On the existence of solutions and causality for relativistic viscous conformal fluids

Disconzi, Marcelo M.

doi:10.3934/cpaa.2019075

Cited by 14 publications

(25 citation statements)

References 37 publications

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“…Therefore, we have verified the hypotheses of [56, Theorem A.23] and we conclude that equations (3) admit a solution in a neighborhood of p. Recall that a solution to Einstein's equations in wave coordinates gives rise to a solution to the full Einstein equations (i.e., Einstein's equations in arbitrary coordinates) if and only if the constraint equations are satisfied, which is the case by assumption. Thus, we have obtained a solution to Einstein's equations coupled to (1) in a neighborhood of p. A standard gluing argument that relies on the causality of solutions already established (see, e.g., [9,Chapter 10] or [56]) gives a solution defined in a neighborhood of Σ. We have therefore obtained a space-time where statements (1)-(5) hold (we notice that statements (2)-(4) are immediate consequences of the 22 above constructions).…”

Section: Appendix A: Kinetic Theory Derivationmentioning

confidence: 99%

Nonlinear causality of general first-order relativistic viscous hydrodynamics

2019

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Effective theory arguments are used to derive the most general energy-momentum tensor of a relativistic viscous fluid with an arbitrary equation of state (in the absence of other conserved currents) that is first-order in the derivatives of the energy density and flow velocity and does not include extended variables such as in Mueller-Israel-Stewart-like theories. This energy-momentum tensor leads to a causal theory, provided one abandons the usual conventions for the out-of-equilibrium hydrodynamic variables put forward by Landau-Lifshitz and Eckart. In particular, causality requires nonzero out-of-equilibrium energy density corrections and heat flow. Conditions are found to ensure linear stability around equilibrium in flat space-time. We also prove local existence and uniqueness of solutions to the equations of motion. Our causality, existence, and uniqueness results hold in the full nonlinear regime, without symmetry assumptions, in four space-time dimensions, with or without coupling to Einstein's equations, and are mathematically rigorously established. Furthermore, a kinetic theory realization of this energy-momentum tensor is also provided. * Electronic address: fabio.bemfica@ect.ufrn.br † Electronic address: marcelo.disconzi@vanderbilt.edu ‡ Electronic address: noronha@if.usp.br 1

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Section: Appendix A: Kinetic Theory Derivationmentioning

confidence: 99%

Nonlinear causality of general first-order relativistic viscous hydrodynamics

2019

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“…For brevity, our proof will omit certain technicalities that might be of interest for more mathematically minded readers but would obfuscate the main ideas. Those interested in such technical aspects can consult [107], where proofs of Theorems 1 and 2 are given for an audience of mathematically inclined readers 18 . Proof of Theorem 1.…”

Section: Well-posedness and Causalitymentioning

confidence: 99%

“…In[107], for simplicity, only the case a 1 = 4, a 2 ≥ 3a 1 a 1 −1 = 4 is treated. The arguments there presented, however, are essentially the same to cover the remaining cases.…”

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confidence: 99%

Causality and existence of solutions of relativistic viscous fluid dynamics with gravity

Bemfica¹,

Disconzi²,

Noronha³

2018

Phys. Rev. D

213

263

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A new approach is described to help improve the foundations of relativistic viscous fluid dynamics and its coupling to general relativity. Focusing on neutral conformal fluids constructed solely in terms of hydrodynamic variables, we derive the most general viscous energy-momentum tensor yielding equations of motion of second order in the derivatives, which is shown to provide a novel type of generalization of the relativistic Navier-Stokes equations for which causality holds. We show how this energy-momentum tensor may be derived from conformal kinetic theory. We rigorously prove existence, uniqueness, and causality of solutions of this theory (in the full nonlinear regime) both in a Minkowski background and also when the fluid is dynamically coupled to Einstein's equations. Linearized disturbances around equilibrium in Minkowski spacetime are stable in this causal theory. A numerical study reveals the presence of an out-of-equilibrium hydrodynamic attractor for a rapidly expanding fluid. Further properties are also studied and a brief discussion of how this approach can be generalized to non-conformal fluids is presented. 9 For geometric equations such as Einstein's equations, uniqueness is understood in a geometric sense, i.e., up to changes by diffeomorphisms. See, e.g., [11, Theorem 10.2.2]. 10 Strictly speaking, we are defining here local well-posedness of the initial value problem, which is the relevant notion of existence and uniqueness for evolution problems. We can also define local well-posedness for boundary value problems, etc. 11 In the mathematical literature, continuity with respect to the initial data is sometimes also referred to as stability, but we stress that this is entirely different from the notion of stability which is discussed in this paper (which follows the notion of stability introduced in [19], see section V). For example, the ordinary differential equationẋ = x, x(0) = x 0 has solution x(t) = x 0 e t , which varies continuously with x 0 . However, the trivial solution x trivial (t) ≡ 0 corresponding to x 0 = 0 is unstable in the terminology of this paper in that for any x 0 = 0, x(t) will diverge exponentially from x trivial .15 Provided the weak energy condition is satisfied, see section VIII B and Ref. [91]. 16 This meaning of the word frame has nothing to do with "rest" and "boosted frames." Unfortunately, these terminologies are too widespread to be changed here. Hence, we use the word frame to refer to both a choice of local temperature and velocity, e.g., the Landau frame, and in the usual sense of relativity, e.g., the rest frame. The difference between both uses will be clear from the context. We also note that frame, in the sense of a choice of local variables, has been used unevenly in the literature. In [92], for instance, frame is used in the same sense as employed here. In [23], the authors employ frame, or, more specifically, hydrodynamic frame, to refer solely to the choice that determines the local flow velocity, while the choices that determine the local temperatur...

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“…The characteristics corresponding to u α ξ α = 0 are simply the flow lines of u α , which are causal as long as u α remains timelike. The characteristics associated with the vanishing of the bracket in (7) are precisely the characteristics of an acoustical metric [55,72] with an effective sound speed squared given by c 2 acoustical = ζ τΠ(ε+P +Π) + α 1 + α2 n ε+P +Π ≥ 0. Thus, the system will be causal as long as u α remains timelike (and normalized) and c 2 acoustical ≤ 1, which is precisely condition (1).…”

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confidence: 99%

Causality of the Einstein-Israel-Stewart Theory with Bulk Viscosity

2019

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We prove that Einstein's equations coupled to equations of Israel-Stewart-type, describing the dynamics of a relativistic fluid with bulk viscosity and nonzero baryon charge (without shear viscosity or baryon diffusion) dynamically coupled to gravity, are causal in the full nonlinear regime. We also show that these equations can be written as a first-order symmetric hyperbolic system, implying local existence and uniqueness of solutions to the equations of motion. We use an arbitrary equation of state and do not make any simplifying symmetry or near-equilibrium assumption, requiring only physically natural conditions on the fields. These results pave the way for the inclusion of bulk viscosity effects in simulations of gravitational-wave signals coming from neutron star mergers.

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On the existence of solutions and causality for relativistic viscous conformal fluids

Cited by 14 publications

References 37 publications

Nonlinear causality of general first-order relativistic viscous hydrodynamics

Nonlinear causality of general first-order relativistic viscous hydrodynamics

Causality and existence of solutions of relativistic viscous fluid dynamics with gravity

Causality of the Einstein-Israel-Stewart Theory with Bulk Viscosity

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