Stationary solutions in AdS

Stationary solutions in AdS [Incomplete]

by renphysics (contact: renphysics@adscft.org)

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}$$

Aspect	Numerical relativity for astrophysics	Numerical relativity for AdS
Physical motivation	Simulate extreme gravitational events such as black holes or neutron stars merging. To understand general relativity including gravitation waves	Better understand strongly coupled systems. By simulating gravitational dynamics in AdS, one can model analogous processes like thermalization in the dual quantum system, probing strongly coupled matter far from equilibrium
Global structure	Globally hyperbolic; Cauchy data on spatial slice	Not globally hyperbolic; must supply boundary data on the timelike AdS boundary (where the CFT lives)
Stationary solutions	Limited due to uniqueness theorems - topology: spherical - rigidity - no hair theorem	Rich landscape: black droplets, black funnels
Fundations	1990s	2009-2013
Examples	Binary black holes mergers	black droplets, black funnels, black
Gauge choices	BSSN, generalized harmonic, etc.	DeTurck gauge Chesler-Yaffe gauge Bondi-Sachs gauge

Comercial coftware: Comsol. Standard equations, highly complex boundaries. Finite element method NR for AdS: non-Standard equations, simple boundaries, pseudospectral method

a strongly coupled quantum field theory can be mapped to a classical gravity system in a higher-dimensional AdS universe.

Symmetries

A basic classification of solutions is by their symmetries, including the group and the orbit.

境自远尘皆入咏，物含妙理总堪寻。 (Far beyond the world’s dust, the cosmic vista itself becomes poetry; within each thing lies a subtle principle, one that merits our profound inquiry.)

形骸已与流年老，诗句犹争造化功。 (My body creaks under the weight of passing years, My poems aim still to rival the perfections of nature. By Lu You, translated by C.N Yang.)

An analytic solution is like a Chinese traditional poem, and a numerical solution is like a modern poem.

Besides the symmetry, there are other ways to classify spacetimes.

• Petrov classification
• Ricci tensor
• Energy-momentum tensor

Stationary solutions in AdS spacetimes

In AdS

An analytic solution

$$ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Sigma_{2,k}^2$$ $$f(r)=k-\frac{2M}{r}-\frac{\Lambda}{3}r^2$$

If $\Lambda=0$, there can be black holes with different topologies in $D\ge 5$. For example, $S^3$ and $S^1\times S^2$. A remark about the hyperbolic black hole. At high temperature, the causal structure is similar to the Schwarzschild-AdS balck hole, while at low temperature, the causal structure is similar to the RN-AdS black hole. At the spacetime is maximally symmetric, and it is called the Rindler-AdS coordinates of the AdS.

Ellipticity and DeTurck gauge fixing

Boundary conditions

Pseuspectral method

Elliptic

$$(\nabla^2+m^2)u=f$$

Hyperbolic

$$(\partial_t^2-\nabla^2)u=f$$

Parabolic

$$(\partial_t-\nabla^2)u=f$$ $$x=(-1.,-0.900969,-0.62349,-0.222521,0.222521,0.62349,0.900969,1.)^T$$ $$\frac{d}{dx}= \left( \begin{array}{cccccccc} -16.5 & 20.1957 & -5.31194 & 2.57242 & -1.63596 & 1.23191 & -1.0521 & 0.5 \\ -5.04892 & 2.39295 & 3.60388 & -1.47395 & 0.890084 & -0.655971 & 0.554958 & -0.263024 \\ 1.32799 & -3.60388 & 0.510003 & 2.49396 & -1.18202 & 0.801938 & -0.655971 & 0.307979 \\ -0.643104 & 1.47395 & -2.49396 & 0.117057 & 2.24698 & -1.18202 & 0.890084 & -0.408991 \\ 0.408991 & -0.890084 & 1.18202 & -2.24698 & -0.117057 & 2.49396 & -1.47395 & 0.643104 \\ -0.307979 & 0.655971 & -0.801938 & 1.18202 & -2.49396 & -0.510003 & 3.60388 & -1.32799 \\ 0.263024 & -0.554958 & 0.655971 & -0.890084 & 1.47395 & -3.60388 & -2.39295 & 5.04892 \\ -0.5 & 1.0521 & -1.23191 & 1.63596 & -2.57242 & 5.31194 & -20.1957 & 16.5 \\ \end{array} \right)$$ $$Lu=f\quad \Rightarrow\quad u=L^{-1}f$$

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Examples

Metric ansatz:

$$ds^2=\frac{L^2}{x\,y(1+x)^2}\bigg\{-x(1-y)(1+x\,y)Tdt^2+\frac{x(1+x)^2Ady^2}{4y(1-y)(1+x\,y)}+\frac{r_0^2B[dx+x(1-x)^2Fdy]^2}{x(1-x)^4}+\frac{r_0^2S}{(1-x)^2}d\Omega_{2}^2\bigg\}$$

The reference metric given by $T=A=B=S=1$ and $F=0$.

{\hfill [J.E. Santos, B. Way, 1208.6291]}

{\bf Planar black hole} at $x=1$. Boundary conditions are $T=A=B=S=1$ and $F=0$.

$$ds^2=\frac{L^2}{y}\left[-\frac{1}{4}(1-y^2)dt^2+\frac{dy^2}{4y(1-y^2)}+\frac{r_0^2dx^2}{4 (1-x)^4}+\frac{r_0^2}{4(1-x)^2}d\Omega_2^2\right]$$

After coordinate transformation $t = 2 \tau$, $x = 1-r_0/2R$, $y= z^2$:

$$ds^2=\frac{L^2}{z^2}\left[-(1-z^4)d\tau^2+\frac{dz^2}{1-z^4}+dR^2+R^2d\Omega_2^2\right]$$

{\bf Conformal boundary} at $y=0$.

$$ds^2=\frac{L^2}{y}\left[\frac{dy^2}{4 y}+\frac{1}{x(1+x)^2}\;ds_\partial^2\right],$$ $$ds_\partial^2=-x\,dt^2+\frac{r_0^2dx^2}{x(1-x)^4}+\frac{r_0^2}{(1-x)^2}d\Omega_2^2$$

After $x= 1-r_0/r$, it becomes the Schwarzschild metric:

$$ds_\partial^2=-\left(1-\frac{r_0}{r}\right)dt^2+\frac{dr^2}{1-\frac{r_0}{r}}+r^2d\Omega_2^2\;.$$

{\bf Horizon} at $y=1$.

Expanding the equations of motion about the horizon will give the condition $T=A$ and other conditions on $\partial_yA|_{y=1}$, $\partial_yB|_{y=1}$, $\partial_yF|_{y=1}$, and $\partial_yS|_{y=1}$.

{\bf Hyperbolic black hole} at $x=0$.

$$ds^2=\frac{L^2}{y}\left[-(1-y)dt^2+\frac{dy^2}{4y(1-y)}+\frac{dx^2}{4x^2}+\frac{1}{4x}d\Omega_2^2\right]$$

After coordinate transformation $y=z^2$ and $x=\exp(-2\eta)$,

$$ds^2=\frac{L^2}{z^2}\left[-(1-z^2)dt^2+\frac{dz^2}{1-z^2}+d\eta^2+\frac{e^{2\eta}}{4}d\Omega_2^2\right]$$

Zero energy hyperbolic black hole:

$$ds^2_{\mathbb{H}}=\frac{L^2}{z^2}\left[-(1-z^2)dt^2+\frac{dz^2}{1-z^2}+d\eta^2+\sinh^2\eta\, d\Omega_{d-3}^2\right]$$

Character of Einstein’s equations

Boundary conditions

A stationary solution is a solution to a well-posed boundary value problem. (Dias-Santos-Way 1510.02804)

Manifolds with a boundary

There is an induced metric and extrinsic curvature ($K_{\mu\nu}$) at the boundary.

Modified Dirichlet

Mixed Dirichlet-Neumann

mixed condition or Neumann condition for simplicity.

Asymptotic boundary

Fictitious boundary

Axis

Killing horizon

Extremal horizon

Methods

• Pseudospectral method
• Newton’s method

The package GRSpectral implements the above method.

References for NR

Stationary

Time evolution

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