Thermodynamic Stability Implies Causality

The stability conditions of a relativistic hydrodynamic theory can be derived directly from the requirement that the entropy should be maximized in equilibrium. Here, we use a simple geometrical argument to prove that, if the hydrodynamic theory is stable according to this entropic criterion, then localized perturbations to the equilibrium state cannot propagate outside their future light cone. In other words, within relativistic hydrodynamics, acausal theories must be thermodynamically unstable, at least close to equilibrium. We show that the physical origin of this deep connection between stability and causality lies in the relationship between entropy and information. Our result may be interpreted as an “equilibrium conservation theorem,” which generalizes the Hawking-Ellis vacuum conservation theorem to finite temperature and chemical potential.

Figure 1

Visualization of the geometric argument. The initial perturbation (in red) is located to the left of the origin. Causality requires it to stay confined in the $t\ge x$ half-plane. We build the triangle $ABC$ in a way that all its edges are spacelike. $B$ and $C$ intersect at the origin. We can regulate $B$ to be arbitrarily close to the line $t=x$. The green arrows are a Euclidean representation of the unit normal vectors to the edges and are taken inward pointing, consistently with our choice of metric signature [35].

Figure 2

Visualization of the geometric argument for the theorem of Bemfica et al. [7]. All the edges of the triangle $ABC$ are spacelike. We create an arbitrary initial perturbation (in red) on the side $C$. Since $A$ is outside the causal future of $C$, we are free to set the perturbation to zero on $A$. The inverse temperature four-vector ${\beta }^a$ (dark green) is aligned with the $t$ axis. The light green arrows are a Euclidean representation of the unit normals to the edges.