A new approach to static numerical relativity, and its application to Kaluza-Klein black holes

TL;DR

<3-5 sentence high-level summary>

Abstract

We propose a general framework for solving the Einstein equation for static and Euclidean metrics. First, we address the issue of gauge-fixing by borrowing from the Ricci-flow literature the so-called DeTurck trick, which renders the Einstein equation strictly elliptic and generalizes the usual harmonic-coordinate gauge. We then study two algorithms, Ricci-flow and Newton's method, for solving the resulting Einstein-DeTurck equation. We illustrate the use of these methods by studying localized black holes and non-uniform black strings in five-dimensional Kaluza-Klein theory, improving on previous calculations of their thermodynamic and geometric properties. We study spectra of various operators for these solutions, in particular finding negative modes of the Lichnerowicz operator. We classify the localized solutions into two branches that meet at a minimum temperature. We find good evidence for a merger between the localized and non-uniform solutions. We also find a narrow window of localized solutions that possess negative modes yet appear to have positive specific heat.

Figures (16)

• Figure 1: The geometry of the two overlapping charts for the localized black holes. The points drawn are for the lowest resolution considered here, $20 \times 60$ with $R = 3 L$ where we have chosen $L = 1$. Boundary points are shown enlarged as are points that must be interpolated from the other chart.
• Figure 2: Behaviour of the area of the horizon (left) and area of the time circle 2-sphere (right) as a function of Ricci-flow time for 3 initial data. In blue, $r_0$ is slightly greater than the critical value, in red, $r_0$ is slightly less, and the yellow gives the critical value. The red and blue curves diverge from the yellow due to the presence of the single negative mode. The dashes horizontal lines give the value of the area at the fixed point, calculated using Newton's method, and thus we see agreement with the critical flow.
• Figure 3: Cross-sections of proper embeddings of the horizon are shown in red for $L = 1$ against Ricci-flow time for the same 3 flows as in the previous figures. The axis of symmetry is also shown in blue with its proper length. The top figure corresponds to the red curves, $r_0$ less than the critical value, the middle figure gives the critical flow, and the bottom figure corresponds the blue curves so $r_0$ is greater than the critical value.
• Figure 4: Brackets for $r_0$ over a range of $\beta$. For the lower end of the bracket the sphere collapses, and for the upper end the time circle shrinks in finite flow time. Outside this range, higher resolution is required to accurately simulate the flows. The value $r_0$ gives the initial radius of the horizon for the flow. We also plot the equatorial radius of the horizon for solutions shown as small dots, together with the small localized approximation for this radius which is rather accurate over this range of $\beta$.
• Figure 5: Thermodynamic behaviour of the various static Kaluza-Klein black holes for $L = 1$. Top left: Plot of mass, $M$, against $\beta$. The small localized branch are shown in red, the large localized branch in orange, and the non-uniform solutions are in light blue. All are for $80\times240$ resolution with $R = 3 L$. The uniform branch is given by the dark blue line, and the small localized analytic approximations are shown by a thin line. The solid blue disc indicates the Gregory-Laflamme marginal point. Similar conventions are used to plot the horizon area $A_{horiz}$ (top right), tension $\tau$ (bottom left) and free energy $F$ (bottom right).