● spherical mass
• --->
particle
Let us assume that the central mass initially is static.
Birkhoff's theorem states that the metric around the mass is the Schwarzschild static external metric for some fixed ADM mass M.
The mass does not stay static, though, when a particle flies past it. The disturbance causes the metric inside the central mass to change and the springs do work or gain more energy. It is like a gravitational wave passing through the central mass.
Let us make a photon or some other fast particle to fly close to the mass. The particle has its own Schwarzschild metric around it.
Or, in the case of a photon, what is the metric around the particle? It does not need to be a photon. It could be a laser pulse. The gravity field of the pulse probably gets length-contracted in the fast movement of the pulse. The length of the pulse in meters determines the length of the potential pit around the pulse.
The particle probably stretches the radial metric around it. When it flies past the mass, pressure does work when the spatial metric is stretched. The particle gains some new kinetic energy. The particle also causes energy to move around inside the mass, which increases the inertia of the particle.
These effects are tidal effects which are not visible in the static Schwarzschild solution around the mass. If we calculate the newtonian gravity field of a static central mass, in the 1687 version of the theory of gravity, tidal effects are not visible there, either.
The little particle changes the metric enough, so that it experiences tidal effects which were not in the Schwarzschild solution.
Tidal effects do not show that anything is wrong with the geometric interpretation of general relativity. We have to calculate the geometry for the complete system, not just for a static central mass.
Should we adjust the Schwarzschild exterior metric to reflect tidal effects on photons? If we define the "geometry" as the paths of rays of light, then tidal effects should be taken into account.
Question. Are there tidal effects around a black hole?
The Schwarzschild interior solution: does pressure stretch the spatial metric, after all?
In this blog we have been wondering why in the Schwarzschild interior solution pressure does not affect the spatial metric inside the mass - only the temporal metric.
The infinite rigidity of the spatial metric seems to open a door for a perpetuum mobile. Make an infinitely rigid vessel and fill it with incompressible fluid. Use a mass m to stretch the radial metric of the vessel and increase its volume. Put some more fluid inside. Then shoot the mass m away at light speed. Do we get infinite energy when the volume of the vessel contracts?
The Schwarzschild interior solution is for a static system, and probably does not include tidal effects on moving particles.
If we have a pressurized spherical vessel, is the inertial mass of an object inside the vessel the same in the radial direction as to the tangential direction?
If we move the object deeper into the vessel, it harvests more energy from a large volume of the vessel when it stretches the spatial metric. The energy has to move over a long distance. This suggests that the inertia is greater in a radial movement.
If the inertia is greater to the radial direction, that suggests that the spatial metric is stretched to that direction. This would imply that the volume of the vessel has grown relative to the surface area of the vessel.
This in turn would imply that the spatial metric is not infinitely rigid. It can be stretched with pressure.
That would mean that general relativity is a rubber membrane model, after all. Pressure can stretch a rubber membrane.
Karl Schwarzschild considered incompressible liquid. Such liquid does not exist. If we replace the liquid with a more realistic model where particles are moving tangentially or radially and bumping into each other, we realize that tidal effects take place. Movements in the radial direction are shorter because the inertia is larger to that direction. This effectively means that radial distances have grown. The spatial metric has effectively changed.
The interior Schwarzschild metric determines the stretching of radial distances based on the mass alone. If pressure does change radial distances, the solution should be modified in some way.
What would the Einstein-Hilbert action look like if we had another universal force which attracts mass-energy?
Assume that besides gravity, we have another force which universally attracts mass-energy. The other force would be written to the lagrangian of matter fields, L_M in the Einstein-Hilbert action.
The new force would essentially modify the "geometry" of spacetime, but that would not be visible in the Einstein-Hilbert metric. A naive physicist would calculate a wrong solution because he would think that the modified geometry is fully captured by the Einstein-Hilbert metric.
Question. If we have a complex interaction, like gravity and pressure interacting, is the modification of particle orbits visible in the metric in the Einstein-Hilbert action?
Birkhoff's theorem states that the metric outside a spherically symmetric system cannot change. But pressure probably affects tidal effects, and they, in turn, affect particle orbits. Pressure affects the "effective" metric.
The Schwarzschild interior solution seems to include the effect of mass-energy stretching the radial metric, and the pressure doing work because of that. But is the interaction included into the metric in all cases? In our October 10, 2021 blog post we remarked that in a complex lagrangian it is hard to split the effect between individual forces.
If pressure changes the orbits of all particles in a way which is not visible in the metric, then the Einstein field equations - naively solved - will produce wrong results. For example, if pressure and gravity combined make hydrogen atoms deformed in the radial direction, and this is not visible in the metric, then a naive physicist will solve the system assuming that they are not deformed. Karl Schwarzschild calculated his interior solution assuming that the only thing which deforms the atoms of the hypothetical incompressible fluid is the spatial Einstein-Hilbert metric.
Hypothesis. Pressure and gravity squeeze atoms in the radial direction, but this is not visible in the metric. Pressure does squeeze the effective metric. The spatial metric of spacetime is not infinitely rigid. General relativity is a "rubber membrane" model, after all, if all effects are included.
Hypothesis 2. The Einstein-Hilbert action does not need the metric components R and sqrt(-g) at all, if we solve the action over L_M including all the effects of the newtonian gravity force.
Hypothesis 2 would remove the privileged status of the newtonian gravity force. It would no longer be the force which determines the "geometry" of spacetime.
Making general relativity a rubber membrane model would make it more flexible, and would increase the probability that it has solutions for realistic matter field lagrangians L_M. We intuitively believe that a rubber membrane can adjust to all kinds of loads.
Suppose that the true theory of gravity is the newtonian gravity force & all its effects. Clearly, the newtonian force cannot "know" whether mass-energy is conserved in the asymptotic Minkowski space. Then there cannot be a Birkhoff's theorem in the true theory of gravity.
A thought experiment: convert a cloud of slow dust into radiation - Tolman's paradox
Suppose that we have a cloud of slow dust, and its gravity is weak. Then we can calculate its gravity from the newtonian force.
Suppose that we convert the cloud suddenly into radiation. The pressure of radiation is 1/3 of its energy density. The pressure is a substantial source of gravitational attraction.
We assume that the photons in the radiation go to random directions. There is no systematic momentum in them. The mass-energy of the cloud remains as it was?
The gravity of the cloud grows suddenly by the contribution of the pressure? This would break Birkhoff's theorem.
This is known as Tolman's paradox.
J. Ehlers et al. (2005) solve Tolman's paradox by putting a membrane around the radiation and showing that the negative pressure of the membrane cancels the effect of the positive pressure of the radiation. But this is a static solution, and we are interested into what happens dynamically. If we suddenly remove the membrane, then the positive pressure inside may remain for some time, but it is no longer canceled by negative pressure?
Weak gravity probably is additive. If we suddenly remove the tension from the membrane, pressure will start accelerating the membrane outward. The gravity field of the accelerating membrane is attractive. The gravity field of the pressure and mass-energy behind it is attracive. Their sum probably is larger than the gravity of the mass-energy alone?
Let us make the membrane so massive that its mass is the same as the mass of the radiation. How long does it take for the membrane to move appreciably?
Alternatively, we may have solid material whose gravity is weak. Suddenly, we release springs which create substantial pressure inside the material. The pressure adds to the gravitational attraction of the material? In principle we can add so much pressure that it overwhelms the gravity of the mass-energy in the material.
Let us analyze how a rubber membrane model would behave in this thought experiment. We may imagine that we have "tiled" a part of the membrane with coins. Suddenly we increase the pressure between the coins. The membrane stretches downward to accommodate the pressure. A wave is born which propagates away from the coins.
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