### BMiSsed

There’s a general mantra that we all repeat to ourselves: gauge transformations are *not symmetries*; they are *redundancies* of our description. There is an exception, of course: gauge transformations that don’t go to the identity at infinity *aren’t* redundancies; they are actual symmetries.

Strominger, rather beautifully showed that BMS supertranslations (or, more precisely, a certain diagonal subgroup of $\text{BMS}^+$ (which act as supertranslations on $\mathcal{I}^+$) and $\text{BMS}^-$ (which act as supertranslations on $\mathcal{I}^-$) are symmetries of the gravitational S-matrix. The corresponding conservation laws are equivalent to Weinberg’s Soft-Graviton Theorem. Similarly, in electromagnetism, the $U(1)$ gauge transformations which don’t go to the identity on $\mathcal{I}^\pm$ give rise to the Soft-Photon Theorem.

A while back, there was considerable brouhaha about Hawking’s claim that BMS symmetry had something to do with resolving the blackhole information paradox. Well, finally, a paper from Hawking, Perry and Strominger has arrived.

In a nutshell, it seems they want to propose that gauge transformations which don’t go to the identity at the blackhole *horizon* are also *not redundancies*, but rather *symmetries* of the theory. And the corresponding conservation laws (they mostly talk about the electromagnetic case — hence soft-photons) provide previously unforeseen hair to the blackhole.

Lots of details (or, at least, the promise of followup work containing said details) follow, but the crux of the matter is the following: two blackholes which differ by a gauge transformation which is not the identity on the horizon are *different* (degenerate) blackholes. Moreover, some diagonal subgroup of $\text{BMS}^H$ and $\text{BMS}^+$ is supposed to be a symmetry of the Hawking process (hence allowing the “hair” to escape, as the blackhole evaporates).

But I’m stuck at the starting point: why are we changing the rules and declaring that gauge transformations at the horizon are symmetries, rather than redundancies?

As I. I. Rabi said (on a different subject), “Who ordered that?”

The second, more subtle, point is how are we supposed to “find” the desired diagonal subgroup of $\text{BMS}^H\times \text{BMS}^+$? In the case that Strominger studied, a crucial fact was that $\mathcal{I}^-$ and $\mathcal{I}^+$ intersect on the $S^2$ at spatial infinity. Studying the action of $\text{BMS}^\pm$ near spatial infinity picked out the desired subgroup of $\text{BMS}^-\times \text{BMS}^+$. But, while the horizon of an *eternal* blackhole does intersect $\mathcal{I}^+$ (a fact HPS use in section 3), the horizon of an evaporating one does not. So I can’t imagine any natural way to relate supertranslations (or $U(1)$ gauge rotations) on the horizon of an evaporating blackhole to supertranslations (gauge rotations) on $\mathcal{I}^+$. Section 6 of their paper is supposed to address this question, but I honestly can’t make heads or tails of it.

In the end, my puzzlement comes down to this: the whole setup is that of local quantum field theory, the context in which the blackhole information paradox originally arose. The “solution” seems to be to change the rules of local quantum field theory (in a seemingly ad-hoc way, at the horizon). But, if we’ve learned anything — from String Theory, AdS/CFT, … —- it’s not that local quantum field theory needs to be modified in some way; it’s that local quantum field theory, in the presence of blackholes, breaks down at something of order the Page Time and this breakdown is *not* some local effect.

## Re: BMiSsed

Welcome back. Very helpful. If you are in the mood for additional public service announcements, do you have any thoughts on arXiv:1601.01800 ?