Carroll black holes

Dear Editor, Please find below our reply to referees’ remarks. We hope that the resubmitted version would now be suitable to be published in SciPost Physics. Best regards, Florian Ecker, Daniel Grumiller, Jelle Hartong, Alfredo Pérez, Stefan Prohazka, Ricardo Troncoso.

---x-----x-----x------- Reply to referee report 1:

We thank the referee for recommending our article to be published in SciPost Physics, once it is supplemented with a summary of the results in the concluding remarks.

Thus, in order to address the request, section "Outlook" was renamed in "Summary and Outlook" and now includes a brief summary of our main results at the beginning, including the following paragraphs: “We have focused on a wide class of Carroll geometries which, despite the absence of a lightcone structure, possess black hole-like behaviour. We identified Carroll black holes as configurations exhibiting an extremal surface together with thermal properties such as finite entropy. The former is the analogue of a Lorentzian extremal surface. A crucial ingredient was incorporating the notion of Carroll thermal manifolds, introduced by relaxing the standard definition of a Carroll manifold so as to allow a vanishing ``clock one-form'' on isolated surfaces. Our strategy consisted of thoroughly analysing various formulations of 2d magnetic Carroll dilaton gravity models, being generic enough so as to accommodate the dimensional reduction of spherically symmetric configurations of higher-dimensional magnetic Carroll gravity. We have also shown that the processes of spherical reduction and taking the magnetic Carroll limit commute. We discussed examples in the context of magnetic Carroll gravity in diverse dimensions, including the Carroll versions of Schwarzschild, Reissner--Nordstr\"om, BTZ, as well as black hole solutions of generic Carroll dilaton gravity, including Carroll JT and Carroll Witten black holes in two spacetime dimensions. Some examples of rotating Carroll black holes were also briefly analyzed. “

We included some brief remarks about similar configurations in Galilean theories, a bit below within the same section. We hope that the resubmitted version would now be suitable to be published in SciPost Physics.

Best regards,

  Florian Ecker, Daniel Grumiller, Jelle Hartong, Alfredo Pérez, Stefan Prohazka, Ricardo Troncoso.

---x---x-x-x-x-xx—x-x-x-x-x-x Reply to referee report 2:

We thank the referee for the thorough inspection of our article, as well as for recommending our article for publication in SciPost Physics, once a number of minor points are addressed. The number of specific remarks is 3 (overall) plus 16 punctual ones, which we address below:

• Overall remarks:

Referee: 1- As the authors themselves more or less point out, there is nothing "black" about Carrollian black holes. To avoid a slightly misleading title, perhaps it would be more appropriate to title the article "Carrollian analogs of black holes". Reply :1- We agree that there is nothing manifestly "black" about Carroll black holes. Indeed, as we pointed out in the abstract: “Despite the absence of a lightcone structure, some solutions of Carroll gravity show black hole-like behaviour.” However, the suggestion for the title: "Carrollian analogs", might introduce some severe confusion for the readers, since "black hole analogs" has a very specific technical meaning, completely unrelated to the entities we discuss. We believe that Carroll thermal properties together with the geometric notion of a Carroll extremal surface justify labelling these states as "Carroll black holes". So, we prefer to stick to the title of the article.

Referee: 2- It would be useful to have a 4d perspective on charged and rotating BHs that go beyond the results in sec. 7. As far as I can tell, it is straightforward to expand the Kerr and RN BHs in powers of $c^{2}. Would the resulting Carrollian geometries qualify as "Carrollian black holes"? Reply : 2- Concerning the Carrollian version of RN, different approaches to deal with it are already described in section 7.2. The rotating case remains as an open issue. Indeed, as pointed out in section “Summary and outlook”: “Although (magnetic) Carroll gravity generically admits configurations with non-vanishing angular momentum [96, 98], finding higher-dimensional rotating Carroll black holes is an open task. The main obstruction comes from the Hamiltonian constraint in magnetic Carroll gravity, which requires a spatial metric with a vanishing Ricci scalar (or constant Ricci scalar, in the presence of a cosmological constant). For example, the Carrollian limit of the Kerr solution in Boyer–Lindquist or Kerr–Schild coordinates exhibits a non-vanishing spatial Ricci scalar, thus failing to satisfy the Hamiltonian constraint. It could be advantageous to seek an appropriate coordinate system that addresses this issue.”

Referee: 3- Related to point (2) in "weaknesses" above, a small discussion of how this definition would apply in Galilean and Aristotelian contexts would improve the paper. Reply : 3- Concerning Aristotelian contexts, due to the absence of boosts, we actually do not have much to say about it. Nevertheless, we have added a paragraph entitled “Galilean black holes”, addressing the requested point in section “Summary and Outlook”.

• Punctual remarks:

Referee: 1- Starting on page 3, the limit considered is termed the "ultrarelativistic" limit. However, the ultrarelativistic limit is $v/c\rightarrow 1$ rather than $v/c\rightarrow 0$, which is the ultralocal limit. This should be changed throughout. Reply : 1-Magnetic Carroll theories generically admit configurations whose fields manifestly depend on time and spatial coordinates, and hence, the term “ultralocal” clearly does not apply for them. (Ultralocality would apply for Electric Carroll theories, but we do not deal with them) We agree with the fact that the term “ultrarelativistic” has been widely used when is v/c→1. Nevertheless, most of the recent literature about Carrollian limits use the same terminology for c->0. We prefer to stick to the term “ultrarelativistic limit” in order to agree with most of the current literature.

Referee: 2- Below eq. (3), $\Theta$ is referred to as the intrinsic torsion. However, the intrinsic torsion of a Carrollian structure is symmetric in general dimension. The relation between $\Theta$ and the intrinsic torsion in 2d is given in-line below eq. (21).

Reply : 2- Our definition of intrinsic torsion is given by the 2-form \Theta, and so the relation with other related quantities can be readily found when expressing them explicitly in terms of tensors.

Referee: 3- The Carroll boost parameter $\lambda$ in eq. (4) has an unconventional sign. Why?

Reply : 3- There is no universally accepted "conventional sign" of the boost parameter. Thus, we prefer to keep our convention.

Referee: 4- It would be useful to include the completeness relation $\delta_{\nu}^{\mu}=-v^{\mu}\tau_{\nu}+h^{\mu\rho}h_{\rho\nu}$ in eq. (17).

Reply : 4- Equation (17) has been changed to include the completeness relation. The new version reads: "For this we introduce the dual vectors $v^\mu $ and $e^\mu $ satisfying \begin{align} v^\mu \tau \mu =-1 \qquad e^\mu e\mu =1 \qquad \delta ^\mu \nu =-v^\mu \tau \nu +e^\mu e_\nu ~. \end{align}"

Referee: 5- Perhaps switch eqs. (26) and (27). This would help define the indices $I,J,\dots$. Reply : 5- Eqs. (26) and (27) have been switched, defining the PSM field content and the Poisson tensor.

Referee: 6- The Poisson tensor in (26) does not appear to be antisymmetric. Why not? Reply: 6- Poisson tensor has been rewritten with all the explicitly given, such that antisymmetry is manifest.

Referee: 7- To avoid a proliferation of terminology, the term "pre-Carrollian", which first appears above eq. (38), should be replaced with "pre-ultralocal (PUL)" throughout.

Reply : 7- As pointed out in our “Reply 1”, we stick to "pre-Carrollian".

Referee: 8- Perhaps include a spoiler below eq. (49) that reveals that $\gamma$ ultimately will not play a role. Reply : 8- It is a good suggestion. Thus, we added a sentence below eq. (49) addressing the eventual irrelevance of $\gamma$: “We will see below that $\gamma$ does not play a role.”

Referee: 9- In eqs. (64) and (66) the identity $\overset{(C)}{\nabla}_{\mu}V^{\mu}=-\mathcal{K}$ is used. Perhaps include this identity as an aid to the reader. Reply : 9- The right-hand side of the relation written by the referee should be zero by the defining relation of the pre-Carrollian derivative (see eq. (42)). This is consistent with eq.(66) since for X^\mu =V^\mu the right hand side is a boundary term itself and boundary terms are dropped in this relation.

Referee: 10- The second equality of eq. (70) is a "weak" equality in the sense that is is only valid on-shell. It should probably be replaced with an "$\approx$". Reply : 10- The suggestion might confuse the readers since in this context, the symbol $\approx$" could be interpreted as an approximation, which is not. For that reason, we stick to the equal sign, since right below it is explicitly explained that the equality holds on-shell (instead of on the constraint surface).

Referee: 11- In sec. 3.1, there is a recurring typo: $\partial M$ should be replaced with $\partial \mathcal{M}$. Reply : 11- The symbol "M" was corrected by "\mathcal{M}", denoting the manifold in section 3.1

Referee: 12- Above eq. (199): "Lagrangean" should be "Lagrangian". Reply : 12.- "Lagrangean" was changed to "Lagrangian" above eq. (199)

Referee: 13- Explicitly state that this is the spherically reduced Carroll-Schwarzschild black hole above eq. (211). Reply : 13- "spherically reduced" has been explicitly Included into the sentence above eq. (211).

Referee: 14- What does "Minkowski coordinates" mean above eq. (276)? Isn't the point rather that there exists Cartesian coordinates in flat space such that the Carrollian structure takes the form of eq. (276)? Reply : 14- "Minkowski" was deleted in the sentence above eq. (276).

Referee: 15- Below eq. (277): the fact that the infinitely many Killing vectors of a flat Carrollian structure reduce to the Carrollian symmetries was first discussed in arXiv:1402.0657, which should be included as a reference. It would be nice, but not necessary, to explicitly demonstrate the statement which is frequently made but seldomly shown. Reply : 15- The reference was added (now appearing as [118]) in penultimate paragraph. We also show explicitly how the symmetries are reduced once the connection is required to be invariant.

Referee: 16- Appendix B could be made more self-contained by, for example, explicitly including eq. (25) and by including references. I recommend a small rewriting of this appendix. Reply : 16- Appendix B was rewritten according to the suggestion, such that it is now more self-contained, and also including some additional references. We hope that the resubmitted version would now be suitable to be published in SciPost Physics.

Best regards,

  Florian Ecker, Daniel Grumiller, Jelle Hartong, Alfredo Pérez, Stefan Prohazka, Ricardo Troncoso.

Detailed in box "Author comments"