Causality of Holographic Hydrodynamics

TL;DR

The paper investigates causality constraints in holographic hydrodynamics with Gauss–Bonnet gravity, linking the GB coupling to the difference of the dual CFT central charges. By analyzing both second-order hydrodynamics and the full GB theory via quasinormal modes, it derives bounds on λ_GB: [-0.711(2), 0.113(0)] from the truncated hydrodynamics and a tighter [-7/36, 9/100] from the exact analysis, the latter aligning with Hofman–Maldacena’s positive-energy constraints. The work demonstrates that exact causality (through all channels) imposes stricter limits on the theory, which translate into tight central-charge difference bounds (−1/2 ≤ (c−a)/c ≤ 1/2). It also highlights the different regimes of validity for the second-order description and the full theory, and discusses potential instabilities near the bounds.

Abstract

We study causality violation in holographic hydrodynamics in the gauge theory/string theory correspondence, focussing on Gauss-Bonnet gravity. The value of the Gauss-Bonnet coupling is related to the difference between the central charges of the dual conformal gauge theory. We show that, when this difference is sufficiently large, causality is violated both in the second-order truncated theory of hydrodynamics, as well as in the exact theory. We find that the latter provides more stringent constraints, which match precisely those appearing in the CFT analysis of Hofman and Maldacena.

Figures (5)

• Figure 1: (Colour online) Causality of the second-order Gauss-Bonnet hydrodynamics is violated once $\tau_\Pi T <2\frac{\eta}{s}$. Thus, $\lambda_\textrm{\tiny GB}\in [\lambda_{min},\lambda_{max}]$, where $\lambda_{min}=-0.711(2)$ and $\lambda_{max}=0.113(0)$.
• Figure 2: (Colour online) Front velocity for the shear (red) and sound (blue) channels for the second-order hydrodynamics, as given in (\ref{['cs']}) and (\ref{['cso']}). The dashed vertical lines indicate $\lambda_{min}$ and $\lambda_{max}$, where $v^{front}_{[\rm sound]}$ reaches one.
• Figure 3: (Colour online) Typical behaviour of $U^0$, the leading contribution to the Schrödinger potential, for both the shear (blue) and sound (red) channels. The solid and dashed curves show the behaviour for large and small $|\lambda_\textrm{\tiny GB}|$, respectively. (Our representative values here are: $\lambda_\textrm{\tiny GB}=-1.5$ and $-0.15$.)
• Figure 4: (Colour online) The phase velocity Re$(\mathfrak{w})/\bm{k}$ for $\lambda_\textrm{\tiny GB}=-2.5$: The blue line shows the behaviour of the lowest quasinormal mode calculated numerically. The red curve corresponds the second-order hydrodynamic approximation (\ref{['sound']}). The green curve corresponds the next order Taylor expansion (\ref{['soundf']}) arising from the second-order dispersion relation.
• Figure 5: (Colour online) The decay width Im$(\mathfrak{w})$ for $\lambda_\textrm{\tiny GB}=-2.5$: The blue line shows the behaviour of the lowest quasinormal mode calculated numerically. The red curve corresponds the second-order hydrodynamic approximation (\ref{['sound']}). The green curve corresponds the next order Taylor expansion (\ref{['soundf']}) arising from the second-order dispersion relation.