Surface Gravity Squared as Gravitational Boundary Pressure

The Einstein null-boundary variation already identifies the regular scalar source. Surface gravity is therefore part of the boundary data selected by the action, not a quantity inserted by dimensional analogy.

Gravity itself tells us which boundary rate is being changed. This is the starting point, so the later pressure is not guessed from its units.

Fixing the generator normalization and the relational cut removes the expansion channel. The source then reduces unambiguously to the normal rate represented by surface gravity.

Fix the horizon’s clock, relational cut, and non-expanding sector. That isolates the surface-gravity rate.

On one nonzero signed branch, the reciprocal source change is invertible. The Einstein symplectic potential then places the response density directly beside the allowed acceleration-length variation.

Rewrite the normal rate as a distance. In the action, the exact coefficient beside $\delta L_s$ is the normal response density. The next step shows why that response has ordinary pressure form.

The independent area-boost corner energy yields the same fixed-area conjugate force. This is a second action-based derivation of the force, separate from merely naming the response density.

Differentiate the boundary energy with respect to the new distance. The result is exactly pressure times area—the ordinary definition of force from pressure.

In signed source variables, the gradient of the established membrane surface stress is the pullback of the source-space response one-form $-P_L\,\mathrm dL_s$. In positive thermodynamic magnitudes, the corresponding one-form is $-P_L\,\mathrm dL_{\kappa}$. The normal response and tangential membrane description are two views of the same boundary data.

On a fixed-sign branch, the magnitude of the membrane stress is $P_LL_{\kappa}$. Pressure itself is not being pulled back. Spatial changes in $L_{\kappa}$ turn the normal response into the already-established magnitude gradient $D_A|\Pi_{\partial}|=-P_LD_AL_{\kappa}$.

The static normal screen constraint is the Young–Laplace balance $\Delta p-\Pi_{\partial}H=\Delta p-P_LL_sH=0$. In the positive-$\kappa$ orientation, $L_s=L_{\kappa}$, so this becomes $\Delta p-P_LL_{\kappa}H=0$.

Curvature turns the surface stress $P_LL_{\kappa}$ into the pressure difference supported across the screen.

For the stationary Schwarzschild family, the thermal area leg and mechanical acceleration-length leg are equal derivatives of one parent boundary energy.

For a Schwarzschild black hole, the thermal contribution and the mechanical scale contribution are exactly equal halves of the total energy.

Under the instantaneous Cai–Kim prescription, the constrained screen-size derivative coincides with the acceleration-length response and closes the compact-screen first law with positive pressure–volume work.

For an expanding-universe horizon, the same pressure becomes the screen's energy-per-volume slope and supplies an ordinary pressure-volume work term.

The Ashtekar–Paraizo–Shu finite balance law rewrites as a constant-force evolution in the preferred equilibrium projection, extending the representation beyond infinitesimal stationary variations.

The sum of matter and gravitational flux equals the change of the complete APS boundary charge. In the nonrotating preferred projection, that charge change is a constant force times $\Delta L_{\mathrm{APS}}$; the individual fluxes do not map separately to the area and scale legs.

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

The same stationary response can be written using Wald entropy in more general theories of gravity, although the local membrane details can change.