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Nonlinear causality of Israel-Stewart theory with diffusion

Ian Cordeiro, Fábio S. Bemfica, Enrico Speranza, Jorge Noronha

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.

Nonlinear causality of Israel-Stewart theory with diffusion

Abstract

We present the first fully nonlinear causality constraints in 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 dimensions, and show that our general constraints reduce to their results upon an appropriate choice of angles. For this equation of state in both and 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.

Paper Structure

This paper contains 17 sections, 5 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hydrodynamic Frames
  3. Causality in the Landau Frame
  4. The characteristic determinant
  5. Nonlinear causality in the Landau frame
  6. Ultrarelativistic ideal gas
  7. Linear causality conditions
  8. Causality in the Eckart Frame
  9. The characteristic determinant
  10. Nonlinear causality in the Eckart frame
  11. Ultrarelativistic ideal gas
  12. Dominant energy condition
  13. Linear causality conditions
  14. Conclusion
  15. ELEMENTARY CONSTRAINTS ON POLYNOMIAL ROOTS
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\psi\in[-1,1]$, $\Delta_L\equiv \Delta[P_4^{(L)}]$ be the discriminant of $P_4^{(L)}$ and define the following quantities: The roots of the characteristic polynomial defined via the characteristic equation $\det(\mathbb{A}^\alpha_L\phi_\alpha) = 0$ are real if and only if one of the following conditions holds. Furthermore, these conditions provide information of the multiplicity of the roots

Figures (1)

  • Figure 1: Flowchart depicting implications between subcases for diffusion in both Landau and Eckart frames. All boxes correspond to nonlinear constraints unless specified as linear. Boxes are color-coded by their adherence to the definition of causality prescribed by (CI) and (CII) in Eq. \ref{['eq:causalitydef']}---cyan signifies that the causality bounds provided in the paper are simultaneously necessary and sufficient, magenta signifies that only sufficient conditions are provided. Note that (A) $\Rightarrow$ (B) is read as "(A) implies (B)," or equivalently, that "(B) is necessary for (A)." Furthermore, "(A) $\Leftrightarrow$ (B)" is read as "(a) if and only if (B)" or "(B) is necessary and sufficient for (A)." For brevity, we write EoS = "equation of state" and UR = "ultrarelativistic (ideal gas)."

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof