# Is there a connection between these two results on soft hair on black holes?

In 2016 Strominger, Hawking and Perry published the paper "Soft Hair on Black Holes" proposing new results that could have importance to the study of the black hole information problem.

One particular result they mention in section 5, that whenever charged matter crosses the horizon, soft photons on the horizon are excited. I believe the same is true for soft gravitons when any matter crosses the horizon.

On the other hand, a series of three papers tackled the problem in more generality from a distinct line of thought. These papers are:

The topic of the three papers is the IR sector and what it has to say about correlations. In the first paper they deal with the usual Bloch-Nordsieck approach of considering inclusive cross sections to avoid IR divergences. In the second paper they talk about the Faddev-Kulish method of dressing asymptotic states with clouds of soft photons or gravitons.

In the third (which is of interest here), they discuss why the dressing approach is needed.

They mention in a comment in the end of the paper the following:

Our results may have implications for the black hole information loss problem. Virtually all discussions of information loss in the black hole context rely on the possibility of localizing particles – from throwing a particle into a black hole to keeping information localized. We argued above that normalizable (and in particular localized) states are necessarily accompanied by soft radiation. It is well known that the absorption cross-section of radiation with frequency $$\omega$$ vanishes as $$\omega\to 0$$ and therefore it seems plausible that, whenever a localized particle is thrown into a black hole, the soft part of its state which is strongly correlated with the hard part remains outside the black hole. If this is true a black hole geometry is always in a mixed state which is purified by radiation outside the horizon.

If I understand where they are going they are basically saying the following:

1. Throwing a particle into a black hole requires one to consider a localized wavepacket as incoming state. By their arguments such a state is necessarily accompanied by a soft cloud of bosons. If the particle is not charged, it will be a soft cloud of gravitons. If it is charged it will be both a soft cloud of gravitons and photons.

2. By the soft factorization of the dynamics, the hard and soft parts of the state evolve independently. Hence the soft part cannot constrain the hard part. But anyway, it is established that the absorption cross section by the black hole vanishes as $$\omega\to 0$$.

Hence while the hard particle will cross the horizon, the soft cloud will be trapped outside.

Now, at least intuitively for me, these two results seem highly related.

In the first matter crossing the horizon excites soft bosons on the horizon. In this one, matter crosses the horizon and leaves upon it a soft cloud of soft bosons. The final effect is to end up with bosons added in the horizon whenever matter crosses the horizon.

So: are these two thing actually connected and if so how? Or are they just two distinct mechanisms, such that both occur when matter falls through the horizon?

I have the feeling (which may be totally wrong), that the second thing is actually the physical mechanism through which the first thing occurs.

Short answer: Neither Hawking, Perry & Strominger's (HPS) approach nor that of Carney et al. is more general. They are talking about different things. Soft gravitons of HPS “at the horizon” are not the same as soft gravitons or photons of “dressed” hard particles. They are representations of different symmetries. HPS soft gravitons arise from diffeomorphisms acting nontrivially on the horizon of the black hole. While the more usual soft gravitons (or photons) of the S-matrix are from the asymptotic symmetries at the null infinities $$\mathscr{I}^\pm$$. For a review, a book [1] is recommended.

Some context.

1. The usual device when introducing S-matrix is to assume that the interactions are “turned off” far away from the origin, so that we can consider in- and out- states as quanta of free fields. However, for long-range interactions such as gravity or electromagnetism we cannot do this consistently (IR divergencies). A sort of fix are dressed states that allow us to use free hard particles in calculations as long as they are accompanied by a cloud of soft, zero energy quanta of long-range fields.

2. Another point of view on the infrared structure of the theory is the asymptotic symmetry. We are writing the expansion of EOM at long range, and trying to find the symmetries of those equations. As a result we get the asymptotic symmetry group (ASG). For gravity this is a group of all diffeomorphisms allowed by a particular boundary conditions modulo trivial diffeomorphisms. For asymptotically flat spacetimes in D=4 this is the BMS$${}_4$$ group, see the book [2] for a pedagogical review.

3. Another ingredient for the IR structure is the memory effect when the passage of gravitational wave produces persistent displacement between different inertial observers. There are also various analogues of it for EM or other long range fields.

The three concepts above have been known for the decades but only recently they were linked into IR triangle (image from [1]):

As example for pure gravity, the Weinberg's soft graviton theorem is just the memory effect in the momentum space, while at the same time it is also Ward identity for the states connected by asymptotic symmetries.

But IR triangle is not unique. One could make many choices: what theory to consider, at what dimension, what types of soft quanta, at what order of asymptotic expansion we stop, classical/tree level vs. quantum etc. As a result we have a combinatorial explosion of different triangles, and calculations for each must be handled on a case by case basis (an image from [1] for illustration):

HPS story. A particle falling into a black hole has a lot in common with a particle flying away at infinity from the point of view of an observer outside of BH: as time goes, the observer's ability to distinguish “features” of the particle diminishes asymptotically. By analogy, one could then obtain asymptotic symmetry group acting on the horizon of the black hole. The action of nontrivial diffeomorphisms of these asymptotic symmetries is interpreted as creation of soft gravitons living at the horizon. Additionally, we expect a whole brand new black hole triangle with a new black hole memory effect. (See this paper by HPS + Haco for a symmetry group of Kerr BH, and this paper for the BH memory effect). Quantitative versions of the soft graviton theorems are still missing, however, for this triangle.

So, for the HPS paper (and various followups) we have the black hole triangle, while Carney et al. papers are dealing with a more traditional QED triangle with ASG of large gauge transformations at null infinity and soft photon theorems (while “memory” corner is not used explicitly). These are not quite unrelated sets of results but both are quite distinct, most obviously the “asymptotic symmetry” is quite different.

Of course, there is hope that in full quantum gravity theory there would be some sort of embedding or transformation between different viewpoints, producing a unified picture, but so far this is a pure conjecture.

1. Strominger, A. (2018). Lectures on the infrared structure of gravity and gauge theory. Princeton University Press, arXiv:1703.05448.

2. Compère, G. (2018). Advanced Lectures on General Relativity. Springer, arXiv:1801.07064.

• Thanks for the answer. So in the end these are indeed two distinct mechanism that both occur on their own when matter crosses the horizon? – user1620696 Mar 1 at 16:14
• There are no two mechanisms, the Carney et al. papers do not deal with black holes at all, except for the two paragraphs of introduction and the discussion. And there we only have “may have implications”. But I will try to edit the answer to include more context. – A.V.S. Mar 1 at 16:25
• I think I get your point. What you mean is that any relation between what Carney et al. are saying and the black hole case is at the moment just conjecture so that it may be the case or may not. Is that what you mean? – user1620696 Mar 1 at 16:36
• Yes. It is like saying that Taylor expansion at one zero of some function may have implications for the problem posed at another zero of that function. If the function is like $\sin x$ then the usefulness would be quite large. But for the generic analytic function by the time usefulness could be understood, chances are, the problem could be solved in other ways. – A.V.S. Mar 1 at 16:45