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arxiv: 2605.29814 · v1 · pith:VJM6TVWInew · submitted 2026-05-28 · 🌀 gr-qc

Modified Entropy from Action Principle

A. Dehyadegari , A. Sheykhi This is my paper

Pith reviewed 2026-06-29 06:34 UTC · model grok-4.3

classification 🌀 gr-qc
keywords modified gravityapparent horizon entropyKretschmann scalarRicci scalarFriedmann equationsgravity-thermodynamics conjectureFRW universe
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The pith

Extending the Einstein-Hilbert action with a function of the Ricci and Kretschmann scalars produces a general expression for apparent horizon entropy.

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

The authors extend the Einstein-Hilbert action by adding an arbitrary function of the Ricci scalar and the Kretschmann scalar. From this they derive modified Friedmann equations for a flat FRW universe that contain no higher-order derivatives and recover the standard equations in the appropriate limit. They then apply the gravity-thermodynamics conjecture at the apparent horizon together with the first law of thermodynamics to obtain one general expression for the modified entropy. This expression can be evaluated for multiple known entropy models and supplies a consistent thermodynamic link between curvature-invariant gravity and horizon entropy.

Core claim

Extending the Einstein-Hilbert action with an arbitrary function of the Ricci scalar and the Kretschmann scalar yields modified Friedmann equations for a spatially flat FRW universe that remain free of higher-order derivatives. The gravity-thermodynamics conjecture applied at the apparent horizon then supplies a general expression for the modified entropy via the first law of thermodynamics; the same expression computes the entropy for various well-known models and establishes a thermodynamic framework that connects modified gravity theories built from curvature invariants to generalized entropy functions on the cosmological apparent horizon.

What carries the argument

The general expression for apparent horizon entropy obtained by combining the gravity-thermodynamics conjecture with the modified Friedmann equations.

If this is right

  • The modified entropy expression applies to various well-known entropy models.
  • The thermodynamic behavior at the apparent horizon is consistent with the new action principle.
  • The modified Friedmann equations reduce to standard ones in the limiting case.

Where Pith is reading between the lines

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

  • This framework could allow testing modified gravity through cosmological horizon entropy measurements.
  • Similar approaches might apply to other curvature invariants beyond Ricci and Kretschmann.
  • Connections to black hole thermodynamics in modified gravity could be explored using the same method.

Load-bearing premise

The gravity-thermodynamics conjecture remains valid for the new action principle and directly supplies the first law at the apparent horizon without additional corrections from the Kretschmann term.

What would settle it

A calculation or observation showing that the first law at the apparent horizon requires corrections due to the Kretschmann term in this modified theory would falsify the central claim.

read the original abstract

We propose a modified gravity theory by extending the Einstein-Hilbert action with an arbitrary function of the Ricci scalar and the Kretschmann scalar invariants. The resulting modified Friedmann equations for a spatially flat FRW universe are derived, which remain free of higher-order derivatives and reduce to the standard Friedmann equations in the limiting case. Employing the gravity-thermodynamics conjecture, we investigate the thermodynamic behavior at the apparent horizon and derive the corresponding modified entropy. Using the first law of thermodynamics together with the modified Friedmann equations, we obtain a general expression for the apparent horizon entropy. This formalism allows us to compute the modified entropy for various well-known entropy models. Our approach establishes a consistent thermodynamic framework linking modified gravity theories constructed from curvature invariants to generalized entropy functions on the cosmological apparent horizon.

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 / 1 minor

Summary. The paper proposes a modified gravity theory by extending the Einstein-Hilbert action with an arbitrary function f(R, K) of the Ricci scalar and Kretschmann scalar. It derives modified Friedmann equations for a spatially flat FRW universe that remain free of higher-order derivatives and reduce to the standard equations in the appropriate limit. Employing the gravity-thermodynamics conjecture, the work investigates thermodynamic behavior at the apparent horizon, combines the first law with the modified Friedmann equations to obtain a general expression for apparent-horizon entropy, and applies the formalism to compute modified entropies for various well-known entropy models.

Significance. If the derivations hold, the manuscript supplies a thermodynamic framework that connects f(R, K) curvature-invariant gravity to generalized entropy functions on the cosmological apparent horizon. The approach extends the gravity-thermodynamics conjecture to a new class of actions and yields explicit entropy expressions for known models. The result would be of interest for cosmological thermodynamics in modified gravity, provided the first-law application is justified without additional corrections.

major comments (2)
  1. [Abstract] Abstract (thermodynamic behavior paragraph): the gravity-thermodynamics conjecture is invoked directly to supply the first law at the apparent horizon. Because the action contains the Kretschmann scalar, its variation produces curvature-squared contributions whose boundary terms at the apparent horizon are not automatically guaranteed to vanish or to be absorbed into a redefinition of entropy; the manuscript must demonstrate explicitly that no such corrections arise, as this assumption is load-bearing for the general entropy expression.
  2. [Abstract] Abstract: the claim that the modified Friedmann equations are free of higher-order derivatives is asserted without intermediate steps shown. The variation of the Kretschmann term in a general metric typically yields fourth-order derivatives; the paper must display the explicit reduction in the flat FRW case (including any cancellations) to support the central claim that the equations remain second-order.
minor comments (1)
  1. The abstract would be clearer if it stated the explicit functional form of the derived general entropy expression rather than only describing its existence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below, indicating the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract (thermodynamic behavior paragraph): the gravity-thermodynamics conjecture is invoked directly to supply the first law at the apparent horizon. Because the action contains the Kretschmann scalar, its variation produces curvature-squared contributions whose boundary terms at the apparent horizon are not automatically guaranteed to vanish or to be absorbed into a redefinition of entropy; the manuscript must demonstrate explicitly that no such corrections arise, as this assumption is load-bearing for the general entropy expression.

    Authors: We agree that an explicit demonstration is necessary given the Kretschmann term. In the revised manuscript we will add a dedicated subsection (in the thermodynamics section) that computes the relevant surface terms arising from the variation of the Kretschmann scalar evaluated on the flat FRW apparent horizon. We show that, due to the symmetries of the metric and the fact that the horizon is a null surface with vanishing expansion, these contributions either cancel identically or are absorbed into the redefinition of the entropy without introducing extra terms beyond those already present in the modified Friedmann equations. This will make the application of the first law fully justified within the gravity-thermodynamics conjecture for this theory. revision: yes

  2. Referee: [Abstract] Abstract: the claim that the modified Friedmann equations are free of higher-order derivatives is asserted without intermediate steps shown. The variation of the Kretschmann term in a general metric typically yields fourth-order derivatives; the paper must display the explicit reduction in the flat FRW case (including any cancellations) to support the central claim that the equations remain second-order.

    Authors: We accept the referee’s point that the reduction must be shown explicitly. In the revised version we will insert the full variation of the f(R,K) action for the flat FRW metric, step by step, before arriving at the modified Friedmann equations. This will include the explicit cancellation of all fourth-order terms that occurs when the Kretschmann scalar is evaluated on the FRW background and the metric variations are taken, confirming that only second-order equations remain. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external conjecture on independent modified equations

full rationale

The paper first derives modified Friedmann equations directly from varying the f(R,K) action (independent step). It then invokes the gravity-thermodynamics conjecture as an external assumption to supply the first law at the apparent horizon and solves for the entropy expression that satisfies it. No quoted step reduces a claimed prediction or result to a fitted input, self-definition, or self-citation chain by construction. The entropy formula is obtained by algebraic rearrangement of the first law plus the already-derived Friedmann equations, which is a standard non-circular procedure when the conjecture is granted. No load-bearing uniqueness theorem or ansatz is smuggled via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review is abstract-only, so the ledger is limited to items explicitly named. The arbitrary function f(R, K) is introduced without a concrete form or independent evidence. The gravity-thermodynamics conjecture is invoked without derivation.

axioms (1)
  • domain assumption The gravity-thermodynamics conjecture holds for the modified action and supplies the first law at the apparent horizon
    Invoked to obtain the modified entropy from the first law and the derived Friedmann equations.
invented entities (1)
  • Arbitrary function f(R, K) of Ricci and Kretschmann scalars no independent evidence
    purpose: Extend the Einstein-Hilbert action to produce modified gravity
    Introduced as the central new ingredient; no independent evidence or falsifiable prediction for specific f is given in the abstract.

pith-pipeline@v0.9.1-grok · 5657 in / 1542 out tokens · 38438 ms · 2026-06-29T06:34:21.301862+00:00 · methodology

discussion (0)

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