Static Axisymmetric Vacuum Solutions and Non-Uniform Black Strings

TL;DR

The paper tackles static axisymmetric vacuum gravity in dimensions greater than four and investigates the Gregory-Laflamme instability for compactified black strings. It introduces a conformal-gauge elliptic boundary-value framework solved by Gauss-Seidel relaxation to obtain fully non-linear non-uniform black string solutions, and validates the method against perturbation theory while exploring large deformation regimes. The key finding is that, at fixed circle length, non-uniform strings have higher mass than the critical uniform string and exhibit lower entropy, ruling them out as end states of GL decay; this has important implications for the phase structure and dynamics of higher-dimensional black holes. The work also outlines consistency checks, potential numerical improvements, and broad avenues for applying the method to other axisymmetric problems and localized horizons in higher dimensions.

Abstract

We describe new numerical methods to solve the static axisymmetric vacuum Einstein equations in more than four dimensions. As an illustration, we study the compactified non-uniform black string phase connected to the uniform strings at the Gregory-Laflamme critical point. We compute solutions with a ratio of maximum to minimum horizon radius up to nine. For a fixed compactification radius, the mass of these solutions is larger than the mass of the classically unstable uniform strings. Thus they cannot be the end state of the instability.

Figures (13)

• Figure 1: Summary of results: We have shown by numerical methods that the non-uniform branch of solutions always has larger mass, for a fixed compactification radius, than the critical uniform string. This implies that the non-uniform strings cannot be the end state of the Gregory-Laflamme instability for the unstable uniform strings. Not knowing the classical stability of these non-uniform solutions there are 3 plausible scenarios. 1) The non-uniform strings are classically stable, and may quantum mechanically decay by Hawking radiation to the critical point, like uniform stable strings. 2) They are classically unstable and decay to stable uniform strings. We have shown the horizon volume, or entropy, of non-uniform strings is always less than that of a uniform string of the same mass, so this is allowed by the second law, although the mass lost in the decay must be small. 3) They are classically unstable and decay in the same way as the unstable uniform strings. If stable, it is likely these solutions may play an important role in black hole formation in compactified theories.
• Figure 2: Contour plots of the metric functions $A, B, C$ for a perturbation with intermediate non-uniformity, $\lambda \simeq 1.2$. $B_{max}$, the value of $B$ on the horizon at $r = 0$, $z = L$ is in the bottom right hand corner in the $B$ plot. Note that the gradients are largest in this corner. (Generated using $m = 1$, $L = 2.4758$, with grid resolution $240*100$, $dr, dz \simeq 0.025$)
• Figure 3: Embeddings of the spatial geometry of the event horizon into 3 dimensional space, projecting out the 2 trivial sphere directions. The top plot is for $\lambda = 0.1$, where the leading order perturbation theory gives good results. Already for $\lambda = 1.2$, the bottom left plot, the deformation is large and in the non-perturbative regime. The bottom right plot is of the most non-uniform string relaxed in this paper, having $\lambda = 3.9$.
• Figure 4: Top left and right: Again for $\lambda = 1.2$, plots of the measure weighted $G^{r}_{~z}$ and $G^{r}_{~r}$ respectively. Note $G^{r}_{~r}$ is equivalent to the constraint $(G^{r}_{~r}-G^{z}_{~z})$ due to the interior equations being satisfied. These should be compared to two components of the measure weighted Weyl tensor, $C^{tr}_{~~tr}$ and $C^{tz}_{~~tz}$ on the bottom left and right, with indices as they would appear in curvature invariants. The weighted curvatures are of order one, but the weighted constraints are much smaller, showing the constraints are well satisfied, the peak constraint values being a few percent of the typical curvatures.
• Figure 5: Plots of the metric function $A$ for $B_{max} = 0.025$. We fix $m = 1$, the mass per unit length of the background metric, and vary the asymptotic $S^1$ length, $L$. The lower plot uses the critical value $L = 2.4758$. The top left uses a value of $L$$20 \%$ smaller than the critical, and the top right uses $20 \%$ larger. One sees that the small perturbation in the critical case is simply superimposed on the static mass mode background for the two top plots. In these cases, the mass mode readjusts the mass per unit length so that the solution has the correct periodic length. We find we may start with any $L$ and have $m = 1$, and the method works. This allows consistency checks to be made, as in the perturbation approach, and is analogous to a non-perturbative 'change of scheme'.