# Physical process version of the first law The physics process version of the first law of black hole thermodynamics compares the quantities between an initial state and a final state of a black hole, both of which are close to equilibrium. As the name suggests, some physical process happens between the initial and final stages, causing quantities such as entropy, mass and angular momentum to change. In contract to this, the other formulation of the first law compares quantities in two different spacetimes, each containing a *stationary* black hole. ## Derivation \[e.g. from [[2007#Amsel, Marolf, Virmani]]\] - [[0408 Raychaudhuri equation|Raychaudhuri]] at linear order => $-\frac{d \hat{\theta}}{d v}+\kappa \hat{\theta}=S(v)$, where $S(v)=8 \pi T_{\lambda \sigma} \chi^{\lambda} \chi^{\sigma}$ (using Einstein equation) - solve => $\hat{\theta}(v)=\int_{v}^{\infty} e^{\kappa\left(v-v^{\prime}\right)} S\left(v^{\prime}\right) d v^{\prime}$ - area increase: $\frac{d(\Delta A)}{d A}=\int_{-\infty}^{\infty} \hat{\theta} d v=\int_{-\infty}^{\infty} d v \int_{v}^{\infty} d v^{\prime} e^{\kappa\left(v-v^{\prime}\right)} S\left(v^{\prime}\right)$ - change order of integration: $\frac{d(\Delta A)}{d A}=\frac{1}{\kappa} \int_{-\infty}^{\infty} d v^{\prime} S\left(v^{\prime}\right)=\frac{8 \pi}{\kappa} \int_{-\infty}^{\infty} d v T_{\mu \nu} \chi^{\mu} \chi^{\nu}$ - => (identify the RHS as Killing energy) $\frac{\kappa \Delta A}{8 \pi}=\Delta E_{\chi}$ - relate to more general generators, say $\chi^{\mu}=t^{\mu}+\Omega \phi^{\mu}$: $\frac{\kappa \Delta A}{8 \pi}=\Delta E-\Omega \Delta J$. QED. ## Horizon shift - $v=-\infty$ is the bifurcate sphere of unperturbed spacetime - it differs from the surface where the generators intersect the (unperturbed) past horizon by first order corrections - but the integrand is also of first order, so the integrated result is only inaccurate at second order ## Comments - dubbed in Wald's book: quantum field theory in curved spacetime and black hole thermodynamics - issues raised in [[JacobsonParentani2003]]and later clarified in [[2007#Amsel, Marolf, Virmani]] ## Connections to other concepts - [[0221 Weak cosmic censorship|weak cosmic censorship]] - [[2022#Lin, Ning, Chen]]