A change in the metric can propagate faster than a photon in general relativity

We have treated this question in earlier blog posts. On February 25, 2023 we suggested that no change in the metric can overtake a photon. It turns out that if the change in the metric is such that it cannot be used to carry a signal, then it may propagate even at an infinite speed.

A spherically symmetric shell of matter

                   _____

                 /           \

               |      o       |    clock

                 \______/

           shell of matter

Let us have a spherically symmetric shell of matter which is kept static with its tangential pressure. We can make the shell to expand or contract. There is a clock at the center. It will tick faster or slower, depending on the radius of the sphere.

In general relativity, the spatial metric inside a static shell is flat and the metric of time is a constant.

Does the speed of the clock change immediately when we expand the radius or is there a delay which would depend on the speed of light?

Suppose that the clock would still tick slower for a while. Then the local speed of light would be slower at the center (slower in global coordinates) than at the edges of the volume inside the shell. Such a metric can focus parallel rays of light. Focusing requires positive Ricci curvature, but the volume is empty and the Ricci curvature has to be zero.

We conclude that the speed of the clock changes immediately when we change the radius of the shell.

                 ____

               /         \  +

                \____/

This is analogous to the electric potential inside a charged shell. We believe that the electric potential changes immediately when we adjust the radius of the charged shell.

Hypothesis. A change in the metric can propagate even infinitely fast if the change does not allow one to send a signal faster than the local speed of light. This is often case in spherically symmetric configurations.

Question. Is it possible to send a signal faster than the local speed of light?

Taking apart a compact object "quickly"

Suppose that our matter shell is so massive that the local speed of light (measured in global coordinates) is very slow.

Let us expand the shell quickly. The local speed of light becomes much faster and we can send a signal to an observer at the center of the shell in a reasonable time.

We were able to "melt" a frozen object quickly and then we can take it apart in a short time (time measured by a distant observer).

However, if we have very many shells nested inside each other, and each is close to being a black hole, then the method we used above will not work quickly.

Disassembling a black hole through overspinning

What about disassembling a Schwarzschild black hole by overspinning it?

Suppose that we can speed up the local flow of time inside the black hole by rotating it fast. That does not necessarily involve any signal which would travel faster than the local speed of light. Once time flows fast inside the black hole, it may be possible to take it apart.

However, spinning is not a spherically symmetric operation. An observer inside the black hole might receive a signal faster than what is the local speed of light.

The centripetal force versus newtonian gravity

On July 13, 2023 we suggested that the "true" radius (in global coordinates) of an extremal spinning black hole is

        R = sqrt(2) G M / c²,

where M is the total energy of the black hole, including the kinetic energy in rotation.

Furthermore, we suggested that the horizon at the equator is moving at a speed

       v = c / sqrt(2).

Let us have a particle, for example, a photon rotating along the horizon. We calculate the centripetal force and compare that to the newtonian gravity force.

https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit

According to Wikipedia, the innermost stable orbit and the photon sphere are located at the horizon for an extremal spinnning black hole.

The centripetal acceleration in special relativity is the same as in newtonian mechanics:

       v² / R = 1 / (2 sqrt(2)) * c⁴ / (G M).

Let us assume that only the irreducible mass of the spinning black hole generates newtonian gravity. Then the acceleration of gravity is

      1 / sqrt(2) G M / R²

      = 1 / (2 sqrt(2)) * c⁴ / (G M).

The accelerations match. Our newtonian model explains why there is a stable orbit at the event horizon.

Conclusions

We showed that a change in the metric of time can propagate at an infinite speed. But that cannot carry any signal.

We are not sure if general relativity allows a signal to propagate faster than the local speed of light, if the speed of light is slowed down by a deep gravitational potential.

Next we will study orbits around an overspun black hole. Does the matter fly out?