There's been a lot of news lately about a possible solution to the black hole information paradox from a presentation given by Stephen Hawking to the KTH Royal Institute of Technology in Stockholm. Many of them mention the term super translation.

This article, for example, states (with emphasis added):

His flash of inspiration came when listening to a lecture in April about what are called super-translations, a bit of the heady branch of mathematics known as group theory. Dr Hawking thinks that incoming particles shed their information like a coat as they pass into a black hole, leaving it draped on the event horizon itself. Super-translations mathematically describe how that information influx can slightly jiggle the fabric of space at the horizon, in turn shifting around when and how the black hole radiates.

What exactly are super-translations, mathematically?

I have been trying to search for this term online, but the results are flooded with news articles. Where can I find some sources that describe super-translations in greater detail?

EDIT: Another description of super-translations is provided in a recent article:

The horizon of a black hole has the weird feature that it’s a sphere and it’s expanding outward at the speed of light. For every point on the sphere, there’s a light ray. So it’s composed of light rays. But it doesn’t get any bigger and that’s because of the force of gravity and the curvature of space. And, by the way, that’s why nothing that is inside a black hole can get out—because the boundary of the black hole itself is already moving at the speed of light.

There’s this symmetry of a black hole that we all knew about in which you move uniformly forward and backward in time along all of the light rays. But there’s another symmetry, which is the new thing in this paper (though various forms of it have been discussed elsewhere). It’s a symmetry in which the individual light rays are moved up and down. See, individual light rays can’t talk to each other—if you’re riding on a light ray, causality prevents you from talking to somebody riding on an adjacent light ray. So these light rays are not tethered together. You can slide them up and down relative to one another. That sliding is called a super-translation.

And in a way, it looks like you're not doing anything. Think of a bundle of infinitely long straws and you move one up and down relative to the other. Are you doing anything, or not? What we showed is that you are doing something. It turns out that adding a soft graviton has an alternate description as a super-translation in which you move some of these light rays back and forth relative to one another.

That’s super-translations on black holes. Super-translations were introduced in the 1960s, and they were talking not about the light rays that comprise the boundary of spacetime at the horizon of a black hole but the light rays that comprise the boundary of spacetime out at infinity. The story started by analyzing those supertranslations.

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    $\begingroup$ I want to say "What you get when a regular translation is bitten by a radioactive grad student," but I don't think that's right... $\endgroup$ – pjs36 Aug 29 '15 at 1:21
  • $\begingroup$ Sabine Hoffsteder has some references to the literature on her blogpost about Hawking's recent work, which probably will lead you to the right direction. (Try searching for "BMS Supertranslations"; also see chapter 11 of Wald's General Relativity.) $\endgroup$ – Alex Nelson Aug 29 '15 at 3:28
  • $\begingroup$ Once the radioactive bite takes affect, the super-translation is strong enough to jiggle space fabric. It uses this fabric to make a red pajama suit to wear as it swings between groups of buildings in an effort to defeat Black Hole. $\endgroup$ – Michael Aug 29 '15 at 4:46
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    $\begingroup$ Related Phys.SE post: physics.stackexchange.com/q/296189/2451 $\endgroup$ – Qmechanic Sep 27 '19 at 10:57

A suprertranslation is a diffeomorphism on an asymptotically-flat Riemannian or pseudo-Riemannian manifold, such that the result is also asymptotically-flat. (The diffeomorphism doesn't need to be an isometry).
(Of course, I didn't define asymptotic-flattness. I don't know the definition, but do know that in an asymptotically-flat manifold if you go far-enough in any direction, the metric becomes close to being flat as you wish).
The group of suprertranslations of a certain manifold is called the BMS group, after Bondi,Van der Burg and Sachs who discussed them in the context of gravitational waves in the following papers:
*Bondi, Hermann, M. G. J. Van der Burg, and A. W. K. Metzner. "Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 269. No. 1336. The Royal Society, 1962.
*Sachs, Rainer K. "Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 270. No. 1340. The Royal Society, 1962.

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  • $\begingroup$ This doesn't sound right. A super translation sounds like it should be a translation in a super direction, in the sense of supermanifolds (en.wikipedia.org/wiki/Supermanifold). $\endgroup$ – Qiaochu Yuan Sep 23 '15 at 20:19
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    $\begingroup$ @Qiaochu Yuan I don't know what Hawking was specifically refering to, but I do know that what I said is indeed a supertranslation. You can see also here arxiv.org/abs/1411.5745. I did not see any reference to your definition of supertranslation. The prefix "super" is overused by physicists... $\endgroup$ – Lior Sep 24 '15 at 5:01
  • $\begingroup$ This answer is my understanding as well, but if you head over to nLab (ncatlab.org/nlab/show/super-translation+group) you will see that they characterize the supertranslations as a super group translation in the sense of supermanifolds. In Hawking's paper he also mentions that "the usual time and space translations form a sub-group of the supertranslations". $\endgroup$ – levitopher Nov 25 '15 at 21:22
  • $\begingroup$ I guess the take away is that asymptotically flat manifolds have additional symmetry directions, characterized by the super-directions, and black hole horizons have this feature as well. $\endgroup$ – levitopher Nov 25 '15 at 21:23

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