# Gravitational flux in GR

According to GR, how do changes in the mass of an energy density become reflected in its gravitational field?

Is the dissemination of the change in mass throughout the gravitational field c-limited?
If so, what geodesic does the dissemination of the change travel. Is the geodesic the same as that of a photon?
If the geodesic is different than that of a photon, then does gravitational flux travel outside the manifold of space-time as described by GR?
If the geodesic is the same, then may the gravitational flux travel faster than c?

Please consider this problem of gravitational flux within the context of a black hole:

In a black hole, photons cannot escape the event horizon due to the curvature of space-time (the geodesic). Can gravitational flux travel beyond the event horizon? If so, how? Does gravitational flux travel a different geodesic from photons, i.e., "outside" of the space-time manifold? Does gravitational flux travel faster than c? What gives?

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It is not very meaningful to talk about gravitational flux. I'll tell you why. Something like electric current satisfies $$\nabla_\mu J^\mu = 0$$, and so you can calculate the net flux of charge leaving a volume from a continuity equation
$$\tfrac{\partial Q}{\partial t}$$ using the divergence theorem, evaluating $$n_i J^i$$ on the boundary.

But the divergence theorem only holds for the divergence of a vector, not a tensor. So even though in GR we have $$\nabla_\mu T^{\mu\nu} = 0$$ to help us keep track of where mass and energy is going, you can't always neatly define the total amount of mass/energy in a system, because you can't meaningfully do that integral over the boundary. There are highly symmetric cases where you can, but in most cases you can't. It basically follows from the fact that the gravitational field itself carries energy.

I realize I may not have answered your question. Gravitational flux as you describe it isn't all that meaningful either, however. The speed of light is something intrinsic to the space, and the "dissemination" is merely a deformation of that space. You can talk about small perturbations of the metric, and those travel at speed c (gravity waves). But it isn't meaningful to ask how fast large perturbations are moving, because that's like asking how fast space is moving. Moving relative to what?

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What is the difference between a "small" and a "large" perturbation of the metric? I don't understand how that's a meaningful distinction.

The thought experiment is a system of two objects, on a black hole and the other a mass a distance from the black hole beyond the event horizon of the black hole. When I referred to speed, I meant relative to the black hole.

I'll rephrase the question: Assume a change in the mass of the black hole. This will cause a change in the gravitational field of the black hole. How does information relating to the mass change of the black hole reach the location in space-time of the second mass? Remember, the second mass sits beyond the event horizon of the black hole. To reach that location, does the "gravity wave" (or "gravitational flux" or "probability current" or whatever you want to call it) travel the same geodesic that a photon would travel to that location? If so, how can the gravitational wave reach the location and not the photon if both travel at c? And if the gravitational wave travels a different path than that which a photon would travel, then how is that possible?

According to GR, how do changes in the mass of an energy density become reflected in its gravitational field?
Mass-energy directly couples to the gravitational field. Changes are often called gravitational waves.

Is the dissemination of the change in mass throughout the gravitational field c-limited?
Yes.

If so, what geodesic does the dissemination of the change travel. Is the geodesic the same as that of a photon?
I think you have a misconception here. Geodesics are spacetime paths not paths in space that change in time due to changes in the field. So in other words the complete past and future of a geodesic is already fixed in the manifold. Of course particular observers "see" those spacetime geodesics in components of space and time and can observe changes.

In a black hole, photons cannot escape the event horizon due to the curvature of space-time (the geodesic). Can gravitational flux travel beyond the event horizon? If so, how? Does gravitational flux travel a different geodesic from photons, i.e., "outside" of the space-time manifold? Does gravitational flux travel faster than c? What gives?
Photons inside the event horizon are on a trapped surface that is contracting, so they cannot "escape". Also here, all the geodesics are all fixed in spacetime, which is the reason that a black hole can only exist in a closed universe.

Errata: Also here, all the geodesics are all fixed in spacetime, which is the reason that a black hole cannot exist in a closed universe.

Haelfix
"Errata: Also here, all the geodesics are all fixed in spacetime, which is the reason that a black hole cannot exist in a closed universe."

Hmm, can you write down your assumptions for this statement.

"Errata: Also here, all the geodesics are all fixed in spacetime, which is the reason that a black hole cannot exist in a closed universe."

Hmm, can you write down your assumptions for this statement.
There would simply be not enough time to form a complete black hole in a closed universe. For instance in the case of a "last drop in the bucket" light ray approaching an almost completed black hole at least one spacetime coordinate would approach infinity in all orthogonal coordinate charts when the ray is about to cross, let alone pass, the attempted formation of an event horizon. By the "time" it is "really close" the universe would collapse already.

Rephrasing the question

I'll rephrase the question again: Assume a change in the mass of the black hole. This will cause a change in the gravitational field of the black hole. How does information relating to the mass change of the black hole reach the location in space-time of the second mass? Remember, the second mass sits beyond the event horizon of the black hole. To reach that location, does the "gravity wave" (or "gravitational flux" or "probability current" or whatever you want to call it) travel the same path that a photon would travel to that location? If so, how can the gravitational wave reach the location and not the photon if both travel at c? And if the gravitational wave travels a different path than that which a photon would travel, then how is that possible?

I'll rephrase the question again: Assume a change in the mass of the black hole. This will cause a change in the gravitational field of the black hole. How does information relating to the mass change of the black hole reach the location in space-time of the second mass?
In steps. The additional mass is coming from somewhere else right? So, when it moves towards the black hole, sub lightspeed, the field is adjusted appropriately. Think you are in the middle of a large trampoline and someone, slowly, approaches you from the edge.

Things are more complicated if case we have the equivalent of a sonic boom for masses traveling faster than light, but these configurations are not considered physical.

Remember, the second mass sits beyond the event horizon of the black hole. To reach that location, does the "gravity wave" (or "gravitational flux" or "probability current" or whatever you want to call it) travel the same path that a photon would travel to that location?
Remember that in addition to coupling to itself, the gravitational field couples to mass-energy. To stay with the trampoline example, each local part of the trampoline knows how to bend, it does not need signals from the center or the edges.

I think your confusion comes from the idea of gravitational waves, don't take the term waves too literally. The spacetime manifolds of all possible universes in general relativity are not in motion, there is notion of a wave in spacetime.

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Haelfix