Horizon boundary mechanics

01 · The mechanical result

One potential, two mechanical responses.

The starting point is the area–rate boundary potential, not a new bulk energy:

For a uniform cut of area $A$, the null-boundary source term fixes the response to a change in surface gravity. On a nonzero branch, $L_s=\frac{c^2}{\kappa}$ converts that rate into its natural signed length coordinate:

The complete two-source differential is

Thus $F_L=A P_L$. At fixed transverse area, define the swept normal-volume variation by $\delta V_{\perp}:=A\,\delta L_s$; then $F_L\,\delta L_s=P_L\,\delta V_{\perp}$. This is the mechanical reason for calling the response pressure. On independent area–length data this is a work one-form, not the full differential of $A L_s$. A literal geometric displacement or volume additionally depends on the realization.

02 · One potential, two descriptions

The same acceleration length is mechanical and thermal.

For positive thermodynamic magnitudes, Hawking–Unruh thermality writes the acceleration length as the reduced light-travel scale of the inverse temperature:

The signed mechanics remains $U_{\partial}=\varepsilon TS=F_L L_s$, where $\varepsilon=\operatorname{sgn}(\kappa)$. Schwarzschild gives $Mc^2=2F_L L_{\kappa}$; uncharged Kerr gives $Mc^2=2F_L L_{\kappa}+2\Omega_HJ$.

03 · Horizon realizations

One response, different geometric meanings.

Acceleration length is the common normal scale. Radius becomes an equivalent coordinate only when the chosen sector supplies the relation.

Area is intrinsic to a cut; a planar Rindler patch can have$A=\Delta y\,\Delta z$ independently of its acceleration length. On a spherical marginal horizon only,$E_{\mathrm{MS}}=c^4R/(2G)$ gives$U_\partial/E_{\mathrm{MS}}=R/L_s$. On the positive branch this ratio is one for Cai–Kim FRW and one half for infinity-normalized Schwarzschild. No spherical projection or enclosed Misner–Sharp mass is assigned to the Rindler plane.

04 · The cosmological payoff

A classical-looking state law for the expanding-universe horizon.

In the Cai–Kim apparent-horizon prescription, the screen pressure is$P_{\mathrm{scr}}=P_L=\rho/3$ and the radius itself is the acceleration length. The horizon state law is

Accelerated expansion is therefore exactly $p_{\mathrm m}<-P_{\mathrm{scr}}$: the negative material pressure exceeds the positive screen pressure in magnitude. This is a rewriting of the standard Friedmann system, not a new dark-energy component.