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

J. T. Tyler

What is horizon mechanics?

Black-hole mechanics first connected horizon area and surface gravity through laws that mirror entropy and temperature. Hawking radiation made that analogy thermodynamic: a horizon carries temperature, while the Bekenstein–Hawking law assigns entropy to its area.

Later null-boundary, membrane, Noether-charge, corner, cosmological, and dynamical-horizon formalisms clarified different parts of this structure. They do not all use the same clock, ensemble, or notion of pressure. Those distinctions matter here.

This work asks what mechanical response is conjugate to the surface-gravity source already present in the Einstein boundary potential. On a prescribed-clock, non-expanding null sector, its signed reciprocal coordinate is $L_s=\frac{c^2}{\kappa}$ . The resulting normal force per area is $P_L=\frac{\kappa^2}{8\pi G}$ .

Technical derivation

Horizon pressure from the Einstein boundary potential

The Einstein boundary action first selects the physical source sector. Within it, $L_s=\frac{c^2}{\kappa}$ is an invertible source-coordinate change that exposes the normal mechanical response. The common area–rate potential gives the mechanical differential; its canonical certification retains the stated boundary data. Literal geometric volume belongs to sectors that supply an appropriate displacement. The separately selected Cai–Kim screen prescription is not a general moving-boundary canonical theorem.

P_L=\frac{\kappa^2}{8\pi G}=\frac{c^4}{8\pi G L_{\kappa}^{2}}

Established input

Einstein null-boundary source structure, the $A\kappa$ potential, membrane stress, Hawking–Unruh temperature, entropy, and standard horizon laws.

Derived here

The acceleration-length source coordinate and response structure, $P_L=\frac{\kappa^2}{8\pi G}$ , its normal force, and its work one-form under the stated boundary conditions.

Sectoral realization

Rindler, Schwarzschild, Cai–Kim FRW, Hayward FRW, and APS each identify the general variables differently.

Not claimed

No new bulk stress tensor, modified field equation, independent energy, universal moving-horizon temperature, or observational prediction.

From the null symplectic potential to force

The generic scalar source on a null boundary is $\mu=\widehat\kappa+\theta/2$ . Surface gravity alone is the scalar source only on the controlled $\theta=\delta\theta=0$ sector with a prescribed physical clock. There the rate-source polarization contains

\Theta_{\kappa}^{(0)}=\frac{c^2}{8\pi G}\int\!\mathrm d\tau\int_S\!\epsilon_S\,\delta\kappa

For a uniform cut and unit elapsed clock interval, replacing $\kappa$ by $L_s=\frac{c^2}{\kappa}$ gives

\frac{\Theta_{\kappa}^{(0)}}{\Delta\tau}=-F_L\,\delta L_s,\qquad F_L=\frac{A\kappa^2}{8\pi G}

Full manuscript, Sec. II ↗

Corner check, clock, and allowed variations

The area–boost corner construction supplies the same energy-valued potential, independently checking the normal response:

U_{\partial}=\frac{A c^2\kappa}{8\pi G}

\left.\delta U_{\partial}\right|_A=\frac{A c^2}{8\pi G}\,\delta\kappa=-F_L\,\delta L_s

The numerical value of $\kappa$ is defined relative to the selected horizon clock. Auxiliary null rescalings that preserve the physical evolution are gauge; changing that physical clock is not. The source variation holds the cut, clock, and transverse area fixed.

Pressure, surface stress, and normal balance

The established signed membrane surface stress and the normal pressure are distinct dimensional quantities related by acceleration length:

\Pi_{\partial}=\frac{c^2\kappa}{8\pi G}=P_L L_s,\qquad P_L=\frac{F_L}{A}=\frac{\kappa^2}{8\pi G}

At fixed area the work variation is defined by $\delta V_{\perp}:=A\,\delta L_s$ , so $F_L\,\delta L_s=P_L\,\delta V_{\perp}$ . In the full two-source space this is not the statement $\mathrm d(AL_s)=A\,\mathrm dL_s$ ; the omitted $L_s\,\mathrm dA$ term belongs to the independent transverse variation.

Normal Young–Laplace form

\Delta p-P_L L_s\,\mathcal H=0

The mechanical–thermal bridge

With $\varepsilon=\operatorname{sgn}(\kappa)$ , the signed and positive-magnitude statements must be kept separate:

U_{\partial}=\varepsilon TS=F_L L_s,\qquad |U_{\partial}|=TS=F_L L_{\kappa}

\mathrm dU_{\partial}=\Pi_{\partial}\,\mathrm dA-F_L\,\mathrm dL_s

\mathrm d|U_{\partial}|=T\,\mathrm dS-F_L\,\mathrm dL_{\kappa}

Where equilibrium thermality applies, $c\Delta\tau_E=2\pi L_{\kappa}$ and $k_{\mathrm B}T=\frac{\hbar|\kappa|}{2\pi c}=\frac{\hbar c}{2\pi L_{\kappa}}$ . Combining the thermal and mechanical normalizations reconstructs the standard entropy density:

\frac{k_{\mathrm B}T}{P_LL_{\kappa}}=4\ell_{\mathrm P}^{2},\qquad \frac{S}{A}=\frac{k_{\mathrm B}}{4\ell_{\mathrm P}^{2}}

Radius, force, and Smarr scaling

For a spherical screen define $A=4\pi R^2$ , $\alpha_R=L_s/R$ , and $M_R=c^2R/(2G)$ . The radial force is $F_R=U_{\partial}/R=M_R\kappa$ . The change of variables exposes the normalization term:

F_L\,\mathrm dL_s=F_R\,\mathrm dR+U_{\partial}\,\frac{\mathrm d\alpha_R}{\alpha_R}

\Pi_{\partial}\,\mathrm dA=2F_R\,\mathrm dR,\qquad \mathrm dU_{\partial}=F_R\,\mathrm dR-U_{\partial}\,\frac{\mathrm d\alpha_R}{\alpha_R}

Rindler

$L_{\kappa}=c^2/a$ is literal proper distance and $P_L=a^2/(8\pi G)$ survives in flat spacetime.

Schwarzschild

$L_{\kappa}=2R_H$ , $U_{\partial}=Mc^2/2$ , $F_L=c^4/(8G)$ , and $F_R=M\kappa=2F_L$ .

Uncharged Kerr

The rotation term completes $Mc^2=2F_LL_{\kappa}+2\Omega_HJ$ .

Schwarzschild work bookkeeping

F_L\,\mathrm dL_{\kappa}=(M\kappa)\,\mathrm dR_H=2P_L\,\mathrm dV_H

Mc^2=2TS=2F_LL_{\kappa}=2M\kappa R_H

Spherical interpretation · current manuscript

Two length scales explain the energy ratio.

On a spherical marginal horizon, and not an arbitrary spherical cut,

E_{\mathrm{MS}}=\frac{c^4R}{2G},\qquad U_\partial=\frac{E_{\mathrm{MS}}R}{L_s},\qquad U_\partial L_s=cJ_\eta=E_{\mathrm{MS}}R

The energy varies with radius. Retaining both radius contributions gives

\mathrm dU_\partial=\frac{R}{L_s}\,\mathrm dE_{\mathrm{MS}}+\frac{E_{\mathrm{MS}}}{L_s}\,\mathrm dR-\frac{E_{\mathrm{MS}}R}{L_s^2}\,\mathrm dL_s=\frac{2U_\partial}{R}\,\mathrm dR-F_L\,\mathrm dL_s

This is the area response plus length work. It is a spherical interpretation of the general potential, not a new independent charge law. Rindler needs neither a spherical radius nor an enclosed Misner–Sharp energy.

Full manuscript, Sec. V.F ↗

The FRW screen state law

R_A=\frac{c}{\sqrt{H^2+k c^2/a^2}},\qquad A_A=4\pi R_A^2,\qquad V_A=\frac{4\pi R_A^3}{3}

Cai–Kim assigns $\kappa_{\mathrm{CK}}=c^2/R_A$ , so $L_{\kappa}=R_A$ . With $E_A=c^4R_A/(2G)$ ,

P_{\mathrm{scr}}=\frac{c^4}{8\pi G R_A^2}=\frac{\rho}{3},\qquad F_L=\frac{c^4}{2G}

T_{\mathrm{CK}}\,\mathrm dS_A=2\,\mathrm dE_A,\qquad P_{\mathrm{scr}}\,\mathrm dV_A=\mathrm dE_A

\mathrm dE_A=T_{\mathrm{CK}}\,\mathrm dS_A-P_{\mathrm{scr}}\,\mathrm dV_A

The Friedmann equations in pressure variables

\frac{1}{R_A^2}=\frac{8\pi G}{c^4}P_{\mathrm{scr}}

\frac{2\ddot a}{a c^2}=-\frac{8\pi G}{c^4}\left(P_{\mathrm{scr}}+p_{\mathrm m}\right)

\dot P_{\mathrm{scr}}+3H\left(P_{\mathrm{scr}}+\frac{p_{\mathrm m}}{3}\right)=0

The equations generate the same solutions as standard Friedmann cosmology. In these variables, $p_{\mathrm m}<-P_{\mathrm{scr}}$ is accelerated expansion; no new dark-energy fluid has been introduced.

FRW comparison

Cai–Kim screen pressure and Hayward cross-focusing answer different questions.

Equation-of-state parameter

w=p_{\mathrm m}/\rho

$w$ -1.00 $P_L^H/P_{\mathrm{scr}}$ 1.000 $L_H^{\mathrm s}/R_A$ -1.00

Acceleration occurs when $w<-1/3$ .

Hayward dynamics and the pressure contrast

Full signed Kodama–Hayward surface gravity measures cross-focusing, not the Cai–Kim screen-size response:

\kappa_H^{\mathrm s}=-\frac{c^2}{R_A}\left(1-\frac{\dot R_A}{2HR_A}\right)=\frac{c^2}{R_A}\frac{3w-1}{4}

\frac{\kappa_H^{\mathrm s}}{\kappa_{\mathrm{CK}}}=-\frac{P_{\mathrm{scr}}-p_{\mathrm m}}{4P_{\mathrm{scr}}},\qquad P_L^H=\frac{(P_{\mathrm{scr}}-p_{\mathrm m})^2}{16P_{\mathrm{scr}}}

L_H^{\mathrm s}=\frac{4R_A}{3w-1},\qquad U_H=\frac{3w-1}{4}E_A=-\frac{3}{4}(P_{\mathrm{scr}}-p_{\mathrm m})V_A

When $w$ varies, the work contains an additional normalization/composition term:

F_L^H\,\mathrm dL_H^{\mathrm s}=\frac{p_{\mathrm m}-P_{\mathrm{scr}}}{4}\,\mathrm dV_A-\frac{3E_A}{4}\,\mathrm dw

Trace, quintessence, and the radiation limit

The contrast $P_{\mathrm{scr}}-p_{\mathrm m}$ is trace-sensitive. For a dominant canonical scalar, $\rho_\phi=K_\phi+V_\phi$ and $p_\phi=K_\phi-V_\phi$ , so acceleration is $V_\phi>2K_\phi$ . At radiation, $w=1/3$ makes $\kappa_H^{\mathrm s}=0$ ; the signed reciprocal Hayward chart moves to infinity while the parent rate variable stays regular.

Integrability and finite dynamical balance

The isolated scale-work one-form need not define a separate stored energy when area and length vary independently:

\alpha_L=-F_L\,\delta L_s,\qquad \mathrm d_{\Gamma}\alpha_L=-P_L\,\delta A\wedge\delta L_s

Its curl cancels against the area leg in the complete exact boundary potential. In the nonrotating APS dynamical-horizon normalization, $L_{\mathrm{APS}}=2R$ , $F_L=c^4/(8G)$ , and $U_{\mathrm{APS}}=c^4R/(4G)$ . The finite APS law governs the complete parent charge; it does not separately assign its fluxes to the area and length legs.

Further consequences and limits

Stationary Wald channel

Replacing Einstein entropy density by stationary Wald density lifts the fixed-entropy scale response, while local membrane coefficients can remain theory- and dimension-dependent.

Inverse-energy length

\bar\lambda_{\partial}=\frac{\hbar c}{|U_{\partial}|},\qquad \frac{S}{k_{\mathrm B}}=\frac{2\pi L_{\kappa}}{\bar\lambda_{\partial}}

This exact inverse-energy construction does not treat a horizon as a particle. A literal quantum phase interpretation would require additional quantum dynamics.

Moving screens and zero surface gravity

At $\kappa=0$ , acceleration length is not a finite coordinate, but the parent $\kappa$ variation can remain regular. General moving screens require a specified clock, lapse, and boundary ensemble.

Prior work and public record

FrameworkEstablished contributionRole here

Black-hole mechanicsArea, surface gravity, Smarr scaling.Parent macroscopic structure.

Membrane / Young–LaplaceSurface stress and normal balance.Independent pressure check and factorization.

Cai–Kim / HaywardFRW thermal and dynamical laws.Distinct screen-size and cross-focusing realizations.

APSFinite dynamical charge–flux balance.Support for the complete parent charge.

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One-page summaryThe result and cosmological pressure geometry in one page.Seven-page paperThe compact derivation and principal realizations.Full working manuscriptThe current 50-page foundations, scope, extensions, and appendices.Explanatory guideA linked, equation-preserving guide to the canonical argument and its limits.Core glossary68 linked entries for the seven-page argument.Full glossary123 searchable entries for the complete manuscript.

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