Context
Horizon mechanics and thermodynamics: connections, scope, and a chronological guide
Horizon mechanics relates gravitational energy to , , and other boundary data. Horizon thermodynamics gives some of those quantities a physical temperature–entropy interpretation. This guide follows the principal developments discussed in the accompanying manuscript.
Start with the overlap map for the connections, use the separate comparison table to distinguish settings from contributions, and explore the history by date. The links a common boundary potential to explicit horizon-scale work. Its place alongside the wider theories is a connection between results, not a replacement for them.
01 · CONNECTIONS
How the main approaches connect
The overlaps show shared themes. Click a framework to open its history; click the equations or headings for a focused explanation.
The common law includes planar Rindler. The separate volume-work line applies to the stated spherical families, including de Sitter in the equilibrium FRW realization. The gold border identifies this website’s work; other approaches also connect the themes.
Text links for the map
- Hawking radiation
- Unruh thermality
- Gibbons–Hawking cosmological thermality
- Euclidean gravitational thermodynamics
- Jacobson’s equation-of-state argument
- Verlinde’s entropic-force proposal
- black-hole mechanics and thermal translation
- Smarr mass scaling
- Wald / Iyer–Wald
- the Frodden–Ghosh–Perez local first law
- Hayward’s unified first law
- Brown–York quasilocal energy and stress
- complementary boundary variations
- Cai–Kim FRW horizon balance
- Akbar–Cai enclosed-energy balance
- Kodama flow
- this work: macroscopic equilibrium boundary mechanics
- Padmanabhan’s radial identity
- Freidel–Yokokura screen mechanics
02 · COMPARISON
Settings and contributions
The table separates where a formulation is used from what it develops. , , and are different contributions.
Click a check, dot, dash, or scope band for its exact meaning; click an approach name to open its history. preserves the equilibrium FRW scope. The work / thermal column states each law’s role instead of treating all work terms as the same.
Author-row scope: the core Einstein construction. The full manuscript also develops a ; that separate result is not included in this row’s higher-curvature mark.
Readable, selectable table
| Approach / cited formulation | Black-hole horizons | Rindler / local acceleration | de Sitter | Evolving FRW | General boundary / variational construction | Higher-curvature gravity | QFT radiation / detector thermality | Work / thermal role |
|---|---|---|---|---|---|---|---|---|
| Black-hole mechanicsBardeen–Carter–Hawking · 1973 [1] | ||||||||
| Smarr relationStationary mass scaling · 1973 [2] | ||||||||
| Hawking · Unruh · Gibbons–Hawking [3] | ||||||||
| Brown–YorkQuasilocal energy and stress · 1992–93 [4] | ||||||||
| Wald / Iyer–WaldNoether charge and entropy · 1993–94 [5] | ||||||||
| Hayward unified first lawSpherical Einstein gravity · 1997–98 [6] | ||||||||
| PadmanabhanStatic-horizon treatment · 2002 [7] | ||||||||
| Cai–KimFRW apparent-horizon balance · 2005 [8] | ||||||||
| Akbar–CaiFRW enclosed-energy balance · 2006–07 [9] | ||||||||
| Frodden–Ghosh–PerezLocal first law · 2011–13 [10] | ||||||||
| Complementary variationsParattu–Majhi–Padmanabhan; Chakraborty [11] | ||||||||
| Freidel–Yokokura screensGeneral timelike-screen mechanics · 2014–15 [12] | ||||||||
| Hopfmüller–Freidel; Odak et al. · 2016–23 [13] | ||||||||
| FRW screen pressureTHIS WORK · J. T. Tyler · May 2025 [14] | ||||||||
| Macroscopic equilibrium boundary mechanicsTHIS WORK · J. T. Tyler · June 2026 [14] | ||||||||
Comparison source key [1]–[14]
- Bardeen, Carter & Hawking — The four laws of black hole mechanics (1973)
- Smarr — Mass formula for Kerr black holes (1973)
- Hawking — Particle creation by black holes (1975)
Unruh — Notes on black-hole evaporation (1976)
Gibbons & Hawking — Cosmological event horizons, thermodynamics, and particle creation (1977)
Cai, Cao & Hu — Hawking radiation of apparent horizon in a FRW universe (2008/2009)
Acquaviva et al. — Unruh–DeWitt detector and horizon temperature in dynamical spacetimes (2011/2012) - Brown & York — Quasilocal energy and conserved charges derived from the gravitational action
Brown & York — The microcanonical functional integral. I. The gravitational field (1992/1993) - Wald — Black hole entropy is Noether charge (1993)
Iyer & Wald — Some properties of Noether charge and a proposal for dynamical black hole entropy (1994) - Hayward — Unified first law of black-hole dynamics and relativistic thermodynamics
- Padmanabhan — Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes
- Cai & Kim — First law of thermodynamics and Friedmann equations of FRW universe
- Akbar & Cai — Thermodynamic behavior of Friedmann equation at apparent horizon of FRW universe
- Frodden, Ghosh & Perez — A local first law for black hole thermodynamics
- Parattu, Majhi & Padmanabhan — Structure of the gravitational action and its relation with horizon thermodynamics (2013)
Chakraborty & Padmanabhan — Geometrical variables with direct thermodynamic significance (2014) - Freidel & Yokokura — Non-equilibrium thermodynamics of gravitational screens
- Hopfmüller & Freidel — Gravity degrees of freedom on a null surface
Odak, Rignon-Bret & Speziale — General gravitational charges on null hypersurfaces - J. T. Tyler — Author-supplied chronology: FRW screen pressure, May 2025; acceleration-length formulation, June 2026
J. T. Tyler — Horizon boundary mechanics: seven-page derivation
J. T. Tyler — Horizon boundary mechanics: full technical manuscript
03 · CHRONOLOGICAL HISTORY
A chronological guide
Read the previews, then open the entries that interest you. An open entry combines the historical result, its equations, and its connection to the boundary-mechanical picture. Several entries can remain open for comparison.
32 entries
Dates identify cited milestones, with preprint/publication dates distinguished. Same-year entries follow a reading order. The two author milestones are May 2025 and June 2026; newer refinements are not retroactively assigned to those dates.
Black-hole mechanics and the thermal translationClassical mass changes are paired with area, angular momentum, and charge. The area coefficient is the mechanical starting point; the later thermal identification gives the same law its thermodynamic meaning.
What the contribution establishes
Bardeen–Carter–Hawking established the stationary black-hole first law. Its horizon term pairs
Later thermal bridge (1974–1975). Bekenstein’s and Hawking’s temperature give
Connection to boundary mechanics
The familiar macroscopic area response is already present as
Scope. The displayed stationary Einstein–Maxwell mass law includes rotational and electric work. The generalized second law concerns horizon entropy together with outside entropy.
Smarr’s integrated scaling relationA scaling identity displays the horizon product together with rotational and electric products. It is an identity at a state, not a first law integrated with constant coefficients.
What the contribution establishes
The Kerr mass formula and its charged extension expose the mechanical horizon product. In four-dimensional asymptotically flat Einstein–Maxwell gravity, its familiar form is
Using the later thermal identification
Connection to boundary mechanics
The product identifies
Scope. The dimension, cosmological constant and gravitational action determine the appropriate scaling relation. The full differential of U is introduced at this work’s chronological entry.
Hawking radiation and the temperature normalizationQuantum fields on a black-hole spacetime yield a thermal emission spectrum. This gives the classical surface-gravity coefficient a physical temperature and fixes the entropy normalization.
What the contribution establishes
Hawking’s calculation established particle creation by a black hole. In the stationary limit, the temperature is set by ; combined with the first law it fixes the Einstein area entropy.
For comparison across settings, the table groups this result with and as . Hawking-type FRW tunneling calculations exist, but a universal thermal detector response in evolving FRW does not follow without specifying the state, observer, and evolution.
Connection to boundary mechanics
These identifications turn the mechanical product into
Scope. Quantum thermality is additional physical content, even though ħ and k_B cancel in TS and T dS.
Unruh thermality for accelerated observersUniform acceleration changes a detector’s response to the vacuum. Rindler geometry supplies a normal horizon distance; thermality assigns a temperature to that acceleration scale.
What the contribution establishes
A uniformly accelerated observer has a . The vacuum has a thermal response with respect to that observer’s time. On the instantaneous rest slice, the normal distance is
Connection to boundary mechanics
The distance is geometric; Unruh supplies its thermal interpretation, not its existence. This is the direct normal-separation realization of the common .
Scope. The Rindler horizon is planar, not a sphere of radius L. Vacuum thermality does not by itself specify a universal total boundary energy.
Sources: [5]
Cosmological event-horizon thermodynamicsTemperature and entropy also belong to cosmological event horizons. This extends the thermal horizon picture beyond black holes without imposing one common global mass.
What the contribution establishes
Gibbons–Hawking established thermality for , including de Sitter. For de Sitter radius
Connection to boundary mechanics
The common product is again
Scope. Generic evolving FRW apparent horizons are not automatically stationary de Sitter event horizons.
Euclidean gravitational thermodynamicsAn imaginary-time period fixes the stationary horizon temperature. Evaluating the gravitational action relates the thermal partition function to entropy and thermodynamic potentials.
What the contribution establishes
Regularity after continuation to fixes the period at a nonextremal stationary horizon. The action then organizes a and its thermodynamic potential.
Connection to boundary mechanics
The relation
Scope. The temperature period is not, by itself, the complete on-shell action or a choice of total energy.
Sources: [7]
Kodama flow and spherical energy fluxSpherical symmetry supplies a preferred flow even when spacetime is not stationary. It supports an energy current without requiring a global timelike Killing vector.
What the contribution establishes
Kodama identified a locally conserved energy flux for spherical systems. The resulting plays a role similar to a stationary time symmetry in a setting where such a symmetry may be absent.
Connection to boundary mechanics
The result supplies geometric flow and energy-flux structure for spherical applications. It does not equate every Kodama boundary charge with the .
Scope. The construction relies on spherical symmetry. Later Kodama-based Noether entropy is a separate dated entry.
Sources: [37]
The membrane paradigmFor exterior calculations, a black hole can be represented by a stretched surface with fluid and electromagnetic properties.
What the contribution establishes
The cited membrane formulation replaces the horizon by a carrying effective surface stress, conductivity, and . Its evolution rewrites gravitational equations in a fluid-mechanical language.
Connection to boundary mechanics
This work’s emphasis is narrower: the potential’s area response and complementary length work form a compact equilibrium state law. Its scope is not a replacement for the membrane’s dissipative dynamics.
Scope. 1986 dates the cited Price–Thorne treatment, not the first appearance of every membrane idea.
Sources: [15]
Quasilocal energy and boundary stressVarying the action with respect to a finite boundary’s geometry produces an energy and stress tensor. The boundary observers and reference prescription are part of the definition.
What the contribution establishes
Brown–York define and conserved charges by varying the gravitational action with respect to the geometry of a finite boundary. The resulting separates energy, momentum, and spatial stress. Their related also recovers black-hole entropy and a thermodynamic first law.
Connection to boundary mechanics
This supplies a general finite-boundary construction, rather than only one scalar state law. Comparing its energy with
Scope. The Brown–York energy is not generally identical to the horizon potential or the mass at infinity.
Noether-charge entropy and covariant phase spaceThe action supplies horizon entropy, symmetry charges, and a covariant first-law derivation. The construction also applies to theories whose entropy is not simply proportional to area.
What the contribution establishes
Wald identifies black-hole entropy with a . Iyer–Wald develop the derivation and analyze entropy beyond the simplest stationary setting.
Connection to boundary mechanics
In Einstein gravity the entropy term becomes
Scope. Higher-curvature entropy need not be A/(4G) in natural units. The simple Einstein-area mechanical translation is not universal across actions.
Einstein’s equation as an equation of stateAn energy–area balance imposed on every local Rindler horizon, together with focusing and conservation, yields Einstein’s equation. The original argument motivates that balance thermodynamically.
What the contribution establishes
Jacobson uses the , area entropy, and Unruh temperature. Equivalently, its Einstein-area balance can be stated directly in mechanical variables:
Connection to boundary mechanics
The mechanical argument can be carried through. One must still impose this normalized crossing-energy/area balance for every local horizon and use and conservation. Knowing only
Scope. The original thermal interpretation is retained as history; a mechanical restatement does not make a full spacetime field equation follow from one finite-patch state differential.
Sources: [9]
The unified first lawThe gradient of spherical Misner–Sharp energy separates into energy supply and matter work. Projection along a trapping horizon gives a dynamical horizon first law.
What the contribution establishes
Einstein’s equations in spherical symmetry yield
Connection to boundary mechanics
This is already a mechanical energy balance, but it follows the spherical geometry and matter supply. Its
Scope. The trapping-horizon projection and orientation determine the sign of the area/surface-gravity contribution.
Sources: [20]
Dynamical Noether entropy with Kodama flowThe Kodama flow can enter a Noether construction for spherical dynamical horizons. This connects a preferred flow with horizon entropy without assuming global stationarity.
What the contribution establishes
The authors use to extend a treatment to dynamical black-hole entropy in spherical symmetry.
Connection to boundary mechanics
The normal-flow dependence helps explain why area and acceleration data deserve separate attention. The resulting charge is not automatically the full Misner–Sharp energy.
Scope. This is a spherical dynamical construction, not a universal stationary Smarr relation.
Sources: [26]
Charges and fluxes with radiationRadiation through a boundary complicates the definition of conserved generators. A boundary prescription separates charges from the fluxes that obstruct ordinary Hamiltonian conservation.
What the contribution establishes
Wald–Zoupas formulate a prescription for charges in settings with flux, using the and boundary data. The central issue is in the presence of radiation.
Connection to boundary mechanics
This clarifies why a selected source-response one-form, an integrated potential, and a Hamiltonian mass variation are distinct objects. The scalar equilibrium response does not replace the full charge-and-flux construction.
Scope. A boundary contribution and its ambiguity must be fixed as part of the definition.
Sources: [19]
Isolated horizons and their first lawA horizon can itself be in equilibrium without the entire exterior spacetime being stationary. Its Hamiltonian evolution yields a first law with appropriately chosen horizon energy.
What the contribution establishes
An imposes equilibrium conditions locally at the horizon. The analysis relates its energy variation to area and charge, with extensions incorporating rotation.
Connection to boundary mechanics
The result supports local horizon mechanics without global stationarity. It still requires a choice of evolution and energy, so its first law is not automatically the complete differential of
Scope. Horizon equilibrium does not assert that all fields outside the horizon are time-independent.
Sources: [24]
Horizon equations as a radial work identityA radial horizon displacement makes an Einstein equation look like an energy–work identity. This is a direct precedent for geometric displacement in macroscopic horizon mechanics.
What the contribution establishes
For the basic static spherical setup, a horizon field equation becomes
The cited 2002 treatment includes black-hole and de Sitter horizons, and a separate planar Rindler entropy/energy calculation. Its Rindler energy assignment is zero; it is not the finite-patch potential used here. The spherical radius-work identity is not asserted to be a planar volume-work law.
Connection to boundary mechanics
This is an established macroscopic radial-work identity with matter-source pressure. The present work identifies a different response: force and pressure conjugate to the acceleration length of the specified boundary potential. In vacuum Schwarzschild the matter pressure vanishes while this potential’s scale work is nonzero; the energies and variations being compared differ.
Scope. The cited field-equation identity and the potential differential refer to different energies and pressures; they are not interchangeable solely because both contain dV.
Sources: [23]
Dynamical-horizon balance lawsFor growing quasilocal horizons, finite area and energy changes are related to matter and gravitational radiation. This makes nonequilibrium horizon mechanics quantitative.
What the contribution establishes
A is built from marginal surfaces. Its balance laws include matter flux and a gravitational contribution involving horizon .
Connection to boundary mechanics
The equations are geometric and mechanical before thermal interpretation. They contain genuine flux and evolving-shape information absent from the uniform two-variable equilibrium potential.
Scope. A finite physical-process balance must be distinguished from a variation between neighboring equilibrium states.
Sources: [25]
Apparent-horizon balance and the Friedmann equationsThe apparent-horizon energy-flux law yields the Friedmann equations when combined with horizon geometry and bulk conservation. The key Einstein-area balance also has a purely mechanical expression.
What the contribution establishes
Cai–Kim use the equilibrium assignment
The same paper also obtains cosmological equations in Gauss–Bonnet and more general , using the corresponding horizon entropy rather than assuming the Einstein area formula universally. De Sitter is included as a special FRW case.
Connection to boundary mechanics
With and the , this mechanical form reproduces the Einstein–FRW argument. The crossing energy is not
Scope. The instantaneous equilibrium-screen assignment L=R_A uses the specified positive κ_eq=c²/R_A. It does not make the evolving history stationary or identify the full geometric Hayward rate with that value.
FRW enclosed-energy balance with work densityThe changing energy enclosed by a cosmological apparent horizon can be organized using a work density and dynamical surface gravity. This complements the crossing-flux treatment.
What the contribution establishes
Akbar–Cai rewrite the Friedmann equations in a horizon first-law form using , and . In a consistent signed convention it is
The treatment explicitly extends to Gauss–Bonnet and . The higher-curvature check in the comparison table credits those developed cases; it does not mean that the Einstein definitions of energy and entropy remain unchanged.
Connection to boundary mechanics
The source of work and its pressure definition matter. The present equilibrium-screen coefficient
Scope. Signed surface gravity and horizon orientation must be handled together; replacing a signed temperature by its magnitude without adjusting the first-law sign changes the equation.
Extended black-hole thermodynamicsVarying the cosmological constant adds a new conjugate pair. The mass energy acts as enthalpy rather than internal energy in this enlarged thermodynamic state space.
What the contribution establishes
The is written as pressure
Connection to boundary mechanics
The extra
Scope. Thermodynamic volume is a conjugate variable and need not equal the spherical areal volume in every geometry.
Sources: [29]
Entropic force and emergent gravityAn entropy–displacement postulate, a screen degree-of-freedom count, and equipartition give force relations. The interpretation addresses emergence, not just a rearranged horizon state law.
What the contribution establishes
Verlinde combines a test-particle entropy gradient with temperature and . Where that entropy is identified with ,
Connection to boundary mechanics
At fixed boundary
Scope. Mechanical variables can express the algebra without establishing the statistical or emergent interpretation.
The local-observer first lawObservers held at a fixed near-horizon proper distance assign a local energy and first law. Their distance and clock normalization are explicit parts of the construction.
What the contribution establishes
For nearby observers at fixed small
Connection to boundary mechanics
The inverse-distance local energy is an important antecedent. The present contribution is the force and pressure for the permitted acceleration-scale variation of the specified boundary potential, developed across the stated horizon realizations. A fixed-distance local-observer area law does not itself make that identification; infinity-normalized Schwarzschild
Scope. The distance relation is the near-horizon local-observer one; its approximation and fixed-distance variation should be preserved.
Sources: [13]
Horizon energy as a boost boundary termA boost boundary term connects area, accelerated evolution, and local horizon energy. It supplies an action-based relation between normal motion and the energy-area product.
What the contribution establishes
Bianchi–Wieland relate horizon energy to the boundary term. Area participates in the boundary generator; describes a normal-plane change rather than a transverse deformation.
Connection to boundary mechanics
This is an established mechanical antecedent for the area/rate distinction. A change to reciprocal acceleration length must carry its conjugate coefficient with it.
Scope. The choice of observer and boost normalization matters for the numerical energy.
Sources: [12]
Surface Hamiltonians and complementary variationsThe gravitational boundary variation separates into the counterparts of T dS and S dT. Both the area response and the rate response are established parts of the action-based picture.
What the contribution establishes
The surface has a
The 2014 Chakraborty–Padmanabhan development extends the conjugate-variable construction to . The table groups these cited developments together. Their established
Connection to boundary mechanics
The contribution developed here is a new length-conjugate force and normal-scale pressure: identifying the geometrically normalized work coordinate
Scope. One must distinguish the full product differential from an individual selected term of the action or symplectic potential.
Gravitational screens as viscous mechanical boundariesTimelike gravitational screens can be described through internal energy, surface tension, and fluid-like force balance. The construction includes dynamical and dissipative effects.
What the contribution establishes
Projected Einstein equations take the forms of a first law, tangential surface-tension balance, and a normal balance for a screen.
Connection to boundary mechanics
The present formulation organizes one compact equilibrium boundary potential through area and acceleration length. Its characteristic emphasis is a displacement-adapted work coefficient, within a wider literature that already includes macroscopic screen mechanics.
Scope. A timelike screen, its stress tensor, and its dissipation are not identical to a restricted nonexpanding null-boundary potential.
Sources: [16]
Canonical degrees of freedom on a null surfaceA null-surface canonical analysis separates gravitational degrees of freedom and their conjugate responses. Scalar data coexist with shape and vector sectors.
What the contribution establishes
The analysis identifies on a , including spin-2, spin-1, and scalar contributions, together with the null boundary and corner action.
Connection to boundary mechanics
The scalar sector supplies a setting for the selected area–rate response. The length-variable formulation uses that response under its prescribed conditions; it does not replace the additional gravitational degrees of freedom.
Scope. A general null-boundary phase space contains more data than uniform A and κ.
Sources: [17]
General gravitational charges on null hypersurfacesBoundary conditions and choices of prescribed variables select charges and fluxes on null hypersurfaces. The variational prescription is part of the definition, not a dispensable convention.
What the contribution establishes
The authors analyze , and charge prescriptions for null boundaries. Different choices affect variational principles and .
Connection to boundary mechanics
This is the relevant setting for specifying which scalar response becomes
Scope. The clock, cuts, source prescription and admissible variations remain essential to a canonical interpretation.
Sources: [18]
Shell work, thermodynamic energy and a spherical pressure alternativeA maximum-force treatment obtains thermodynamic E=M/2 using a shell volume and also records a spherical pressure alternative. The overlap is a limited antecedent, with different energy and variation choices.
What the contribution establishes
Ong’s Section IV uses shell volume 4πR²ε and adopted pressure 1/(32πRε), with ε fixed in the displayed derivative. It gives E=M/2=TS in G=c=1 units. Before Section V it also gives spherical P=1/(16πR²), but declines that interpretation because it conflicts with the chosen Smarr relation.
Connection to boundary mechanics
These are relevant special-case antecedents. The distinguishes shell energy and a maximum-force premise from the specified boundary potential and its length-conjugate response. Matching a formula does not establish an equivalent cross-horizon construction; declining an interpretation does not remove the published antecedent.
Scope. The adopted shell pressure is not generally one half of the spherical alternative. Do not turn a force scale into a universal force-bound proof.
The initial preprint is September 2023; the cited publication-matching v3 is December 2023.
Sources: [43]
Entropy for nonstationary black-hole perturbationsFirst-law entropy on general horizon cuts requires attention to nonstationary perturbations. The appropriate variation is not always obtained by using the area of an arbitrary event-horizon cut.
What the contribution establishes
Visser–Yan study dynamical black-hole entropy and first-law relations away from the , including perturbations of .
Connection to boundary mechanics
This distinguishes perturbative dynamical entropy from the equilibrium potential differential. A mechanically written area term does not erase the choice of horizon cut or the additional nonstationary structure.
Scope. The cited perturbative construction is distinct from a general finite-process law far from equilibrium.
Sources: [27]
Finite physical-process laws far from equilibriumUsing dynamical horizon segments, finite changes can be related to physical fluxes even far from equilibrium. The short presentation appeared in 2025, followed by the detailed analysis in 2026.
What the contribution establishes
This work develops first- and second-law statements for that can be far from equilibrium. Finite changes are caused by physical processes, rather than representing only tangent variations among equilibrium states.
Connection to boundary mechanics
The comparison is complementary: a simple equilibrium potential makes neighboring-state work explicit; this program provides finite-process dynamics and flux laws in a different regime.
Scope. The detailed treatment was first submitted in March 2026; the linked arXiv identifier belongs to the April listing. The short preprint predates it.
No matching entries. Clear the search or choose All approaches.
The common thread is geometry and energy, not one interchangeable energy definition. Mechanical variables can express many of the balances directly; the energy, clock and permitted variation give those balances their meaning.
References and chronology
Primary references and current manuscripts (43)
- Bardeen, Carter & Hawking — The four laws of black hole mechanics (1973)
- Bekenstein — Black holes and entropy (1973)
- Hawking — Particle creation by black holes (1975)
- Smarr — Mass formula for Kerr black holes (1973)
- Unruh — Notes on black-hole evaporation (1976)
- Gibbons & Hawking — Cosmological event horizons, thermodynamics, and particle creation (1977)
- Gibbons & Hawking — Action integrals and partition functions in quantum gravity (1977)
- Iyer & Wald — Some properties of Noether charge and a proposal for dynamical black hole entropy (1994)
- Jacobson — Thermodynamics of spacetime: The Einstein equation of state (1995)
- Parattu, Majhi & Padmanabhan — Structure of the gravitational action and its relation with horizon thermodynamics (2013)
- Chakraborty & Padmanabhan — Geometrical variables with direct thermodynamic significance (2014)
- Bianchi & Wieland — Horizon energy as the boost boundary term (2012)
- Frodden, Ghosh & Perez — A local first law for black hole thermodynamics
- Brown & York — Quasilocal energy and conserved charges derived from the gravitational action
- Price & Thorne — Membrane viewpoint on black holes (1986)
- Freidel & Yokokura — Non-equilibrium thermodynamics of gravitational screens
- Hopfmüller & Freidel — Gravity degrees of freedom on a null surface
- Odak, Rignon-Bret & Speziale — General gravitational charges on null hypersurfaces
- Wald & Zoupas — A general definition of conserved quantities in general relativity
- Hayward — Unified first law of black-hole dynamics and relativistic thermodynamics
- Cai & Kim — First law of thermodynamics and Friedmann equations of FRW universe
- Akbar & Cai — Thermodynamic behavior of Friedmann equation at apparent horizon of FRW universe
- Padmanabhan — Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes
- Ashtekar, Fairhurst & Krishnan — Isolated horizons: Hamiltonian evolution and the first law
- Ashtekar & Krishnan — Dynamical horizons and their properties
- Hayward, Mukohyama & Ashworth — Dynamic black-hole entropy
- Visser & Yan — Properties of dynamical black hole entropy (2024)
- Ashtekar, Paraizo & Shu — Thermodynamics of dynamical black holes beyond perturbation theory (2026)
- Kastor, Ray & Traschen — Enthalpy and the mechanics of AdS black holes
- Verlinde — On the origin of gravity and the laws of Newton
- Faraoni — Cosmological apparent and trapping horizons
- Dey, Liberati & Pranzetti — The black hole quantum atmosphere
- J. T. Tyler — Horizon boundary mechanics: seven-page derivation
- J. T. Tyler — Horizon boundary mechanics: full technical manuscript
- Hawking — Black hole explosions? (1974)
- Wald — Black hole entropy is Noether charge (1993)
- Kodama — Conserved energy flux for the spherically symmetric system (1980)
- Ashtekar, Paraizo & Shu — Thermodynamics of black holes, far from equilibrium (preprint December 2025; publication 2026)
- J. T. Tyler — Author-supplied chronology: FRW screen pressure, May 2025; acceleration-length formulation, June 2026
- Brown & York — The microcanonical functional integral. I. The gravitational field (1992/1993)
- Cai, Cao & Hu — Hawking radiation of apparent horizon in a FRW universe (2008/2009)
- Acquaviva et al. — Unruh–DeWitt detector and horizon temperature in dynamical spacetimes (2011/2012)
- Yen Chin Ong — A Maximum Force Perspective on Black Hole Thermodynamics, Quantum Pressure, and Near-Extremality (2023), EPJC 83, 1068; arXiv:2309.04110v3