arxiv: 2604.10877 · v1 · submitted 2026-04-13 · 🌀 gr-qc · hep-th· math.DG

Recognition: unknown

Holographic is Hamiltonian, relatively

Piotr T. Chru\'sciel , Raphaela Wutte

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:23 UTC · model grok-4.3

classification 🌀 gr-qc hep-thmath.DG

keywords relative holographic energyrelative Hamiltonian energygeneral relativityenergy in gravityholography

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The pith

Relative holographic energy coincides with relative Hamiltonian energy.

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

The paper shows that a relative holographic energy equals the relative Hamiltonian energy in general relativity. This holds for spacetimes and boundary conditions where both quantities can be defined rigorously. A reader would care because the result unifies two distinct ways of assigning energy to gravitational systems. It allows one definition to stand in for the other when the setup permits comparison. The equivalence rests on the shared structure of relative quantities rather than absolute ones.

Core claim

We show that a relative holographic energy coincides with the relative Hamiltonian energy. The demonstration proceeds by direct identification once both expressions are formulated relative to a common reference and the required regularity conditions on the spacetime are met.

What carries the argument

The relative holographic energy, shown to equal the relative Hamiltonian energy through explicit comparison.

If this is right

Energy calculations can use either the holographic or Hamiltonian route interchangeably when both are available.
Consistency between holographic and classical Hamiltonian methods is confirmed for relative energies.
The result applies only where the relative definitions are simultaneously well-posed.

Where Pith is reading between the lines

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

The equality may simplify checks of energy conservation in numerical simulations of asymptotically flat or AdS spacetimes.
Similar relative constructions could be tested for other conserved quantities such as angular momentum.
The coincidence supplies a cross-check for holographic calculations that lack an independent Hamiltonian verification.

Load-bearing premise

The spacetimes and boundary conditions allow both the relative holographic energy and relative Hamiltonian energy to be rigorously defined and compared.

What would settle it

A concrete spacetime with well-defined boundary conditions in which the two relative energies evaluate to numerically distinct values.

read the original abstract

We show that a relative holographic energy coincides with the relative Hamiltonian energy.

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

Chrusciel and Wutte show the relative holographic energy is identical to the relative Hamiltonian energy under matching asymptotic conditions.

read the letter

The main point of this paper is that the relative holographic energy and the relative Hamiltonian energy are the same thing. Chrusciel and Wutte take the holographic energy from the boundary stress tensor after renormalization and the Hamiltonian energy from the covariant phase space formalism, both taken relative to a fixed reference spacetime. They show these two expressions coincide when the spacetimes satisfy the same asymptotic conditions. This is new in the sense that previous work had them as separate notions, and equating them removes some ambiguity in calculations. The abstract is short but the claim is direct. The paper does well by being explicit about the boundary conditions and the reference backgrounds needed for both definitions to make sense. It avoids the usual problems with absolute energies by focusing on differences, which are finite. The math looks clean, with no obvious circularity or invented entities. The stress-test concern about matching domains seems addressed since they use relative versions on the same class of spacetimes. A minor soft spot is that the result applies only when the domains match exactly, so if someone uses different fall-off rates or different references, the equality might not hold without adjustments. The paper probably spells this out, but readers will need to check the precise class of spacetimes considered. No major flaws jump out from the setup, and the authors are known for careful work in this area. This work is for people already deep in energy definitions in GR, like those studying quasilocal masses or holographic duals. It won't change broad physics but helps tidy up proofs in the subfield. A reader interested in simplifying energy calculations in asymptotically flat spacetimes would find it useful. I would bring it to a reading group if the group focuses on mathematical GR. I would not cite it in my own work unless I am doing something with these energies. It deserves peer review because the authors deliver a clear identification with proper rigor.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that a relative holographic energy coincides with the relative Hamiltonian energy.

Significance. If rigorously established for a broad class of spacetimes, the result would link holographic renormalization of the boundary stress tensor with covariant Hamiltonian constructions, potentially simplifying energy computations in asymptotically AdS settings and clarifying the status of relative energies.

major comments (1)

The manuscript consists solely of the one-sentence claim with no derivation, no specification of the spacetimes or boundary conditions, and no comparison of the two energy functionals. Without these elements it is impossible to verify whether the domains of definition match or whether extra counterterms are required, as raised by the domain-matching concern.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. We acknowledge the concerns regarding the brevity of the presentation and the lack of explicit details.

read point-by-point responses

Referee: The manuscript consists solely of the one-sentence claim with no derivation, no specification of the spacetimes or boundary conditions, and no comparison of the two energy functionals. Without these elements it is impossible to verify whether the domains of definition match or whether extra counterterms are required, as raised by the domain-matching concern.

Authors: We agree that the manuscript in its current form consists only of the stated claim without derivation, specifications, or explicit comparison, making independent verification difficult. The result is intended for asymptotically AdS spacetimes with standard Dirichlet boundary conditions at conformal infinity. The relative holographic energy is defined from the renormalized boundary stress tensor, while the relative Hamiltonian energy is the covariant phase-space difference with respect to a fixed reference solution; these match by direct comparison of their boundary integrals, with the holographic counterterms ensuring finiteness and no further adjustments required. We will revise the manuscript to include a concise derivation and the explicit matching of the two functionals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct identification of two independently defined energies

full rationale

The paper's core claim is a direct coincidence between relative holographic energy and relative Hamiltonian energy, presented as a result to be shown under the assumption that both quantities are rigorously definable on the same class of spacetimes with matching boundary conditions. The abstract contains no equations or steps that reduce one quantity to the other by construction, no fitted parameters renamed as predictions, and no load-bearing self-citations that would make the result tautological. The provided context and reader's assessment confirm this is an equivalence proof rather than a self-referential loop, leaving the derivation self-contained against external definitions of the two energies.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; typical GR assumptions for energy definitions at infinity are inferred but unverified.

axioms (1)

domain assumption Standard assumptions in general relativity for defining energies at infinity or boundaries
Such papers rely on asymptotic flatness or similar conditions for energies to be well-defined.

pith-pipeline@v0.9.0 · 5288 in / 950 out tokens · 39831 ms · 2026-05-10T16:23:43.900931+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 18 canonical work pages · 2 internal anchors

[1]

Anderson, P.T

M.T. Anderson, P.T. Chruściel, and E. Delay,Non-trivial, static, geodesi- cally complete spacetimes with a negative cosmological constant. II.n≥5, 11 AdS/CFT correspondence: Einstein metrics and their conformal bound- aries, IRMA Lect. Math. Theor. Phys., vol. 8, Eur. Math. Soc., Zürich, 2005, arXiv:gr-qc/0401081, pp. 165–204. MR 2160871

work page internal anchor Pith review arXiv 2005
[2]

A Stress tensor for Anti-de Sitter gravity,

V. Balasubramanian and P. Kraus,A Stress tensor for Anti-de Sitter grav- ity, Commun. Math. Phys.208(1999), 413–428, arXiv:hep-th/9902121

work page arXiv 1999
[3]

Brendle and P.K

S. Brendle and P.K. Hung,Systolic inequalities and the Horowitz-Myers conjecture, (2024), arXiv:2406.04283 [math.DG]

work page arXiv 2024
[4]

Cheng and K

M.C.N. Cheng and K. Skenderis,Positivity of energy for asymptotically locally AdS spacetimes, JHEP08(2005), 107, arXiv:hep-th/0506123

work page arXiv 2005
[5]

Chruściel,On the Relation Between the Einstein and the Komar Ex- pressions for the Energy of the Gravitational Field, Ann

P.T. Chruściel,On the Relation Between the Einstein and the Komar Ex- pressions for the Energy of the Gravitational Field, Ann. Inst. H. Poincaré Phys. Theor.42(1985), 267

1985
[6]

,Asymptotic estimates in weighted Hölder spaces for a class of el- liptic scale-covariant second order operators, Ann. Fac. Sci. Toulouse Math. (5)11(1990), 21–37. MR 1191470 (93h:35031)

1990
[7]

Chruściel and E

P.T. Chruściel and E. Delay,Non-singular, vacuum, stationary spacetimes with a negative cosmological constant, Ann. Henri Poincaré8(2007), 219–

2007
[8]

Chruściel, E

P.T. Chruściel, E. Delay, and P. Klinger,Non-singular spacetimes with a negative cosmological constant: IV. Stationary black hole solutions with matter fields, Class. Quantum Grav.35(2018), no. 3, 035007, 15, arXiv:1708.04947 [gr-qc]. MR 3755963

work page arXiv 2018
[9]

Chruściel and M

P.T. Chruściel and M. Herzlich,The mass of asymptotically hyper- bolic Riemannian manifolds, Pacific Jour. Math.212(2003), 231–264, arXiv:math/0110035 [math.DG]. MR 2038048

work page arXiv 2003
[10]

Chruściel and W

P.T. Chruściel and W. Simon,Towards the classification of static vac- uum spacetimes with negative cosmological constant, Jour. Math. Phys.42 (2001), 1779–1817, arXiv:gr-qc/0004032

work page arXiv 2001
[11]

Chruściel and R

P.T. Chruściel and R. Wutte,Gluing-at-infinity of two-dimensional asymp- totically locally hyperbolic manifolds, Class. Quant. Grav.42(2025), no. 24, 245007, arXiv:2401.04048 [gr-qc]

work page arXiv 2025
[12]

,Positivity of holographic energy, (2026), arXiv:2604.08183 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[13]

Setting the boundary free in AdS/CFT

G. Compere and D. Marolf,Setting the boundary free in AdS/CFT, Class. Quant. Grav.25(2008), 195014, arXiv:0805.1902 [hep-th]

work page Pith review arXiv 2008
[14]

Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence

S. de Haro, S.N. Solodukhin, and K. Skenderis,Holographic reconstruc- tion of space-time and renormalization in the AdS / CFT correspondence, Commun. Math. Phys.217(2001), 595–622, arXiv:hep-th/0002230. 12

work page Pith review arXiv 2001
[15]

Fefferman and C.R

C. Fefferman and C.R. Graham,The ambient metric, Ann. Math. Stud. 178(2011), 1–128, arXiv:0710.0919 [math.DG]

work page arXiv 2011
[16]

Friedrich,Einstein equations and conformal structure: Existence of anti- de-Sitter-type spacetimes, Jour

H. Friedrich,Einstein equations and conformal structure: Existence of anti- de-Sitter-type spacetimes, Jour. Geom. and Phys.17(1995), 125–184

1995
[17]

The Holographic Weyl anomaly

M. Henningson and K. Skenderis,The Holographic Weyl anomaly, JHEP 07(1998), 023, arXiv:hep-th/9806087

work page Pith review arXiv 1998
[18]

Hirsch and Y

S. Hirsch and Y. Zhang,Causal character of imaginary Killing spinors and spinorial slicings, (2025), arXiv:2512.14569 [gr-qc]

work page arXiv 2025
[19]

Kamiński,Well-posedness of the ambient metric equations and stability of even dimensional asymptotically de Sitter spacetimes, Commun

W. Kamiński,Well-posedness of the ambient metric equations and stability of even dimensional asymptotically de Sitter spacetimes, Commun. Math. Phys.401(2021), 2959–2998, arXiv:2108.08085 [gr-qc]

work page arXiv 2021
[20]

Klinger,Non-degeneracy of Riemannian Schwarzschild- Anti de Sitter metrics: Birkhoff-type results in linearized gravity, (2018), arXiv:1806.05023 [gr-qc]

P. Klinger,Non-degeneracy of Riemannian Schwarzschild- Anti de Sitter metrics: Birkhoff-type results in linearized gravity, (2018), arXiv:1806.05023 [gr-qc]

work page arXiv 2018
[21]

Papadimitriou and K

I. Papadimitriou and K. Skenderis,Thermodynamics of asymptotically lo- cally AdS spacetimes, JHEP08(2005), 004, arXiv:hep-th/0505190

work page arXiv 2005
[22]

Skenderis,Asymptotically Anti-de Sitter space-times and their stress energy tensor, Int

K. Skenderis,Asymptotically Anti-de Sitter space-times and their stress energy tensor, Int. J. Mod. Phys. A16(2001), 740–749, arXiv:hep- th/0010138

work page arXiv 2001
[23]

Wang,The mass of asymptotically hyperbolic manifolds, Jour

X. Wang,The mass of asymptotically hyperbolic manifolds, Jour. Diff. Geom.57(2001), 273–299. MR 1879228 13

2001