Spacetime mappings of the Brown-York quasilocal energy

Jeremy C\^ot\'e; Marianne Lapierre-L\'eonard; Valerio Faraoni

REVIEW 1 major objections 3 minor 30 references

Spacetime mappings of the Brown-York quasilocal energy

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Pith's one-line read The Brown-York quasilocal energy transforms under spacetime mappings by explicit gauge-restored formulas, with no simple scaling law.

desk verdict Useful tool-building paper: the Brown-York transformation formulas under conformal and Kerr-Schild maps are new and correct, but the Reissner-Nordstrom validation example has a factor error that should be fixed. read the letter →

arxiv 1908.02595 v1 pith:HFLX4BZZ submitted 2019-08-07 gr-qc hep-th

Jeremy Côté , Marianne Lapierre-Léonard , Valerio Faraoni This is my paper · ORCID

classification gr-qchep-th PACS 04.20.-q04.70.-s

keywords Brown-YorkquasilocalenergyconformaltransformationKerr-SchildsphericalsymmetrygaugedependenceMisner-Sharp-Hernandezmassarealradiusblackholethermodynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how the Brown-York quasilocal energy, a boundary-integral notion of mass for a finite region of a curved spacetime, changes when the metric is rescaled conformally or deformed by a generalized Kerr-Schild transformation. In spherical symmetry, and after restoring the diagonal areal-radius gauge by a time redefinition, it derives closed-form transformation laws for the Brown-York mass: Eq. (15) for conformal maps and Eq. (34) for Kerr-Schild maps. These matter because many dynamical black-hole solutions are generated from static seeds by exactly such maps, and because the alternative Misner-Sharp-Hernandez mass already has known transformation laws. The paper also shows that the Brown-York mass obeys no simple scaling relation analogous to the Misner-Sharp-Hernandez law, and that its gauge dependence must be handled explicitly for any comparison to be meaningful.

What carries the argument

The central object is the Brown-York quasilocal energy evaluated on a sphere of areal radius $R$, expressed in the diagonal areal-radius gauge (1) as $M_{BY}=R(1-1/\sqrt{B})$. The machinery that carries the argument is the gauge-restoring time redefinition $d\tilde t = (1/F)(dt+\beta d\tilde R)$: choosing $\beta$ to cancel the cross term in the mapped line element, with an integrating factor $F$ satisfying the integrability condition (10), brings every conformally or Kerr-Schild-mapped spherical metric back into the form in which formula (2) applies. For Kerr-Schild maps the null, geodesic character of $l_a$ is used to normalize $l^0=-1$, which collapses the transformed $B$-component to the compact expression in Eq. (34).

What would settle it

Compute the Brown-York mass for a spherical metric mapped with a conformal factor $\Omega(t,R)=e^{tR}$ on Minkowski space, solving the integrability condition (10) for the integrating factor $F$. If no global smooth $F$ exists, Eq. (15) does not define a unique $\tilde M_{BY}$, which would show the transformation law is valid only in patches; conversely, if several $F$ satisfy (10) and give different values, the gauge non-uniqueness is visible directly.

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Extended reading notes

Core claim

Working in spherical symmetry and in the areal-radius gauge $ds^2 = -A dt^2 + B dR^2 + R^2 d\Omega^2$, where the Brown-York mass is $M_{BY}=R(1-1/\sqrt{B})$, the paper derives how $M_{BY}$ changes when the metric is conformally rescaled, $g\to \tilde g=\Omega^2 g$, and when it is deformed by a generalized Kerr-Schild term $g\to \bar g = g + 2\lambda l_a l_b$ with $l_a$ null and geodesic. In each case the new line element is first brought back to the same areal-radius gauge by introducing a new time coordinate $d\tilde t = (1/F)(dt+\beta\, d\tilde R)$, which is the step that makes the comparison meaningful. The results are $\tilde M_{BY} = \Omega R[1 - \sqrt{A(\Omega_{,R}R+\Omega)^2 - \Omega_{,t}^2 B R^2}/(\Omega\sqrt{A B})]$ for conformal transformations and $\bar M_{BY} = R(1-\sqrt{A-2\lambda}/\sqrt{A B})$ for Kerr-Schild maps with the null vector normalized to $l^0=-1$. The paper verifies these formulas by recovering the known Brown-York expressions for FLRW space and for Reissner-Nordstrom, and it emphasizes that unlike the Misner-Sharp-Hernandez mass, no simple one-term transformation law exists.

Load-bearing premise

The load-bearing premise is that after each mapping one can choose a new time coordinate, globally, that removes the cross term and restores the metric to the same diagonal form used before the map; if such a coordinate exists only locally, the transformed Brown-York mass is only a local quantity and cannot be compared with the original.

Editorial extensions

If this is right

The Brown-York mass of a conformally rescaled spherical spacetime is not a simple multiple of the original mass, so conformal-frame calculations of quasilocal energy must carry the full expression (15).
For Kerr-Schild-generated solutions such as Reissner-Nordstrom from Minkowski space, Eq. (34) reproduces the standard Brown-York mass, confirming the rule's consistency with direct computation.
Comparisons of Brown-York mass across conformal or Kerr-Schild frames are meaningful only after restoring the same areal-radius gauge; the transformed mass inherits the integrating factor's non-uniqueness.
The Misner-Sharp-Hernandez mass remains gauge invariant and has simpler mapping laws, so in spherical practical calculations it is the more convenient construct.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct corollary the paper leaves implicit: any choice of integrating factor $F$ that satisfies (10) gives an equally valid transformed mass, so the spread of $\tilde M_{BY}$ over allowed $F$ quantifies the genuine gauge ambiguity of the Brown-York mass under these maps.
The same time-redefinition machinery could be applied to scalar-tensor or other alternative gravity theories by composing a conformal frame change with the Kerr-Schild step, giving testable expressions for apparent-horizon thermodynamics in those theories; the paper does not do that.
For a dynamical horizon obtained by conformally transforming a static seed, Eq. (15) predicts a time-dependent Brown-York mass whose horizon value need not equal twice the Misner-Sharp-Hernandez mass; checking this in a specific cosmological black-hole solution would be a direct numerical test of the rule.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here is my take on 1908.02595. The paper does something genuinely useful: it works out how the Brown-York quasilocal mass transforms under conformal rescalings and generalized Kerr-Schild maps in spherical symmetry, where the gauge is restored by a time redefinition. The main formulas, Eq. (15)/(16) and Eq. (34), are new — I checked the cited literature and these are not there. I re-derived them from the gauge (1) and they are algebraically consistent. This is tool-building, but it is honest tool-building: the paper cannot give a simple map from the old BY mass to the new one, and it says so.

What it does well: the derivation is straightforward and the contrast with Misner-Sharp-Hernandez is clear. The paper is also honest about the gauge-dependence of Brown-York, so the whole approach only makes sense when the same areal-radius gauge is available before and after the map. That caveat is stated. The Schwarzschild and FLRW examples are fine. Citation pattern is appropriate; no self-citation problem.

Soft spots. The Reissner-Nordstrom example in Sec. 4.2 has an internal inconsistency. The stated lambda = m/r - q^2/r^2 with l^mu=(1,-1,0,0), when inserted into Eq. (34), gives r(1 - sqrt(1 - 2m/r + 2q^2/r^2)) rather than the claimed Eq. (47). To get the RN Brown-York mass you need lambda = m/r - q^2/(2r^2). This is correctable and does not touch the transformation law, but it should be fixed. The FLRW example also has a small-looking typo in the density term (8π/3 vs 8πG/3, depending on units), which is minor.

The worry about global integrating factor F is, I think, a non-issue. For a local calculation on a 2-sphere, you only need local coordinate data, and a smooth beta always has a local integrating factor. So the central claim survives.

Who is this for? People working on black hole thermodynamics or scalar-tensor gravity who want to know whether a particular conformal or Kerr-Schild trick changes the Brown-York mass. It will not settle any big debate, but it fills a small gap and it is reproducible. With the example fixed, I would send it out; the right referee will spend an afternoon on it and accept.

Referee Report

1 major / 3 minor

Summary. The paper derives how the Brown-York quasilocal mass in spherical symmetry transforms under conformal rescalings and generalized Kerr-Schild mappings, starting from the areal-radius gauge (1) and the standard expression (2). For conformal maps the result is Eq. (15)/(16); for generalized Kerr-Schild maps it is Eq. (31)/(34). In each case the diagonal gauge is restored by introducing a new time coordinate with an integrating factor. The results are contrasted with the known transformation of the Misner-Sharp-Hernandez mass, and two applications are given: a conformal map from Minkowski to FLRW and a Kerr-Schild map from Minkowski to Reissner-Nordström.

Significance. If correct, these formulas fill a concrete gap in the quasilocal-energy toolbox: they show how a gauge-dependent quantity, the Brown-York mass, behaves under spacetime maps that are widely used to generate dynamical black-hole solutions. The central derivation is transparent and self-contained; I independently re-derived Eqs. (14), (15), (31), and (34) from Eqs. (1) and (2) and found them algebraically consistent. The paper contains no fitted parameters, and the examples are checkable consistency tests. The practical relevance is mainly in black-hole thermodynamics and in informed choices among quasilocal energy definitions.

major comments (1)

[Section 4.2, Eqs. (45)-(47)] The Reissner-Nordström validation example is internally inconsistent. Inserting the stated λ = m/r - q^2/r^2 and l_µ = (1,-1,0,0) into the Kerr-Schild transformation law (34) gives Mbar_BY = r(1 - sqrt(1 - 2m/r + 2q^2/r^2)), not the quoted Reissner-Nordström value in Eq. (47). The correct Kerr-Schild parameter for the line element (44) is λ = m/r - q^2/(2r^2), so that 2λ = 2m/r - q^2/r^2. The same error appears in the time redefinition (46), which should read dT = dt + [(2m/r - q^2/r^2)/(1 - 2m/r + q^2/r^2)] dr. This is a local error in an illustrative check and does not affect the derivation of Eq. (34), but the example is presented as a consistency test and must be corrected.

minor comments (3)

[Section 3, Eqs. (25)-(34)] The index convention for l^a is inconsistent: Eq. (25) writes l^µ with an upper index, but Eqs. (26)-(29) and (31) use l0 and l1 as covariant components, while Eq. (33) returns to contravariant components. Please state explicitly that l0 and l1 in the intermediate formulas are covariant components.
[Section 2, Eqs. (9)-(10)] The integrability condition (10) is written with partial derivatives with respect to t and tilde R, but the dependence of β on (t,R) and R = tilde R/Ω means the independent coordinates should be stated. A short sentence clarifying that (t, tilde R) are the independent coordinates after the change R -> tilde R = ΩR would remove ambiguity.
[General] There are several typos: "thermodunamics" in Section 2 should be "thermodynamics", and "intepretation" should be "interpretation".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Brown-York transformation laws are computed from the metric components and the standard Brown-York formula, with no fitted inputs or load-bearing self-citation.

full rationale

The derivation is self-contained. The paper starts from the accepted gauge expression MBY = R(1 - 1/sqrt(B)) (Eq. 2), applies the conformal transformation (4) or the generalized Kerr-Schild map (21), computes the transformed metric components in the restored areal-radius gauge, and substitutes the resulting tilde B or bar B into Eq. (2) to obtain Eqs. (15) and (34). No fitted parameter is introduced, and no part of the final formula is assumed as input. The examples (FLRW from conformal Minkowski, Reissner-Nordström from Kerr-Schild Minkowski) are independent checks rather than inputs. The prior transformation laws for the Misner-Sharp-Hernandez mass from Ref. [10] are cited only for comparison, not as load-bearing support for the Brown-York results. The residual gauge freedom in the integrating factor F and the need to restore the gauge are limitations of interpretation, not circular steps. One internal check is imperfect: Eq. (45) sets lambda = m/r - q^2/r^2, but the claimed Reissner-Nordström result (47) is obtained when lambda = m/r - q^2/(2r^2); this is an inconsistency in the example, not circularity in the transformation law itself.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the standard spherically symmetric Brown-York formula, the assumed gauge (1), and the local possibility of restoring that gauge after each mapping. No free parameters are fitted and no new entities are postulated. The examples use standard cosmology and Reissner-Nordstrom geometry, though the RN example contains an apparent typo in lambda; that typo does not support the general derivation.

assumptions (6)

domain assumption The spherically symmetric line element can be written in the areal-radius gauge ds^2 = -A(t,R)dt^2 + B(t,R)dR^2 + R^2 dOmega^2.
Eq. (1); this gauge is the starting point and is also the gauge to which the mapped metrics are restored before reading off the Brown-York mass.
standard math M_BY = R (1 - 1/sqrt(B)) is the correct Brown-York quasilocal mass in this gauge.
Eq. (2), citing Refs. [24,25,26]. The whole derivation is a computation of how this expression changes under metric mappings.
domain assumption The conformal factor Omega(t,R) is positive and preserves spherical symmetry.
Eqs. (4)-(5); the scope is restricted to this class of conformal rescalings.
domain assumption The Kerr-Schild vector l_a is null and geodesic with respect to the seed metric.
Eqs. (21)-(22); standard definition of generalized Kerr-Schild transformations. The null condition is what makes the inverse metric simple; the geodesic condition is part of the stated class but not used in the coordinate computation.
standard math A local integrating factor F exists for the time redefinition, restoring diagonal form.
Eqs. (9)-(10) and (27); any 1-form on a 2D manifold has a local integrating factor, and the final diagonal metric components are independent of the choice of F.
domain assumption The Friedmann equation H^2 = 8 pi rho / 3 is used in the FLRW example.
Section 4.1, Eq. (41); this is an example from standard cosmology, not a premise of the general transformation laws.

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Cite this review

Pith. "Pith review of Spacetime mappings of the Brown-York quasilocal energy." pith.science (2026). https://pith.science/paper/HFLX4BZZ

@misc{pith2026190802595,
  author       = {Pith},
  title        = {Pith review of: Spacetime mappings of the Brown-York quasilocal energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFLX4BZZ}},
  note         = {Machine review of arXiv:1908.02595}
}

read the original abstract

In several areas of theoretical physics it is useful to know how a quasilocal energy transforms under conformal rescalings or generalized Kerr-Schild mappings. We derive the transformation properties of the Brown-York quasilocal energy in spherical symmetry and we contrast them with those of the Misner-Sharp-Hernandez energy.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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