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

J. T. Tyler

ONE-PAGE SUMMARY14 September 2026

Horizon boundary mechanics

Force, pressure, and work from the Einstein boundary potential

resolves the into a to and a to . and stationary realize its ; supplies a cosmological .

1. Canonical boundary mechanics: force, pressure and work

For on a , Einstein $(\theta=\delta\theta=0)$ , the selected gives [1]

\frac{\Theta^{(0)}_{\kappa}}{\Delta\tau}=\frac{Ac^2}{8\pi G}\,\delta\kappa=-F_L\,\delta L_\kappa,\qquad L_\kappa=\frac{c^2}{\kappa},\qquad\kappa>0.

Here $L_\kappa$ is the light-travel length associated with timescale $c/\kappa$ , not a universal . The established $J_\eta=Ac^3/(8\pi G)$ and $\Omega_\kappa=\kappa/c$ give [2]

\boxed{U_\partial=\Omega_\kappa J_\eta=\frac{Ac^2\kappa}{8\pi G},\qquad U_\partial L_\kappa=cJ_\eta.}

Their : $\Omega_\kappa\,dJ_\eta=\Pi_\partial\,dA$ and $J_\eta\,d\Omega_\kappa=-F_L\,dL_\kappa$ , with $\Pi_\partial=U_\partial/A$ . Hence

dU_\partial=\Pi_\partial\,dA-F_L\,dL_\kappa,\qquad F_L=-\left(\frac{\partial U_\partial}{\partial L_\kappa}\right)_A=\frac{U_\partial}{c^2}\kappa,\qquad\boxed{P_L\equiv\frac{F_L}{A}=\frac{\kappa^2}{8\pi G}}.

is intrinsic to the : a needs no radius or spherical projection. On a , $E_{\mathrm{MS}}=c^4R/(2G)$ gives $U_\partial=E_{\mathrm{MS}}R/L_\kappa$ ; thus $U_\partial/E_{\mathrm{MS}}=R/L_\kappa$ is $1/2$ for and $1$ for . give $U_\partial=TS=F_LL_\kappa$ ; Schwarzschild reads $Mc^2=TS+F_LL_\kappa$ , each product $Mc^2/2$ .

2. Cosmological application: the FRW apparent horizon

For , expanding , with $H=\dot a/a>0$ , select the instantaneous [3]:

R_A=\frac{c}{H}=L_\kappa,\qquad\kappa=cH,\qquad A=4\pi R_A^2,\qquad V=\frac{4\pi R_A^3}{3}.

Use the standard Einstein–FRW $E_{\mathrm{MS}}=\rho V$ [4], where $\rho$ is . The gives

E_{\mathrm{MS}}=\frac{c^4R_A}{2G}=\rho V=U_\partial=TS=F_LL_\kappa=3P_LV.

Dividing by $V$ relates the to the usual :

\boxed{\frac{c^2H^2}{8\pi G}=\frac{\kappa^2}{8\pi G}=P_L=\frac{\rho}{3}},\qquad H^2=\frac{8\pi G}{c^2}P_L=\frac{8\pi G}{3c^2}\rho.

Since $dV=A\,dR_A=A\,dL_\kappa$ , the becomes

dE_{\mathrm{MS}}=T\,dS-P_L\,dV,\qquad F_L=\frac{E_{\mathrm{MS}}}{c^2}\kappa=\frac{c^4}{2G}.

This is a , not a ; $P_L$ is distinct from the .

Curvature, acceleration and conservation

With $p_m$ and $\kappa_E=8\pi G/c^4$ ,

\boxed{\frac{1}{R_A^2}=\kappa_EP_L},\qquad\boxed{{}^{(2)}\mathcal R_\perp\equiv\frac{2\ddot a}{ac^2}=-\kappa_E(P_L+p_m)}.

Here $1/R_A^2$ is the sphere’s and ${}^{(2)}\mathcal R_\perp$ is the . sets the horizon-curvature scale; its sum with governs , which begins when $p_m<-P_L$ . becomes

\boxed{\dot P_L+H(\rho+p_m)=0},\qquad\rho=3P_L.

The $\rho+p_m$ controls the evolution of screen pressure.

. The flat case is shown for clarity. For general FRW use $R_A=c/\sqrt{H^2+kc^2/a^2}$ and $\kappa=c^2/R_A$ . Then $P_L=\rho/3$ and Eqs. (5), (7)–(9) retain their form; $V=4\pi R_A^3/3$ is , not generally .

. The null response fixes , , and other in the selected . An evolving FRW screen is a separate calibrated realization, not an application of the . use $T=\hbar\kappa/(2\pi ck_B)$ and $S=k_Bc^3A/(4G\hbar)$ .