Horizon Boundary Mechanics: Force, Pressure, and Work from the Einstein Boundary Potential

J. T. Tyler

HORIZON BOUNDARY MECHANICS20 October 2026

Technical summary

Force, pressure, and work from the Einstein boundary potential

Identifying as the gives a new and 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 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 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. 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 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 , $\kappa=a_{\mathrm{obs}}$ and $L_\kappa=c^2/a_{\mathrm{obs}}$ ; a finite planar area is specified independently. 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 $L=2R$ , : $dL/L=dR/R$ . The work describes the same change of the same horizon sphere. Here $R$ is its , $V=4\pi R^3/3$ , 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 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 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.

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.

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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