Horizon pressure from acceleration length

01 · The mechanical result

The action-selected rate becomes a force.

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.

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.

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.