# Black hole thermodynamics Black holes are thermodynamic objects. They (Hawking) radiate at temperature $T$ like any ordinary object at temperature $T$ does. Their thermodynamic variables (such as energy, entropy, and chemical potential) satisfy the laws of thermodynamics like any other physical system does. ## Refs - review/notes - [[2024#Witten (Notes)]] - 0th law - [[2020#Ghosh, Sarkar]] - [[2022#Bhattacharyya, Biswas, Dinda, Kundu]] - 1st law - the expressions are essentially [[0004 Black hole entropy|black hole entropy]] or [[0145 Generalised area|generalised area]] - 2nd law - see [[0005 Black hole second law]] - 3rd law - The third law for black holes has been shown to be violated. See e.g. [[2024#Kehle, Unger]] and refs therein. ## Basic of thermodynamics in holography ### Energies ![[0316_sum.png|300]] - $dE=TdS-pdV+\mu dN$ - $dF=d(E-TS)=dE-TdS-SdT-pdV=-SdT-pdV+\mu dN$ ### Dependent variables - $E=E(S,V,N)$ - $F(T,V,N)=E(S,V,N)-TS$ ### Statistical relations (in canonical ensemble) - $F=-T\ln Z=-\frac{1}{\beta}\ln Z$ - $Z=\sum_i e^{-\beta E_i}$ - => average energy $E=-\partial_\beta Z/Z$ - $S=-\frac{\partial F}{\partial T}=\beta^2\frac{\partial F}{\partial \beta}=\beta^2(\frac{1}{\beta^2}\ln Z-\frac{1}{\beta}\partial_\beta \ln Z)$ - => ==$S=\left(1-\beta \partial_{\beta}\right) \log Z$== ### Holographic relations - $Z=e^{-I}$, $F=I/\beta$ - $S=\left(\beta \partial_{\beta}-1\right) I$ - => if $I=\beta X$ - then $S=0$, then $F=E$, then $E=-T\ln Z$ => $I=\beta E$ - gravity perspective: $F=E$ also when $T=0$, so when the time circle does not shrink, it behaves almost like there is no temperature; gravitationally this reflects the fact that it does not matter whether you compactify the time direction or not - $E=F+TS=I/\beta+S/\beta=(I+S)/\beta$ ### Grand canonical ensemble In the presence of charge, we define the grand canonical ensemble by$W(T,\Phi)=F(T,Q)-\Phi Q.$Then the partition function $Z(T,\Phi)$ is related to $W$ via$W=-T\ln Z$so that$W=I/\beta,$where $I$ is the (Euclidean) action of the saddle in the grand canonical ensemble. ## Interesting directions - some interesting geometric relations: for extremal BHs, the deformation away from extremality is also the radial derivative of the extremal solution itself. (see [[0394 Attractor mechanism]]) <!-- talk by Finnlarsen --> - treating $\Lambda$ as a charge: see e.g. [[2021#Hajian, Ozsahin, Tekin]]