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arxiv: 2605.22434 · v1 · pith:FJQ6PUVHnew · submitted 2026-05-21 · 🌀 gr-qc · hep-th

Entropic route to Brown-York tensor: A unified framework for null and timelike hypersurfaces

Krishnakanta Bhattacharya , Bhera Ram , Bibhas Ranjan Majhi This is my paper

Pith reviewed 2026-05-22 05:08 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Brown-York tensornull hypersurfacestimelike hypersurfacesentropy functionalquasi-local energyscalar-tensor theoriesemergent gravity
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0 comments X

The pith

The Brown-York tensor emerges uniformly from the projection of canonical momentum derived from a shared entropy density on both timelike and null hypersurfaces.

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

The paper applies Padmanabhan's entropy functional to construct the Brown-York tensor in a way that works identically for timelike and null hypersurfaces. It treats the tensor as the projection of the momentum conjugate to the normal vector, obtained directly from the entropy density. This approach also accounts for the non-symmetric form of the null version and extends without change to scalar-tensor theories, where the scalar field produces non-conservation of the tensor. The result links bulk gravitational dynamics to boundary quantities through a single variational principle based on entropy.

Core claim

Using the same entropy density, the Brown-York tensor arises naturally as the projection of the canonical momentum conjugate to the normal vectors on the relevant hypersurface, thereby providing a common construction applicable to both timelike and null hypersurfaces. This perspective also offers insight into the structural differences of the null BY tensor, including its non-symmetric character. The formulation reproduces the expected equations of motion and the corresponding BY tensor in scalar-tensor theories while clarifying its non-conservation when the scalar field is non-minimally coupled.

What carries the argument

Padmanabhan's entropy functional, from which the canonical momentum conjugate to the normal is obtained and then projected to define the Brown-York tensor.

If this is right

  • The identical entropy density supplies the Brown-York tensor for both timelike and null hypersurfaces.
  • The non-symmetric character of the null Brown-York tensor follows directly from the projection structure.
  • In scalar-tensor theories the same entropy route reproduces both the equations of motion and the associated Brown-York tensor.
  • Non-conservation of the Brown-York tensor appears when the scalar field is non-minimally coupled.

Where Pith is reading between the lines

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

  • The same entropy-based projection might generate other quasi-local quantities such as the Misner-Sharp mass or Komar integrals.
  • Boundary terms in gravitational actions could be reinterpreted as entropy variations across a wider class of modified theories.
  • Numerical checks on simple black-hole spacetimes with null boundaries could test whether the derived tensor satisfies the expected conservation properties.

Load-bearing premise

Padmanabhan's entropy functional remains valid and directly yields the conjugate momentum projection that defines the Brown-York tensor on both classes of hypersurface.

What would settle it

An explicit computation on a known null hypersurface in general relativity where the projected conjugate momentum from the entropy density fails to equal the standard Brown-York tensor expression would falsify the unified construction.

read the original abstract

Building on Padmanabhan's entropy functional, originally introduced to derive Einstein's equations and highlight the emergent nature of gravity, we demonstrate its robustness in a broader context. Using the same entropy density, we show that the Brown-York (BY) tensor arises naturally as the projection of the canonical momentum conjugate to the normal vectors on the relevant hypersurface, thereby providing a common construction applicable to both timelike and null hypersurfaces. This perspective also offers insight into the structural differences of the null BY tensor, including its non-symmetric character. We further extend the analysis to scalar-tensor theories, showing that the entropy-based formulation reproduces the expected equations of motion along with the corresponding BY tensor, and, clarifies its non-conservation in the presence of additional scalar field which is non-minimally coupled. Our results provide a coherent variational interpretation of quasi-local gravitational quantities and reveal a common underlying structure linking bulk dynamics and boundary momentum.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that Padmanabhan's entropy functional, previously used to derive Einstein's equations, directly yields the Brown-York tensor as the projection of the canonical momentum conjugate to the normal vectors. This construction is asserted to apply uniformly to both timelike and null hypersurfaces, to account for the non-symmetric character of the null BY tensor, and to extend to scalar-tensor theories where it reproduces the equations of motion while clarifying the non-conservation of the boundary term due to a non-minimally coupled scalar field.

Significance. If the central derivation holds without additional counterterms or redefinitions, the work would supply a thermodynamic route to quasi-local gravitational quantities that unifies bulk dynamics with boundary momentum across different hypersurface types. It would also strengthen the entropic interpretation of gravity by showing that the same functional produces both the field equations and the associated boundary tensor in modified theories.

major comments (2)
  1. [null hypersurface derivation] The central step for null hypersurfaces requires that the identical entropy density produces a well-defined conjugate momentum whose projection gives the BY tensor even though the induced metric is degenerate and n^a n_a = 0. The manuscript must supply the explicit variation and projection formulas (presumably in the section deriving the null case) to demonstrate that no extra renormalization is introduced by hand; otherwise the claim of a common construction without modification is not yet secured.
  2. [scalar-tensor theories] In the scalar-tensor extension, the non-conservation of the BY tensor is attributed to the non-minimal coupling. The manuscript should exhibit the precise term arising from the entropy variation that produces this non-conservation (e.g., the contribution proportional to the scalar gradient or its coupling function) so that the result can be checked against the standard field-equation derivation.
minor comments (2)
  1. [abstract] The abstract states that the BY tensor 'arises naturally' but does not indicate whether the projection operator is the same for timelike and null normals or whether a limiting procedure is used; a short clarifying sentence would help readers.
  2. [notation] Notation for the normal vector and its conjugate momentum should be checked for consistency between the timelike and null sections to avoid ambiguity when the same symbols are reused.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. These remarks help clarify the presentation of the unified entropic construction. We respond to each major comment below.

read point-by-point responses

  1. Referee: [null hypersurface derivation] The central step for null hypersurfaces requires that the identical entropy density produces a well-defined conjugate momentum whose projection gives the BY tensor even though the induced metric is degenerate and n^a n_a = 0. The manuscript must supply the explicit variation and projection formulas (presumably in the section deriving the null case) to demonstrate that no extra renormalization is introduced by hand; otherwise the claim of a common construction without modification is not yet secured.

    Authors: We thank the referee for highlighting the need for greater explicitness in the null case. The derivation proceeds by varying the same entropy functional with respect to the normal vector, obtaining the conjugate momentum, and projecting it onto the hypersurface using the (degenerate) induced metric. In the revised manuscript we will insert the explicit variation formula together with the projection step, confirming that the construction employs the identical entropy density and introduces no additional counterterms or renormalizations by hand. revision: yes

  2. Referee: [scalar-tensor theories] In the scalar-tensor extension, the non-conservation of the BY tensor is attributed to the non-minimal coupling. The manuscript should exhibit the precise term arising from the entropy variation that produces this non-conservation (e.g., the contribution proportional to the scalar gradient or its coupling function) so that the result can be checked against the standard field-equation derivation.

    Authors: We agree that displaying the explicit contribution will strengthen the exposition. The non-conservation originates from the variation of the entropy functional in the presence of the non-minimal coupling; the relevant term is proportional to the scalar-field gradient contracted with the coupling function. In the revised version we will isolate and write this term explicitly, allowing direct comparison with the divergence obtained from the bulk field equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; entropic derivation applies external functional to new boundary quantity

full rationale

The paper takes Padmanabhan's entropy functional (an external, previously published construction used to obtain Einstein equations) as given input and applies it to define the conjugate momentum whose projection yields the Brown-York tensor on both timelike and null hypersurfaces. This constitutes a variational reinterpretation rather than a redefinition of the input by construction. The extension to scalar-tensor theories follows the same logic without introducing fitted parameters or self-referential loops. No quoted step reduces the claimed result to a prior fit or to a self-citation whose validity is presupposed inside the paper. The framework is therefore self-contained against the external benchmark of Padmanabhan's entropy and standard hypersurface projections.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the continued applicability of Padmanabhan's entropy density to the conjugate-momentum projection without new parameters or entities.

axioms (1)
  • domain assumption Padmanabhan's entropy functional remains valid for deriving the Brown-York tensor as conjugate momentum on hypersurfaces
    The paper explicitly builds on this functional, originally introduced to derive Einstein's equations.

pith-pipeline@v0.9.0 · 5699 in / 1207 out tokens · 58726 ms · 2026-05-22T05:08:37.034661+00:00 · methodology

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

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Reference graph

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