Our claim was too simple. Gravity has nonlinear effects. Even if a single particle moving at a constant velocity would satisfy Gauss's law, that does not prove that the law would hold for a collapsing dust ball.
The pressure term in the stress-energy tensor of the dust ball grows as it contracts. The stress-energy tensor T looks schematically like this for a single dust particle.
m m v ...
m v m v² / c²
...
If the momentum components m v were zero, then we would have a "pure" pressure besides the mass density. The interior Schwarzschild solution (1916),
as well the result by Charles W. Misner and Peter Putnam (1955),
and the paper by Jürgen Ehlers et al. (2005) show that pressure "gravitates": a positive pressure tends to pull a test mass to the center of a pressurized spherical vessel. The force comes from the fact that the metric of time is slower at the center. Pressure "focuses" a cube of test masses.
But these three results are for a "pure pressure" with no momentum components in the stress-energy tensor.
The bounce-back mechanism of how pressure gravitates?
• | • --> <-- • |
m wall dm dm wall
Consider the above configuration where particles dm bounce between two walls back and forth. We believe that dm "holds" some inertia dI of m, and when dm bounces back from the wall close to the test mass m, then m receives an impulse
2 v dI
to the right. When dm bounces from the wall far from m, then m receives a smaller impulse
2 v dI'
to the left.
Conjecture. If the pressure term in the stress-energy tensor T comes from a locally uniform mass flow, where all the matter flows at a certain velocity v and does not "bounce back", then the pressure term does not "gravitate". That is, we cannot increase the gravity of a system by making its parts to move in a uniform fashion.
The conjecture prohibits Tolman's paradox in such cases.
How "uniform" the flow has to be? Is it enough that each particle is moving uniformly with itself? In the early stages of the Oppenheimer-Snyder collapse, the movement is extremely uniform, such that the dust cloud contracts through uniform contraction of all distances. If we have a single particle moving at a constant velocity, that is very uniform, too.
Corollary. The Oppenheimer-Snyder collapse does not clash with Birkhoff's theorem. The gravity, measured from outside the collapsing dust ball, stays constant.
However, just before the collapse, a positive "pure" pressure is removed from the dust ball, allowing the collapse to start. We believe that this pressure change reduces the gravity outside the dust ball, and does clash with Birkhoff's theorem.
Could we remove the concept of pressure altogether from a model of gravity? It is not possible to model negative pressures with an ideal gas of particles bouncing around. We need a concept of pressure in a theory of gravity.
Does the bounce-back always involve an impulse which COMPENSATES the gravity of pressure? No, this model does not work
In the diagram above, when the particle dm bounces from a wall, the wall bounces from the particle. The wall "holds" some inertia of the test mass m. Thus, the wall gives an opposite impulse
-2 v dI
to m.
Is it so that the impulse always compensates the "gravity" of pressure? Then pressure would not gravitate at all if all pressure really "consists" of particles bouncing back and forth!
That would solve Tolman's paradox and preserve Birkhoff's theorem.
This may hold for a test mass m which is static relative to the pressurized system. But what if m moves closer? We believe that m stretches the spatial metric of the pressurized system and "frees" pressure energy. We would expect m to be attracted toward the pressurized system.
Also, the results of Schwarzschild, Misner, Putnam, Ehlers, et al. show that "pure pressure" gravitates in general relativity. The bounce-back mechanism cannot describe the pressure of general relativity.
The bounce-back is a different thing from pressure. Note that if we have an electric test charge q which shares some inertia with another charge Q, and Q bounces from an electrically neutral wall, then the bounce does give an impulse to q.
The role of "privacy"
If we have a single particle moving at a constant velocity v, we believe that Gauss's law holds for its gravity. If we would be allowed to linearly sum the metric perturbations caused by such particles, then there would be no gravity effect of pressure at all. Thus, pressure must produce gravity through a nonlinear, or "public" effect.
Our derivation of the Biot-Savart law for electromagnetism on November 14, 2023 required "privacy" with respect to a charge which has the opposite sign. Could it be that privacy does not apply when the charges have the same sign?
Gravity by a pressure is a "reaction" in the pressurized matter?
If the test mass m approaches pressurized gas, m stretches the spatial metric and the pressure is lowered. This frees energy which is used to accelerate m toward the gas. This is a "reaction" which m causes on the gas.
But if we calculate the interaction of m and a gas particle privately, we assume that m causes no reaction on the gas particle whatsoever. This is an incorrect assumption? If the reaction is
~ m,
then we cannot ignore it even if we set m very small.
We still face the problem why the pressure term in the Oppenheimer-Snyder collapse does not seem to cause gravity.
Varying the metric in the presence of a mass-flow element in the stress-energy tensor
If Gauss's law holds for a moving particle in gravity, then there must be some reason why the Einstein-Hilbert action "resists" the attraction that the pressure element in the stress-energy tensor causes on a test mass m.
• <-- •
m v dm
-----> x
If we move m closer to the moving particle dm, then the stretching of the radial metric increases the kinetic energy of dm. If the matter lagrangian is of the form
kinetic energy - potential energy,
the effect of increasing kinetic energy is like the effect of decreasing potential energy: the system tends to move to that direction. If the system would contain ordinary pressure, then moving m closer would reduce the potential energy in the pressure, because of the stretching. An equivalent effect comes from a kinetic energy term: the stretching increases the kinetic energy.
But is there an effect from moving m closer, such that it somehow reduces the kinetic energy of a moving particle or particles?
In the diagram, the stress-energy tensor of the particle looks something like this:
dm -dm v ...
-dm v dm v²
...
The metric around m does not contain cross terms between dt and dx if m is static, but it does contain them if m moves. If we move m closer to dm, does that counteract the attraction that the pressure element dm v² causes on m?
If yes, that might explain why in the Oppenheimer-Snyder collapse, the pressure does not increase the gravity.
The metric around a moving particle is complicated
• m test mass 2
• ● --> v
m test mass 1 M particle
Consider the following configuration. Initially the particle M is static. The test masses m are accelerated uniformly toward M, and from their acceleration we can deduce that Gauss's law for gravity holds in this case.
Let us then use some of the mass-energy of M to give it kinetic energy and M starts to move right in the diagram. We believe that Gauss's law still holds: the "flux" of the lines of force of gravity through a surface is still equivalent to the mass-energy enclosed inside the surface. We ignore the steepening of strong gravity fields here.
The acceleration of the test mass 1 becomes smaller and the acceleration of the test mass 2 becomes larger. We can easily calculate this behavior in the comoving frame of M, but we have hard time finding an intuitive explanation for the behavior in the laboratory frame. How does the momentum flow component in the stress-energy tensor cause the gravity field to become flattened like this?
Maybe we simply have to accept this as a fact without a further intuitive explanation: the momentum components in the stress-energy tensor "cancel" the gravity of the pressure component.
Momentum means that the gravitating system is moving, and that we can in many cases cancel the pressure component by switching to a comoving frame of the gravitating system. Then the pressure does not seem to cause extra gravity in any frame.
If there is a "pure" pressure, then we cannot cancel the pressure component in any frame. In this case, the pressure seems to cause extra gravity.
The metric around "pure" pressure
Let us have a volume of "pure" pressure p where the momentum flow is zero. The stress-energy tensor looks schematically like this:
M 0 ...
0 p
...
The metric is such that both the mass-energy M and the positive pressure p tend to slow down the metric of time at the center of the system. That is, the pressure p "gravitates". The pressure "focuses" a cube of test masses so that the volume of the cube starts to shrink.
On October 28, 2023 we argued that the focusing effect inside a pressurized volume must necessarily cause gravity attraction also outside that volume. To cancel the attraction, we would need an equivalent negative pressure wrapped around the volume. But in the preceding section we noted that also momentum components can cancel the extra gravity of pressure.
The problem of particle granularity. Suppose that a pure pressure is caused by microscopic particles, like in an ideal gas. Can we do the following: switch to the comoving frame for each particle, calculate the gravity, and then linearly sum the gravity effects? Then the pure pressure would not cause extra gravity, since for each individual particle, it is not pure pressure. It is just a moving particle.
If the granularity would significantly affect the gravity of matter, then general relativity would be a very badly behaving theory. We expect granularity at a fine level not to have much effect on the gravity field. Proving this property for general relativity is very hard, though, since it is a nonlinear theory.
Conjecture. Fine granularity does not affect the gravity field of a system much in general relativity. One cannot reduce pure pressure into a sum of moving particles.
Why the Oppenheimer-Snyder collapse maybe can be reduced to moving particles?
The pressure component in the Oppenheimer-Snyder collapse comes from a very uniform movement of mass. It is not a fine granularity thing. We can cancel the pressure component by switching to a comoving frame of the dust particles. This may be the reason why the pressure in this case does not seem to cause extra gravity.
Conclusions
Pure pressure certainly produces extra gravity attraction in general relativity. This is proved by the results Schwarzschild and others. But why the pressure component in the Oppenheimer-Snyder collapse does not cause extra gravity?
The reason may be that we can cancel the pressure term by switching to the comoving frame of the dust particles. However, we were not able to prove that this is the reason. It is just a guess.
No comments:
Post a Comment