## January 10, 2016

### 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.

Cue further brouhaha

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.

Posted by distler at January 10, 2016 11:39 AM

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### 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 ?

Posted by: Thomas on January 10, 2016 8:47 PM | Permalink | Reply to this

### Re: 1601.01800

Since my previous post was about asymptotic safety, I don’t particularly feel drawn to write another one anytime soon.

But I will leave you to puzzle over the logic in the key sentence of the paper:

The second functional derivative of (5) results in a sum of terms containing at least one power of the Weyl tensor. Since $C$ is trace-free by construction, all its contractions with the metric vanish. This entails that there is no feedback of the Goroff-Sagnotti term on the renormalization flow of Newton’s constant and the cosmological constant. [emphasis mine]

If you replace “at least” with “exactly”, then the conclusion would follow. But you can’t, so it doesn’t.

Posted by: Jacques Distler on January 10, 2016 11:41 PM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

“But, if we’ve learned anything - from String Theory, AdS/CFT…”

But what if they’re wrong?

Posted by: hellbent on January 10, 2016 9:24 PM | Permalink | Reply to this

### Re: BMiSsed

Then we lack the intellectual tools to successfully address any of these questions and the whole discussion is a complete waste of time.

Posted by: Jacques Distler on January 10, 2016 11:58 PM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

Ah, one of my favorite posts of yours.

Posted by: hellbent on January 11, 2016 12:58 AM | Permalink | Reply to this

### Re: 1601.01800

Let me resolve the puzzle around the “key sentence” of the paper:

The whole construction uses the background field formalism. Computing the second variation of the C³-term around a background, one finds that the Hessian contains terms proportional to C, C^2 and C³ (omitting tensor structures for simplicity). This can be verified quickly from the chain rule.

The beta-function of Newtons constant is encoded in the coefficient proportional to the Ricci-scalar of the background metric. None of the terms appearing above has the structure to give such a contribution. Formally this can be proven using off-diagonal heat-kernel techniques. Thus the conclusion made in the key sentence is correct.

Posted by: Frank Saueressig on January 12, 2016 2:43 AM | Permalink | Reply to this

### Re: 1601.01800

The whole construction uses the background field formalism. Computing the second variation of the $C^3$-term around a background, one finds that the Hessian contains terms proportional to $C$, $C^2$ and $C^3$ (omitting tensor structures for simplicity).

And is obviously nonzero.

The beta-function of Newtons constant is encoded in the coefficient proportional to the Ricci-scalar of the background metric. None of the terms appearing above has the structure to give such a contribution.

Umh. No. You need to stick that Hessian into (2), where it surely contributes to the $\beta$-function for Newton’s constant.

Formally this can be proven using off-diagonal heat-kernel techniques.

I would like to see that proof, as the result (that sticking the aforementioned Hessian into equation (2), of your paper, yields no contribution to the running of the Ricci scalar term in the action) surprises me greatly.

Thus the conclusion made in the key sentence is correct.

That key sentence is the crux of the whole paper, so actually presenting the argument (which you have alluded to in your comment, but not given) would be an important thing to include in the paper.

Posted by: Jacques Distler on January 12, 2016 8:46 AM | Permalink | PGP Sig | Reply to this

### Re: 1601.01800

Following up on the discussion, let me sketch the argument. We agree that the GS term contributes to the Hessian $\Gamma^{(2)}$ with structures containing $C$, $C^2$ and $C^3$.

Eq. (2) contains the inverse of this Hessian. This inverse can be constructed as an expansion in the background curvature tensors. In order to isolate the contributions to the running Newtons constant, we collect all terms of first order in the background curvature. At this order all terms originating from the GS-Term are of the form $Tr C$ where Tr means that all indices of $C$ are contracted, e.g., $C_{\mu\nu}^{\mu\nu}$, etc. Since C is trace-free by construction, none of the terms gives a contribution. Based on this, we concluded that the beta-function for Newtons constant is not altered by adding the GS-term. I hope this clarifies the matter.

Posted by: Frank Saueressig on January 13, 2016 12:42 PM | Permalink | Reply to this

### Re: 1601.01800

Hmmm. Thank you, Frank. I will think about that.

In the standard expansion in powers of derivatives and background curvature, it is, of course, true that the coefficient of the GS term does not feed back into the $\beta$-function for the Einstein-Hilbert term. Instead, it feeds into its own $\beta$-function and into those of yet-higher order terms in the expansion.

Naïvely, I would not have expected that to remain true in the ERGE (and, indeed, it wouldn’t have been true if I were including a $C^2$ term in the action).

Posted by: Jacques Distler on January 13, 2016 4:12 PM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

Hi Jacques,

Might I suggest that http://arxiv.org/abs/1510.07038 might shed some light on your questions? In that paper, we show how to write these kinds of Ward identities in a path integral language, and argue that one can understand the antipodal map as coming from one’s residual gauge equation (in whatever gauge one decides to use). In the black hole background, one can choose the region R to be outside the (apparent) horizon or the entire spacetime after the black hole unitarily evaporates (assuming that happens) and find the same Ward identity Hawking-Perry-Strominger find.

Of course, HPS may not agree with everything I am saying, but it might at least be an useful alternative point of view. I should also say, I am still digesting their comments on how this identity relates to the information paradox, since that is a bit less clear to me. In particular, unless the hope is that once one includes superrotations in the story that quantum gravity will be integrable, then I would still want a more concrete mechanism for information transmission, etc.

Posted by: Steve Avery on January 12, 2016 12:27 PM | Permalink | Reply to this

### Re: BMiSsed

Thanks for the reference. I’ll take a look.

Let me, at least try to sketch a bit more of their argument, so that we can see whether your considerations shed some light. Following them, let’s use the electromagnetic case as a foil.

First, let’s review the Minkowski case. Let $\varepsilon(z,\overline{z})$ be an arbitrary function on $S^2$. We can extend it to a function $\varepsilon_+$ on $\mathcal{I}^+$, in a canonical way, by taking $\varepsilon_+$ to be independent of $u$ (the retarded time coordinate on $\mathcal{I}^+$). The associated charge, $Q_+ = \int_{\mathcal{I}^+} \mathcal{J}_+$ where $\mathcal{J}_+ = d\varepsilon_+\wedge *F + \varepsilon_+ j$ (and Maxwell’s equation is $d*F=j$). We can do a similar extension of $\varepsilon$ to a function on $\mathcal{I}^-$, and the soft photon theorem is the statement that the matrix elements of $Q_+-Q_-$ vanish, for any choice of $\varepsilon(z,\overline{z})$.

Now let’s turn to the case of interest, a blackhole formed by a collapsing spherical shell (denoted in the Penrose diagram, below, in red), which subsequently evaporates.

$H_\lt$

They want to say the following

1. There is a canonical way to extend $\varepsilon$ to a function $\hat{\varepsilon}$, defined on the Cauchy surface $R= H_\lt \cup S \cup \mathcal{I}^+_\lt$, denoted in blue in the Penrose diagram above.
2. The associated charge, $Q_R = \int_R \hat{\mathcal{J}}$, satisfies $Q_+-Q_R=0$.
3. The contribution to $Q_R$ from the integral over the spacelike surface, $S$, is somehow negligible.

One then concludes that $\int_{H_\lt} \hat{\mathcal{J}} = \int_{\mathcal{I}^+_\gt} \mathcal{J}_+$

I already said that I don’t see (1).

I also don’t see (3). Recall that, in Feynman diagrams, the soft theorem is obtained by attaching soft-photon (or soft-graviton) lines (both incoming and outgoing) to the external lines (in this case, the infalling matter denoted in red and the outgoing Hawking quanta) in all possible ways. “Most” of those soft lines cross the surface $S$, so I don’t see how you can ever get the correct Ward identity by neglecting the contribution from $S$.

Posted by: Jacques Distler on January 12, 2016 2:26 PM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

I’ll focus on your point (1). (I also don’t understand the approximation in point 3.)

I don’t think there is such a canonical extension, however, I think they address this in Section 6, in which they argue that if one puts in \alpha it cancels out of the Ward identity.

In the language that I prefer to use, I would say that in the path integral one fixes a gauge to get a well-defined measure. The gauge-fixing condition (along with boundary conditions on \scri) gives the extension you are looking for. For the Minkowski case, the antipodal identification is independent of the gauge-fixing condition, but this will no longer be true on the horizon. However, the physical content of the Ward identity (just as they argue) should be independent of the gauge condition (or the extension). Maybe that clarifies things??

Posted by: Steve Avery on January 12, 2016 6:25 PM | Permalink | Reply to this

### Re: BMiSsed

I definitely don’t see them saying that there is a canonical way to extend $\varepsilon$ away from $\mathcal{I}^+_-$ (their equations (2.7) and (2.9) are equal for any choice of extension). It seems sufficient for their purpose to note that for any extension you choose, the integral over the yellow Cauchy surface in their Figure (2) gives the same result as the integral over the green Cauchy surface; the result of that integral is what they call $Q^+_\varepsilon$. But then that makes me worry whether you couldn’t choose an extension that is smooth with compact support outside $H_\lt$; then the second term in their (4.4) would be zero but the second term in their (4.2) could be nonzero. This looks like a gauge choice where everything sneaks out through the spacelike surface $S$ Jaques is worried about neglecting.

### Re: BMiSsed

But then that makes me worry whether you couldn’t choose an extension that is smooth with compact support outside $H_\lt$; then the second term in their (4.4) would be zero but the second term in their (4.2) could be nonzero. This looks like a gauge choice where everything sneaks out through the spacelike surface $S$ Jaques is worried about neglecting.

Thank you, Marcus; that’s exactly what I was going to write. The only thing I would add is that the lack of dependence on the choice of extension, $\hat{\varepsilon}$ (or, indeed, on the choice of spacelike surface, $S$) is a manifestation of bulk gauge invariance.

Which is why it’s hard to understand why one should attach physical significance to some particular choice of extension.

Posted by: Jacques Distler on January 13, 2016 12:45 AM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

Hi Mark!

I won’t try to address this question within the framework of Strominger et al., but let me point out that this issue is something Burkhard and I thought about a lot even for Minkowski space. The key point, is that one must fix a gauge in order for the path integral to be well-defined, and in Sec. 3 of our paper we describe what a “good” gauge condition is. In particular, there must _not_ be residual gauge transformations with bounded support, of the kind (I think) you are describing. If there are then one can derive nonsensical Ward identities that imply, for instance, that the gauge propagator vanishes. I’m not sure exactly how Strominger et al. think about this kind of issue.

Posted by: Steve Avery on January 13, 2016 9:44 AM | Permalink | Reply to this

### Re: BMiSsed

So, if I understand your argument, $\hat{\varepsilon}$ is entirely determined by its value, $\varepsilon(z,\overline z)$, at spatial infinity, once a “good” gauge is chosen.

In particular, its restriction to $H_\lt$ is determined (as is its restriction to the surface $S$, which we have been discussing).

So, in principle we could:

• Choose, say, Lorentz gauge and compute the corresponding $\hat{\varepsilon}$ and ask whether it has the properties demanded by HPS.
• Conversely, we could demand that $\hat{\varepsilon}$ behave as HPS wish it to, and ask whether there is a “good” gauge in which that is true.
Posted by: Jacques Distler on January 13, 2016 11:22 AM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

Yes, I think I agree with that statement.

I would also point out, however, that I don’t think HPS put too much weight on the approximation that the surface S contribution is negligible. I think they mostly wanted to demonstrate that there is some physical statement (which in principle includes something happening on S). At least that’s my reading of the relevant section.

Posted by: Steve Avery on January 13, 2016 11:28 AM | Permalink | Reply to this

### Re: BMiSsed

I would also point out, however, that I don’t think HPS put too much weight on the approximation that the surface $S$ contribution is negligible.

Well, clearly, in your prescription, $\hat{\varepsilon}$ ought to be nonvanishing along $S$, so whether the contribution to the charge is negligible depends on the process under consideration.

For instance, there ought to be scattering processes, taking place far away from the blackhole, where the dominant contribution should come from $S$ (and negligible contribution from $H_\lt$).

Presumably, HPS are not interested in processes like that, and want to focus on the (“connected”) scattering amplitude for the formation/evaporation process.

Even there, I would expect significant (maybe even the dominant) contributions to come from $S$ (for the reasons I stated previously).

And, as I’ll explain, I think that’s a problem for them.

I think we can agree that (under the stated conditions on the extension $\hat{\varepsilon}$), there should be operator identity of the form

(1)$\int_{H_\lt} \hat{\mathcal{J}} + \int_{S} \hat{\mathcal{J}}= \int_{\mathcal{I}^+_\gt} \mathcal{J}_+$

assuming (of course) that the formation/evaporation process is unitary.

So far, I think this is more-or-less tautological, as we could have drawn any Cauchy surface which contains $\mathcal{I}^+_\lt$ and made the same statement.

My understanding of HPS is that

1. They would like to interpret either

• the first term on the LHS, $\int_{H_\lt} \hat{\mathcal{J}}$ or
• the full LHS, $\int_{H_\lt} \hat{\mathcal{J}}+\int_{S} \hat{\mathcal{J}}$

as “soft hair”, encoding information about the infalling matter that formed the blackhole.

2. They would like to “recover” the information, encoded in that soft hair, by measurements made on the outgoing Hawking radiation, i.e., through the matrix elements of $\int_{\mathcal{I}^+_\gt} \mathcal{J}_+$.

The reason why they drew the Cauchy surface the way they did was to avoid the contribution (which would have been detected by an integral over $H_\gt$) from attaching inward-directed soft-photon/soft-graviton lines to the outgoing Hawking quanta. As drawn, the only inward-directed soft lines that cross $H_\lt$ are the ones attached to the incoming particles that formed the blackhole.

But the contribution to (1) from the integral over $S$ includes both the effects from attaching (outgoing, future-directed) soft lines to the incoming matter (which we want) and from attaching (outgoing, past-directed) soft lines to the outgoing Hawking quanta.

That means that the LHS of (1) isn’t actually telling us about the infalling matter (as some new “hair” would do), but about some inextricable scrambled combination of the infalling matter and the outgoing Hawking radiation.

That’s why (I think) they want the contribution from $S$ to be negligible (so that the LHS of (1) contains only contributions related to the infalling matter). But I don’t see any reason to think it should be negligible.

Posted by: Jacques Distler on January 13, 2016 3:55 PM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

I think I agree with most of what you are saying. Just to be clear, the reason why I said that I think they are just focussed on the existence of some identity, are the following quotes from Sec. 4 of HPS:
“The preceding section gave a mathematical description of soft electric hair. In this section show that it is physically measurable and therefore not a gauge artefact.”
and
“In this section we have made several simplifying assumptions and approximations…The purpose of the approximations and assumptions was simply to find a context in which a concise and simple statement of the consequences of charge conservation could be made.”

I agree with your summary up to the end of your point 2. After that I am a little confused. They chose the surface S at some early time, say a scrambling time after the black hole formed from the null matter. There should then be little to no Hawking radiation passing through S. If I am reading your argument correctly, you are worrying about the soft modes passing through S?

I think they want to say that H + S + I is a Cauchy slice and thus has a Hilbert space associated with it, and the state should be roughly M\otimes 0 \otimes 0. Then a key point in Strominger et al’s story is that we may break these charges into “hard” and “soft” pieces.

The hard piece is responsible for transforming the various hard modes on the slice and the soft part inserts a soft photon. Assuming this still holds for the slicing they have, and the decomposition into three subfactors, then I think they can argue that the contribution from S is negligible. But this would all be something I would want to work out more carefully, before sticking my neck out for their calculation. And at this point you might be better off getting it from the horse’s mouth, if the horse is available.

(Sorry about the poorly typeset comments. I’m assuming you value a faster response over me spending time learning the system.)

Posted by: Steve Avery on January 13, 2016 4:45 PM | Permalink | Reply to this

### Re: BMiSsed

Well, I’ve learned a huge amount from this discussion.

The main point which I certainly didn’t get from their paper is that the extension, $\hat{\varepsilon}$ is canonical, in the sense that, given any (“good”) gauge choice for the system (e.g. Lorentz gauge), then there is a unique $\hat{\varepsilon}$, for any choice of $\varepsilon$ at spatial infinity.

There should then be little to no Hawking radiation passing through $S$. If I am reading your argument correctly, you are worrying about the soft modes passing through $S$?

Correct.

Even more specifically, I am worried that the soft modes passing through $S$ involve both the soft modes attached to the infalling matter (which would be OK – they’re part of what HPS would call the “soft hair”) and the soft modes attached to the outgoing Hawking radiation (which musses up the soft hair we are trying to recover).

Then a key point in Strominger et al’s story is that we may break these charges into “hard” and “soft” pieces.

But it’s certainly not true that the equality $\int_{H_\lt} \hat\mathcal{J}+ \int_{S} \hat\mathcal{J} = \int_{\mathcal{I}^+_\gt}$ holds separately for the hard and soft pieces. If it did, we wouldn’t be talking about all this “soft hair” mumbo jumbo, we’d just compute the “hard” piece of $\int_{\mathcal{I}^+_\gt}$ and recover the “hard” piece of $\int_{H_\lt} \hat\mathcal{J}$ — which is all that would be would be needed to “solve” the information paradox.

But that’s what the “No Hair” theorem prevents us from doing. The hope is that, by including all of the soft contributions (on both sides of the equality), we can recover more information from the measurements on $\mathcal{I}^+_\gt$ that we could by measuring the “hard” contributions alone.

Assuming this still holds for the slicing they have, and the decomposition into three subfactors, then I think they can argue that the contribution from $S$ is negligible.

The hard contribution from $S$ is certainly negligible. But the soft contribution isn’t (indeed, my intuition — not backed up by any calculation — is that the dominant soft contribution comes from $S$).

(Sorry about the poorly typeset comments. I’m assuming you value a faster response over me spending time learning the system.)

I appreciate thoughtful comments, delivered at any speed.

If you want to utilize the system more effectively, choose (my favourite) “Markdown with itex to MathML” from the “Text Filter” menu and type your equations in TeX (more precisely, itex).

Hints:

1. You can get the source-code for any equation here, by double-clicking on it.
2. Markdown has handy shortcuts for many textual niceties (italics, bold, blockquotes, hyperlinks, …).
3. There’s even a built-in WYSIWYG drawing program for including figures in your comments, but maybe that’s getting a little too fancy …
Posted by: Jacques Distler on January 13, 2016 9:52 PM | Permalink | PGP Sig | Reply to this

### Re: BMiSsed

I agree. When one decomposes the Hilbert space the soft modes get all mixed up (since they span all three surfaces, the physics should be similar to breaking a box in half), and this suggests that one cannot ignore the soft charge contribution from any piece.

My goal, however, was only to convince you that there is a Ward identity that relates physical observables; I won’t try to convince you of its relevance to the information paradox or the process they describe, since I still haven’t convinced myself.

Regarding the canonical \epsilon, I would just emphasize that this is how it works in Burkhard and my path integral formalism. I don’t know a clear way of seeing this in what HPS have done. Strominger et al seem to prefer an S-matrix language from canonically quantizing a symplectic structure. Presumably, one can address the issue in that language, although I don’t directly see how, and arrive at an equivalent statement. Regardless, I think the path integral language is a much more natural way to discuss these sorts of identities, but I may be biased ;).

Posted by: Steve Avery on January 15, 2016 10:49 AM | Permalink | Reply to this

### Re: BMiSsed

This approach to use the BMS group seems rather reminiscent of Carlip’s attempts to understand the BTZ black hole entropy in AdS3 using the structure of diffeomorphisms. The latter effort did not succeed – if I recall correctly, Carlip himself eventually concluded that you could get any answer you wanted this way. Quantization of the dof’s of asympototically AdS3 gravity around smooth geometries leads to a central charge of order one, whereas one needs a central charge that is larger by a factor of the AdS3 radius relative the 3d Planck scale. If on the other hand, you cut a hole out of the space at the BTZ horizon and try to build something out of gravitational dof’s there, the exercise is ill-defined because there is no preferred Virasoro-like subalgebra of near-horizon diffeomorphisms, let alone a prescription for what the allowed representations are.

Does changing the arena to 4d asymptotically flat spacetime make life simpler somehow? It would be quite surprising.

Posted by: Emil Martinec on January 13, 2016 1:10 AM | Permalink | Reply to this

### Re: BMiSsed

Even if hair is found it doesn’t seem to change the fact that the QFT vacuum is correlated nontrivially across any boundary, which is the cause of the loss of unitarity and hence the information loss problem…this is of course the observation refined by AMPSS.

Posted by: Will Nelson on January 18, 2016 11:20 AM | Permalink | Reply to this

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