Nonlinear causality of Israel-Stewart theory with diffusion

TL;DR

This paper establishes the first fully nonlinear causality constraints for Israel-Stewart hydrodynamics with diffusion in 3+1 dimensions, revealing new physical bounds that linearized theory cannot capture.

Contribution

It derives algebraic nonlinear causality constraints for Israel-Stewart theory with diffusion in 3+1 dimensions, extending previous 1+1 dimensional results and highlighting differences between hydrodynamic frames.

Findings

Existence of a causally allowed region with spacelike baryon current in Landau frame.

Solutions in Eckart frame do not violate the dominant energy condition under causality.

Linearized causality bounds are insufficient to capture the full physical constraints.

Abstract

We present the first fully nonlinear causality constraints in $D = 3 + 1$ dimensions for Israel-Stewart theory in the presence of energy and number diffusion in the Eckart and Landau hydrodynamic frames, respectively. These constraints are algebraic inequalities that make no assumption on the underlying geometry of the spacetime or the equation of state. In order to highlight the distinct physical and structural behavior of the two hydrodynamic frames, we discuss the special ultrarelativistic ideal gas equation of state considered in earlier literature in $D = 1 + 1$ dimensions, and show that our general $D = 3 + 1$ constraints reduce to their results upon an appropriate choice of angles. For this equation of state in both $D = 1 + 1$ and $D = 3 + 1$ dimensions one can show that: (i) there exists a region allowed by nonlinear causality in which the baryon current transitions into a spacelike vector in the Landau frame, and (ii) an analogous argument shows that the solutions of the Eckart frame equations of motion never violate the dominant energy condition, assuming nonlinear causality holds. We then compare our results with those from linearized Israel-Stewart theory and show that the linear causality bounds fail to capture the new physical constraints on energy and number diffusion that are successfully obtained through our nonlinear causality approach.