All stationary axi-symmetric local solutions of topologically massive gravity

TL;DR

This work systematically classifies all stationary axi-symmetric solutions of 3D topologically massive gravity (TMG). By reducing TMG to a 0+1D topologically massive mechanics (TMM) problem with a Lorentz-vector $\boldsymbol{X}$, the authors perform a Hamiltonian analysis revealing a 6D physical phase space and a four-dimensional invariant subspace in which the Einstein, Schrödinger, and warped sectors emerge in closed form. The generic sector, containing all remaining solutions, is constructed numerically via explicit algorithms that require three constants of motion, and the authors illustrate rich phenomena including solitons and naked singularities, some of which approach warped AdS asymptotically with nonanalytic subleading terms. Generalisations to positive, zero, or vanishing cosmological constant are discussed, and the approach is suggested to extend to other 3D massive gravity theories. Overall, the paper provides a comprehensive map of stationary axi-symmetric TMG solutions and introduces practical numerical tools to explore the elusive generic sector.

Abstract

We classify all stationary axi-symmetric solutions of topologically massive gravity into Einstein, Schrödinger, warped and generic solutions. We construct explicitly all local solutions in the first three sectors and present an algorithm for the numerical construction of all local solutions in the generic sector. The only input for this algorithm is the value of one constant of motion if the solution has an analytic centre, and three constants of motion otherwise. We present several examples, including soliton solutions that asymptote to warped AdS.

Figures (10)

• Figure 1: The plots show a solution in the generic sector without a centre. Here $\mu = 4$ and $\ell = 1$. Evolution starts at $\rho = 0$ where the initial values $\boldsymbol{X}(0) = (-1, 7/3, -2/3)$, $\dot{\boldsymbol{X}}(0) = (-1/5,-1/10,-8/3)$ and $J^+ = -11/4$ were chosen.
• Figure 2: The same solution as in Fig. \ref{['fig:Geon1']} but with a much larger range in $\rho$. We have let it evolve until the numerics becomes dubious. The facts that $\ddot{\boldsymbol{X}}$ approaches a constant and that $K$ tends to the exact value for warped AdS at $\mu \ell = 4$ (straight line) indicate that the solution is asymptotically warped AdS.
• Figure 3: This figure shows the same solution as in Fig. \ref{['fig:Geon1']} but Lorentz rotated to stabilise the numerics. We see that the numerical errors are small as compared to Fig. \ref{['fig:Geon2']}. Note that the scale on the $\rho$-axis is logarithmic and that we have multiplied three quantities by half-integer powers of $\rho$. Asymptotically these quantities exhibit sinusoidal dependence on $\log \rho$.
• Figure 4: The same solution as in Fig. \ref{['fig:AnalCenters']} is plotted in a more favourable Lorentz frame. We observe the same oscillating behaviour as in the case without a centre. For this case we also include a plot of $|\rho|^{3/2}\partial_{\rho} (X^+/\rho^2)$ from which the sub-leading behaviour of $X^+$ can be deduced. Note that the errors are small and that we plot against $|\rho| = -\rho$, so the asymptotic region is to the right in the plots.
• Figure 5: Flow chart for the decision which algorithm to use starts in upper left corner