Caged Black Holes: Black Holes in Compactified Spacetimes II - 5d Numerical Implementation

TL;DR

The paper develops a convergent numerical approach to construct static black holes with S^3 horizons in 5D spacetime compactified on a circle, uncovering a continuous family of solutions parameterized by x and showing that small black holes recover the 5D Schwarzschild limit. By solving the 5D vacuum Einstein equations with a conformal ansatz and two overlapping patches, the authors extract thermodynamic and geometric observables and test the integrated first law via the Smarr relation. They find evidence of a tachyonic instability as masses approach the Gregory-Laflamme scale, and speculate a possible first-order phase transition to the black string. The work provides the first strong numerical evidence for static caged black holes in higher dimensions and lays groundwork for exploring stability and phase structure in compactified spacetimes.

Abstract

We describe the first convergent numerical method to determine static black hole solutions (with S^3 horizon) in 5d compactified spacetime. We obtain a family of solutions parametrized by the ratio of the black hole size and the size of the compact extra dimension. The solutions satisfy the demanding integrated first law. For small black holes our solutions approach the 5d Schwarzschild solution and agree very well with new theoretical predictions for the small corrections to thermodynamics and geometry. The existence of such black holes is thus established. We report on thermodynamical (temperature, entropy, mass and tension along the compact dimension) and geometrical measurements. Most interestingly, for large masses (close to the Gregory-Laflamme critical mass) the scheme destabilizes. We interpret this as evidence for an approach to a physical tachyonic instability. Using extrapolation we speculate that the system undergoes a first order phase transition.

Figures (19)

• Figure 1: A spacelike slice of the black-hole spacetime. $(a)$ In the $\{r,z\}$ plane the black hole's horizon is a curve with a spherical $S^3$ topology. $(b)$ There is a conformal freedom to transform the domain to $\{(r,z): |z| \le L, ~r^2+z^2 \ge \rho_h^{~2} \}$. By fixing $\rho_h/L$ the domain is uniquely specified KudohTanakaNakamura.
• Figure 2: 'The nearby region' of the integration domain covered by the polar coordinates $\rho,\xi$. The thin dashed lines mark the location of the false grid points used for numerical implementation of the Neumann or mixed Neumann-Dirichlet boundary conditions.
• Figure 3: The asymptotic region is glued to the 'nearby patch'. The two patches overlap in order to exchange information about the functions during relaxation.
• Figure 4: A log-log plot of the normalized residuals, $Res \ \psi$ vs. number of iterations, for $x=1/7$, implying convergence.
• Figure 5: A log-log plot of the normalized residuals vs. number of iterations for $x>0.20$. After an initial convergence the solution diverges.