A variational formulation of relativistic hydrodynamics

Generalized relativistic hydrodynamics with a convex extension

We propose a new model which describes relativistic hydrodynamics and generalizes the standard Euler system of isentropic perfect fluids. Remarkably, our system admits a convex extension which allows us to transform it to a symmetric hyperbolic form. This result sheds new light even on the relativistic Euler system.

Variational Principle of Hydrodynamics and Quantization by Stochastic Process

The well-known hydrodynamical representation of the Schrödinger equation is reformulated by extending the idea of Nelson-Yasue's stochastic variational method. The fluid flow is composed by the two stochastic processes from the past and the future, which are unified naturally by the principle of maximum entropy. We show that this formulation is easily applicable to the quantization of scalar fields.

Relativistic hydrodynamic accretion onto a spherical black hole

This paper has been withdrawn by the author. A revised and expanded version is gr-qc/9907028 (Phys.Rev. D60 (1999) 104043).

General Relativistic Hydrodynamics with Special Relativistic Riemann Solvers

We present a general and practical procedure to solve the general relativistic hydrodynamic equations by using any of the special relativistic Riemann solvers recently developed for describing the evolution of special relativistic flows. Our proposal relies on a local change of coordinates in terms of which the spacetime metric is locally Minkowskian and permits accurate numerical calculations of general relativistic hydrodynamics problems using the numerical tools developed for the special relativistic case with negligible computational cost. The feasibility of the method has been confirmed b…

New Relativistic Effects in the Dynamics of Nonlinear Hydrodynamical Waves

In Newtonian and relativistic hydrodynamics the Riemann problem consists of calculating the evolution of a fluid which is initially characterized by two states having different values of uniform rest-mass density, pressure and velocity. When the fluid is allowed to relax, one of three possible wave-patterns is produced, corresponding to the propagation in opposite directions of two nonlinear hydrodynamical waves. New effects emerge in a special relativistic Riemann problem when velocities tangential to the initial discontinuity surface are present. We show that a smooth transition from one wav…

Code Development of Three-Dimensional General Relativistic Hydrodynamics with AMR(Adaptive-Mesh Refinement) and Results From Special and General Relativistic Hydrodynamic

In this paper, the general procedure to solve the General Relativistic Hydrodynamical(GRH) equations with Adaptive-Mesh Refinement (AMR) is presented. In order to achieve, the GRH equations are written in the conservation form to exploit their hyperbolic character. The numerical solutions of general relativistic hydrodynamic equations are done by High Resolution Shock Capturing schemes (HRSC), specifically designed to solve non-linear hyperbolic systems of conservation laws. These schemes depend on the characteristic information of the system. The Marquina fluxes with MUSCL left and right stat…

Cosmological hydrodynamics with relativistic pressure and velocity

We present hydrodynamic equations with relativistic pressure and velocity in the presence of weak gravity, in a cosmological context. Previously we consistently derived special relativistic hydrodynamic equations with weak gravity in Minkowski background. With the relativistic pressure and velocity one cannot derive the cosmological counterpart by a simple transformation from equations in the Minkowski background. Here we present a proper derivation. We point out the potential importance of relativistic pressure and velocity in gravitational lensing.

First-order relativistic hydrodynamics is stable

We study linearized stability in first-order relativistic viscous hydrodynamics in the most general frame. There is a region in the parameter space of transport coefficients where the perturbations of the equilibrium state are stable. This defines a class of stable frames, with the Landau-Lifshitz frame falling outside the class. The existence of stable frames suggests that viscous relativistic fluids may admit a sensible hydrodynamic description in terms of temperature, fluid velocity, and the chemical potential only, i.e. in terms of the same hydrodynamic variables as non-relativistic fluids…

Scaling Relations in Two-Dimensional Relativistic Hydrodynamic Turbulence

We derive exact scaling relations for two-dimensional relativistic hydrodynamic turbulence in the inertial range of scales. We consider both the energy cascade towards large scales and the enstrophy cascade towards small scales. We illustrate these relations by numerical simulations of turbulent weakly compressible flows. Intriguingly, the fluid-gravity correspondence implies that the gravitational field in spacetimes with anti-de Sitter asymptotics should exhibit similar scaling relations.

Relativistic Hydrodynamics a brief review of classical and quantum fluids in relativistic astrophysics

The objective of this work is to revisit fundamental aspects of relativistic hydrodynamics, aiming at the construction of a first course in relativistic hydrodynamics and its applications to astrophysics at the level of end of undergraduate course and beginning of graduate course. We aim to introduce more basic concepts of basic hydrodynamics, going through models analogous to gravity to the theory of superfluids, applying mainly to astrophysics and the cosmology of the dark universe. We review the classical hydrodynamics, Galileo symmetry and its extension to Lorentz Symmetry applied to fluid…

A new three-dimensional general-relativistic hydrodynamics code

We present a new three-dimensional general relativistic hydrodynamics code, the Whisky code. This code incorporates the expertise developed over the past years in the numerical solution of Einstein equations and of the hydrodynamics equations in a curved spacetime, and is the result of a collaboration of several European Institutes. We here discuss the ability of the code to carry out long-term accurate evolutions of the linear and nonlinear dynamics of isolated relativistic stars.

On the convexity of Relativistic Hydrodynamics

The relativistic hydrodynamic system of equations for a perfect fluid obeying a causal equation of state is hyperbolic (Anile 1989 {\it Relativistic Fluids and Magneto-Fluids} (Cambridge: Cambridge University Press)). In this report, we derive the conditions for this system to be convex in terms of the fundamental derivative of the equation of state (Menikoff and Plohr 1989 {\it Rev. Mod. Phys.} {\bf 61} 75). The classical limit is recovered.

Relativistic cosmological hydrodynamics

We investigate the relativistic cosmological hydrodynamic perturbations. We present the general large scale solutions of the perturbation variables valid for the general sign of three space curvature, the cosmological constant, and generally evolving background equation of state. The large scale evolution is characterized by a conserved gauge invariant quantity which is the same as a perturbed potential (or three-space curvature) in the comoving gauge.

Special relativistic hydrodynamics with gravitation

The special relativistic hydrodynamics with weak gravity is hitherto unknown in the literature. Whether such an asymmetric combination is possible was unclear. Here, the hydrodynamic equations with Poisson-type gravity considering fully relativistic velocity and pressure under the weak gravity and the action-at-a-distance limit are consistently derived from Einstein's general relativity. Analysis is made in the maximal slicing where the Poisson's equation becomes much simpler than our previous study in the zero-shear gauge. Also presented is the hydrodynamic equations in the first post-Newtoni…

Variational Principle for Relativistic Fluid Dynamics

The variational principle for the special and general relativistic hydrodynamics are discussed in view of its application to obtain approximate solutions to these problems. We show that effective Lagrangians can be obtained for suitable ansatz for the dynamical variables such as density profile of the system. As an example, the relativistic version of spherical droplet motion (Rayleigh-Plesset equation) is derived from a simple Lagrangian. For the general relativistic case the most general Lagrangian for spherically symmetric systems is given.

Flux-variational formulation of relativistic perfect fluids

We give a variational formulation of perfect fluids on a general pseudoriemannian manifold by variating tangent fields according the flux produced by them. In this approach no constraints are needed. As a result, Euler and continuity equations are obtained quite directly.

Relativistic formulation of the Hall-Vinen-Bekarevich-Khalatnikov superfluid hydrodynamics

The relativistic analogue of the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) hydrodynamics is derived making use of the phenomenological method similar to that used by Bekarevich and Khalatnikov [1] in their derivation of HVBK-hydrodynamics. The resulting equations describe a finite-temperature superfluid liquid with the distributed vorticity. The main dissipative effects, including mutual friction, are taken into account. The proposed hydrodynamics is needed for reliable modeling of the dynamical properties of superfluid neutron stars.

General-relativistic hydrodynamics of non-perfect fluids: 3+1 conservative formulation and application to viscous black-hole accretion

We consider the relativistic hydrodynamics of non-perfect fluids with the goal of determining a formulation that is suited for numerical integration in special-relativistic and general-relativistic scenarios. To this end, we review the various formulations of relativistic second-order dissipative hydrodynamics proposed so far and present in detail a particular formulation that is fully general, causal, and can be cast into a 3+1 flux-conservative form, as the one employed in modern numerical-relativity codes. As an example, we employ a variant of this formulation restricted to a relaxation-typ…

Mathematical Clarification of General Relativistic Variational Principles

In this paper a mathematically precise global (i.e. not the usual local) approach is presented to the variational principles of general relativistic classical field theories. Problems of the classic (usual) approaches are also discussed in comparison. The aim of developing a global approach is to provide a possible tool for future efforts on proving global existence theorems of field theoretical solutions.

Implementing fully relativistic hydrodynamics in three dimensions

We report on our numerical implementation of fully relativistic hydrodynamics coupled to Einstein's field equations in three spatial dimensions. We briefly review several steps in our code development, including our recasting of Einstein's equations and several tests which demonstrate its advantages for numerical integrations. We outline our implementation of relativistic hydrodynamics, and present numerical results for the evolution of both stable and unstable Oppenheimer-Volkov equilibrium stars, which represent a very promising first test of our code.

A powerful hydrodynamic booster for relativistic jets

Velocities close to the speed of light are a robust observational property of the jets observed in microquasars and AGNs, and are expected to be behind much of the phenomenology of GRBs. Yet, the mechanism boosting relativistic jets to such large Lorentz factors is still essentially unknown. Building on recent general-relativistic, multidimensional simulations of progenitors of short GRBs, we discuss a new effect in relativistic hydrodynamics which can act as an efficient booster in jets. This effect is purely hydrodynamical and occurs when large velocities tangential to a discontinuity are pr…

Transient Relativistic Fluid Dynamics in a General Hydrodynamic Frame

We propose a new theory of second-order viscous relativistic hydrodynamics which does not impose any frame conditions on the choice of the hydrodynamic variables. It differs from Mueller-Israel-Stewart theory by including additional transient degrees of freedom, and its first-order truncation reduces to Bemfica-Disconzi-Noronha-Kovtun theory. Conditions for causality and stability are explicitly given in the conformal regime. As an illustrative example, we consider Bjorken flow solutions to our equations and identify variables which make a hydrodynamic attractor manifest.

From polytropic to barotropic perfect fluids in general-relativistic hydrodynamics

Two general-relativistic hydrodynamical models are considered: a model of self-gravitating static configurations of perfect fluid and a model of steady accretion of fluid onto a black hole. We generalise analytic results obtained for the original polytropic versions of these models onto a wider class of barotropic equations of state. The knowledge about the polytropic solutions is used to establish bounds on certain characteristic quantities appearing in both cases.

Numerical simulations of interfaces in relativistic hydrodynamics

We consider models of relativistic matter containing sharp interfaces across which the matter model changes. These models will be relevant for neutron stars with crusts, phase transitions, or for viscous boundaries where the length scale is too short to be modelled smoothly. In particular we look at numerical techniques that allow us to evolve stable interfaces, for the interfaces to merge, and for strong waves and shocks to interact with the interfaces. We test these techniques for ideal hydrodynamics in special and general relativity for simple equations of state, finding that simple level s…

Relativistic Hydrodynamics in Close Binary Systems: Analysis of Neutron-Star Collapse

We discuss the underlying relativistic physics which causes neutron stars to compress and collapse in close binary systems as has recently been observed in numerical (3+1) dimensional general relativistic hydrodynamic simulations. We show that compression is driven by velocity-dependent relativistic hydrodynamic terms which increase the self gravity of the stars. They also produce fluid motion with respect to the corotating frame of the binary. We present numerical and analytic results which confirm that such terms are insignificant for uniform translation or when the hydrodynamics is constrai…

Luminosity, redshift and gas abundance in general relativistic radiation hydrodynamics

Quasi-stationary flows of gas accreting onto a compact center are analyzed in the framework of general-relativistic radiation hydrodynamics, under assumptions of spherical symmetry and thin gas approximation. Numerical investigation shows that luminosity, redshift and gas abundance are correlated. The gas can constitute up to one third of the total mass of brightest low-redshift sources, but its abundance goes down to 1/30 for sources with luminosities close to the Eddington limit.

Relativistic Hydrodynamic Attractors with Broken Symmetries: Non-Conformal and Non-Homogeneous

Standard textbooks will state that hydrodynamics requires near-equilibrium to be applicable. Recently, however, out-of-equilibrium attractor solutions for hydrodynamics have been found in kinetic theory and holography in systems with a high degree of symmetry, suggesting the possibility of a genuine out-of-equilibrium formulation of hydrodynamics. This work demonstrates that attractor solutions also occur in non-conformal kinetic theory and spatially non-homogeneous systems, potentially having important implications for the interpretation of experimental data in heavy-ion and proton-proton col…

密度框架下的相对论粘性流体力学：数值检验与比较

Relativistic Viscous Hydrodynamics in the Density Frame: Numerical Tests and Comparisons

本文对密度框架下的一维相对论粘性流体动力学进行数值研究，验证其与QCD动理学理论模拟的一致性及边界稳健性，并与BDNK理论和MUSIC程序进行比较。

Solving 3D relativistic hydrodynamical problems with WENO discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods coupled to WENO algorithms allow high order convergence for smooth problems and for the simulation of discontinuities and shocks. In this work, we investigate WENO-DG algorithms in the context of numerical general relativity, in particular for general relativistic hydrodynamics. We implement the standard WENO method at different orders, a compact (simple) WENO scheme, as well as an alternative subcell evolution algorithm. To evaluate the performance of the different numerical schemes, we study non-relativistic, special relativistic, and general relativistic …

Newtonian hydrodynamic equations with relativistic pressure and velocity

We present a new approximation to include fully general relativistic pressure and velocity in Newtonian hydrodynamics. The energy conservation, momentum conservation and two Poisson's equations are consistently derived from Einstein's gravity in the zero-shear gauge assuming weak gravity and action-at-a-distance limit. The equations show proper special relativity limit in the absence of gravity. Our approximation is complementary to the post-Newtonian approximation and the equations are valid in fully nonlinear situations.

Riemann Solvers in General Relativistic Hydrodynamics

Our contribution concerns with the numerical solution of the 3D general relativistic hydrodynamical system of equations within the framework of the 3+1 formalism. We summarize the theoretical ingredients which are necessary in order to build up a numerical scheme based on the solution of local Riemann problems. Hence, the full spectral decomposition of the Jacobian matrices of the system, i.e., the eigenvalues and the right and left eigenvectors, is explicitly shown. An alternative approach consists in using any of the special relativistic Riemann solvers recently developed for describing the …

Explosion of a Minimum-Mass Neutron Star within Relativistic Hydrodynamics

The relativistic hydrodynamics equations are adapted for the spherically symmetric case and the Lagrangian form. They are used to model the explosive disruption of a minimum-mass neutron star: a key ingredient of the stripping model for short gamma-ray bursts. The shock breakout from the neutron star surface accompanied by the acceleration of matter to ultrarelativistic velocities is studied. A comparison with the results of previously published nonrelativistic calculations is made.

New theories of relativistic hydrodynamics in the LHC era

The success of relativistic hydrodynamics as an essential part of the phenomenological description of heavy-ion collisions at RHIC and the LHC has motivated a significant body of theoretical work concerning its fundamental aspects. Our review presents these developments from the perspective of the underlying microscopic physics, using the language of quantum field theory, relativistic kinetic theory, and holography. We discuss the gradient expansion, the phenomenon of hydrodynamization, as well as several models of hydrodynamic evolution equations, highlighting the interplay between collective…

Newtonian Hydrodynamics with General Relativistic Pressure

We present the general relativistic pressure correction terms in Newtonian hydrodynamic equations to the nonlinear order: these are equations (\ref{mass-conservation-Mink})-(\ref{Poisson-eq-Mink}). The derivation is made in the zero-shear gauge based on the fully nonlinear formulation of cosmological perturbation in Einstein's gravity. The correction terms {\it differ} from many of the previously suggested forms in the literature based on hand-waving manners. We confirm our results by comparing with (i) the nonlinear perturbation theory, (ii) the first order post-Newtonian approximation, and (…