Causality and Hyperbolicity of Lovelock Theories

TL;DR

The paper investigates causality and well-posedness in Lovelock gravity by analyzing characteristic hypersurfaces derived from the principal symbol. It proves that Killing horizons are characteristic for all gravitational degrees of freedom, ensuring that gravitational signals cannot escape a stationary black-hole interior. In Ricci-flat type N spacetimes, the characteristic structure reduces to null surfaces of a family of effective metrics, guaranteeing hyperbolicity in these backgrounds, while in static black-hole spacetimes hyperbolicity holds for large black holes but can fail near the horizon for small ones, implying ill-posed evolution in those regimes. These results clarify the causal structure of Lovelock theories and have implications for black-hole stability, nonlinear evolution, and holographic constraints on theory parameters.

Abstract

In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalise a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hypersurfaces are generically all non-null and that Lovelock theories are hyperbolic in any such spacetime. Next we consider static, maximally symmetric black hole solutions of Lovelock theories. Again, characteristic surfaces are generically non-null. For some small black holes, hyperbolicity is violated near the horizon. This implies that the stability of such black holes is not a well-posed problem.

Figures (4)

• Figure 1: Cross-section of null cones of the effective metrics, and the light cone of the physical metric, with axes scaled so that the light cone appears as a circle. The solid black curve shows the light cone with respect to the physical metric. The dotted red curve, the dot - dashed green curve, and the dashed blue curve give the null cones for tensor, vector, and scalar perturbation sectors, respectively. Here we give results for a spherically symmetric solution ($\kappa=1$) with $\Lambda=0$, $r_0=1$, $k_2=-1/4$ in $d=7$ at $r=1.5, 3, 4$. We consider a vector lying in the equatorial plane, with components $v^t,v^r,v^\phi$ (with $\phi \sim \phi+2\pi$ an angular coordinate). The plots show a cross-section of the null cones with $v^t = f^{-1}$, which implies $(v^r)^2 +f r^{-2} c_A^{-1}(v^\phi)^2=1$.
• Figure 2: Effective potentials for $\Lambda=0$, $\kappa=1$, $r_0=1$, $k_2=-1/4$ in $d=7$. The solid black curve corresponds to the physical metric. The dotted red curve, the dot - dashed green curve, and the dashed blue curve give the tensor, vector, and scalar perturbation sectors, respectively.
• Figure 3: Effective potentials for $\Lambda=0$, $\kappa=1$, $r_0=1$, $k_2=-1/4$ in $d=5$ (left) and $d=6$ (right). Same colour scheme as figure \ref{['Fig:Veff1']}.
• Figure 4: Right: effective potential for AdS length $\ell=1$, $\kappa=0$, $r_0=1$, $k_2=-1/80$ in $d=5$. Same colour scheme as figure \ref{['Fig:Veff1']}.