# Would an interior inflation affect the size of the event horizon of a black hole?

Lets imagine that the interior of black holes are not singular, and they undergo an inflation. I know no information can escape a black hole, but the event horizon can grow in size, for instance, by accretion. Could this inflation inside make the event horizon increase in size?

• When you say "interior of black holes" are you referring to the inside of the event horizon? Because that region is not singular. Aug 23, 2020 at 21:17
• I was talking about just GR, in which they are singular. Aug 25, 2020 at 9:47

The case you are referring to in a way is similar to the formulation of Wheeler's bags of gold paradox. I will discuss two issues here and clarify why such situations don't influence the horizon.

Case 1: Consider a spacelike slicing of an eternal black hole (you can consider single sided as well, eternal in AdS are easy to visualize and have nice properties) which end at Kruskal time $$(u_L, 0)$$ and on $$(0, v_R)$$ on the horizons. This slice stays away from the singularity at all points, and does not have large curvature invariants. Let us consider "maximal volume slices" in the interior of the black hole, i.e. you impose that the volume of such slices is maximized. This calculation is performed in Appendix A of this paper and the expression for the volume grows as:

$$V \propto \frac{\beta}{2\pi} \log{(u_Lv_R)}.$$

This is a slicing whose Kruskal time evolved volume keeps increasing in the interior, and can be thus thought of as an inflating spacetime. Note that there is absolutely no reason here why the horizon should change, because you are considering different spacelike slicings of the eternal spacetime.

Case 2: Consider an inflating spacetime which is directly glued to the black hole interior using Israel junction conditions. This is nicely discussed in this paper and leads to the bags of gold problem. Here since the interior region by construction itself undergoes inflation, the volume of the spacelike slicings going inside the interior increase (analogous to but not the same as in Case 1). Again since you have imposed junction conditions properly in the interior, there is no reason for the horizon to increase.

The basic reason for why the interior can undergo inflationary behaviour while the horizon remains unchanged is because any changes in the interior won't be able to influence the horizon via causality. This is obeyed in both Case 1 and 2, where any sources or excitations in the interior on these slices lead to bags of gold like scenario, but their effect is restricted to the interior itself as they cannot causally propagate to influence the horizon.

Aside: Note that even in the formulation of the bags of gold paradox, the coarse grained Bekenstein Hawking entropy remains the same even if slices in the interior on which the excitations live grow with time. This is again due to the fact that excitations in the interior cannot causally influence the future horizon(s).

No, it could not.

The horizon, is not in the casual future of any point in the black hole interior (by definition). Consequently, nothing that happens in the interior can affect the horizon.

• What if the interior inflates? Aug 31, 2020 at 15:08
• @Manuel Doesn't matter, it still would not be able to affect the horizon. Aug 31, 2020 at 16:03