Entropy Bound and Causality Violation in Higher Curvature Gravity

TL;DR

The paper investigates how curvature-squared corrections in higher-curvature gravity theories modify fundamental holographic bounds and black hole thermodynamics. Using Gauss-Bonnet and $(\mathrm{Riemann})^2$ terms in AdS$_5$ gravity, the authors derive how the shear viscosity to entropy density ratio $\eta/s$ can depart from the universal Einstein-gravity value $1/(4\pi)$ and relate these deviations to entropy bounds of AdS black holes. They show that consistency (positive extremal entropy and causality) imposes constraints: for AdS$_5$ Gauss-Bonnet, $\lambda_{GB}<1/12$ (among other bounds), and for $(\mathrm{Riemann})^2$ gravity, $\lambda_{\mathrm{Riem}}<1/8$, with additional $\epsilon$-dependent limits giving $0<\eta/s\le (3/2)(1/4\pi)$. These results link bulk thermodynamics and boundary causality to transport properties of strongly coupled gauge theories, offering insights into quantum gravity corrections and possible phenomenology for real-world strongly coupled fluids. They also highlight that higher-derivative corrections can modestly lower or raise $\eta/s$ depending on topology and coupling, underscoring the need to understand full higher-curvature theories beyond GB and $R^2$ terms.

Abstract

In any quantum theory of gravity we do expect corrections to Einstein gravity to occur. Yet, at fundamental level, it is not apparent what the most relevant corrections are. We argue that the generic curvature square corrections present in lower dimensional actions of various compactified string theories provide a natural passage between the classical and quantum realms of gravity. The Gauss-Bonnet and $({\rm Riemann})^2$ gravities, in particular, provide concrete examples in which inconsistency of a theory, such as, a violation of microcausality, and a classical limit on black hole entropy are correlated. In such theories the ratio of the shear viscosity to the entropy density, $η/s$, can be smaller than for a boundary conformal field theory with Einstein gravity dual. This result is interesting from the viewpoint that the nuclear matter or quark-gluon plasma produced (such as at RHIC) under extreme densities and temperatures may violate the conjectured bound $η/s\ge 1/4π$, {\it albeit} marginally so.

Figures (10)

• Figure 1: The Hawking temperature as a function of $x$ (left plot) with $d=5$, $\lambda_{\hbox{$\rm GB$}}=0.1$ and (right plot) with $d=6$, $\lambda_{\rm GB}=0.25$. The solid (green), short-dash (purple) and long-dash (red) lines correspond, respectively, to $\epsilon=+1$, $\epsilon=0$ and $\epsilon=-1$.
• Figure 2: The Hawking temperature vs entropy with $\epsilon=+1$ (spherical black hole). (Left plot) $d=5$, $\lambda_{\rm GB}=1/12, 0.15, 0.25$ (top to bottom) and (right plot) $d=6$, $\lambda_{\rm GB}= 0.135, 0.25, 0.5$ (top to bottom). In each plot the dotted line corresponds to $\lambda_{\hbox{$\rm GB$}}=0$.
• Figure 3: The Hawking temperature vs entropy with $\epsilon=-1$. (Left plot) $d=5$, $\lambda_{\rm GB}=0, 0.0834, 0.25$ and (right plot) $d=6$, $\lambda_{\rm GB}=0, 0.138, 0.25$ (top to bottom).
• Figure 4: The speed of graviton with $\lambda_{\hbox{$\rm GB$}}=0.084$ (left plot) and $\lambda_{\hbox{$\rm GB$}}=0.09$ (right plot), with $\tilde{\epsilon} =0.3$ (solid, green), $\tilde{\epsilon}=0$ (dash, purple) and $\tilde{\epsilon}=-0.3$ (long-dash, red). Here it is assumed that $\tilde{\epsilon}/q^2 \ll 1/2$ (i.e. $q\to \infty$).
• Figure 5: The speed of graviton $c_g^2$ (left plot) and the local speed of light $b_g^2$ (right plot) with $\lambda_{\hbox{$\rm GB$}}=0.1$, and $\tilde{\epsilon} =0.3$ (solid, green), $\tilde{\epsilon}=0$ (dash, purple), $\tilde{\epsilon}=-0.3$ (long-dash, red).