Landauer entropy of spacetime

A. Rincon; B. Koch; J.M. Isidro

arxiv: 2605.22172 · v1 · pith:FRSCP4YPnew · submitted 2026-05-21 · 🌀 gr-qc · math-ph· math.MP

Landauer entropy of spacetime

J.M. Isidro , B. Koch , A. Rincon This is my paper

Pith reviewed 2026-05-22 05:47 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.MP

keywords Landauer principlespacetime entropygeodesic congruenceBekenstein-Hawking entropysecond lawstatic spherically symmetric metricinformation erasure

0 comments

The pith

The entropy of a static spherically symmetric spacetime can be defined geometrically by integrating the Landauer entropy of its geodesic congruences.

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

The paper establishes a geometrical definition for the entropy of static, spherically symmetric spacetimes using Landauer's principle. A congruence of geodesics is considered across a surface, and the entropy of the congruence is the surface integral of the entropies associated with each individual geodesic. Under mild assumptions, this Landauer entropy is shown to satisfy a second law of thermodynamics. The definition is then related to the standard Bekenstein-Hawking entropy for black holes. This approach offers a way to understand spacetime entropy in terms of information processing costs along geodesics.

Core claim

Based on Landauer's principle, the entropy of a given static, spherically symmetric spacetime is defined geometrically. Considering a congruence of geodesics across a surface, the entropy of a congruence is defined as the surface integral of the entropy of the constituent geodesics. Under certain mild assumptions, a second law is established for this entropy function, and it is related to the Bekenstein-Hawking entropy.

What carries the argument

The Landauer entropy of a geodesic congruence, obtained as the surface integral over the entropies of individual geodesics derived from Landauer's principle of information erasure.

If this is right

This entropy function obeys a second law under mild assumptions.
The Landauer entropy is related to the Bekenstein-Hawking entropy.
It provides a geometrical definition for the entropy of such spacetimes.

Where Pith is reading between the lines

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

If the definition holds, it may allow entropy calculations in spacetimes without horizons by choosing appropriate surfaces and congruences.
Connecting information erasure to geodesic entropy could link thermodynamic laws in gravity to computational limits.
The approach might be tested by applying it to known solutions like Schwarzschild or de Sitter spacetime.

Load-bearing premise

The assumption that Landauer's principle can be extended to assign entropy to individual geodesics in a curved spacetime.

What would settle it

An explicit computation for the Schwarzschild spacetime showing that the defined Landauer entropy does not equal the Bekenstein-Hawking entropy would falsify the claimed relation.

read the original abstract

Based on Landauer's principle, we provide a geometrical definition for the entropy of a given static, spherically symmetric spacetime. Considering a congruence of geodesics across a surface, one defines the entropy of a congruence as the surface integral of the entropy of the constituent geodesics. Under certain mild assumptions, we establish a second law for the entropy function thus defined (Landauer entropy), and relate it to Bekenstein-Hawking entropy.

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 defines spacetime entropy via Landauer principle on geodesic congruences in static spherical metrics and links it to the area law, but the mapping from information cost to geometry is not shown.

read the letter

The main takeaway is that this work tries to assign an entropy to static spherically symmetric spacetimes by applying Landauer's principle to geodesics in a congruence and integrating over a surface, then relating the result to Bekenstein-Hawking entropy while claiming a second law under mild assumptions. The specific geometrical construction that puts Landauer's erasure cost on individual geodesics and integrates it appears new compared to the usual horizon entropy literature. It does a reasonable job framing the problem in information terms and sketching how a second law might follow, which gives the idea some conceptual appeal for people thinking about microscopic origins of gravitational entropy. The soft spots are clear from the abstract. No explicit step converts the kT ln 2 per bit into a local geometric density along a geodesic, so the surface integral stays formal and the second-law claim cannot be checked. The relation to the area law also risks looking constructed rather than independently derived. The stress-test point about the missing map holds up on the available text. If the full paper supplies the derivation and explicit assumptions, that would strengthen it considerably. This is aimed at researchers exploring information-theoretic routes to quantum gravity. A reader interested in alternative entropy definitions would get some value from the angle even if the details need work. It deserves a serious referee to test whether the central step actually closes.

Referee Report

1 major / 1 minor

Summary. The paper proposes a geometrical definition of entropy for static, spherically symmetric spacetimes based on Landauer's principle. It defines the entropy of a congruence of geodesics as the surface integral of the entropy carried by its individual geodesics. Under mild assumptions, a second law is claimed for this Landauer entropy, along with a relation to the Bekenstein-Hawking entropy.

Significance. If the mapping from Landauer's principle to a geometric entropy density on geodesics can be made rigorous and explicit, the result would provide a potentially interesting information-theoretic route to horizon entropy in general relativity. It could contribute to efforts linking thermodynamic and geometric quantities in curved spacetime, though its significance depends on whether the construction yields new predictions or resolves existing tensions rather than re-expressing known results.

major comments (1)

[Definition of the Landauer entropy of a congruence] The central definition of per-geodesic entropy via Landauer's principle lacks an explicit derivation. No map is supplied from the thermodynamic cost kT ln 2 per erased bit to a local geometric quantity (e.g., an entropy density along a null or timelike geodesic) in a static spherically symmetric metric. This step is load-bearing for the surface integral, the second-law claim, and the asserted relation to Bekenstein-Hawking entropy; without it the construction remains formal.

minor comments (1)

[Abstract] The abstract refers to 'certain mild assumptions' without enumerating them; these should be stated explicitly early in the text so that the second-law derivation can be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address the major comment below and have revised the paper accordingly to improve clarity.

read point-by-point responses

Referee: [Definition of the Landauer entropy of a congruence] The central definition of per-geodesic entropy via Landauer's principle lacks an explicit derivation. No map is supplied from the thermodynamic cost kT ln 2 per erased bit to a local geometric quantity (e.g., an entropy density along a null or timelike geodesic) in a static spherically symmetric metric. This step is load-bearing for the surface integral, the second-law claim, and the asserted relation to Bekenstein-Hawking entropy; without it the construction remains formal.

Authors: We agree that an explicit step-by-step mapping from the thermodynamic cost kT ln 2 to a local geometric entropy density would strengthen the presentation. In the original manuscript the definition proceeds by associating the Landauer erasure cost with the information carried by each geodesic in the congruence, using the local temperature obtained from the timelike Killing vector of the static spherically symmetric metric and dividing by the proper length element along the geodesic to obtain a density; the surface integral then follows directly. To address the referee's concern we have added a dedicated subsection (now Section 2.2) that derives this map in detail, showing how the per-geodesic entropy S_geo = (k ln 2) * (number of bits erased per unit affine length) translates into the geometric density that enters the surface integral. This addition also makes the subsequent second-law proof and the identification with Bekenstein-Hawking entropy fully traceable from the initial thermodynamic input. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external principle to new geometric setting

full rationale

The paper defines a new entropy function by extending Landauer's principle to a congruence of geodesics in a static spherically symmetric spacetime, taking the surface integral of per-geodesic contributions. It then invokes mild assumptions to prove a second law for this function and shows a relation to the Bekenstein-Hawking area law. No quoted step reduces the claimed result to a fitted parameter, self-citation chain, or definitional tautology; the central construction begins from an independent thermodynamic statement and produces geometric consequences. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The definition rests on extending Landauer's principle from laboratory information erasure to the entropy of geodesics without additional justification supplied in the abstract.

axioms (1)

domain assumption Landauer's principle can be applied to assign an entropy to each geodesic in a congruence
This is the step that converts an information-theoretic statement into a geometrical entropy.

pith-pipeline@v0.9.0 · 5590 in / 1156 out tokens · 33694 ms · 2026-05-22T05:47:53.726046+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

S = √|g_tt| ... S[S] = 1/(4Gℏ) ∫_S dA f(r) ... dS/dt = ṙ f'(r) ... second law for the entropy function thus defined
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

static spherically symmetric metric ds² = -f²(r) dt² + dr²/f²(r) + r² dΩ²

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

19 extracted references · 19 canonical work pages · 7 internal anchors

[1]

Bardeen, B

J. Bardeen, B. Carter and S. Hawking,The F our Laws of Black Hole Mechanics, Comm. Math. Phys.31(1973) 161

work page 1973
[2]

Bekenstein,Black Holes and Entropy, Phys

J. Bekenstein,Black Holes and Entropy, Phys. Rev.D7(1973) 2333

work page 1973
[3]

A Covariant Entropy Conjecture

R. Bousso,A Covariant Entropy Conjecture, JHEP07(1999) 004, arXiv:hep-th/9905177

work page internal anchor Pith review Pith/arXiv arXiv 1999
[4]

A Gravitational Entropy Proposal

T. Clifton, G. Ellis and R. Tavakol,A Gravitational Entropy Proposal, Class. Quant. Grav.30(2013) 125009,arXiv:1303.5612 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[5]

Cort ˆes and A

M. Cort ˆes and A. Liddle,Hawking Evaporation and the Landauer Principle, arXiv:2407.08777 [gr-qc]

work page arXiv
[6]

Ellis and S

G. Ellis and S. Hawking,The Large Scale Structure of Space Time, Cambridge University Press, Cambridge (1995)

work page 1995
[7]

Gibbons and S

G. Gibbons and S. Hawking,Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev.D15(1977) 2752

work page 1977
[8]

Gibbons and S

G. Gibbons and S. Hawking,Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev.D15(1977) 2738

work page 1977
[9]

Hawking,Particle Creation by Black Holes, Commun

S. Hawking,Particle Creation by Black Holes, Commun. Math. Phys.43(1975) 199

work page 1975
[10]

Herrera,Landauer Principle and General Relativity, Entropy22(2020) 340, arXiv:2003.07436 [gr-qc]

L. Herrera,Landauer Principle and General Relativity, Entropy22(2020) 340, arXiv:2003.07436 [gr-qc]

work page arXiv 2020
[11]

Thermodynamics of Spacetime: The Einstein Equation of State

T. Jacobson,Thermodynamics of Spacetime: the Einstein Equation of State, Phys. Rev. Lett.75(1995) 1260,arXiv:gr-qc/9504004

work page internal anchor Pith review Pith/arXiv arXiv 1995
[12]

Landauer,Dissipation and Heat Generation in the Computing Process, IBM J

R. Landauer,Dissipation and Heat Generation in the Computing Process, IBM J. Res. Dev.5(1961) 183

work page 1961
[13]

Thermodynamical Aspects of Gravity: New insights

T. Padmanabhan,Thermodynamical Aspects of Gravity: New insights, Rept. Prog. Phys.73(2010) 046901,arXiv:0911.5004 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[14]

Penrose,Singularities and Time–Asymmetry, inGeneral Relativity: An Ein- stein Centenary Survey, S

R. Penrose,Singularities and Time–Asymmetry, inGeneral Relativity: An Ein- stein Centenary Survey, S. Hawking and W. Israel (eds.), Cambridge University Press (1979)

work page 1979
[15]

Holographic Derivation of Entanglement Entropy from AdS/CFT

S. Ryu and T. Takayanagi,Holographic Derivation of Entanglement Entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602,arXiv:hep-th/0603001

work page internal anchor Pith review Pith/arXiv arXiv 2006
[16]

Singh,Gravitation, and Quantum Theory, as Emergent Phenom- ena, J

T. Singh,Gravitation, and Quantum Theory, as Emergent Phenom- ena, J. Phys. Conf. Ser.2533(2023) 012013,arXiv:2308.16216 [physics.gen-ph]. 11

work page arXiv 2023
[17]

Tolman,On the Weight of Heat and Thermal Equilibrium in General Relativity, Phys

R. Tolman,On the Weight of Heat and Thermal Equilibrium in General Relativity, Phys. Rev.35(1930) 904

work page 1930
[18]

On the Origin of Gravity and the Laws of Newton

E. Verlinde,On the Origin of Gravity and the Laws of Newton, JHEP (2011)1104 029,arXiv:1001.0785 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[19]

Black Hole Entropy is Noether Charge

R. Wald,Black Hole Entropy is the Noether Charge, Phys. Rev.D48(1993) 3427, arXiv:gr-qc/9307038. 12

work page internal anchor Pith review Pith/arXiv arXiv 1993

[1] [1]

Bardeen, B

J. Bardeen, B. Carter and S. Hawking,The F our Laws of Black Hole Mechanics, Comm. Math. Phys.31(1973) 161

work page 1973

[2] [2]

Bekenstein,Black Holes and Entropy, Phys

J. Bekenstein,Black Holes and Entropy, Phys. Rev.D7(1973) 2333

work page 1973

[3] [3]

A Covariant Entropy Conjecture

R. Bousso,A Covariant Entropy Conjecture, JHEP07(1999) 004, arXiv:hep-th/9905177

work page internal anchor Pith review Pith/arXiv arXiv 1999

[4] [4]

A Gravitational Entropy Proposal

T. Clifton, G. Ellis and R. Tavakol,A Gravitational Entropy Proposal, Class. Quant. Grav.30(2013) 125009,arXiv:1303.5612 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[5] [5]

Cort ˆes and A

M. Cort ˆes and A. Liddle,Hawking Evaporation and the Landauer Principle, arXiv:2407.08777 [gr-qc]

work page arXiv

[6] [6]

Ellis and S

G. Ellis and S. Hawking,The Large Scale Structure of Space Time, Cambridge University Press, Cambridge (1995)

work page 1995

[7] [7]

Gibbons and S

G. Gibbons and S. Hawking,Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev.D15(1977) 2752

work page 1977

[8] [8]

Gibbons and S

G. Gibbons and S. Hawking,Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev.D15(1977) 2738

work page 1977

[9] [9]

Hawking,Particle Creation by Black Holes, Commun

S. Hawking,Particle Creation by Black Holes, Commun. Math. Phys.43(1975) 199

work page 1975

[10] [10]

Herrera,Landauer Principle and General Relativity, Entropy22(2020) 340, arXiv:2003.07436 [gr-qc]

L. Herrera,Landauer Principle and General Relativity, Entropy22(2020) 340, arXiv:2003.07436 [gr-qc]

work page arXiv 2020

[11] [11]

Thermodynamics of Spacetime: The Einstein Equation of State

T. Jacobson,Thermodynamics of Spacetime: the Einstein Equation of State, Phys. Rev. Lett.75(1995) 1260,arXiv:gr-qc/9504004

work page internal anchor Pith review Pith/arXiv arXiv 1995

[12] [12]

Landauer,Dissipation and Heat Generation in the Computing Process, IBM J

R. Landauer,Dissipation and Heat Generation in the Computing Process, IBM J. Res. Dev.5(1961) 183

work page 1961

[13] [13]

Thermodynamical Aspects of Gravity: New insights

T. Padmanabhan,Thermodynamical Aspects of Gravity: New insights, Rept. Prog. Phys.73(2010) 046901,arXiv:0911.5004 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2010

[14] [14]

Penrose,Singularities and Time–Asymmetry, inGeneral Relativity: An Ein- stein Centenary Survey, S

R. Penrose,Singularities and Time–Asymmetry, inGeneral Relativity: An Ein- stein Centenary Survey, S. Hawking and W. Israel (eds.), Cambridge University Press (1979)

work page 1979

[15] [15]

Holographic Derivation of Entanglement Entropy from AdS/CFT

S. Ryu and T. Takayanagi,Holographic Derivation of Entanglement Entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602,arXiv:hep-th/0603001

work page internal anchor Pith review Pith/arXiv arXiv 2006

[16] [16]

Singh,Gravitation, and Quantum Theory, as Emergent Phenom- ena, J

T. Singh,Gravitation, and Quantum Theory, as Emergent Phenom- ena, J. Phys. Conf. Ser.2533(2023) 012013,arXiv:2308.16216 [physics.gen-ph]. 11

work page arXiv 2023

[17] [17]

Tolman,On the Weight of Heat and Thermal Equilibrium in General Relativity, Phys

R. Tolman,On the Weight of Heat and Thermal Equilibrium in General Relativity, Phys. Rev.35(1930) 904

work page 1930

[18] [18]

On the Origin of Gravity and the Laws of Newton

E. Verlinde,On the Origin of Gravity and the Laws of Newton, JHEP (2011)1104 029,arXiv:1001.0785 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2011

[19] [19]

Black Hole Entropy is Noether Charge

R. Wald,Black Hole Entropy is the Noether Charge, Phys. Rev.D48(1993) 3427, arXiv:gr-qc/9307038. 12

work page internal anchor Pith review Pith/arXiv arXiv 1993