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

J. T. Tyler

HORIZON BOUNDARY MECHANICS · AN INTUITIVE SUMMARY

All About L

is the distance light travels during the of a horizon:

L=c\,t_\kappa=\frac{c^2}{\kappa},\qquad t_\kappa=\frac{c}{\kappa}

The fixes how the evolution is measured. Its $\kappa$ supplies the rate $\kappa/c$ ; $t_\kappa$ is the reciprocal time scale. A stronger acceleration means a shorter time and a shorter length.

1. See the distance in Rindler

Imagine an observer who forever. Light sent forward from sufficiently far behind never catches the observer. The limiting light ray marks the . On the observer’s instantaneous rest slice, its closest point is exactly $L=c^2/a_{\mathrm{obs}}$ away. The acceleration time sets this distance scale; it is not a photon’s travel time from the horizon to the accelerating observer.

2. Separate size from acceleration scale

A rate depends on geometry and on the . Schwarzschild’s usual rate is evaluated at the horizon and normalized at infinity. A different clock redshifts rate and potential; it does not change the geometry into FRW. The in evolving FRW differs from the . Exact de Sitter is the stationary overlap. The keeps those settings separate.

describes how much transverse surface there is. describes the scale of its normalized evolution. These are . A particular spherical geometry can connect them through its radius, but the common mechanics needs both.

Why this length? Built only from $c$ and the prescribed $\kappa$ , with no extra scale, a length must be proportional to $c^2/\kappa$ . The Rindler distance fixes the factor. ; $L$ is preferred for this normalized geometric interpretation.

3. Surface stress and horizon-scale work

The established combines $\Pi$ with $A$ :

U=\Pi A=\frac{Ac^4}{8\pi G L}

Identifying $L$ as the geometrically normalized gives a new and . The potential's change has two mechanical contributions: surface stress paired with changing area, and force paired with changing horizon scale:

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

F_L=\frac{U}{L},\qquad P_L=\frac{F_L}{A}=\frac{\Pi}{L}=\frac{\kappa^2}{8\pi G}

The is the force–pressure identification and its geometric work interpretation. The work is force times length change; pressure is force per area. The established area–rate differential supplies the input, and the selected gravitational action supplies the same coefficient under the stated permitted variations. The product-rule explanation is available in Explain.

4. The same length also sets the temperature

The Hawking–Unruh uses exactly this scale. With Einstein , the thermal and mechanical energy products agree:

T=\frac{\hbar c}{2\pi k_B L},\qquad U=TS

The same mechanical law therefore reads $dU=T\,dS-F_L\,dL$ , since $T\,dS=\Pi\,dA$ . Its has the complementary thermal form $S\,dT=-F_L\,dL$ .

5. In equilibrium FRW, the scale is the radius

For the , $\kappa=c^2/R_A$ . The acceleration scale is therefore the actual , and its length work is directly spherical :

L=R_A,\qquad F_L\,dL=P_L\,dV_A

Here $dV_A=A\,dR_A$ , so the acceleration-length and radial pressures coincide. The area and work terms refer to the same changing horizon sphere.

6. In Schwarzschild, use the radius to show the work

For a , with the , the relation is just as definite:

\kappa=\frac{c^2}{2R},\qquad L=2R,\qquad U=\frac{Mc^2}{2}

Changing $L$ now means changing the horizon radius in fixed proportion. They describe the :

\frac{dL}{L}=\frac{dR}{R}

Writing the work with makes its mechanical meaning immediate:

F_L\,dL=F_R\,dR=M\kappa\,dR=P_R\,dV

The is $F_R=2F_L=M\kappa$ . Its transforms with the coordinate:

P_R=\frac{F_R}{A}=2P_L=\frac{\kappa^2}{4\pi G}

The first law for $U=Mc^2/2$ now follows the changing horizon sphere explicitly:

dU=\Pi\,dA-M\kappa\,dR=T\,dS-P_R\,dV

The volume here is the actual $V=4\pi R^3/3$ , for which $U=3P_R V$ . The ordinary Schwarzschild remains $d(Mc^2)=T\,dS$ .

keeps the common meaning of the boundary’s acceleration scale. makes the Schwarzschild realization concrete: the work describes the same change of horizon size that changes its area. The conversion changes the force and pressure coefficients; the work itself is unchanged.

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