Exploring Electron-Positron Annihilation & Gravitation

[Eric Walker]

I have been trying to understand the implications of a thought experiment and am interested to know either where it's going off course or what those implications might be. Through some reading of earlier threads on this forum I have verified my starting hunch that photons bend spacetime despite having zero rest mass.

Assume you have an electron and a positron approaching the event horizon of a black hole. Assume that matter can pass over the event horizon of the black hole without "drama" (i.e., there is no firewall that obliterates everything). Place the electron and positron on a trajectory to annihilate almost immediately after crossing the event horizon. Hopefully this can be made to occur on a timescale overlapping our own timeline by moving the electron and positron arbitrarily close to one another before they slip over the event horizon.

Once the pair are past the event horizon, the 511 keV annihilation photons will not escape the gravitational well of the black hole. I assume that one of the following statements applies:

1. The net gravitational effect for the boundary surrounding the black hole before and after the electron-positron annihilation on the rest of the universe will be equivalent to the addition of the relativistic masses of the electron-positron pair prior to their slipping over the event horizon.
2. The net gravitational effect will differ, and a gravitational influence traveling at or near the speed of light will escape the boundary around the black hole, possibly imparting information for a sensitive enough detector to pick up. (Since this is a thought experiment, assume there is a detector sensitive enough to pick up the influence of gravity bent by 1.022 MeV/c^2 of matter in comparison to the mass of the black hole.)
3. The assumption about there being no firewall at the event horizon is bad.
   
4. There is something not even wrong about this thought experiment which prevents one from drawing any conclusions.

My guess is that (1) would be what would happen. What do people think?

 

[PeterDonis]

My guess is that (1) would be what would happen.

Rather than use your language to describe what would happen, since it seems a little fuzzy to me, here is how I would describe what would happen: the black hole would gain mass, and the mass it would gain is equal to the energy at infinity of the electron-positron pair. Since the pair does not annihilate until it is beneath the horizon, the annihilation has no observable effect outside the horizon.

This assumes that the no firewall assumption is correct; more generally, it assumes that classical GR is correct for this domain, which implies the no firewall assumption. Whether the no firewall assumption is actually correct is an open topic of research; I think it is, but we won't know for sure until we can do experiments close enough to the horizon of a black hole.

 

[Dale]

Through some reading of earlier threads on this forum I have verified my starting hunch that photons bend spacetime despite having zero rest mass.

A classical pulse of light bends spacetime, so it is not too big a leap to assume a photon would also. Of course, the details are not worked out yet.

 

[PeterDonis]

Of course, the details are not worked out yet.

Actually the stress-energy tensor of an EM field is one of the easier ones to work out. It's a common exercise in GR textbooks. For a brief summary of its properties, see here:

https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor

 

[Dale]

Actually the stress-energy tensor of an EM field is one of the easier ones to work out. It's a common exercise in GR textbooks.

Yes, for classical EM fields

 

[PeterDonis]

for classical EM fields

In the relativity context, "photon" just means a particular kind of classical EM field (basically one that is confined to a narrow "pulse" traveling along a null worldline). The SET for "photons" is just a special case of the SET for classical EM fields in general.

 

[Dale]

In the relativity context, "photon" just means a particular kind of classical EM field

Well, I try to avoid that language, but that is what I mentioned at the beginning of post 3. But here I think the OP was really talking about QM photons.

 

[Eric Walker]

But here I think the OP was really talking about QM photons.

In my blissful naivety about physics, I did not think to make a distinction. (Hopefully in this case not a distinction without difference.)