SciPost logo

SciPost Submission Page

Generalized Komar charges and Smarr formulas for black holes and boson stars

by Romina Ballesteros and Tomas Ortin

Submission summary

Authors (as registered SciPost users): Tomás Ortín
Submission information
Preprint Link: scipost_202411_00021v1  (pdf)
Date submitted: 2024-11-11 11:35
Submitted by: Ortín, Tomás
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
Specialties:
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical

Abstract

The standard Komar charge is a (d−2)(d−2)-form that can be defined in spacetimes admitting a Killing vector and which is closed when the vacuum Einstein equations are satisfied. Its integral at spatial infinity (the Komar integral) gives the conserved charge associated to the Killing vector, and, due to its on-shell closedness, the same value (expressed in terms of other physical variables) is obtained integrating over the event horizon (if any). This equality is the basis of the Smarr formula. This charge can be generalized so that it still is closed on-shell in presence of matter and its integrals give generalizations of the Smarr formula. We show how the Komar charge and other closed (d−2)(d−2)-form charges can be used to prove non-existence theorems for gravitational solitons and boson stars. In particular, we show how one can deal with generalized symmetric fields (invariant under a combination of isometries and other global symmetries) and how the geralized symmetric ansatz permits to evade the non-existence theorems.

Current status:
Awaiting resubmission

Reports on this Submission

Report #2 by Anonymous (Referee 2) on 2025-1-3 (Invited Report)

Report

The authors set out to investigate the question of existence of solitons in General Relativity minimally coupled to various (bosonic) matter fields. They constrain the space of possible black hole and solitonic solutions by deriving generalized Komar charges and Smarr-like formulae in these theories. The standard Smarr relations applicable to stationary solutions of vacuum General Relativity are modified by additional terms due to the presence of the matter fields and associated charges. The manuscript also considers the case when the stationarity condition for matter fields is implemented in a more general (than the usual) way.

Given that there is significant interest in understanding the space of stationary solutions in gravitational theories, the research question discussed in the manuscript is worthy of consideration. The present work is also is motivated by the fact that an application of Smarr's relation (together with the positive energy theorem) yields nonexistence of gravitational solitons in vacuum General Relativity. However, it appears that in a more general class of theories the generalized Smarr relation is not quite as powerful tool as in vacuum General Relativity. Nevertheless, Smarr-like relations may provide important consistency checks for finding black hole and solitonic solutions in various theories.

The methodology and the presentation of the results is also sound and therefore I can recommend the manuscript for publication, modulo a minor revision.

Requested changes

1. The authors briefly outline some of their notational conventions in the beginning of section 2 (and footnote 18, in particular). However, they already use some non-trivial notation early on in the Introduction which may be a bit confusing for the reader at first. I think it would be helpful to fix notations and conventions already in the Introduction. For example, the vierbein notation is first discussed before eq. (2.2) but it already appears in eq. (1.6). It appears that latin indices are tetrad labels rather than abstract indices, a point which might also be worth clarifying. Furthermore, there are some undefined notations in the manuscript later on: e.g. {\cal D} is not defined in eq. (2.10).

2. The authors may consider studying the review Chrusciel et al., Living Rev.Rel. 15 (2012) 7 in the context of black hole solutions. Section 7.2 may be particularly relevant since additional Smarr-type relations are discussed. The methods presented there could be applicable to a more general class of theories and may be helpful to further constrain the space of possible stationary solutions in those theories.

Recommendation

Ask for minor revision

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Report #1 by Anonymous (Referee 1) on 2024-12-26 (Invited Report)

Report

The authors use the generalized Komar charge to demonstrate the (non-)existence of solitonic solutions in theories where General Relativity is minimally coupled to bosonic degrees of freedom, as well as to derive Smarr-like relations for black hole solutions in such theories. Over the past decades, the issue of the existence of (non-topological) solitons in different field theories, in both flat and curved spacetime, has been thoroughly discussed. Examples presented by the authors do not appear to contradict or introduce significant novelty compared to results already known in the literature. However, the method outlined in the paper could provide a mechanism to partially automate the process of answering questions related to the existence of solitonic solutions. As noted in the discussion around Eq. 1.22, this promise relies on having an explicit expression for the form $\omega$, which is not necessarily available in general.

Similarly, regarding the derivation of Smarr-like formulas in hairy black hole scenarios, alternative methods have been developed and successfully applied—for instance, the results of Herdeiro et al., cited by the authors. Thus, looking forward, it may be beneficial to understand the comparative advantage of the proposed approach with respect to the methods used so far, both regarding solitonic solutions (e.g., arguments based on Derrick's theorem) as well as the Smarr formula.

Requested changes

1. Usefulness of the method: Regarding the last point of the report, the authors may provide an additional discussion of the advantage of their method compared to the alternatives. For example, on pg. 4, the authors claim that in the case of other methods for deriving the Smarr-like relations in the theories under consideration, "the results are more difficult to understand because they involve unphysical quantities." However, Herdeiro et al., in 1403.2757 and 2406.03552, seem to successfully connect the horizon mass/angular momentum to ADM quantities and use the generalized Smarr formula to validate their numerical results—see the section on "Physical relations and checks" in 1403.2757 and the discussion around Eq. 2.70 in 2406.03552.

2. References: It would be appropriate for the authors to reference the original results regarding the (non-)existence of non-topological solitons in bosonic field theories (both in flat and curved spacetime) and to situate their specific conclusions, as well as the broader aim of the paper, within the context of these results. In particular, I would suggest citing the seminal works of Derrick, Kaup, Coleman, and Lee

https://doi.org/10.1063/1.1704233
https://journals.aps.org/pr/abstract/10.1103/PhysRev.172.1331
https://www.sciencedirect.com/science/article/abs/pii/055032138590286X?via%3Dihub
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.35.3658

3. Terminology: The authors choose to use the loaded term "boson star" instead of the more standard terms "non-topological soliton" or "soliton star" (e.g., doi.org/10.1016/0370-1573(92)90064-7). While harmonizing terminology in the literature is challenging, the authors could consider adding a comment on this matter. Specifically, it is widely assumed that a "boson star" solution already implies a self-gravitating complex scalar or vector field protected by $U(1)$ or some other global
symmetry (e.g. https://arxiv.org/abs/1202.5809). Thus, when the authors claim that "there are no boson star solutions in [real, massless scalar] theory [without the potential]" (pg. 10), they are merely stating a well-known (and trivial) fact: such a theory does not admit non-topological solitons.

Recommendation

Ask for minor revision

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Login to report or comment