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

J. T. Tyler

THE SUMMARY20 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 .

Force, pressure, and work.

1. Gravitational input

Fix $c$ , $G$ , the , and the . For , $(\theta=\delta\theta=0)$ , the gravitational input is the and its selected :

\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 selects this length without an extra scale. A spherical family then supplies its relation to radius.

The therefore becomes

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

is the of the original . Its is inherited through the .

The same substitution in the gives

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

and hence, before restricting and to a particular ,

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

Thus and are the two responses of one ; 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 is therefore

U=\Pi A=FL=PAL

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

3. Thermodynamics

Where the standard applies, using the ,

\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 $P_R$ .

4. Horizon realizations

Rindler

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

Schwarzschild: work on the horizon sphere

With the , $L=2R$ 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 $L=2R$ , : $dL/L=dR/R$ . 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.

Equilibrium FRW: length and radius coincide

The separately justified 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 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 and . These are the separately justified —not an extension of the by substitution alone.

The established , expressed through , is . The supplies alongside that same ; 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.

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