Catalog of AdS black holes

Pure AdS gravity

The action for the pure AdS $_{d+1}$ gravity is

S=\int d^{d+1}x\sqrt{-g}\,(R-2\Lambda),\qquad \Lambda=-\frac{d(d-1)}{2L^2},

where $\Lambda$ is the cosmological constant, and $L$ is the AdS radius. The equations of motion are Einstein’s equations

R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=0 \quad\Leftrightarrow\quad R_{\mu\nu}+\frac{d}{L^2}g_{\mu\nu}=0

Black hole

C-metric

Black droplet, black funnel

Black tunnel, black hammock

Black ring

Black binaries

Entanglement islands in higher dimensions

With matter fields

AdS gravity with matter fields can be used to construct various holographic quantum matter.

RN-AdS black hole

Holographic superconductors

A “minimal” holographic superconductor model includes gravity, a $U(1)$ gauge field, and a charged scalar:

S = \int d^{d+1}x\,\sqrt{-g}\left[ \frac{1}{16\pi G}\left(R+\frac{d(d-1)}{L^2}\right) -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} -|D\Psi|^2-m^2|\Psi|^2 \right],

with $D_\mu = \nabla_\mu - i q A_\mu$ . Below a critical temperature, the scalar can condense, spontaneously breaking the boundary $U(1)$ symmetry and producing superconducting behavior.

To model charge density waves (CDW) and related orders (such as pair density waves, PDW), one needs to allow for solutions that spontaneously break translation invariance.