Graviton time delay and a speed limit for small black holes in Einstein-Gauss-Bonnet theory

TL;DR

This work demonstrates that in Einstein-Gauss-Bonnet gravity, gravitons of tensor polarization can experience Shapiro time advances when scattering off small black holes, while scalar and vector gravitons remain time-delay dominated. Using a framework of hyperbolicity, effective metrics, and bicharacteristic curves, the authors show a velocity/causality speed limit for small black holes and compute graviton trajectories and optical observables. Perturbative and numerical analyses reveal a regime where the time advance coexists with deflection angles $\Delta\phi_\infty<\pi$, with the finite-time-advance scale set by the GB length $L$ and a mass-parameter dependence $|D_\infty|\sim L$; for $d\ge7$, tensor-time-advance is robust, while in $d=5,6$ small BH unphysicality precludes the most extreme limits but time advances still appear in suitable configurations. The study further argues that proposed time-machine constructions based on time advances do not arise from well-posed Cauchy evolution, owing to the speed limit and hyperbolicity considerations, which preserves causal consistency in EGB theory within the explored regime.

Abstract

Camanho, Edelstein, Maldacena and Zhiboedov have shown that gravitons can experience a negative Shapiro time delay, i.e. a time advance, in Einstein-Gauss-Bonnet theory. They studied gravitons propagating in singular "shock-wave" geometries. We study this effect for gravitons propagating in smooth black hole spacetimes. For a small enough black hole, we find that gravitons of appropriate polarisation, and small impact parameter, can experience time advance. Such gravitons can also exhibit a deflection angle less than $π$, characteristic of a repulsive short-distance gravitational interaction. We discuss problems with the suggestion that the time advance can be used to build a "time machine". In particular, we argue that a small black hole cannot be boosted to a speed arbitrarily close to the speed of light, as would be required in such a construction.

Figures (7)

• Figure 1: Effective potentials for a black hole with $r_{H}=1$ in $d=5,6,7,8$ dimensions. We fix the Gauss-Bonnet coupling $\lambda_{\rm GB}=2$. The red curve corresponds to the effective potential for photons, i.e., null geodesics of the physical metric ($c=1$). Superluminal propagation ($c_A>1$) corresponds to an effective potential which is larger than that for photons. This happens only for tensor polarizations. The violation of hyperbolicity is associated with the region in which one of the effective potentials becomes negative. This happens near the horizon for small black holes in five and six dimensions.
• Figure 2: Effective potential (left) and deflection angle (right) for tensor polarized gravitons scattered by a small black hole in $d=8$. We set $r_{H}=0.03 \sqrt{\alpha}$, which gives $\mu\approx1.4\times10^{-5}\alpha^{5/2}$ and $L\approx0.2\sqrt{\alpha}$.
• Figure 3: Deflection angle for small black holes in $d=7,8$. We set $r_{H}=0.03 \sqrt{\alpha}$ which gives $\mu\approx4.5\times10^{-4}\alpha^{2}$, $L\approx0.28\sqrt{\alpha}$ in $d=7$, and $\mu\approx1.4\times10^{-5}\alpha^{5/2}$, $L\approx0.2\sqrt{\alpha}$ in $d=8$. The dashed line represents the perturbative approximation \ref{['eq:dphi_approx']}.
• Figure 4: Time delay for small black holes in $d=7,8$. We set $r_{H}=0.03 \sqrt{\alpha}$ as before. The solid, dashed, dot-dashed and dotted lines correspond to $R=2.5\sqrt{\alpha}$, $R=5\sqrt{\alpha}$, $R=10\sqrt{\alpha}$ and $R=50\sqrt{\alpha}$ respectively. The black dashed line corresponds to the perturbative approximation \ref{['advance']}.
• Figure 5: Time delay for fixed $b$ (such that $\Delta\phi_{\infty}=\pi$) expressed as a function of the radius of the cavity for a small black hole in $d=7$. We set $r_{H}=0.03 \sqrt{\alpha}$ as before. The dashed line corresponds to the limit $R \rightarrow \infty$.

Theorems & Definitions (12)

• Definition A.1
• Definition A.2
• Definition A.3
• Definition A.4
• Remark
• Definition A.5
• Remark
• Definition A.6
• Remark
• Definition A.7