arxiv: 2512.10343 · v2 · submitted 2025-12-11 · 🌀 gr-qc · astro-ph.HE· astro-ph.SR· hep-th

Recognition: 2 theorem links

· Lean Theorem

Stationary Stars Are Axisymmetric in Higher Curvature Gravity

Nitesh K. Dubey , Sanved Kolekar , Sudipta Sarkar

Authors on Pith no claims yet

Pith reviewed 2026-05-16 23:41 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEastro-ph.SRhep-th

keywords axisymmetrystationary starshigher curvature gravityKilling symmetrythermodynamic equilibriumdiffeomorphism invarianceasymptotic flatness

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The pith

Stationary stars in higher curvature gravity are axisymmetric because their interior Killing symmetry extends uniquely to the exterior.

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

This paper aims to prove that stationary stellar configurations remain axisymmetric in a wide range of modified gravity theories that include higher curvature terms. It does so by showing how the symmetry from thermodynamic equilibrium inside the star extends to the exterior spacetime. The extension relies on asymptotic flatness and smoothness conditions. A reader would care as it indicates that axisymmetry is a general property of covariant gravitational theories, not limited to Einstein's general relativity. This applies to the equilibrium states of stars like neutron stars or white dwarfs.

Core claim

Assuming asymptotic flatness and standard smoothness requirements, the Killing symmetry implied by thermodynamic equilibrium inside the star uniquely extends to the exterior region, thereby enforcing rotational invariance. This demonstrates that axisymmetry of stationary stellar configurations is not a feature peculiar to Einstein gravity but a universal property of generally covariant gravitational theories, persisting even in the presence of higher curvature corrections.

What carries the argument

The unique extension of the interior Killing symmetry to the exterior spacetime.

If this is right

Stationary stars are axisymmetric in higher curvature theories of gravity.
The axisymmetry theorem for stars extends beyond general relativity.
Thermodynamic equilibrium enforces rotational invariance through symmetry extension.
Compact objects in equilibrium are axisymmetric in diffeomorphism invariant metric theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Non-axisymmetric stationary stars, if found, would indicate a violation of the smoothness or asymptotic flatness assumptions.
The approach could be extended to analyze symmetry in other modified gravity scenarios for stars.
Numerical relativity simulations in higher curvature theories should yield axisymmetric solutions for stationary cases.

Load-bearing premise

The assumption that the interior Killing symmetry from thermodynamic equilibrium extends uniquely to the exterior under asymptotic flatness and smoothness conditions.

What would settle it

Constructing or observing a stationary non-axisymmetric star in a higher curvature gravity theory while maintaining asymptotic flatness and smoothness would falsify the central claim.

read the original abstract

The final equilibrium stage of stellar evolution can result in either a black hole or a compact object such as a white dwarf or neutron star. In general relativity, both stationary black holes and stationary stellar configurations are known to be axisymmetric, and black hole rigidity has been extended to several higher curvature modifications of gravity. In contrast, no comparable result had previously been established for stationary stars beyond general relativity. In this work we extend the stellar axisymmetry theorem to a broad class of diffeomorphism invariant metric theories. Assuming asymptotic flatness and standard smoothness requirements, we show that the Killing symmetry implied by thermodynamic equilibrium inside the star uniquely extends to the exterior region, thereby enforcing rotational invariance. This demonstrates that axisymmetry of stationary stellar configurations is not a feature peculiar to Einstein gravity but a universal property of generally covariant gravitational theories, persisting even in the presence of higher curvature corrections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the stellar axisymmetry theorem to higher-curvature gravity by showing the interior Killing vector extends uniquely to the exterior, but the higher-order field equations make that uniqueness step the main point to verify.

read the letter

The main point from this paper is that stationary stars in diffeomorphism invariant higher curvature gravity theories are axisymmetric, just as they are in general relativity. The authors show that the Killing symmetry coming from thermodynamic equilibrium inside the star extends uniquely to the exterior region under asymptotic flatness and smoothness assumptions, which enforces the rotational invariance. They do well in framing the result as a generalization that applies broadly rather than being limited to Einstein gravity. The abstract makes clear that while black hole rigidity has been extended before, the stellar case had not, so this fills that gap. The logic follows the standard path of using equilibrium to get a Killing vector and then extending it. The soft spot is in the extension mechanism itself. Higher curvature theories lead to field equations that are fourth order or higher, unlike the second order Einstein equations. This changes the character of the system for the Killing form, and it is not obvious that smoothness and asymptotic flatness alone guarantee a unique extension without additional boundary data at the star's surface. The stress test note points this out, and if the paper does not provide explicit matching conditions for the higher derivatives, that could be a place where the claim needs strengthening. Otherwise the central argument appears to hold up on its own terms. There are no free parameters or invented entities here, and the assumptions are the usual ones for these theorems. The work is formal and mathematical. This is for colleagues interested in modified gravity and the properties of compact stars. Someone modeling stationary configurations in these theories would get value from knowing axisymmetry can be assumed without loss. It deserves a serious referee to go through the detailed derivation and check whether the uniqueness really follows in the higher order case.

Referee Report

1 major / 2 minor

Summary. The paper proves that stationary stellar configurations are axisymmetric in a broad class of diffeomorphism-invariant metric theories of gravity that include higher-curvature corrections. Assuming asymptotic flatness and standard smoothness, the authors show that the timelike Killing vector implied by thermodynamic equilibrium in the stellar interior extends uniquely across the surface to the exterior vacuum region, thereby enforcing axisymmetry. This generalizes the corresponding result from general relativity.

Significance. If the central extension argument holds, the result establishes axisymmetry of stationary stars as a universal feature of generally covariant gravitational theories rather than a special property of Einstein gravity. It supplies a rigorous theorem that can underpin modeling assumptions for compact objects in modified gravity and complements existing rigidity results for black holes in higher-curvature theories.

major comments (1)

[Proof of the main theorem (likely §3 or §4)] The load-bearing step is the claimed unique extension of the interior Killing vector to the exterior. In higher-curvature theories the vacuum field equations are fourth-order (or higher), so the system satisfied by a putative Killing vector is no longer strictly elliptic at the same differential order as the second-order Einstein case. The manuscript must demonstrate explicitly (e.g., via the reduced equations or a maximum-principle argument) that asymptotic flatness plus interior data still fix the exterior solution uniquely; otherwise the axisymmetry conclusion does not follow.

minor comments (2)

[Introduction] The precise subclass of diffeomorphism-invariant theories (e.g., which curvature invariants are allowed) should be stated at the outset with an explicit Lagrangian or field-equation form.
[Notation and preliminaries] Notation for the Killing vector, its norm, and the surface-matching conditions should be introduced once and used consistently; a short table of symbols would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for explicit justification of uniqueness in the extension argument. We address the major comment below and will revise the manuscript to strengthen the presentation of the proof.

read point-by-point responses

Referee: The load-bearing step is the claimed unique extension of the interior Killing vector to the exterior. In higher-curvature theories the vacuum field equations are fourth-order (or higher), so the system satisfied by a putative Killing vector is no longer strictly elliptic at the same differential order as the second-order Einstein case. The manuscript must demonstrate explicitly (e.g., via the reduced equations or a maximum-principle argument) that asymptotic flatness plus interior data still fix the exterior solution uniquely; otherwise the axisymmetry conclusion does not follow.

Authors: We agree that the higher differential order of the vacuum equations in these theories requires an explicit demonstration of uniqueness rather than an appeal to the second-order GR case. In the manuscript, the interior timelike Killing vector is fixed by the thermodynamic equilibrium condition (vanishing heat flux and rigid rotation), and the matching conditions across the stellar surface supply the full set of Cauchy data (the vector and its derivatives up to the requisite order) for the exterior vacuum problem. Under asymptotic flatness, this data uniquely determines the exterior solution because the higher-order equations for the Killing vector can be reduced to a quasilinear elliptic system whose principal part admits a maximum principle (after suitable gauge fixing). We will revise §3 to include: (i) the explicit reduction of the higher-curvature field equations to the equation satisfied by the Killing vector in the exterior, (ii) a statement of the adapted maximum-principle argument with the necessary boundary conditions at the surface and at infinity, and (iii) a brief reference to the relevant uniqueness results for higher-order elliptic operators in asymptotically flat settings. This addition makes the extension step fully rigorous while leaving the overall theorem unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; uniqueness extension is a standard mathematical claim

full rationale

The paper's derivation chain rests on showing that a timelike Killing vector from interior thermodynamic equilibrium extends uniquely to the exterior under asymptotic flatness and smoothness, thereby enforcing axisymmetry in diffeomorphism-invariant higher-curvature theories. This is presented as an extension of known GR rigidity results rather than a self-referential definition or fitted prediction. No equations or steps in the abstract reduce the claimed result to its inputs by construction, and no load-bearing self-citation of an unverified uniqueness theorem is quoted. The argument is self-contained against external mathematical benchmarks of elliptic systems and boundary-value uniqueness; the skeptic concern about higher-order equations pertains to correctness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on standard background assumptions in gravitational physics rather than new free parameters or invented entities.

axioms (3)

domain assumption Diffeomorphism invariance of the metric theory
Invoked to define the broad class of theories considered.
domain assumption Asymptotic flatness
Required for the exterior region and uniqueness of the Killing extension.
domain assumption Standard smoothness requirements
Needed to guarantee unique extension of the interior Killing symmetry.

pith-pipeline@v0.9.0 · 5457 in / 1285 out tokens · 35426 ms · 2026-05-16T23:41:18.027011+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the Killing symmetry implied by thermodynamic equilibrium inside the star uniquely extends to the exterior region, thereby enforcing rotational invariance
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Holmgren’s uniqueness theorem guarantees that the vanishing initial data lead to the unique analytic solution tab = 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

[1]

(3) Here, P cdef ≡ m∑ i=0 (− 1)i ∇ (e1 · · · ∇ei)Q e1···eicdef , (4) with Q e1···eicdef ≡ ∂L ∂ ( ∇ (e1 · · · ∇ei)Rcdef ) , i = 0,

take the form [ 19] Eab = − 1 2 gabL + Pa cdeRbcde − 2∇ c∇ dPcabd + Wab = 0. (3) Here, P cdef ≡ m∑ i=0 (− 1)i ∇ (e1 · · · ∇ei)Q e1···eicdef , (4) with Q e1···eicdef ≡ ∂L ∂ ( ∇ (e1 · · · ∇ei)Rcdef ) , i = 0, . . . , m. (5) The tensor Wab consists of terms built from covariant derivatives of Q e1···eicdef together with curvature ten- sors. It appears only w...

work page
[2]

Assuming the coeﬃcients in Eq

also holds on the stellar surface. Assuming the coeﬃcients in Eq. ( 6) are smooth on the surface of the star, taking ξb and ∂aξb on stellar surface as initial Cauchy data, the well-posedness of the Cauchy problem for normally hyperbolic equations implies the existence of a unique solution just outside the surface of the star [ 20–23]. B. Properties of the...

work page
[3]

Lovelock theory We begin with Lovelock theories of gravity. The ﬁeld equations of a general diﬀeomorphism invariant metric theory typically involve higher order derivatives and may exhibit various pathologies, including perturbative ghost modes. A well motivated and distinguished subclass is provided by the Lovelock theories, whose equations of motion rem...

work page
[4]

The tensor Rab thus serves as a generalized Ricci tensor, whose vanishing characterizes the vacuum sector of Lovelock gravity

remain strictly second order in the metric, thereby avoiding Ostrograd- sky instabilities and ensuring a well posed classical the- ory. The tensor Rab thus serves as a generalized Ricci tensor, whose vanishing characterizes the vacuum sector of Lovelock gravity. Since LξRab = 0, it leads to a diﬀerential equation of the deformation tensor tab ≡ ∇ aξb + ∇ ...

work page
[5]

( 11) yields an explicit expression for the second derivatives of ξb, gcd∂c∂dξb = − ξd∂d ( gceΓ b ce ) − F b(g, ∂g, ξ, ∂ξ ), (12) whose right-hand side is continuous at Σ

with Eq. ( 11) yields an explicit expression for the second derivatives of ξb, gcd∂c∂dξb = − ξd∂d ( gceΓ b ce ) − F b(g, ∂g, ξ, ∂ξ ), (12) whose right-hand side is continuous at Σ. Thus all terms appearing in the ﬁrst derivatives of tab are continuous across Σ, and ∂ctab also vanishes on Σ. The tensors tab and ∂ctab on Σ therefore supply Cauchy data for E...

work page
[6]

Hence, the extension deﬁned by Eq

forms a linear system with analytic coeﬃcients, Holmgren’s uniqueness theorem [ 17, 23] guarantees that the vanishing initial data lead to the unique analytic so- lution tab = 0. Hence, the extension deﬁned by Eq. ( 6) must be a Killing vector ﬁeld in the exterior region. Stationarity implies the existence of an additional Killing vector ﬁeld ηa. Denoting...

work page
[7]

In general, in higher-curvature or eﬀective models, the ﬁeld equations acquire additional dynamical features

Gravity beyond second order The situation becomes more involved once one goes be- yond Lovelock’s theories. In general, in higher-curvature or eﬀective models, the ﬁeld equations acquire additional dynamical features. Setting LξEab = 0 in Eq.(

work page
[8]

In the generic situation—absent Lovelock-type cancellations— this bound is saturated, so the principal part of LξEab has diﬀerential order m + 4 acting on tab

gives the linear diﬀerential equation with at most ( m + 4) derivatives of tab , with tab = 0 as a solution (see Eq.(I.42) of the Supplementary Materials). In the generic situation—absent Lovelock-type cancellations— this bound is saturated, so the principal part of LξEab has diﬀerential order m + 4 acting on tab. To prove the uniqueness of the solution t...

work page
[9]

So, the Killing vector ﬁelds ξb and ηb commute with each other

holds, and lb is a unique solution for the general theory. So, the Killing vector ﬁelds ξb and ηb commute with each other. C. Axisymmetry of the exterior spacetime By analytic continuation the Killing vector ﬁelds dis- cussed in the preceding subsections can be extended to cover the full spacetime outside the star. Consequently, the manifold admits two gl...

work page 2023
[10]

S. W. Hawking. Black holes in general relativity. Com- munications in Mathematical Physics , 25:152–166, 1972

work page 1972
[11]

B. Carter. Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. , 26:331–333, Feb 1971. 1 The theorem on axisymmetry will also hold for a non-vacuum exterior if the matter distribution outside has suﬃcient de cay property, and such that Lξ Tab = 0 in the exterior

work page 1971
[12]

Stationary black holes: Uniqueness and beyond

Markus Heusler. Stationary black holes: Uniqueness and beyond. Living Reviews in Relativity , 1(6), 1998

work page 1998
[13]

Lindblom

L. Lindblom. Stationary stars are axisymmetric. Astro- phys. J. , 208:873–880, September 1976

work page 1976
[14]

Stefan Hollands, Akihiro Ishibashi, and Harvey S. Reall. A Stationary Black Hole Must be Axisymmetric in Eﬀec- tive Field Theory. Commun. Math. Phys. , 401(3):2757– 2791, 2023

work page 2023
[15]

Lovelock

D. Lovelock. The Einstein tensor and its generalizations. 6 J. Math. Phys. , 12:498–501, 1971

work page 1971
[16]

Padmanabhan and D

T. Padmanabhan and D. Kothawala. Lanczos-Lovelock models of gravity. Phys. Rept. , 531:115–171, 2013

work page 2013
[17]

Curvature Squared Terms and String Theories

Barton Zwiebach. Curvature Squared Terms and String Theories. Phys. Lett. B , 156:315–317, 1985

work page 1985
[18]

Gross and John H

David J. Gross and John H. Sloan. The Quartic Eﬀective Action for the Heterotic String. Nucl. Phys. B , 291:41– 89, 1987

work page 1987
[19]

Myers and Jonathan Z

Robert C. Myers and Jonathan Z. Simon. Black-hole thermodynamics in lovelock gravity. Phys. Rev. D , 38:2434–2444, Oct 1988

work page 1988
[20]

A note on thermodynamics of black holes in lovelock gravity

Rong-Gen Cai. A note on thermodynamics of black holes in lovelock gravity. Physics Letters B , 582(3–4):237–242, 2004

work page 2004
[21]

Padmanabhan, and Sudipta Sarkar

Sanved Kolekar, T. Padmanabhan, and Sudipta Sarkar. Entropy Increase during Physical Processes for Black Holes in Lanczos-Lovelock Gravity. Phys. Rev. D , 86:021501, 2012

work page 2012
[22]

Black Hole Ze- roth Law in Higher Curvature Gravity

Rajes Ghosh and Sudipta Sarkar. Black Hole Ze- roth Law in Higher Curvature Gravity. Phys. Rev. D , 102(10):101503, 2020

work page 2020
[23]

Black Hole Thermodynamics: General Relativity and Beyond

Sudipta Sarkar. Black Hole Thermodynamics: General Relativity and Beyond. Gen. Rel. Grav. , 51(5):63, 2019

work page 2019
[24]

The thermodynamics of irreversible pro- cesses

Carl Eckart. The thermodynamics of irreversible pro- cesses. iii. relativistic theory of the simple ﬂuid. Phys. Rev., 58:919–924, Nov 1940

work page 1940
[25]

Gerald B. Folland. Introduction to Partial Diﬀeren- tial Equations , volume 102. Princeton University Press, Princeton, NJ, 2nd edition, 1995. New edition (NED)

work page 1995
[26]

Methods of Math- ematical Physics, Vol

Richard Courant and David Hilbert. Methods of Math- ematical Physics, Vol. II: Partial Diﬀerential Equations . Interscience Publishers (Wiley), New York, 1962

work page 1962
[27]

Vivek Iyer and Robert M. Wald. Some properties of the noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D , 50:846–864, Jul 1994

work page 1994
[28]

A note on ﬁeld equations in generalized theories of gravity

Jun-Jin Peng. A note on ﬁeld equations in generalized theories of gravity. Phys. Scripta , 99(10):105229, 2024

work page 2024
[29]

Initial value prob- lems for wave equations on manifolds

Christian B¨ ar and Roger Tagne Wafo. Initial value prob- lems for wave equations on manifolds. Mathematical Physics, Analysis and Geometry , 18(1):7, 2015

work page 2015
[30]

Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Mars- den. Well-posed quasi-linear second-order hyperbolic sys- tems with applications to nonlinear elastodynamics and general relativity. Archive for Rational Mechanics and Analysis, 63(3):273–294, 1977

work page 1977
[31]

Analysis, Manifolds and Physics

Yvonne Choquet-Bruhat, C´ ecile DeWitt-Morette, and Margaret Dillard-Bleick. Analysis, Manifolds and Physics. Revised Edition, Volume 1 . North-Holland Pub- lishing Company, Amsterdam; New York, revised edition, 1982

work page 1982
[32]

Nandakumaran and P.S

A.K. Nandakumaran and P.S. Datti. Partial Diﬀerential Equations: Classical Theory with a Modern Touch . Cam- bridge IISc Series. Cambridge University Press, 2020

work page 2020
[33]

Gravity Theories in More Than Four- Dimensions

Bruno Zumino. Gravity Theories in More Than Four- Dimensions. Phys. Rept. , 137:109, 1986

work page 1986
[34]

Boulware and S

David G. Boulware and S. Deser. String-generated grav- ity models. Phys. Rev. Lett. , 55:2656–2660, Dec 1985

work page 1985
[35]

Physics of Black Holes, volume 769 of Lecture Notes in Physics

Eleftherios Papantonopoulos, editor. Physics of Black Holes, volume 769 of Lecture Notes in Physics . Springer Berlin, Heidelberg, 1 edition, 2009

work page 2009
[36]

Brassel, Sunil D

Byron P. Brassel, Sunil D. Maharaj, and Rituparno Goswami. Stars and junction conditions in Ein- stein–Gauss–Bonnet gravity. Class. Quant. Grav. , 40(12):125004, 2023

work page 2023
[37]

Thin shell dynamics in Lovelock gravity

Pablo Guilleminot, Nelson Merino, and Rodrigo Olea. Thin shell dynamics in Lovelock gravity. Eur. Phys. J. C, 82(11):1025, 2022

work page 2022
[38]

Intersecting hyper- surfaces, topological densities and Lovelock gravity

Elias Gravanis and Steven Willison. Intersecting hyper- surfaces, topological densities and Lovelock gravity. J. Geom. Phys. , 57:1861–1882, 2007

work page 2007
[39]

Brane versus shell cosmologies in einstein and einstein-gauss-bonnet theo- ries

Nathalie Deruelle and Tom´ a ˇ s Doleˇ zel. Brane versus shell cosmologies in einstein and einstein-gauss-bonnet theo- ries. Phys. Rev. D , 62:103502, Oct 2000. Supplementary Materials I. FIELD EQUA TION FOR THE GENERAL THEOR Y Let us consider a local diﬀeomorphism-invariant metric theory of gravi ty described by a Lagrangian density that is built from th...

work page 2000
[40]

Iyer and R

V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994), URL https://link.aps.org/doi/10.1103/PhysRevD.50.846

work page doi:10.1103/physrevd.50.846 1994
[41]

Peng, Phys

J.-J. Peng, Phys. Scripta 99, 105229 (2024), 2306.11561

work page arXiv 2024
[42]

Yano, The Theory of Lie Derivatives and Its Applications (North-Holland Publishing Company, Amsterdam, 1957)

K. Yano, The Theory of Lie Derivatives and Its Applications (North-Holland Publishing Company, Amsterdam, 1957)

work page 1957
[43]

Tanski, On commutation of covariant derivative and lie derivative (2020)

I. Tanski, On commutation of covariant derivative and lie derivative (2020)

work page 2020
[44]

R. R. Lompay and A. N. Petrov, J. Math. Phys. 54, 102504 (2013), 1309.5620

work page internal anchor Pith review Pith/arXiv arXiv 2013
[45]

where the following has been used in the ﬁrst line: δ(∇ λ Rµνρσ ) = ∇ λ (δRµνρσ ) − δΓ α λµ Rανρσ − δΓ α λν Rµαρσ − δΓ α λρ Rµνασ − δΓ α λσ Rµνρα

For instance, in order to separate out a term proportional to δRµνρσ from Qλµνρσ δ∇ λ Rµνρσ , one is able to achieve this by expressing such a term as Qλµνρσ δ∇ λ Rµνρσ = Qλµνρσ ∇ λ δRµνρσ − 4Qαβνρσ Rλνρσ δΓ λ αβ = ∇ µ ( Qµαβρσ δRαβρσ ) − (∇ λ Qλµνρσ ) δRµνρσ − 4Qαβνρσ Rλνρσ δΓ λ αβ . where the following has been used in the ﬁrst line: δ(∇ λ Rµνρσ ) = ∇ λ...

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[46]

Applying this identity repeatedly for i symmetrised derivatives yields the overall factor ( − 1)i

Each integration by parts on a covariant derivative produces a minus sign, ∫ dDx √ − g Aa∇ aB = − ∫ dDx √ − g (∇ aAa) B + (boundary) . Applying this identity repeatedly for i symmetrised derivatives yields the overall factor ( − 1)i

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