Carroll Expansion of General Relativity

We thank the referee for their useful remarks and apologize for our late reply.

First, as a minor technical point, we should emphasize that the magnetic Carroll limit of GR is only a subset of the full NLO theory we construct in our expansion. Following the comments of Report 1, we have clarified this further in our Introduction. Next, on the requested changes in Report 2:

1. While it indeed serves as a major motivation for our work, it is important to note that the precise identification between the Galilean Newton--Cartan expansion of GR and the post-Newtonian expansion is still a subject of ongoing work. To clarify this, we have added a comment to this extent to our Discussion.
2. It is true that there are technical similarities between the Carroll and Galilei expansion. However, we feel that any helpful summary of the Galilean expansion would be too long to be appropriate to include here. Instead, to aid the reader we have included more detailed references to specific sections and equations of 2001.10277.
3. It is an interesting question whether there are any underlying reasons for the physical differences between the magnetic and electric limits. Apart from their different technical origins, the fact that the electric limit only depends on the intrinsic Carroll data $v^\mu$ naturally makes it more tuned for the intuitive Carroll behavior described by independent worldlines. In the magnetic case, since the time evolution $L_v h_{\mu\nu}$ of the spatial metric vanishes due to the constraint $K_{\mu\nu}=0$, it deals with the Carroll symmetry in a different way. Physically, perhaps the most straightforward interpretation of this dichotomy is as the two distinct ways to solve the Ward idenitity for Carroll boosts, which constrains scalar two-point functions $\langle \phi(t,x) \phi(0,0) \rangle$ to be $f(t)\delta(x)$ or $g(x)$, where the former case is 'electric' and the latter case is 'magnetic'. However, since the generalization of this statement to spinning propagators has not yet been developed, we have refrained from commenting on this in the present draft. Furthermore, we do not know of an argument that would make it clear why mass arises in the 'magnetic' and not in the 'electric' sector of the Carroll expansion of gravity.
4. This typo has been fixed in v3, in fact the previous expression $\Phi^{\mu\nu} = \delta^{ab} \pi^\mu{}_a \pi^\nu{}_b$ was wrong, it should instead be $\Phi^{\mu\nu} = \delta^{ab} \left(\theta^\mu{}_a \pi^\nu{}_b + \pi^\mu{}_a \theta^\nu{}_b\right)$.
5. The typo (in the second equality of Equation (2.9)) has been fixed in v3.
6. As far as we know, there is deep reason for why the $S^2$ terms in the PUL expansion of the Levi-Civita Ricci tensor at $c^0$ vanish, it essentially arises due to a contraction $T_\nu\Pi^{\nu\sigma}=0$ contraction. We have pointed out this fact in a small comment below Equation (2.25).
7. This comment corresponds to the specific comment 11 of Report 1, which we have now addressed with an extra paragraph below Equation (3.6).
8. This is an interesting question. Based on its relation to the BKL limit, the electric limit appears to focus on the near-singularity limit of the Schwarzschild metric, rather than the near-horizon limit. On the other hand, it appears that the magnetic limit describes the asymptotic region of a black hole in Schwarzschild-type coordinates. (For this, we should also credit conversations with Stefan Vandoren at the 2022 Carroll Workshop in Vienna.) We would like to explore this connection in the future.

We thank the referee for their detailed report and their constructive comments.

The three general comments will be addressed in an upcoming revision by clarifying and adding additional emphasis to our Introduction and Discussion sections as well as the introductory texts at the start of each section. Briefly, our response is as follows:

1. Our motivations for studying the Carroll expansion of general relativity lie in the covariant description of ultra-local gravity effects that it provides, zooming in on a particular tractable yet nontrivial subsector of the dynamics of the full relativistic theory, as exemplified by the solutions we find in this paper. Additionally, the tools we develop here (in particular the relation to the 3+1 decomposition) are also expected to be useful for the non-relativistic Galilean expansion.
2. Our main result is the development of a systematic Carroll expansion of general relativity. As an example of this general formalism, we show that it can easily provide a Lagrangian description for the 'electric' and 'magnetic' Hamiltonian limits described by Henneaux and Salgado. These theories are special subcases arising from our general Carroll expansion, corresponding to the leading-order and a truncation of the next-to-leading-order theories, respectively. Our identifications of these theories as a subset of our general framework serves both to make contact with the existing literature and to identify particular tractable subsectors where first examples of explicit solutions can be obtained.
3. It is true that the specification of our expansion in Equations (2.4a)-(2.4c) contains important physical information. However, this is not so much the fact that this expansion starts at order $c^0$ (since we have already split off the factors of $c$ dictated by dimensional analysis in (2.1) or (2.3)), but rather the assumption that these vielbeine admit an analytic expansion in powers of $c^2$. As we already explained in Footnote 2, this restricts the class of metrics and in particular the coordinates in which our expansion can be performed. In particular, $c$-dependent symmetry transformations of the relativistic theory such as local boosts will naturally lead to different starting points for our covariant expansions. Similar considerations enter in the Galilean non-relativistic expansion of general relativity.

On the more specific comments:

1. Additional references will be added in a new version.
2. Additional references will be added in a new version.
3. We will add a 'road map' of the paper at the end of the introduction in a new version.
4. Correct, this will be fixed in a new version.
5. Correct, this will be fixed in a new version.
6. The boost-invariance $\delta_\lambda v^\mu = 0$ is indeed compatible with the relation $v^\mu \tau_\mu = 0$, since $\tau_\mu$ transforms under boosts with a spatial vector which vanishes upon contraction with $v^\mu$. We will add a comment emphasizing the latter in a new version.
7. Correct, this will be fixed in a new version.
8. The fiber bundle description of Carroll manifolds in the paper quoted by the referee can be obtained from our description of Carroll geometry by identifying the fibers with the integral curves of our $v^\mu$ vector field. The base space of this fiber bundle then corresponds to our notion of the 'spatial submanifold'. However, as remarked in our Section 2.5, one can typically not define spatial hypersurfaces in a Carroll boost-invariant way since $\tau_\mu$ transforms. This ambiguity is parametrized by the spatial components $b_i$ of $\tau_\mu$ in our Equation (2.30). On the fiber bundle side, it is reflected by the choice in Ehresmann connection $b_i$. (Likewise, the quantity $\omega$ on the fiber bundle side is related to our 'lapse' parameter $\alpha$.) We will add a footnote discussing this relation in a new version.
9. Following up on this point, in a boost-fixed frame, one can use the projector operators (2.32) to separate spatial and timelike components of tensors. In the fiber bundle language, this would correspond to fixing a particular Ehresmann connection, which likewise splits its tangent spaces in a sum of vertical (timelike) and horizontal (spacelike) subspaces. In particular one can use Carroll boosts to set $b_i=0$, so that $\tau_\mu$ is purely temporal. As a result, it satisfies the Frobenius condition $\tau\wedge d\tau=0$, and the spatial subspaces designated by the spatial projector can be integrated to a spatial hypersurface in the full spacetime manifold. (In the fiber bundle language, this means that the Ehresmann connection vanishes and hence the fiber structure is trivial. Our possibly non-constant lapse is mapped to the separate variable $\omega$. We will also add a footnote on this in a new version.) The main point of our Section 2.5 is that working in such a boost-fixed frame with a well-defined spatial foliation allows us to describe the pullback of various tensors to these spatial surfaces. In particular in the context of the subsector of the NLO theory corresponding to the 'magnetic' limit, it is useful to have access to the spatial Riemann tensor and its contractions. However, at the end of the day, while they may be obtained in a particular convenient boost frame, our boost-covariant results (and in particular the solutions discussed in Section 4) can be considered in any boost frame.
10. The Carroll-compatible connection in our (2.14) appears to be closely related to the connections introduced for the fiber bundle perspective in (3.8) and (3.12) of 1802.06809. However, the language is sufficiently different that a comparison is not straightforward. We hope to clarify this relation in future work.
11. Even though $h^{\mu\nu}$ transforms non-trivially under boost transformations and $h_{\mu\nu}$ does not, we consider variations with respect to $v^\mu$ and $h^{\mu\nu}$ instead of deriving equations because we want to keep the Carroll boost gauge symmetry explicit. The resulting equations of motion then transform covariantly under boosts. Relatedly, note that one cannot solve $(\tau_\mu, h^{\mu\nu})$ from $(v^\mu, h_{\mu\nu})$ precisely because of the ambiguities due to boosts that are present in the former pair but absent from the latter. In contrast, the pairs $(v^\mu, h^{\mu\nu})$ and $(\tau_\mu, h_{\mu\nu})$ can be expressed in terms of each other, and either choice therefore fully specifies the Carroll metric data including its boost frame.
12. Following up on the previous points, the actions that we obtain at each order are invariant under boosts, whereas the equations of motions are covariant, in analogy to the diffeomorphism gauge symmetry. Furthermore, boost transformations lead to Noether/Ward identities satisfied by the equations of motion. For example, the nontrivial (timelike) transformation of $\delta h^{\mu\nu}$ under boosts implies that the corresponding equation of motion is purely spatial.
13. Our reason for varying with respect to $\Phi^{\mu\nu}$ instead of $\Phi_{\mu\nu}$ at NLO is the same as the difference between varying with respect to $h^{\mu\nu}$ and $h_{\mu\nu}$ discussed in point 11. In most of our paper, we consider only a truncated sector of the NLO dynamics where these fields are turned off, but if they would be present, their transformations would imply similar Noether/Ward identities on the corresponding equations of motion.
14. As mentioned in the text the solution (4.3) hinges on the 'ultra-local' nature of the leading-order equations which decouples spatially separated points. Such a simplification of the evolution equation is therefore typically not present in the full relativistic evolution equation. However, the LO/electric Carroll theory does appear to be closely related to the Belinsky-Khalatnikov-Lifshitz limit of general relativity, which we will point out in a new version.