is probably the energy which is then radiated away as gravitational waves.
Since the rod is infinitely rigid, we cannot store elastic energy into its deformation. All the extra energy went into deforming space.
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