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Context and central results

The classical laws of black-hole mechanics were established in the early 1970s, relating changes in a black hole’s energy to surface gravity and changes in horizon area , recognized as the classical mechanical analogues of temperature and entropy , respectively. Bekenstein proposed that horizon area is proportional to black-hole entropy, and Hawking’s radiation calculation soon supplied a surface-gravity-based physical temperature, fixing the proportionality constant in Bekenstein’s area-entropy proposal and extending black-hole mechanics into thermodynamics . Later work further extended these connections to accelerated observers and cosmological horizons , and showed how gravitational field equations can be expressed through thermodynamic relations.

The contribution developed here is a new boundary force \( F_L=U/L \) and normal-scale pressure \( P_L=F_L/A \) , obtained by identifying acceleration length \( L=c^2/\kappa \) as the geometrically normalized work coordinate of the established Einstein boundary potential . Its complete mechanical law pairs surface stress with area change and force with horizon-scale change. The selected nonexpanding-null action response supplies the canonical justification in its stated sector. Rindler, Schwarzschild and stationary de Sitter realize that sector; evolving equilibrium FRW uses a separate screen calibration, giving \( U=E \) , \( P=\rho/3 \) , and \( dE=T\,dS-P\,dV \) . The force–pressure identification and geometric work picture—not the already established potential or product rule—are the advance claimed here .

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THE SUMMARY 20 October 2026

Horizon mechanics: a concise derivation

This work identifies a new generalized boundary force and normal-scale pressure by selecting the surface-gravity-defined length as the geometrically normalized work coordinate. The established potential is the input; the force–pressure structure and its physical realization are the contribution .

Force, pressure, and work.

1. Gravitational input

Fix \( c \) , \( G \) , the clock normalization , and the Einstein boundary prescription . For uniform stationary , nonexpanding-null data \( (\theta=\delta\theta=0) \) , the gravitational input is the boundary potential and its selected canonical rate response :

\( \begin{gathered}U=\frac{A\kappa c^2}{8\pi G}\\[1.1em]\frac{\Theta_\kappa^{(0)}}{\Delta\tau}=\frac{Ac^2}{8\pi G}\,\delta\kappa\end{gathered} \)

2. Mechanics

Express the same nonzero acceleration scale as a length:

\( \begin{gathered}L=\frac{c^2}{\kappa}\\[1.1em]\delta\kappa=-\frac{c^2}{L^2}\,\delta L\end{gathered} \)

The characteristic time is \( t_\kappa=c/\kappa \) ; its light-travel scale is \( L=c\,t_\kappa \) . The Rindler distance normalization selects this length without an extra scale. A spherical family then supplies its relation to radius.

The canonical response therefore becomes

\( \frac{\Theta_\kappa^{(0)}}{\Delta\tau}=-\underbrace{\frac{Ac^4}{8\pi GL^2}}_{F}\,\delta L \)

Force is the length-conjugate response of the original gravitational action . Its canonical justification is inherited through the invertible coordinate change .

The same substitution in the potential gives

\( U(A,L)=\frac{Ac^4}{8\pi GL} \)

and hence, before restricting area and length to a particular horizon family ,

\( dU=\underbrace{\frac{c^4}{8\pi GL}}_{\Pi}\,dA-\underbrace{\frac{Ac^4}{8\pi GL^2}}_{F}\,dL \)

Thus surface stress and force are the two responses of one potential ; pressure is force per area:

\( \Pi=\left(\frac{\partial U}{\partial A}\right)_L=\frac{U}{A} \)

\( \begin{gathered}F=-\left(\frac{\partial U}{\partial L}\right)_A\\[1.1em]F=\frac{U}{L}=M_U\kappa,\qquad M_U=\frac{U}{c^2}\end{gathered} \)

\( P=\frac{F}{A}=\frac{\Pi}{L}=-\frac{d\Pi}{dL}=\frac{\kappa^2}{8\pi G} \)

The mechanical structure is therefore

\( U=\Pi A=FL=PAL \)

\( dU=\Pi\,dA-F\,dL \)

3. Thermodynamics

Where the standard thermal identification applies, using the positive rate ,

\( \begin{gathered}T=\frac{\hbar\kappa}{2\pi k_Bc}\\[1.1em]S=\frac{k_Bc^3A}{4\hbar G}\end{gathered} \)

direct substitution gives \( TS=U \) and \( T\,dS=\Pi\,dA \) . Consequently,

\( U=TS=\Pi A=FL=PAL \)

\( dU=T\,dS-F\,dL \)

Here \( F=F_L \) and \( P=P_L \) name the common acceleration-length responses. Where \( dV=A\,dL \) , the work is \( P_L\,dV \) . A spherical family with a different relation between \( L \) and \( R \) instead uses the converted radial pressure \( P_R \) .

4. Horizon realizations

Rindler

Uniform acceleration supplies \( \kappa=a_{\mathrm{obs}} \) and \( L=c^2/a_{\mathrm{obs}} \) , the normal observer–horizon distance on the instantaneous rest slice. A finite planar area is chosen independently.

Schwarzschild: work on the horizon sphere

With the clock normalized at infinity , \( L=2R \) and \( U=Mc^2/2 \) , so \( Mc^2=TS+F_LL \) . Express the work through the radius of the same horizon whose area appears in the first term:

\( F_L\,dL=M\kappa\,dR=P_R\,dV \)

\( \begin{gathered}F_R=M\kappa=2F_L\\[1.1em]P_R=\frac{F_R}{A}=2P_L=\frac{\kappa^2}{4\pi G}\end{gathered} \)

Because \( L=2R \) , the fractional changes agree : \( dL/L=dR/R \) . The work describes the same change of the same horizon sphere. Here \( R \) is its areal radius , \( V=4\pi R^3/3 \) , and \( dV=A\,dR \) .

\( dU=\Pi\,dA-F_R\,dR=\Pi\,dA-P_R\,dV \)

This is the law for \( U=Mc^2/2 \) . The ordinary mass law remains \( d(Mc^2)=T\,dS \) ; area and radius change together along the Schwarzschild family.

Equilibrium FRW: length and radius coincide

The separately justified equilibrium FRW assignment has \( L=R_A \) , so \( F_R=F_L \) , \( P_R=P_L \) , and \( F_L\,dL=F_R\,dR_A=P_R\,dV \) . Use

\( \begin{gathered}L=R_A,\qquad A=4\pi R_A^2\\[1.1em]E_{\mathrm{MS}}=\rho V=\frac{c^4R_A}{2G}\\[1.1em]V=\frac{4\pi R_A^3}{3}\end{gathered} \)

Then

\( \begin{gathered}U=E_{\mathrm{MS}}\\[1.1em]P=\frac{E_{\mathrm{MS}}}{AR_A}=\frac{\rho}{3}\\[1.1em]dE_{\mathrm{MS}}=T\,dS-P\,dV\end{gathered} \)

With \( \kappa_E=8\pi G/c^4 \) , spatially flat Einstein–FRW dynamics consequently reads

\( H^2=\frac{8\pi G}{c^2}P \)

\( \frac{2\ddot a}{ac^2}=-\kappa_E(P+p_m) \)

\( \dot P+H(3P+p_m)=0 \)

where \( H=\dot a/a \) , and \( \rho,p_m \) are total bulk energy density and pressure . These are the separately justified cosmological realization —not an extension of the stationary-null canonical theorem by substitution alone.

The established gravitational rate response , expressed through acceleration length , is force–displacement work . The potential’s complete differential supplies surface stress alongside that same force ; pressure is its area density. The construction identifies a new generalized boundary force and normal-scale pressure from the specified Einstein boundary response. It does not postulate an additional interaction or modify the gravitational field equations.

Questions and objections: what is new, why L, and the scope of the result →

The evolving cosmological screen uses its stated instantaneous equilibrium assignment. Realizations, assumptions and support .

J. T. Tyler Force, pressure, and work.

Click a word, symbol, or meaningful expression for its explanation; click an equation number for the complete equation.

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Context and central results

The classical laws of black-hole mechanics were established in the early 1970s, relating changes in a black hole’s energy to surface gravity and changes in horizon area , recognized as the classical mechanical analogues of temperature and entropy , respectively. Bekenstein proposed that horizon area is proportional to black-hole entropy, and Hawking’s radiation calculation soon supplied a surface-gravity-based physical temperature, fixing the proportionality constant in Bekenstein’s area-entropy proposal and extending black-hole mechanics into thermodynamics . Later work further extended these connections to accelerated observers and cosmological horizons , and showed how gravitational field equations can be expressed through thermodynamic relations.

The contribution developed here is a new boundary force \( F_L=U/L \) and normal-scale pressure \( P_L=F_L/A \) , obtained by identifying acceleration length \( L=c^2/\kappa \) as the geometrically normalized work coordinate of the established Einstein boundary potential . Its complete mechanical law pairs surface stress with area change and force with horizon-scale change. The selected nonexpanding-null action response supplies the canonical justification in its stated sector. Rindler, Schwarzschild and stationary de Sitter realize that sector; evolving equilibrium FRW uses a separate screen calibration, giving \( U=E \) , \( P=\rho/3 \) , and \( dE=T\,dS-P\,dV \) . The force–pressure identification and geometric work picture—not the already established potential or product rule—are the advance claimed here .

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HORIZON BOUNDARY MECHANICS 20 October 2026

Technical summary

Force, pressure, and work from the Einstein boundary potential

Identifying acceleration length as the normalized work coordinate gives a new boundary force and normal-scale pressure from the specified Einstein potential. One differential pairs area with surface stress and scale change with force; the selected null response supplies the action-level justification. The stationary canonical sector and equilibrium FRW realization retain their distinct geometric inputs.

1. Geometry and the specified boundary potential

Fix Einstein gravity, the physical clock normalization and the boundary prescription. Displayed coefficients use the positive surface-gravity convention; the signed-branch explanation retains both signs. Write the horizon area as \( A \) . The reciprocal acceleration coordinate is

\( L_\kappa=\frac{c^2}{\kappa},\qquad\kappa\ne0 \)

The selected energy-valued boundary potential is written first in surface gravity and then in acceleration length. Here \( U_\partial \) denotes the same quantity called \( U \) in Summary:

\( U_\partial=\frac{A\kappa c^2}{8\pi G}=\frac{Ac^4}{8\pi GL_\kappa} \)

\( A \) and \( L_\kappa \) separate the partial responses before a physical horizon family relates them. The inverse Einstein coupling \( c^4/(8\pi G) \) has energy-per-length units; the boundary action fixes its coefficient, not dimensional analysis alone. The stationary area–boost interpretation is given below.

2. One differential, common mechanical responses

With the coupling fixed, differentiate the potential with respect to both geometric variables.

\( \begin{aligned}dU_\partial&=\frac{c^4}{8\pi GL_\kappa}\,dA-\frac{Ac^4}{8\pi GL_\kappa^2}\,dL_\kappa\\[0.7em]&=\frac{U_\partial}{A}\,dA-\frac{U_\partial}{L_\kappa}\,dL_\kappa.\end{aligned} \)

The two coefficients identify surface stress and force together. Define the potential’s mass equivalent by \( M_U=U_\partial/c^2 \) :

\( \begin{aligned}\Pi_\partial&=\left(\frac{\partial U_\partial}{\partial A}\right)_{L_\kappa}=\frac{U_\partial}{A}=\frac{c^4}{8\pi GL_\kappa},\\[0.9em]F_L&=-\left(\frac{\partial U_\partial}{\partial L_\kappa}\right)_A=\frac{U_\partial}{L_\kappa}=M_U\kappa.\end{aligned} \)

Force per area gives the normal-scale pressure:

\( P_L=\frac{F_L}{A}=\frac{\Pi_\partial}{L_\kappa}=\frac{c^4}{8\pi GL_\kappa^2}=\frac{\kappa^2}{8\pi G} \)

\( \begin{gathered}U_\partial=\Pi_\partial A=F_LL_\kappa=P_LAL_\kappa,\\[0.9em]dU_\partial=\Pi_\partial\,dA-F_L\,dL_\kappa.\end{gathered} \)

The area contribution changes how much surface is counted; the length contribution changes its boundary energy per area. The equivalent stress change is \( A\,d\Pi_\partial=-F_L\,dL_\kappa \) . No second independent work term is added.

3. The canonical length-source response

Use the selected Einstein null-boundary action, prescribed clock, fixed cut identifications and controlled remaining sources. The scalar rate-source convention is fixed, and allowed variations preserve \( \theta=\delta\theta=0 \) . In the general null scalar pair, the source is \( \mu=\widehat\kappa+\theta/2 \) ; the nonexpanding restriction reduces it to the normalized rate [1].

Restoring acceleration units and the boundary-time label \( \tau \) , the supplied rate response and the substitution \( \delta\kappa=-c^2\delta L_\kappa/L_\kappa^2 \) give

\( \begin{aligned}\Theta^{(0)}_\kappa&=\frac{c^2}{8\pi G}\int d\tau\int_{\mathcal S}\epsilon_{\mathcal S}\,\delta\kappa\\[0.8em]&=-\int d\tau\int_{\mathcal S}\epsilon_{\mathcal S}P_L\,\delta L_\kappa.\end{aligned} \)

The area element \( \epsilon_{\mathcal S} \) remains part of the response density. For uniform data constant over the fixed interval, area integration and division by \( \Delta\tau \) yield

\( \begin{gathered}\frac{\Theta^{(0)}_\kappa}{\Delta\tau}=\frac{Ac^2}{8\pi G}\,\delta\kappa=-F_L\,\delta L_\kappa,\\[0.8em]F_L=\int_{\mathcal S}P_L\,dA=AP_L.\end{gathered} \)

Equivalently, the specified boundary Legendre change exchanges the displayed area and stress sources. Write the reduced scalar one-form per time as \( \vartheta=\Theta^{(0)}/\Delta\tau \) :

\( \begin{gathered}\vartheta_A=-\Pi_\partial\,\delta A,\\[0.7em]\vartheta_\Pi=\vartheta_A+\delta(A\Pi_\partial)=A\,\delta\Pi_\partial=-F_L\,\delta L_\kappa.\end{gathered} \)

These are representations of one scalar pair, \( \Omega^{(0)}/\Delta\tau=\delta A\wedge\delta\Pi_\partial=\delta L_\kappa\wedge\delta F_L \) , with the stated orientation. They are not two independent pairs or the complete exact differential of \( U_\partial \) . Stationary Rindler, Schwarzschild and consistently normalized de Sitter realize this sector. A generic evolving FRW worldtube does not satisfy these hypotheses merely because each cut is marginal.

4. Stationary corner interpretation and thermal translation

At a stationary corner, the Einstein area–boost coefficient \( J_\eta \) has action units. With the same prescribed clock as above, rapidity \( \eta \) accumulates at the rate \( \omega=\kappa/c=c/L_\kappa \) . This gives an action divided by its characteristic time [2]:

\( \begin{gathered}J_\eta=\frac{Ac^3}{8\pi G},\qquad\omega=\frac{\kappa}{c}=\frac{c}{L_\kappa},\\[0.8em]U_\partial=J_\eta\omega=\frac{J_\eta}{L_\kappa/c}=\frac{Ac^4}{8\pi GL_\kappa}.\end{gathered} \)

For the chosen area–boost one-form \( J_\eta\delta\eta \) , a fixed-interval source change gives the same length coefficient:

\( \eta=\frac{c\Delta\tau}{L_\kappa}\quad\Longrightarrow\quad\frac{J_\eta\delta\eta}{\Delta\tau}=-\frac{cJ_\eta}{L_\kappa^2}\,\delta L_\kappa=-F_L\,\delta L_\kappa \)

The matching temperature and Einstein area entropy translate the mechanics into thermodynamics:

\( \begin{gathered}T=\frac{\hbar\kappa}{2\pi c k_B},\qquad S=\frac{k_Bc^3A}{4G\hbar},\\[0.8em]U_\partial=TS=\Pi_\partial A=F_LL_\kappa=P_LAL_\kappa,\\[0.8em]dU_\partial=T\,dS-F_L\,dL_\kappa.\end{gathered} \)

For Rindler , \( \kappa=a_{\mathrm{obs}} \) and \( L_\kappa=c^2/a_{\mathrm{obs}} \) ; a finite planar area is specified independently. Schwarzschild has \( L_\kappa=2R \) , \( U_\partial=Mc^2/2 \) , and \( Mc^2=TS+F_LL_\kappa \) . Its radial work connects the acceleration-scale response to the same horizon sphere.

\( F_L\,dL=M\kappa\,dR=P_R\,dV \)

\( \begin{gathered}F_R=M\kappa=2F_L\\[1.1em]P_R=\frac{F_R}{A}=2P_L=\frac{\kappa^2}{4\pi G}\end{gathered} \)

Because \( L=2R \) , the fractional changes agree : \( dL/L=dR/R \) . The work describes the same change of the same horizon sphere. Here \( R \) is its areal radius , \( V=4\pi R^3/3 \) , and \( dV=A\,dR \) .

\( dU=\Pi\,dA-F_R\,dR=\Pi\,dA-P_R\,dV \)

This is the law for \( U=Mc^2/2 \) . The ordinary mass law remains \( d(Mc^2)=T\,dS \) ; area and radius change together along the Schwarzschild family.

5. Equilibrium FRW: the screen law and cosmological payoff

For spatially flat expanding equilibrium FRW, \( H=\dot a/a>0 \) , take the stated instantaneous equilibrium rate [3]:

\( \begin{gathered}R_A=\frac cH=L_\kappa,\qquad\kappa=cH=\frac{c^2}{R_A},\\[0.7em]A=4\pi R_A^2,\qquad V=\frac{4\pi R_A^3}{3}.\end{gathered} \)

The marginal-horizon condition gives the geometric Misner–Sharp energy; the Einstein–FRW constraint identifies it with \( \rho V \) , where \( \rho \) is total bulk energy density [4]. The potential then has the same value:

\( \begin{gathered}E_{\mathrm{MS}}=\frac{c^4R_A}{2G}=\rho V=U_\partial,\\[0.7em]U_\partial=TS=F_LL_\kappa=3P_LV.\end{gathered} \)

Here \( dV=A\,dR_A=A\,dL_\kappa \) , so the boundary mechanical law becomes the screen first law:

\( \begin{gathered}dE_{\mathrm{MS}}=T\,dS-P_L\,dV,\qquad P_L=\frac{\rho}{3},\\[0.7em]F_L=\frac{E_{\mathrm{MS}}}{c^2}\kappa=\frac{c^4}{2G}.\end{gathered} \)

The equilibrium FRW relation \( L_\kappa=R_A \) makes \( F_R=F_L \) and \( P_R=P_L \) . In the ordinary spherical volume, \( U_\partial=3P_RV \) ; the same identity holds for Schwarzschild using its converted \( P_R \) .

Expansion, acceleration and conservation

The pressure relation rewrites the first Friedmann constraint:

\( \frac{1}{R_A^2}=\frac{H^2}{c^2}=\kappa_EP_L,\qquad H^2=\frac{8\pi G}{c^2}P_L \)

With \( \kappa_E=8\pi G/c^4 \) and total bulk pressure \( p_m \) , cosmic acceleration and bulk conservation are

\( \frac{2\ddot a}{ac^2}=-\kappa_E(P_L+p_m) \)

\( \dot P_L+H(3P_L+p_m)=0,\qquad\rho=3P_L \)

Spatial curvature. For general FRW, use \( R_A=c/\sqrt{H^2+kc^2/a^2} \) and the same equilibrium assignment \( \kappa=c^2/R_A \) . Then \( P_L=\rho/3 \) and the screen identity, acceleration and conservation relations retain their forms. Here \( k \) labels spatial curvature, not surface gravity; \( V=4\pi R_A^3/3 \) is areal volume.

Scope. The primary canonical proof concerns the specified nonexpanding null sector. Stationary de Sitter is included; generic moving FRW uses the exact equilibrium-screen calibration instead. The dynamical Hayward construction retains its rate, matter work and energy supply. No complete moving-FRW canonical reduction or removal of the physical cosmological energy-flow sign is claimed.

[1] Hopfmüller–Freidel, 1611.03096 and 1802.06135; Odak et al., 2309.03854.

[2] Bianchi–Wieland, 1205.5325.

[3] Cai–Kim, hep-th/0501055.

[4] Akbar–Cai, hep-th/0609128.

Questions and objections: what is new, why L, and the scope of the result →

Realizations, assumptions and support

The common work law uses different rates, clocks and energies. Geometry and energy belong in one dictionary; the level of action-source support belongs in another. Canonical describes a specified action and source variation, which can compare equilibrium states.

Null-source correspondences · Full changing-ratio work

Geometry and energy dictionary

Realization

Rate and length

Potential and energy

Work interpretation

Rindler, finite planar patch

Observer acceleration a_obs>0; L=c²/a_obs is normal separation

U=Ac²a_obs/(8πG); no enclosed spherical mass

Independent patch area and length; no forced spherical volume

Schwarzschild

Infinity-normalized stationary generator; L=2R

U=Mc²/2; total ADM energy is Mc²

F_L dL=Mκ dR=P_R dV; P_R=2P_L

Exact stationary de Sitter

Specified static-patch clock; positive magnitude L=R_A

U_+=E_MS; signed orientation must match

Stationary overlap; changing radius generally changes the cosmological source

Evolving FRW: instantaneous equilibrium screen

Assigned positive κ_eq=c²/R_A; L=R_A

U_eq=E_MS; ρ=3P_scr

P_R=P_L=P_scr; exact screen-state law, distinct from crossing heat

Evolving FRW: geometric Hayward rate

Signed κ_H^s; L_H^s=c²/κ_H^s only on a nonzero branch

U_H^s=χE_MS

P_R^H=(p_m−P_scr)/4; full work retains −3E_MS dw/4; P_L^H differs

What is supported, and what remains separate

Setting

Support supplied

Not thereby established

Selected nonexpanding Einstein null sector

Length-source response under the mixed prescription, fixed clock/cuts, tangent variations and controlled remaining sources

Unrestricted moving-boundary theorem or counterterm invariance

Stationary Rindler / Schwarzschild / de Sitter

Matched stationary realizations and boost interpretation

Independent variation of every geometric datum in one fixed-source problem

Equilibrium screen on evolving FRW

Exact identities from the assigned rate and Einstein–Misner–Sharp relation

The sole canonical source of the actual evolving worldtube

Full Hayward FRW

Geometric rate, parent potential, work conversion and on-shell pressure reconstruction

Complete matched moving-boundary response with embedding and transport controlled

Appendix C timelike ensemble

Definite source response and positive restricted gravitational charges

Generic equality of its stress source with either cosmological rate

Expanding null boundary

Rate-length contribution with a separate expansion response

Removal of shear/matter/corners, or automatic extension to a non-null worldtube

Two pressures and matter-pressure reconstruction · Full dictionary and assumptions

Papers and reading editions

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Context and central results

The classical laws of black-hole mechanics were established in the early 1970s, relating changes in a black hole’s energy to surface gravity and changes in horizon area , recognized as the classical mechanical analogues of temperature and entropy , respectively. Bekenstein proposed that horizon area is proportional to black-hole entropy, and Hawking’s radiation calculation soon supplied a surface-gravity-based physical temperature, fixing the proportionality constant in Bekenstein’s area-entropy proposal and extending black-hole mechanics into thermodynamics . Later work further extended these connections to accelerated observers and cosmological horizons , and showed how gravitational field equations can be expressed through thermodynamic relations.

The contribution developed here is a new boundary force \( F_L=U/L \) and normal-scale pressure \( P_L=F_L/A \) , obtained by identifying acceleration length \( L=c^2/\kappa \) as the geometrically normalized work coordinate of the established Einstein boundary potential . Its complete mechanical law pairs surface stress with area change and force with horizon-scale change. The selected nonexpanding-null action response supplies the canonical justification in its stated sector. Rindler, Schwarzschild and stationary de Sitter realize that sector; evolving equilibrium FRW uses a separate screen calibration, giving \( U=E \) , \( P=\rho/3 \) , and \( dE=T\,dS-P\,dV \) . The force–pressure identification and geometric work picture—not the already established potential or product rule—are the advance claimed here .

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HORIZON BOUNDARY MECHANICS 20 October 2026

The intuitive path to the mechanics

From two horizon questions to force, pressure, and work

Equilibrium FRW horizons and stationary Schwarzschild black holes share the same functional definitions for energy , radius , horizon surface stress , temperature , and entropy . FRW describes a universe that is smooth on large scales , with no preferred place or direction .

Here \( \Pi \) denotes the horizon surface stress, or tension : the mechanical coefficient multiplying a change in horizon area .

\( E=\frac{c^4}{2G}R,\qquad R=\frac{2GM}{c^2} \)

\( \Pi=\frac{\kappa c^2}{8\pi G} \)

\( T=\frac{\hbar\kappa}{2\pi k_Bc} \)

\( S=\frac{Ak_B}{4\ell_P^2} \)

The geometry and constants in these formulas use \( M=E/c^2 \) , a prescribed physical clock , and the positive surface-gravity value.

1. The factor of two that raises the questions

We can see a factor of two difference in their surface gravity .

Equilibrium FRW

\( \kappa=\frac{c^4}{2GM}=\frac{c^2}{R} \)

Schwarzschild

\( \kappa=\frac{c^4}{4GM}=\frac{c^2}{2R} \)

And this results in a factor of two difference in how their mechanical and thermal products \( \Pi A \) and \( TS \) relate to energy \( E \) :

Energy identities and changes for equilibrium FRW and Schwarzschild horizons

Horizon

Energy identity

Changes

Equilibrium FRW

\( E=TS=\Pi A \)

\( 2\,dE=T\,dS=\Pi\,dA \)

Schwarzschild

\( \frac{E}{2}=TS=\Pi A \)

\( dE=T\,dS=\Pi\,dA \)

Where is the pressure–volume work term in \( dE=T\,dS-P\,dV \) for FRW , given that the area contribution appears as \( 2\,dE \) ?

What accounts for the other half of the Schwarzschild energy when \( TS=E/2 \) ?

2. Treating the changing FRW horizon as a whole

Cai–Kim relates the entropy change to energy crossing the apparent horizon , using an equilibrium-form horizon temperature . Other formulations include matter work and the changing enclosed volume. The question we want to ask is more specific: since the energy , area , surface stress , temperature , and entropy of the entire changing horizon sphere can all be written in terms of the radius or surface gravity , can a pressure in the familiar force-per-area form described by the radius or surface gravity also be found?

Looking at the horizon through this macroscopic description , we see that given \( E=(c^4/2G)R \) and \( \kappa=c^4/(2GM) \) ,

\( M\kappa=\frac{c^4}{2G}=\frac{E}{R} \)

We can ask how this force may play into the mechanics and thermodynamics, and how it may handle the FRW relation \( 2\,dE=T\,dS \) . Since every quantity except pressure is already specified, we can first ask which pressure makes the familiar first-law form hold for a changing horizon sphere:

\( dE=T\,dS-P\,dV \)

Since \( 2\,dE=T\,dS \) , we immediately find

\( dE=P\,dV \)

Solving for \( P \) , where \( \rho=E/V \) is energy density , gives

\( P=\frac{c^4}{8\pi G R^2}=\frac{\kappa^2}{8\pi G}=\frac{\rho}{3} \)

The same pressure follows directly by differentiating the Misner–Sharp horizon energy and areal volume with respect to radius. This Einstein–FRW derivation does not depend on a stationary null horizon.

Given \( P=F/A \) , we can solve for the force and find

\( F=PA=\frac{c^4}{8\pi G R^2}\times A=\frac{c^4}{2G}=\frac{E}{R}=M\kappa \)

This is the force we found above. The pressure and force now have a direct place in the changing horizon ’s first law .

3. The FRW results in pressure variables

This relation between horizon pressure and energy density also simplifies the presentation of the Friedmann equations . In the spatially flat case, the apparent-horizon radius is \( R=c/H \) , giving

\( H^2=\frac{8\pi G}{c^2}P=\frac{8\pi G}{3c^2}\rho \)

Using the Einstein coupling \( k_G=8\pi G/c^4 \) , the horizon-curvature and cosmic-acceleration relations become

\( \frac{1}{R^2}=k_GP,\qquad\frac{2\ddot a}{ac^2}=-k_G(P+p_m) \)

Here \( a(t) \) is the cosmic scale factor , \( H=\dot a/a \) , and \( p_m \) is the pressure of the bulk matter and fields , distinct from the horizon pressure \( P \) . Horizon pressure sets the inverse-square radius of the horizon , while its combination with bulk pressure determines cosmic acceleration . Expansion accelerates when \( p_m<-P \) .

Bulk energy conservation ,

\( \dot\rho+3H(\rho+p_m)=0 \)

becomes

\( \dot P+H(\rho+p_m)=0 \)

The combination \( \rho+p_m \) , the bulk enthalpy density , therefore controls how horizon pressure evolves. These are the familiar cosmological relations expressed in horizon-pressure variables. Only the identification \( R=c/H \) assumes spatial flatness ; the relations written in terms of the apparent-horizon radius also apply with spatial curvature .

4. What is shared by the different horizons?

So this pressure and force clearly play a role in equilibrium FRW horizons , but how do they relate to Schwarzschild horizons , or the even simpler Rindler acceleration horizons , given their different energy descriptions? Schwarzschild ’s \( dE=\Pi\,dA \) does not need an additional work term in its mass first law , though \( \frac{E}{2}=\Pi A=TS \) raises the question of what mechanical product accompanies the thermal half in the energy identity .

The contribution \( \Pi\,dA \) describes the area response at a given horizon surface stress . The Einstein boundary potential gives a common mechanical description through the shared area and surface-gravity structure:

\( U=\frac{A\kappa c^2}{8\pi G}=\Pi A \)

For equilibrium FRW , this value is exactly the Misner–Sharp horizon energy . The dynamical Kodama–Noether construction has the same area–rate structure with its own rate and work allocation. Matching values on a horizon family does not automatically identify their independent source variations.

With the Bekenstein–Hawking temperature and entropy , this gives

\( U=\Pi A=TS \)

As \( U \) is the common mechanical boundary potential , we can take its complete differential to see how it changes:

\( dU=\frac{U}{A}\,dA+\frac{U}{\kappa}\,d\kappa=T\,dS+S\,dT \)

The coefficient \( U/A=\Pi \) is the familiar horizon surface stress , and \( T\,dS \) is the area contribution familiar from horizon thermodynamics. The other contribution, \( S\,dT \) , describes a change in the temperature scale. But \( d\kappa \) does not yet display the length or volume displacement we expect in mechanical work .

5. The natural displacement coordinate

We can look for a natural displacement coordinate associated with this complementary contribution. Rindler horizons have no spherical radius like FRW and Schwarzschild horizons . Instead, a uniformly accelerated observer in flat spacetime has a horizon a proper distance \( c^2/a_{\mathrm{obs}} \) away, measured on the observer’s instantaneous rest slice . With \( \kappa=a_{\mathrm{obs}} \) , this gives the acceleration length :

\( L=\frac{c^2}{\kappa} \)

Surface gravity defines a characteristic time \( t_\kappa=c/\kappa \) . Its corresponding light-travel length is \( L=ct_\kappa=c^2/\kappa \) . In Rindler geometry , this length is also the observer–horizon proper distance on the observer’s instantaneous rest slice.

This makes \( L \) a natural mechanical coordinate . It expresses the normal acceleration scale as a length, so the potential’s response to a change in it can take the familiar force–displacement form . Using only \( c \) and \( \kappa \) , it is the available length apart from a numerical factor, which the Rindler distance fixes. We can use that same coordinate in the other examples— \( L=R \) for equilibrium FRW and \( L=2R \) for Schwarzschild —without requiring a universal proper-distance interpretation. The differential below identifies its force coefficient; the stationary boundary action supplies the canonical check in its stated setting.

Rewriting the boundary potential using this length gives

\( U=\frac{Ac^4}{8\pi GL} \)

Differentiating \( U \) , we find

\( dU=\frac{U}{A}\,dA-\frac{U}{L}\,dL=T\,dS+S\,dT \)

As \( L \) is now the mechanical displacement coordinate , we recognize the contribution \( -(U/L)\,dL \) as the length-work term , with

\( F_L=\frac{U}{L}=M_U\kappa,\qquad M_U=\frac{U}{c^2} \)

This generalizes the FRW definition \( F=E/R=M\kappa \) to the three horizon descriptions using the boundary potential . Our common boundary-potential first law becomes

\( dU=\Pi\,dA-F_L\,dL=T\,dS-F_L\,dL \)

\( -F_L\,dL=S\,dT \)

The negative length-work contribution is therefore exactly \( S\,dT \) : increasing the acceleration length appears thermally as a decrease in horizon temperature .

Two GR foundations: stationary action and cosmological screen

The Einstein null-boundary response can be written directly in acceleration length. For uniform stationary data and a prescribed clock , it gives the same force coefficient after dividing by the chosen duration. Rindler and stationary Schwarzschild realize this canonical setting . The FRW pressure and screen law instead follow directly from the Einstein constraint and Misner–Sharp horizon energy. Both have a GR foundation; the complete independent-source response of a moving FRW boundary is a further question.

6. The pressure and energy identity

We can now derive the pressure as

\( P=\frac{F_L}{A}=\frac{U}{LA}=\frac{c^4}{8\pi G L^2}=\frac{\kappa^2}{8\pi G} \)

This reveals the pressure as

the familiar macroscopic force-per-area form , \( P=F_L/A \) ;

the horizon surface stress divided by acceleration length , \( P=\Pi/L \) , or \( \Pi=PL \) ;

part of the boundary state relation \( PL^2=\frac{c^4}{8\pi G} \) ;

the same quadratic acceleration–stress form as the Newtonian longitudinal field-stress magnitude , \( g^2/(8\pi G) \) ;

one third of the bulk energy density in equilibrium FRW , \( P=\rho/3 \) .

The common energy identity is

\( U=F_LL=\Pi A=TS=PAL \)

Where the geometry supplies \( dV=A\,dL \) , as it does for equilibrium FRW , the boundary-potential first law becomes

\( dU=T\,dS-P\,dV \)

In this equilibrium FRW realization , \( L=R_A \) , so \( F_R=F_L \) and \( P_R=P_L=P \) . The length work is already the work associated with the radius of the changing horizon sphere.

7. Returning to the black-hole question

As \( U=E/2 \) for Schwarzschild , we have

\( E=F_LL+TS,\qquad F_LL=TS=\frac{E}{2} \)

The new force product supplies a mechanical representation of the second equal product in the Schwarzschild mass identity. Both \( TS \) and \( F_LL \) equal \( U \) ; they are not two independent energy reservoirs. This answers the original “other half” question at the level of the energy identity , while retaining the standard mass first law.

The Schwarzschild work can now refer directly to the same horizon sphere. Since \( L=2R \) , the fractional changes agree : \( dL/L=dR/R \) . The same work is

\( F_L\,dL=F_R\,dR=M\kappa\,dR=P_R\,dV \)

\( \begin{gathered}F_R=M\kappa=2F_L=\frac{c^4}{4G}=\frac{F_{\mathrm{FRW}}}{2}\\[1.1em]P_R=\frac{F_R}{A}=2P_L=\frac{\kappa^2}{4\pi G}\end{gathered} \)

Here \( M=E/c^2 \) is the full black-hole mass, while \( M_U=M/2 \) . The common pressure is still \( P=P_L \) ; the radial pressure is \( P_R=2P_L \) . Since \( V=4\pi R^3/3 \) and \( dV=A\,dR \) , both area and work now describe the same changing horizon sphere.

\( dU=\Pi\,dA-F_R\,dR=\Pi\,dA-P_R\,dV \)

The potential remains \( U=E/2 \) , so the mass first law remains \( dE=T\,dS \) . Area and volume are linked by the same areal radius . Changing \( L \) in fixed proportion to \( R \) describes this horizon-size change; it does not move another surface at \( 2R \) .

8. Entropy in mechanical and quantum lengths

Further, we can define the reduced Compton-equivalent wavelength associated with the boundary potential ,

\( \bar\lambda_{\partial}=\frac{\hbar}{M_Uc}=\frac{\hbar c}{U}=\frac{\hbar c}{TS} \)

Given

\( T=\frac{\hbar\kappa}{2\pi k_Bc}=\frac{\hbar c}{2\pi Lk_B} \)

we have

\( \frac{S}{k_B}=\frac{U}{k_BT}=\frac{2\pi L}{\bar\lambda_{\partial}} \)

Entropy is therefore \( 2\pi \) times the ratio of the horizon ’s acceleration length to the reduced Compton-equivalent wavelength of its boundary energy . Equivalently, it measures the boundary energy in units of the horizon ’s thermal energy scale , \( k_BT \) . This expresses the usual area entropy through the mechanical length and the quantum length associated with the same boundary potential .

Questions and objections: what is new, why L, and the scope of the result →

Supporting papers One-page summary Seven-page paper Full technical manuscript

Two questions. One mechanical framework. 398 authored concept and equation entries · Physical explanations, not automatic algebra descriptions.

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Context and central results

The classical laws of black-hole mechanics were established in the early 1970s, relating changes in a black hole’s energy to surface gravity and changes in horizon area , recognized as the classical mechanical analogues of temperature and entropy , respectively. Bekenstein proposed that horizon area is proportional to black-hole entropy, and Hawking’s radiation calculation soon supplied a surface-gravity-based physical temperature, fixing the proportionality constant in Bekenstein’s area-entropy proposal and extending black-hole mechanics into thermodynamics . Later work further extended these connections to accelerated observers and cosmological horizons , and showed how gravitational field equations can be expressed through thermodynamic relations.

The contribution developed here is a new boundary force \( F_L=U/L \) and normal-scale pressure \( P_L=F_L/A \) , obtained by identifying acceleration length \( L=c^2/\kappa \) as the geometrically normalized work coordinate of the established Einstein boundary potential . Its complete mechanical law pairs surface stress with area change and force with horizon-scale change. The selected nonexpanding-null action response supplies the canonical justification in its stated sector. Rindler, Schwarzschild and stationary de Sitter realize that sector; evolving equilibrium FRW uses a separate screen calibration, giving \( U=E \) , \( P=\rho/3 \) , and \( dE=T\,dS-P\,dV \) . The force–pressure identification and geometric work picture—not the already established potential or product rule—are the advance claimed here .

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HORIZON BOUNDARY MECHANICS · AN INTUITIVE SUMMARY

All About L

Acceleration length is the distance light travels during the characteristic acceleration time of a horizon:

\( L=c\,t_\kappa=\frac{c^2}{\kappa},\qquad t_\kappa=\frac{c}{\kappa} \)

The chosen clock fixes how the horizon’s evolution is measured. Its surface gravity \( \kappa \) supplies the rate \( \kappa/c \) ; \( t_\kappa \) is the reciprocal time scale. A stronger acceleration means a shorter time and a shorter length.

1. See the distance in Rindler

Imagine an observer who accelerates uniformly forever. Light sent forward from sufficiently far behind never catches the observer. The limiting light ray marks the Rindler horizon . On the observer’s instantaneous rest slice, its closest point is exactly \( L=c^2/a_{\mathrm{obs}} \) away. The acceleration time sets this distance scale; it is not a photon’s travel time from the horizon to the accelerating observer.

2. Separate size from acceleration scale

A rate depends on geometry and on the clock used to normalize evolution . Schwarzschild’s usual rate is evaluated at the horizon and normalized at infinity. A different clock redshifts rate and potential; it does not change the geometry into FRW. The instantaneous equilibrium-screen assignment in evolving FRW differs from the full geometric Hayward rate . Exact de Sitter is the stationary overlap. The realization dictionary keeps those settings separate.

Area describes how much transverse surface there is. Acceleration length describes the scale of its normalized evolution. These are distinct boundary data . A particular spherical geometry can connect them through its radius, but the common mechanics needs both.

Why this length? Built only from \( c \) and the prescribed \( \kappa \) , with no extra scale, a length must be proportional to \( c^2/\kappa \) . The Rindler distance fixes the factor. Other coordinates preserve the work when their coefficients transform too ; \( L \) is preferred for this normalized geometric interpretation.

3. Surface stress and horizon-scale work

The established boundary potential combines horizon surface stress \( \Pi \) with horizon area \( A \) :

\( U=\Pi A=\frac{Ac^4}{8\pi G L} \)

Identifying \( L \) as the geometrically normalized work coordinate gives a new length-conjugate force and normal-scale pressure . The potential's change has two mechanical contributions: surface stress paired with changing area, and force paired with changing horizon scale:

\( dU=\Pi\,dA-F_L\,dL \)

\( F_L=\frac{U}{L},\qquad P_L=\frac{F_L}{A}=\frac{\Pi}{L}=\frac{\kappa^2}{8\pi G} \)

The contribution is the force–pressure identification and its geometric work interpretation. The work is force times length change; pressure is force per area. The established area–rate differential supplies the input, and the selected gravitational action supplies the same coefficient under the stated permitted variations. The product-rule explanation is available in Explain.

4. The same length also sets the temperature

The Hawking–Unruh temperature uses exactly this scale. With Einstein area entropy , the thermal and mechanical energy products agree:

\( T=\frac{\hbar c}{2\pi k_B L},\qquad U=TS \)

The same mechanical law therefore reads \( dU=T\,dS-F_L\,dL \) , since \( T\,dS=\Pi\,dA \) . Its length-work contribution has the complementary thermal form \( S\,dT=-F_L\,dL \) .

5. In equilibrium FRW, the scale is the radius

For the equilibrium FRW apparent horizon , \( \kappa=c^2/R_A \) . The acceleration scale is therefore the actual horizon radius , and its length work is directly spherical pressure–volume work :

\( L=R_A,\qquad F_L\,dL=P_L\,dV_A \)

Here \( dV_A=A\,dR_A \) , so the acceleration-length and radial pressures coincide. The area and work terms refer to the same changing horizon sphere.

6. In Schwarzschild, use the radius to show the work

For a Schwarzschild black hole , with the standard clock normalized at infinity , the relation is just as definite:

\( \kappa=\frac{c^2}{2R},\qquad L=2R,\qquad U=\frac{Mc^2}{2} \)

Changing \( L \) now means changing the horizon radius in fixed proportion. They describe the same fractional change of the same black hole :

\( \frac{dL}{L}=\frac{dR}{R} \)

Writing the work with the actual horizon radius makes its mechanical meaning immediate:

\( F_L\,dL=F_R\,dR=M\kappa\,dR=P_R\,dV \)

The radial force is \( F_R=2F_L=M\kappa \) . Its pressure per area transforms with the coordinate:

\( P_R=\frac{F_R}{A}=2P_L=\frac{\kappa^2}{4\pi G} \)

The first law for \( U=Mc^2/2 \) now follows the changing horizon sphere explicitly:

\( dU=\Pi\,dA-M\kappa\,dR=T\,dS-P_R\,dV \)

The volume here is the actual spherical areal volume \( V=4\pi R^3/3 \) , for which \( U=3P_R V \) . The ordinary Schwarzschild mass law remains \( d(Mc^2)=T\,dS \) .

Acceleration length keeps the common meaning of the boundary’s acceleration scale. Radius makes the Schwarzschild realization concrete: the work describes the same change of horizon size that changes its area. The conversion changes the force and pressure coefficients; the work itself is unchanged.

Questions and objections: the contribution, choice of length, and scope →

One framework. Connected explanations. Concise derivation · Intuitive path to the mechanics

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HORIZON BOUNDARY MECHANICS · A FIELD GUIDE

Context

Horizon mechanics and thermodynamics: connections, scope, and a chronological guide

Horizon mechanics relates gravitational energy to area , surface gravity , 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 macroscopic equilibrium formulation developed here 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

Show map

The overlaps show shared themes. Click a framework to open its history; click the equations or headings for a focused explanation. How to read the overlaps

thermal identification: U = TSWald / Iyer–WaldFrodden–Ghosh–PerezLocal-observer first lawHayward unified first lawIsolated / dynamical horizonsBoundary energy &mechanical responsesBrown–YorkGeneral finite-boundary stress and energyNull-boundary formalismsCanonical variables, charges and fluxesComplementary variationsAction-level T δS and S δTSpherical & cosmologicalhorizonsCai–KimAkbar–CaiKodama flowClassical horizon structureTHIS WORK · J. T. TYLERMacroscopic equilibriumboundary mechanicsOne potential · area change · horizon-scale workFOR THE FIXED-RATIO SPHERICAL REALIZATIONSRindler · Schwarzschild · equilibrium FRWde Sitter includedPadmanabhan radial identityFreidel–YokokuraTimelike-screen mechanicsOther approaches also span these themes. Placement emphasizes connections, not unique ownership of an intersection.Rindler uses a normal separation; the spherical cases use areal-radius work. General boundary formalisms are not subsets of this state law.Temperature and entropy: overview Open Hawking radiation Open Unruh thermality Open Gibbons–Hawking cosmological thermality Open Euclidean gravitational thermodynamics Thermodynamic reconstruction: overview Open Jacobson’s equation-of-state argument Open Verlinde’s entropic-force proposal Classical laws and scaling: overview Open black-hole mechanics and thermal translation Open Smarr mass scaling Explain the boundary potential U = ΠA Explain the thermal identification U = TS Open Wald / Iyer–Wald Open the Frodden–Ghosh–Perez local first law Open Hayward’s unified first law Explore isolated and dynamical horizons Boundary energy and mechanical responses: overview Open Brown–York quasilocal energy and stress Explore general null-boundary formalisms Open complementary boundary variations Spherical and cosmological horizons: overview Open Cai–Kim FRW horizon balance Open Akbar–Cai enclosed-energy balance Open Kodama flow Explain classical horizon structure Open this work: macroscopic equilibrium boundary mechanics Explain dU = Π dA − F_L dL Explain the fixed-ratio spherical law dU = T dS − P_R dV Open Padmanabhan’s radial identity Open Freidel–Yokokura screen mechanics

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

Temperature and entropy: overview

Hawking radiation

Unruh thermality

Gibbons–Hawking cosmological thermality

Euclidean gravitational thermodynamics

Thermodynamic reconstruction: overview

Jacobson’s equation-of-state argument

Verlinde’s entropic-force proposal

Classical laws and scaling: overview

black-hole mechanics and thermal translation

Smarr mass scaling

the boundary potential U = ΠA

the thermal identification U = TS

Wald / Iyer–Wald

the Frodden–Ghosh–Perez local first law

Hayward’s unified first law

Explore isolated and dynamical horizons

Boundary energy and mechanical responses: overview

Brown–York quasilocal energy and stress

Explore general null-boundary formalisms

complementary boundary variations

Spherical and cosmological horizons: overview

Cai–Kim FRW horizon balance

Akbar–Cai enclosed-energy balance

Kodama flow

this work: macroscopic equilibrium boundary mechanics

dU = Π dA − F_L dL

the fixed-ratio spherical law dU = T dS − P_R dV

Padmanabhan’s radial identity

Freidel–Yokokura screen mechanics

02 · COMPARISON

Settings and contributions

Show table

The table separates where a formulation is used from what it develops. General boundary constructions , higher-curvature results , and quantum-field thermality are different contributions. Read the symbols

Click a check, dot, dash, or scope band for its exact meaning; click an approach name to open its history. The E qualification 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 stationary Wald-entropy scale-response extension ; that separate result is not included in this row’s higher-curvature mark.

Readable, selectable table

Same marks and qualifications as the reviewed graphic

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 mechanics Bardeen–Carter–Hawking · 1973 [1]

✓

—

—

—

—

—

—

Area; rotation/charge work (no independent radius term)

Smarr relation Stationary mass scaling · 1973 [2]

✓

—

—

—

—

—

—

Integrated products (not a differential work law)

Quantum horizon thermality Hawking · Unruh · Gibbons–Hawking [3]

✓

✓

✓

●

—

—

✓

Predicts thermal radiation or detector response

Brown–York Quasilocal energy and stress · 1992–93 [4]

General finite timelike boundaries: not restricted to one horizon geometry.

✓

—

—

Boundary stress, energy, charges; gravitational thermal ensembles

Wald / Iyer–Wald Noether charge and entropy · 1993–94 [5]

General covariant action/charge framework; the first law uses stationary horizons.

✓

✓

—

Symplectic structure, Noether charges, entropy and the first law

Hayward unified first law Spherical Einstein gravity · 1997–98 [6]

✓

—

✓

✓

—

—

—

Dynamical energy supply plus volume work: W dV

Padmanabhan Static-horizon treatment · 2002 [7]

✓

✓

✓

—

—

—

—

Spherical radial pressure work; Rindler entropy/energy also treated

Cai–Kim FRW apparent-horizon balance · 2005 [8]

—

—

✓

✓

—

✓

—

Crossing-energy balance; horizon temperature/entropy inputs

Akbar–Cai FRW enclosed-energy balance · 2006–07 [9]

—

—

✓

✓

—

✓

—

Changing enclosed energy with work density W dV

Frodden–Ghosh–Perez Local first law · 2011–13 [10]

Accelerated near-horizon observers; stationary and isolated-horizon setting.

—

—

—

Local area law at fixed observer–horizon distance

Complementary variations Parattu–Majhi–Padmanabhan; Chakraborty [11]

Conjugate action variables; thermodynamic evaluation on local horizons.

✓

✓

—

Action-level interpretation of both T δS and S δT

Freidel–Yokokura screens General timelike-screen mechanics · 2014–15 [12]

General timelike screens: not restricted to one named horizon.

✓

—

—

Screen stress, work, viscosity; thermodynamic correspondence

Null-boundary formalisms Hopfmüller–Freidel; Odak et al. · 2016–23 [13]

General null-boundary data; an evolving FRW horizon need not be null.

✓

—

—

Canonical pairs, boundary prescriptions, charges and fluxes

FRW screen pressure THIS WORK · J. T. Tyler · May 2025 [14]

—

—

✓

✓ E

—

—

—

P = ρ/3; calibrated equilibrium screen-state law

Macroscopic equilibrium boundary mechanics THIS WORK · J. T. Tyler · June 2026 [14]

✓

✓

✓

✓ E

—

—

—

One-potential equilibrium state law; length work → spherical radius work.

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

Show history

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.

Neighboring states Boundary-potential change Crossing energy Dynamical balance

All approaches Classical laws Thermality Boundary mechanics Spherical / cosmological Reconstruction

Search the history32 entries

Open visible entries Close all

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. About the dates

1973; thermal identification 1974–1975 Classical laws Thermality Black-hole mechanics and the thermal translation Bardeen–Carter–Hawking; Bekenstein; Hawking Classical 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 surface stress \( \Pi=\kappa c^2/(8\pi G) \) with area change. The stationary time symmetry and clock normalization fix which energy is being varied.

\( \delta(Mc^2)=\frac{\kappa c^2}{8\pi G}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q \)

Later thermal bridge (1974–1975). Bekenstein’s area-entropy proposal and Hawking’s temperature give \( \Pi A=TS \) and \( \Pi\,\delta A=T\,\delta S \) . The quantum constants cancel in these products.

\( \Pi A=\frac{\kappa c^2A}{8\pi G}=TS,\qquad \Pi\,\delta A=T\,\delta S \)

\( \delta(Mc^2)=T\,\delta S+\Omega_H\,\delta J+\Phi_H\,\delta Q \)

Connection to boundary mechanics

The familiar macroscopic area response is already present as \( \Pi\,\delta A \) . The question pursued here is how the complementary acceleration-scale response enters the differential of the boundary potential , not how to amend this mass law.

Scope. The displayed stationary Einstein–Maxwell mass law includes rotational and electric work. The generalized second law concerns horizon entropy together with outside entropy.

Sources: [1] [2] [3]

Close this entry Back to the map ↑

1973 Classical laws Boundary mechanics Smarr’s integrated scaling relation Smarr A 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

\( Mc^2=2\Pi A+2\Omega_HJ+\Phi_HQ \)

Using the later thermal identification \( \Pi A=TS \) gives the thermal scaling identity:

\( Mc^2=2TS+2\Omega_HJ+\Phi_HQ \)

\( U\equiv\Pi A=\frac{A\kappa c^2}{8\pi G}=TS \)

Connection to boundary mechanics

The product identifies \( U\equiv\Pi A=A\kappa c^2/(8\pi G)=TS \) . Its surface-Hamiltonian interpretation is developed in the action-based work below. Schwarzschild has \( U=Mc^2/2 \) , so \( Mc^2=2U \) ; this is not a universal total-energy assignment for every horizon.

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.

Sources: [4] [1] [10]

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1974 announcement; 1975 full calculation Thermality Hawking radiation and the temperature normalization Hawking Quantum 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 semiclassical calculation established particle creation by a black hole. In the stationary limit, the temperature is set by surface gravity ; combined with the first law it fixes the Einstein area entropy.

\( T_H=\frac{\hbar\kappa}{2\pi k_Bc},\qquad S=\frac{k_BAc^3}{4G\hbar} \)

For comparison across settings, the table groups this result with Unruh thermality and Gibbons–Hawking thermality as quantum horizon thermality . 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 \( U=TS \) . They are not needed to differentiate \( U(A,L) \) , but the mechanical differential alone does not predict the radiation spectrum.

Scope. Quantum thermality is additional physical content, even though ħ and k_B cancel in TS and T dS.

Sources: [35] [3] [41] [42]

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1976 Thermality Unruh thermality for accelerated observers Unruh Uniform 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 Rindler horizon . The vacuum has a thermal response with respect to that observer’s time. On the instantaneous rest slice, the normal distance is \( L=c^2/a_{\rm obs} \) , independently of the chosen planar patch area.

\( T_U=\frac{\hbar a_{\rm obs}}{2\pi k_Bc}=\frac{\hbar c}{2\pi k_BL} \)

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 acceleration length .

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]

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1977 Thermality Spherical / cosmological Cosmological event-horizon thermodynamics Gibbons–Hawking Temperature 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 cosmological event horizons , including de Sitter. For de Sitter radius \( R=c/H \) , the positive surface-gravity magnitude is \( c^2/R \) , with the usual area entropy.

\( T_{\rm dS}=\frac{\hbar H}{2\pi k_B} \)

Connection to boundary mechanics

The common product is again \( U=\Pi A=TS \) . Equating it to a particular energy requires an additional energy definition; with the spherical Misner–Sharp assignment it equals \( c^4R/(2G) \) .

Scope. Generic evolving FRW apparent horizons are not automatically stationary de Sitter event horizons.

Sources: [6] [21]

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1977 Thermality Boundary mechanics Euclidean gravitational thermodynamics Gibbons–Hawking An 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 Euclidean time fixes the period at a nonextremal stationary horizon. The action then organizes a partition function and its thermodynamic potential.

\( \Delta\tau_E=\frac{2\pi c}{\kappa}=\frac{\hbar}{k_BT} \)

Connection to boundary mechanics

The relation \( c\Delta\tau_E=2\pi L \) gives the acceleration length a thermal interpretation. The geometric length and its mechanical force can be introduced beforehand.

Scope. The temperature period is not, by itself, the complete on-shell action or a choice of total energy.

Sources: [7]

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1980 Spherical / cosmological Boundary mechanics Kodama flow and spherical energy flux Kodama Spherical 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 preferred flow 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 potential U .

Scope. The construction relies on spherical symmetry. Later Kodama-based Noether entropy is a separate dated entry.

Sources: [37]

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1986 cited formulation; earlier Damour–Znajek roots Boundary mechanics Spherical / cosmological The membrane paradigm Price–Thorne; building on Damour and Znajek For 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 stretched horizon carrying effective surface stress, conductivity, and viscosity . 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]

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1992 preprint; 1993 publication Boundary mechanics Thermality Quasilocal energy and boundary stress Brown–York Varying 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 quasilocal energy and conserved charges by varying the gravitational action with respect to the geometry of a finite boundary. The resulting boundary stress tensor separates energy, momentum, and spatial stress. Their related ensemble construction 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 \( U=\Pi A \) requires matching the clock, boundary, and reference prescription. The formulation developed here uses a specified Einstein horizon sector; it does not reconstruct the full boundary stress tensor.

Scope. The Brown–York energy is not generally identical to the horizon potential or the mass at infinity.

Sources: [14] [40]

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1993 Noether entropy; 1994 covariant refinement Boundary mechanics Classical laws Thermality Noether-charge entropy and covariant phase space Wald; Iyer–Wald The 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 Noether charge . Iyer–Wald develop the covariant phase-space derivation and analyze entropy beyond the simplest stationary setting.

\( \delta(Mc^2)-\Omega_H\,\delta J-\Phi_H\,\delta Q=T\,\delta S_{\rm Wald} \)

Connection to boundary mechanics

In Einstein gravity the entropy term becomes \( \Pi\,\delta A \) . The broader achievement is a covariant variational and charge construction , including higher-curvature gravity . A selected area–acceleration response uses part of this machinery; it does not supply a new general construction of all its charges and fields.

Scope. Higher-curvature entropy need not be A/(4G) in natural units. The simple Einstein-area mechanical translation is not universal across actions.

Sources: [36] [8]

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1995 Reconstruction Thermality Einstein’s equation as an equation of state Jacobson An 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 Clausius relation , area entropy, and Unruh temperature. Equivalently, its Einstein-area balance can be stated directly in mechanical variables:

\( \delta E_{\rm cross}=\Pi\,\delta A\quad\longleftrightarrow\quad\delta Q_{\rm heat}=T\,\delta S \)

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 focusing and conservation. Knowing only \( U=\Pi A \) does not supply that extra flux premise.

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]

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1997 preprint; 1998 publication Spherical / cosmological Boundary mechanics The unified first law Hayward The 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 \( dE=A\psi+W\,dV \) . Here \( \psi \) is the energy-supply one-form and \( W \) the work density . The law exists before a temperature–entropy translation.

\( dE=A\psi+W\,dV,\qquad W=\frac{\rho-p_m}{2} \)

Connection to boundary mechanics

This is already a mechanical energy balance, but it follows the spherical geometry and matter supply. Its \( W \) and \( E \) are not interchangeable with \( P_L \) and \( U \) without the relevant horizon dictionary.

Scope. The trapping-horizon projection and orientation determine the sign of the area/surface-gravity contribution.

Sources: [20]

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1998 preprint; 1999 publication Spherical / cosmological Boundary mechanics Dynamical Noether entropy with Kodama flow Hayward–Mukohyama–Ashworth The 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 Kodama flow to extend a Noether-charge 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]

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1999 preprint; 2000 publication Boundary mechanics Spherical / cosmological Charges and fluxes with radiation Wald–Zoupas Radiation 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 symplectic potential and boundary data. The central issue is integrability 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]

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2000 preprint; 2001 publication Spherical / cosmological Boundary mechanics Classical laws Isolated horizons and their first law Ashtekar–Fairhurst–Krishnan A 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 isolated horizon imposes equilibrium conditions locally at the horizon. The Hamiltonian 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 \( U \) .

Scope. Horizon equilibrium does not assert that all fields outside the horizon are time-independent.

Sources: [24]

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2002 Spherical / cosmological Boundary mechanics Thermality Horizon equations as a radial work identity Padmanabhan A 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 \( T\,dS=dE+p_r\,dV \) . Here \( p_r \) is matter radial stress . Its area term may be written \( \Pi\,dA \) from the outset.

\( \Pi\,dA=dE+p_r\,dV \)

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]

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2003 Spherical / cosmological Boundary mechanics Dynamical-horizon balance laws Ashtekar–Krishnan For 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 dynamical horizon is built from marginal surfaces. Its balance laws include matter flux and a gravitational contribution involving horizon shear .

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]

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2005 Spherical / cosmological Reconstruction Thermality Apparent-horizon balance and the Friedmann equations Cai–Kim The 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 \( T_A=\hbar c/(2\pi k_BR_A) \) and area entropy. Writing \( \Pi_{\rm eq}=c^4/(8\pi GR_A) \) makes the same balance

\( \delta E_{\rm cross}=\Pi_{\rm eq}\,dA=T_A\,dS \)

The same paper also obtains cosmological equations in Gauss–Bonnet and more general Lovelock gravity , 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 bulk conservation and the apparent-horizon geometry , this mechanical form reproduces the Einstein–FRW argument. The crossing energy is not \( dE_{\rm MS} \) for a moving horizon; the screen-state work term concerns that different state comparison.

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.

Sources: [21] [31]

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2006 preprint; 2007 publication Spherical / cosmological Boundary mechanics Thermality FRW enclosed-energy balance with work density Akbar–Cai The 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 enclosed energy , work density and dynamical surface gravity . In a consistent signed convention it is \( dE=T_s\,dS+W\,dV \) .

The treatment explicitly extends to Gauss–Bonnet and Lovelock gravity . 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 \( P=\rho/3 \) is derived from the potential and is not Hayward’s \( W=(\rho-p_m)/2 \) .

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.

Sources: [22] [20]

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2009 Boundary mechanics Classical laws Thermality Extended black-hole thermodynamics Kastor–Ray–Traschen Varying 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 cosmological constant is written as pressure \( P_\Lambda=-\Lambda c^4/(8\pi G) \) ; its variation has a conjugate thermodynamic volume. This motivates the enthalpy interpretation of the mass.

\( P_\Lambda=-\frac{\Lambda c^4}{8\pi G},\qquad d(Mc^2)=T\,dS+\Omega_H\,dJ+\Phi_H\,dQ+V_\Lambda\,dP_\Lambda \)

Connection to boundary mechanics

The extra \( V_\Lambda\,dP_\Lambda \) pair is a classical extended-mechanics direction as well as a thermodynamic one. It differs from \( F_L\,dL \) , and from radius work at fixed gravitational couplings.

Scope. Thermodynamic volume is a conjugate variable and need not equal the spherical areal volume in every geometry.

Sources: [29]

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2010 preprint; 2011 publication Reconstruction Thermality Entropic force and emergent gravity Verlinde An 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 equipartition . Where that entropy is identified with Einstein area entropy , \( F\,\delta x=T\,\delta S \) can be written \( F\,\delta x=\Pi\,\delta A \) ; the area response to particle displacement still needs justification.

\( F_{\rm ent}\,\delta x=T\,\delta S,\qquad E_{\rm eq}=\frac12N_{\rm dof}k_BT \)

Connection to boundary mechanics

At fixed boundary \( U \) , the present state law gives \( T\,dS=F_L\,dL \) and \( dS=2\pi k_B(U/(\hbar c))\,dL \) . This is a boundary counterpart of the entropy gradient, not a derivation of arbitrary-screen equipartition or a test particle’s displacement law.

Scope. Mechanical variables can express the algebra without establishing the statistical or emergent interpretation.

Sources: [30] [34]

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2011 preprint; 2013 publication Boundary mechanics Classical laws Thermality The local-observer first law Frodden–Ghosh–Perez Observers 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 proper distance \( \ell \) , the leading local energy takes the form \( E_\ell=Ac^4/(8\pi G\ell) \) . Its fixed-distance area variation is mechanical before the local thermal interpretation.

\( E_\ell=\frac{Ac^4}{8\pi G\ell},\qquad\delta E_\ell=\frac{c^4}{8\pi G\ell}\,\delta A \)

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 \( L=2R \) is also not that local proper distance.

Scope. The distance relation is the near-horizon local-observer one; its approximation and fixed-distance variation should be preserved.

Sources: [13]

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2012 Boundary mechanics Thermality Horizon energy as a boost boundary term Bianchi–Wieland A 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 boost boundary term. Area participates in the boundary generator; boost angle 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]

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2013; further development 2014 Boundary mechanics Thermality Surface Hamiltonians and complementary variations Parattu–Majhi–Padmanabhan; Chakraborty–Padmanabhan The 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 Hamiltonian has a \( TS \) interpretation, and its variation exhibits complementary contributions . In Einstein variables, the same structure reads

\( d(\Pi A)=\Pi\,dA+A\,d\Pi=T\,dS+S\,dT \)

The 2014 Chakraborty–Padmanabhan development extends the conjugate-variable construction to Lovelock gravity . The table groups these cited developments together. Their established \( T\,\delta S \) and \( S\,\delta T \) interpretation is explicit, not merely a partial thermal analogy.

Connection to boundary mechanics

The contribution developed here is a new length-conjugate force and normal-scale pressure: identifying the geometrically normalized work coordinate \( L=c^2/\kappa \) gives \( A\,d\Pi=-F_L\,dL \) , with \( P_L=F_L/A \) . The established differential is the input; the force–pressure identification, its selected action-source role, and its geometric work realizations are the advance claimed here.

Scope. One must distinguish the full product differential from an individual selected term of the action or symplectic potential.

Sources: [10] [11] [12]

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2014 preprint; 2015 publication Boundary mechanics Spherical / cosmological Thermality Gravitational screens as viscous mechanical boundaries Freidel–Yokokura Timelike 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 Young–Laplace-type normal balance for a viscous 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]

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2016 preprint; 2017 publication Boundary mechanics Spherical / cosmological Canonical degrees of freedom on a null surface Hopfmüller–Freidel A 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 canonical pairs on a null surface , 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]

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2023 Boundary mechanics Spherical / cosmological General gravitational charges on null hypersurfaces Odak–Rignon-Bret–Speziale Boundary 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 boundary conditions , polarizations and charge prescriptions for null boundaries. Different choices affect variational principles and integrability .

Connection to boundary mechanics

This is the relevant setting for specifying which scalar response becomes \( -F_L\,\delta L \) . The same coordinate change preserves a selected one-form; it does not prove independence from every other boundary prescription.

Scope. The clock, cuts, source prescription and admissible variations remain essential to a canonical interpretation.

Sources: [18]

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2023 (v1 September; v3 December) Boundary mechanics Classical laws Thermality Shell work, thermodynamic energy and a spherical pressure alternative Yen Chin Ong A 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 precise comparison 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]

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2024 Spherical / cosmological Boundary mechanics Thermality Entropy for nonstationary black-hole perturbations Visser–Yan First-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 bifurcation surface , including perturbations of stationary horizons .

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]

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May 2025 Spherical / cosmological Boundary mechanics THIS WORK · J. T. TYLER FRW screen pressure: the initial radius-work result J. T. Tyler · first public result The initial screen-pressure result pairs the changing horizon area with radius work in an equilibrium FRW state law. It motivates the later common force–pressure formulation.

What the contribution establishes

For the equilibrium FRW screen , \( E=c^4R_A/(2G) \) and \( A=4\pi R_A^2 \) . The radial force is \( F_R=M\kappa \) , and the screen pressure is force per unit area, \( P=F_R/A \) . The radius response yields a screen pressure and a full screen-state differential:

\( F_R=M\kappa,\qquad P=\frac{F_R}{A}=\frac{\rho}{3},\qquad dE=\Pi\,dA-P\,dV=T\,dS-P\,dV \)

The mechanical point

The positive screen response \( P=\rho/3 \) and its work allocation are the initial contribution. The June 2026 formulation identifies the common geometrically normalized work coordinate and the new length-conjugate force and pressure. The FRW result is an exact constrained-screen realization, distinct from crossing heat and bulk matter pressure.

Scope. \( \rho \) is total bulk energy density; P is not the bulk matter pressure. This compares calibrated screen states, not just energy crossing a horizon. Here \( M=E/c^2 \) , \( \kappa=c^2/R_A \) is the equilibrium-screen acceleration, and \( F_R \) is the radial force.

First public FRW screen-pressure result: May 2025, according to the author’s supplied chronology. Linked manuscripts are current revisions.

Read the discovery path →

Sources: [39] [33] [34]

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December 2025 preprint; detailed treatment 2026 Spherical / cosmological Boundary mechanics Finite physical-process laws far from equilibrium Ashtekar–Paraizo–Shu Using 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 dynamical horizon segments 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.

Sources: [38] [28]

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June 2026 · current presentation refined October 2026 Boundary mechanics Spherical / cosmological THIS WORK · J. T. TYLER Macroscopic equilibrium boundary mechanics J. T. Tyler · acceleration-length formulation Identifying acceleration length as the normalized work coordinate gives a new boundary force and pressure. One potential pairs area change with horizon-scale work and yields exact radius work in the stated spherical families.

What the contribution establishes

The motivating question. If a spherical horizon’s energy and area are controlled by its radius, can its work describe change of that same radius? The contribution is a new length-conjugate force and normal-scale pressure , obtained by identifying \( L=c^2/\kappa \) as the geometrically normalized work coordinate of the established boundary potential \( U=\Pi A \) . This is a force–pressure identification with a selected action-source justification, not a claim to have discovered the potential or its product rule.

\( dU=\Pi\,dA-F_L\,dL,\qquad F_L=\frac UL,\quad P_L=\frac{F_L}{A}=\frac{\kappa^2}{8\pi G} \)

For Rindler , L is the normal observer–horizon separation. For equilibrium FRW , \( L=R_A \) . For Schwarzschild , \( L=2R \) , and the same work becomes radius work on the same horizon sphere:

\( F_L\,dL=F_R\,dR=M\kappa\,dR=P_R\,dV \)

For Schwarzschild, \( F_R=2F_L=M\kappa \) and \( P_R=2P_L \) ; for equilibrium FRW, the length and radial coefficients coincide. On these fixed-ratio spherical families, \( dU=\Pi\,dA-P_R\,dV=T\,dS-P_R\,dV \) . The work refers to the same sphere whose area appears in the other term—not an additional surface at \( 2R \) .

The common product is \( U=\Pi A=F_LL=TS \) . Schwarzschild reveals \( Mc^2=TS+F_LL \) , while retaining \( d(Mc^2)=T\,dS \) . Equilibrium FRW has \( U=E_{\rm MS} \) , including de Sitter; Rindler has a finite-patch boundary potential, not a global mass or spherical volume.

For evolving Einstein–FRW, the signed Hayward radial coefficient and positive screen pressure give density and matter-pressure reconstruction . The fixed-w radial work and the changing-w correction remain distinct; this is an on-shell relation, not a new field equation.

The mechanical point

One potential, the changing area, and work through a geometrically identified length. The new responses supply a simple macroscopic organization of equilibrium boundary mechanics. The established area–rate differential supplies the input; the new force–pressure identification supplies its geometric work interpretation, with the spherical work referring to the same changing horizon sphere. The ordinary mass first law is retained.

Scope. The stated canonical prescription governs the nonexpanding null response; evolving FRW has a separate equilibrium calibration. R is an areal radius , not one universal proper distance. Radius-only work assumes a fixed \( L/R \) ; a variable ratio requires the full length response.

Acceleration-length description introduced June 2026, following the May 2025 public screen-pressure result, according to the author’s supplied chronology. This entry presents the current refined form.

Explore the concise derivation →

Sources: [39] [33] [34] [10] [17] [18]

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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. Mechanical and thermal forms

References and chronology

Author-supplied chronology: the FRW screen-pressure result was first made public in May 2025; the acceleration-length description followed in June 2026. The linked papers are revised manuscripts. These dates do not assert that every later canonical or radial-work clarification appeared in the original releases.

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

Horizon thermodynamics. Connected explanations. Concise derivation · All About L · Questions and objections

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Context and central results

The classical laws of black-hole mechanics were established in the early 1970s, relating changes in a black hole’s energy to surface gravity and changes in horizon area , recognized as the classical mechanical analogues of temperature and entropy , respectively. Bekenstein proposed that horizon area is proportional to black-hole entropy, and Hawking’s radiation calculation soon supplied a surface-gravity-based physical temperature, fixing the proportionality constant in Bekenstein’s area-entropy proposal and extending black-hole mechanics into thermodynamics . Later work further extended these connections to accelerated observers and cosmological horizons , and showed how gravitational field equations can be expressed through thermodynamic relations.

The contribution developed here is a new boundary force \( F_L=U/L \) and normal-scale pressure \( P_L=F_L/A \) , obtained by identifying acceleration length \( L=c^2/\kappa \) as the geometrically normalized work coordinate of the established Einstein boundary potential . Its complete mechanical law pairs surface stress with area change and force with horizon-scale change. The selected nonexpanding-null action response supplies the canonical justification in its stated sector. Rindler, Schwarzschild and stationary de Sitter realize that sector; evolving equilibrium FRW uses a separate screen calibration, giving \( U=E \) , \( P=\rho/3 \) , and \( dE=T\,dS-P\,dV \) . The force–pressure identification and geometric work picture—not the already established potential or product rule—are the advance claimed here .

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HORIZON BOUNDARY MECHANICS · CONTRIBUTION AND SCOPE

Questions and objections

What is new, why acceleration length, and how far the result extends.

The contribution is a new length-conjugate boundary force and normal-scale pressure, developed from the established Einstein boundary potential. These questions separate that claim from its inputs, its geometric realizations, and claims the formulation does not make.

Open a question for the argument and its scope. Underlined terms and equations open the same definitions used in the concise derivation. This page is a guide to the scientific distinctions, not a substitute for the derivations in the papers.

1. Was the differential of the boundary potential already known? The established differential is the starting point. The contribution claimed here is a new length-conjugate force and pressure, with a geometric work interpretation.

Yes—the cited surface-Hamiltonian work establishes the complementary area and surface-gravity contributions, including their \( T\,dS \) and \( S\,dT \) interpretations. This work’s novelty claim is different: identifying the geometrically normalized acceleration length as the work coordinate gives a new generalized boundary force and normal-scale pressure , and organizes their physical and canonical meaning.

The potential, the product rule and acceleration length individually are established ingredients. What is developed here is the force–pressure structure resulting from their specific combination, its selected action-source response, and the exact geometric work realizations. The claim is not that no one could have differentiated the known expression; it is that this mechanical identification and its consequences constitute a contribution to horizon boundary mechanics.

\( F_L=-\left(\frac{\partial U}{\partial L}\right)_A=\frac UL,\qquad P_L=\frac{F_L}{A}=\frac{\kappa^2}{8\pi G} \)

Sources and related explanations: Parattu–Majhi–Padmanabhan (2013)

2. Is this merely a change of variables? The change of variables is elementary; the claim concerns the response quantities and physical interpretation it makes possible.

An invertible substitution cannot add a new bulk degree of freedom or modify Einstein’s equations. That is a strength of a formulation intended to explain established gravity. Its additional content must instead be assessed in the physical identification: which coordinate has operational meaning, which coefficient is conjugate to it, and what the chosen action and geometry justify.

Here the normalized length supplies a force–displacement description; the selected action response identifies the same coefficient; the horizon geometry makes that displacement concrete. Calling this “only reorganization” does not evaluate those steps. Conversely, the substitution alone would not prove them. The contribution is the developed mechanical structure, not the chain rule taken in isolation.

No new predictions—isn’t this just a reorganization? A scientific contribution need not change the predictions of an established theory. It can also identify physically meaningful quantities, explain relationships, and connect different physical settings through a common structure. Prediction and explanation are complementary aims of science.

Horizon physics provides important examples. Bardeen–Carter–Hawking established the laws of black-hole mechanics within general relativity, relating mass, horizon area , surface gravity , and other mechanical quantities. Their contribution did not require replacing Einstein’s equations. [1 ] Jacobson derived Einstein’s equation as a local-horizon equation of state, providing a different account of its physical meaning. [2 ] Cai–Kim recovered the Friedmann equations from apparent-horizon thermodynamics, connecting cosmological dynamics with horizon quantities. [3 ] These examples show why a new interpretation or derivation can matter even when the resulting dynamics are already established.

The contribution claimed here is the identification of a new generalized boundary force and normal-scale pressure through a geometrically normalized work coordinate . Expressing the established Einstein boundary potential in terms of acceleration length ,

\( \begin{gathered}L=\frac{c^2}{\kappa},\\[0.7em]U(A,L)=\frac{Ac^4}{8\pi GL}\end{gathered} \)

gives

\( \begin{gathered}dU=\Pi\,dA-F_L\,dL,\qquad F_L=\frac{U}{L},\\[0.7em]P_L=\frac{F_L}{A}=\frac{\kappa^2}{8\pi G}\end{gathered} \)

Within the stated boundary prescription , the force and pressure are conjugate response quantities with a specified action-source interpretation —not additional interactions postulated alongside gravity. The known potential and its rate-form differential are the starting point; the force–pressure identification and geometric work description are the contribution being assessed. [4 , 5 ]

The explanatory gain is concrete. Surface stress pairs with changing area, while force pairs with changing horizon scale. In the Schwarzschild realization , the same work becomes

\( F_L\,dL=F_R\,dR=M\kappa\,dR=P_R\,dV \)

The work then refers explicitly to the changing radius of the same horizon sphere whose area appears in the other term. The common length formulation and its radius representation explain different aspects of the same mechanics; neither requires modifying the underlying gravitational equations. [4 , 5 ]

Calling this a “reorganization” describes its mathematical relationship to established gravity; it does not settle whether the resulting physical understanding is new or useful. A change of notation alone would not establish a contribution. The relevant questions are whether the work coordinate is physically motivated, whether its conjugate responses are justified, and whether their geometric realizations provide a clearer and more unified mechanical account.

The claim is therefore not that this framework predicts a departure from general relativity. It is that identifying the force, pressure, and geometrically adapted work supplies new mechanical understanding within general relativity. That claim should be evaluated on its derivation, its distinction from prior constructions, and its explanatory usefulness—not rejected solely because the underlying predictions remain unchanged.

J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics 31, 161–170 (1973).

T. Jacobson, “Thermodynamics of Spacetime: The Einstein Equation of State,” Physical Review Letters 75, 1260–1263 (1995).

R.-G. Cai and S. P. Kim, “First Law of Thermodynamics and Friedmann Equations of Friedmann–Robertson–Walker Universe,” Journal of High Energy Physics 02, 050 (2005).

J. T. Tyler, current seven-page horizon-boundary-mechanics paper.

J. T. Tyler, current full horizon-boundary-mechanics manuscript, focused results and supporting appendices edition.

3. Why L rather than surface gravity, radius, or another coordinate? A length from c and the prescribed surface gravity, with no extra scale and the Rindler distance normalization, selects L = c²/κ.

The goal is a macroscopic force–displacement description. Within the stated class—lengths constructed only from \( c \) and acceleration-valued \( \kappa \) , without an additional scale—dimensional consistency gives \( q=Cc^2/\kappa \) . The Rindler proper-distance normalization fixes \( C=1 \) . The stationary boost and thermal descriptions provide compatible interpretations of this same scale.

This is a conditional selection result , not the claim that every other coordinate is invalid. Surface gravity is the regular rate variable and is preferable at \( \kappa=0 \) . A spherical radius becomes an exact work coordinate when its relation to \( L \) is retained. The general length formulation also applies to a planar Rindler patch without introducing a spherical radius.

\( L=\frac{c^2}{\kappa},\qquad t_\kappa=\frac c\kappa,\qquad L=ct_\kappa \)

4. Does changing the coordinate change the force? Yes, its coefficient transforms. The invariant object within a fixed prescription is the work, not the numerical coefficient alone.

For a monotone coordinate \( q=f(L) \) , transform the response with it. For example \( q=2L \) gives \( F_q=F_L/2 \) , preserving the work. Nothing is physically broken; the rescaled coordinate simply loses the direct normalized Rindler-distance interpretation.

Schwarzschild provides a useful physical instance: \( L=2R \) , so \( F_R=2F_L=M\kappa \) and \( P_R=2P_L \) . Those are the same selected work in the actual horizon’s radius and areal-volume variables. A coordinate change is distinct from changing the boundary action or its prescribed sources .

New here refers to the identification and development of the length-conjugate response quantities in the stated prescription. Their transformation under a change of work coordinate does not introduce a new interaction or change the work one-form.

\( F_q\,dq=F_L\,dL=F_R\,dR=P_R\,dV \)

5. Is this a physical work term, or just a quantity with force units? The claim combines an energy derivative, a selected gravitational action response, and a specified geometric displacement.

Dimensions alone would not suffice. The potential identifies the generalized force as a negative length derivative at fixed source area. In the stated nonexpanding-null prescription, the coefficient of the permitted length variation in the action agrees with that force. In Rindler, the length is an observer–horizon normal separation; in the fixed-ratio spherical families, its work is radius work.

This is generalized boundary work in a comparison of configurations. It is not an extra external force on a material horizon, and \( P_L \) is not claimed as a local bulk gravitational stress tensor. The potential’s full differential contains the area response as well: neither work alone nor a partial derivative is the whole energy change on every physical family.

\( dU=\Pi\,dA-F_L\,dL \)

6. Don’t Padmanabhan, the membrane paradigm, and local first laws already have mechanics? They do. Their established mechanical content is an antecedent, not the same identification of energy, work coordinate, and pressure.

Padmanabhan’s radial identity pairs matter-source pressure with horizon-volume change. The membrane description has a two-dimensional surface stress and dissipative responses. Frodden–Ghosh–Perez assign near-horizon observers a local energy with inverse-distance dependence and an area law at fixed observer distance. Each has genuine mechanical content.

The distinction here is the new acceleration-length force and pressure of the specified boundary potential. For vacuum Schwarzschild, matter-pressure work vanishes while this potential’s scale work is nonzero; the potential is \( Mc^2/2 \) , not the mass energy appearing in the ordinary first law. Allowing the stated scale-source variation is also different from a fixed-distance local-observer variation. These comparisons sharpen the claim without denying the earlier macroscopic mechanics.

Ong’s 2023 treatment uses a shell volume and obtains thermodynamic energy E=M/2=TS in G=c=1 units. His adopted pressure is 1/(32πRε); he also gives the spherical alternative 1/(16πR²), but declines it because it conflicts with the Smarr expression used there. The shell pressure is not generally half the spherical value. Here U=Mc²/2 is the specified boundary potential, total ADM energy remains Mc², and the radial response follows from acceleration length without a shell thickness or maximum-force premise. Formula overlap is a relevant antecedent, not equivalence of energies, variations or the complete construction.

Sources and related explanations: Padmanabhan (2002) · Frodden–Ghosh–Perez · Freidel–Yokokura · Ong (2023), §IV · Ong in Context

7. Does the separate FRW calibration undermine the common formulation? No. A common state-law form is distinct from a single unrestricted canonical theorem for every horizon.

The potential supplies one mechanical differential. The specified nonexpanding-null action supplies its canonical source-response interpretation. Stationary Rindler, Schwarzschild and consistently normalized de Sitter realize that sector. A general evolving FRW apparent-horizon worldtube need not be null, so the same proof cannot simply be transferred to it.

The FRW result instead uses the Einstein–Misner–Sharp relation and the equilibrium assignment \( \kappa=c^2/R_A \) . Those inputs give \( U=E_{\rm MS} \) , \( L=R_A \) , and \( P_L=\rho/3 \) . This exact sectoral realization does not depend on pretending that its worldtube satisfies the stationary theorem. The shared mechanics and differing justifications should both be stated.

The realization dictionary separates rates, energies and support. An instantaneous equilibrium-screen assignment does not make an evolving worldtube stationary. A marginal cut and its history have different tangent directions. The Appendix C comparison gives a definite action response to its own stress source, but not the intended cosmological rate in the tested ensemble; its positive restricted charge results remain. Complete moving-boundary source matching is separate.

Sources and related explanations: Realizations, assumptions and support

\( P_{\rm scr}=\frac{c^4}{8\pi G R_A^2}=\frac{\rho}{3},\qquad dE=T\,dS-P_{\rm scr}\,dV \)

Sources and related explanations: Cai–Kim

8. Is every work coordinate literally a distance to a spherical horizon? No. Rindler gives normal separation; the spherical realizations give areal-radius work, with the coefficient transformed.

The motivating radius-work question is answered directly in Schwarzschild and equilibrium FRW: their scale work can use the same horizon radius that determines their area. The full geometric identity also shows exactly when a radius-only description needs another contribution.

Writing \( \alpha=L/R \) gives the formula below. At fixed \( \alpha \) , the second term vanishes; Schwarzschild has \( \alpha=2 \) and equilibrium FRW \( \alpha=1 \) . A changing ratio must be retained. Ordinary Rindler remains planar, and an areal radius is not a universal observer’s proper distance.

\( F_L\,dL=\frac UR\,dR+U\,d\ln|\alpha|,\qquad \alpha=\frac LR \)

When radius-only work is insufficient

Changing the scale ratio matters. With \( \alpha_R=L_s/R \) , the full pullback is \( F_L\,dL_s=F_R\,dR+U\,d\alpha_R/\alpha_R \) . For signed Hayward FRW on a nonzero branch, \( \alpha_R=4/(3w-1) \) , so the extra term is \( -3E_A\,dw/4 \) . The convention here writes the scale work as \( +F_L\,dL_s \) ; it enters the potential differential with a minus sign. For positive-magnitude variables, use \( f=|\chi| \) , \( \chi=(3w-1)/4 \) , and keep the correction \( -E_A\,df \) . At fixed \( w \) the signed radius-only reduction applies. See the existing Hayward answer.

Reissner–Nordström, at fixed geometrized charge \( Q \) , uses \( G=c=1 \) and the outer areal radius \( R \) . The full paper gives \( M=(R^2+Q^2)/(2R) \) , \( M_R=R/2 \) , and \( \kappa=(1-Q^2/R^2)/(2R) \) . The scale-work coefficient along this fixed- \( Q \) variation is \( -[A/(8\pi)](\partial\kappa/\partial R)_Q=[1-3Q^2/R^2]/4 \) , generally neither \( M_R\kappa \) nor \( M\kappa \) . At fixed \( Q/R \) , instead, the ratio is constant and the coefficient is \( M_R\kappa \) . The mass definition and what is held fixed must both be specified.

For nonextremal, uncharged Kerr with the standard infinity-normalized generator, \( r_+ \) is the outer Boyer–Lindquist coordinate, not the areal radius or a proper radial distance. With \( a_{\rm K}=J/(Mc) \) , \( A=4\pi(r_+^2+a_{\rm K}^2) \) and \( U=M\kappa r_+ \) . For \( \alpha_+=L_s/r_+ \) , the work is \( F_L\,dL_s=M\kappa\,dr_++U\,d\alpha_+/\alpha_+ \) . The correction vanishes along fixed-dimensionless-spin \( Jc/(GM^2) \) scaling families; fixed \( J \) or fixed \( a_{\rm K} \) is generally different. Using the areal radius \( R=\sqrt{A/(4\pi)} \) instead gives \( U=M_{\rm irr}\kappa R \) , with \( M_{\rm irr}=c^2R/(2G) \) , not the total Kerr mass. The rotational term in the mass first law remains.

These already-derived examples state when the simple radius-work representation is adequate and when additional variation terms are needed. They are not new empirical predictions or failures of other theories.

Sources and related explanations: Full paper: radius work and its qualifications · Changing the prescribed Hayward rate · Choice and transformation of the work coordinate

9. What happens with the full dynamical Hayward surface gravity? The rate prescription changes the response, and a changing length-to-radius ratio contributes work.

The equilibrium FRW rate and the signed Hayward rate are not interchangeable. In four-dimensional Einstein FRW, the supplied manuscript gives \( \chi=\kappa_H^s/\kappa_{\rm eq}=(3w-1)/4 \) , with \( w=p_m/\rho \) . Hence \( P_L^H=\chi^2 P_{\rm scr} \) . The same pressure rule has a different value because its prescribed rate is different.

On the nonzero signed branch, \( U_H=\chi E \) and \( L_H^s=R_A/\chi \) give \( F_L^H dL_H^s=\chi P_{\rm scr}dV-E\,d\chi \) . The last term is \( -3E\,dw/4 \) . For positive-magnitude variables use \( f=|\chi| \) and retain \( -E\,df \) . This is a diagnostic of the radius-only restriction, not an independent empirical confirmation or a completed moving-boundary action theorem.

Can horizon geometry determine the matter pressure? On Einstein–FRW solutions, the positive screen coefficient and the signed Hayward radial response reconstruct total density and pressure. This is an inverse gravitational identity, not a new matter equation of state.

The screen pressure fixes the homogeneous energy-density scale. The signed Hayward radial response additionally records the pressure contrast of the evolving normal geometry. On Einstein–FRW solutions:

\( \begin{gathered}\rho=3P_{\rm scr},\\[0.6em]p_m=P_{\rm scr}+4P_R^H,\\[0.6em]P_R^H=\frac{c^2\kappa_H^s}{8\pi G R_A}=\frac{p_m-P_{\rm scr}}4\end{gathered} \)

Here \( P_R^H \) is signed. It differs from the nonnegative length-response density \( P_L^H=(P_R^H)^2/P_{\rm scr} \) and from Hayward’s work density \( W=(\rho-p_m)/2 \) . Squaring surface gravity loses the radial sign. The mass in \( F_R^H=M_A\kappa_H^s \) is M_A=E_A/c² , not the mass equivalent of the signed Hayward potential.

\( \begin{aligned}F_L^H dL_H^s&=P_R^H dV_A\\&\quad-\frac34E_A dw\end{aligned} \)

The instantaneous reconstruction does not require fixed w. The radial term is the entire scale work only when w, equivalently the appropriate length/radius ratio, is fixed. At zero signed rate use the regular pressure and rate variables: the reciprocal length chart is undefined, and a zero coefficient does not discard an indeterminate force–length variation.

Radius and the geometric Hayward rate can be evaluated without a matter equation of state. Identifying their combination with total bulk pressure uses Einstein’s equations . For H nonzero and finite horizon radius:

\( \begin{gathered}P_{\rm scr}=\frac{c^4}{8\pi G R_A^2},\\[0.6em]P_R^H=-P_{\rm scr}\left(1-\frac{\dot R_A}{2HR_A}\right),\\[0.6em]p_m=P_{\rm scr}\left(\frac{2\dot R_A}{HR_A}-3\right)\end{gathered} \)

In spatially flat expanding FRW, HR_A=c, so radius and its rate suffice. For nonzero curvature, the displayed formula also uses H. This is an on-shell inverse reconstruction of total density and pressure, including vacuum stress; it is not a material shell balance or a derivation of matter microphysics. A matter model is still needed to predict a history from initial data, and a proposed geometric history must be compatible with that model.

How the same coefficients describe the horizon’s causal history

For spatially flat expanding FRW, the cosmic-time tangent norm along the apparent-horizon history is:

\( \begin{gathered}-c^2+(\dot R_A-HR_A)^2\\[0.5em]=4c^2\frac{P_R^H(P_{\rm scr}+P_R^H)}{P_{\rm scr}^2}\end{gathered} \)

This is a tangent norm, not a local material speed or new evolution equation. The history is timelike for −P_scr<P_R^H<0, null at either endpoint, and spacelike outside. Persistent vacuum domination gives de Sitter’s null nonexpanding history; radiation gives a null but area-growing one. Thus marginality, nullness and nonexpansion are distinct. The pressure variables organize the established causal classification without completing a moving-boundary canonical theorem.

Sources and related explanations: Hayward, geometric rate and unified first law · Cai–Kim, instantaneous equilibrium assignment · Faraoni, apparent-horizon causal classification

10. Doesn’t clock or boundary-term dependence make the pressure arbitrary? The response is definite within a specified physical clock and boundary prescription. Different prescriptions are different comparisons.

Surface gravity is normalized by a selected evolution. Under a constant positive rescaling of that generator, \( \kappa\mapsto b\kappa \) , the pressure transforms as \( P_L\mapsto b^2P_L \) ; the energy and length transform consistently. This inherited dependence is not removed by squaring the rate. A fixed physical clock still allows surface gravity to change between geometries.

Schwarzschild’s standard surface gravity is evaluated at the horizon and normalized by the stationary clock at infinity; it is not measured at infinity. A local clock redshifts both rate and potential. Geometry/evolution and clock normalization are separate: a clock change does not turn Schwarzschild into cosmological geometry. Exact de Sitter is the stationary overlap; generic FRW is evolving.

Boundary terms and polarization determine which data are prescribed. Adding a source-dependent boundary functional can shift a response; the construction does not claim otherwise. Its canonical statement fixes those choices and compares allowed neighboring source problems. The substantive test is whether the chosen variation is admissible and the other responses are handled, not whether every conceivable prescription yields one identical number.

Sources and related explanations: Odak–Rignon-Bret–Speziale

12. Are the matching formulas independent confirmations? They demonstrate a common organization, but algebraically equivalent identities are not independent tests.

For \( U\propto A/L \) at fixed source area, \( F_L=U/L \) follows from the inverse-length dependence. Since \( M_U=U/c^2 \) , \( F_L=M_U\kappa \) is the same identity in mass–acceleration form. The geometric Euler relation is a weighted scaling statement; generic degree-one homogeneity alone would not fix the force.

The explanatory economy is that one response rule accommodates the distinct geometric and energy assignments without a separately fitted force law for each horizon. Its evidence is the explicit matching under specified inputs and the controlled failure of simplifications outside their scope. It is not a count of independent confirmations from several rearrangements of one formula.

The null-source derivation is the principal action-level argument. The stationary area–boost route is a complementary geometric interpretation and check. Both share Einstein area normalization, the physical clock and the stationary boost–surface-gravity relation. Their agreement is not an independent empirical confirmation. The source trail retains generator, time-measure, spin-two and corner restrictions.

13. Does this add missing work to the black-hole mass law or split mass into two reservoirs? It gives the complete differential of the boundary potential, while preserving the mass first law.

For Schwarzschild, \( U=Mc^2/2 \) . The potential obeys \( dU=T\,dS-F_L\,dL \) , while the standard total-mass law remains \( d(Mc^2)=T\,dS \) . These describe different energy functions; adding another subtraction to the mass law would double-count the variation.

Identifying the force product reveals \( Mc^2=TS+F_LL \) . Both products equal the same boundary potential , so this is a mechanical interpretation of the Smarr structure, not evidence for two independently variable reservoirs. The proposed kinetic/potential and virial reading is retained in Appendix G , with no separate derivation claimed.

\( Mc^2=TS+F_LL,\qquad TS=F_LL=U,\qquad d(Mc^2)=T\,dS \)

14. Is the construction restricted to Einstein gravity? The main canonical force/pressure result is Einstein-based; a stationary Wald-entropy extension is developed separately.

The full manuscript derives the stationary \( S_WdT \) scale response, giving \( F_L^{(W)}=TS_W/L \) . It then compares this with the membrane pressure in higher-curvature gravity, including the theory- and dimension-dependent factors. This is a stated extension, not merely a promise that one could be attempted.

The Einstein coefficient \( \kappa^2/(8\pi G) \) is not asserted unchanged in every theory. Fixed area need not fix Wald entropy , and coupling or entropy-density changes can add terms. A complete higher-curvature moving-null-source theorem is not claimed. Generality across horizons and generality across actions are different merits; neither is a universal ranking of scientific importance.

The cited Lovelock surface-term variations have order-dependent normalization and restricted variations. The stationary Wald-channel lift is the response of the stated thermal channel; it is not an unqualified generic-null variational theorem.

Sources and related explanations: Chakraborty–Padmanabhan

15. Could the thermodynamic arguments be written mechanically, and what about Verlinde? Many Einstein-area products admit a mechanical translation. A force-law derivation must still retain the original balance and physical inputs.

The Einstein dictionary gives \( T\,dS=\Pi\,dA \) and \( S\,dT=A\,d\Pi \) ; quantum constants cancel in these products. Jacobson’s and Cai–Kim’s arguments can therefore be expressed using mechanical area coefficients, while keeping their crossing-energy balance and geometric assumptions . The potential identity alone does not establish an arbitrary matter-flux balance or a detector spectrum.

At fixed boundary potential, the length-work law gives \( T\,dS=F_L\,dL \) and the same mathematical form and normalization as Verlinde’s entropy–displacement relation, with boundary mass-equivalent \( U/c^2 \) . That is a horizon-boundary counterpart: it is not a derivation of external test-particle displacement, arbitrary screens, or equipartition. The full manuscript states this distinction explicitly.

Sources and related explanations: Jacobson · Verlinde

16. What supports the novelty and chronology claims? The claimed contribution is stated precisely and compared with named antecedents; dates and independent validation are separate questions.

The author’s chronology places the public FRW screen-pressure result in May 2025 and the acceleration-length formulation in June 2026. Current manuscripts include later refinements. A reader should use the dated original record for a priority question rather than infer every later claim from the first announcement.

The contribution claim concerns the identified force, pressure, normalized work coordinate and their developed gravitational interpretation. Its assessment rests on the arguments and the cited comparisons ; an absence of a located antecedent is not itself a proof of universal historical priority. Archival documentation and independent specialist scrutiny complement that scientific assessment. Neither affiliation nor a prior publication decision settles it.

One framework. Connected explanations. Concise derivation · Intuitive path to the mechanics

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