Equations

2.1 Equations

Using the Einstein summation convention the equations describing the motion of a relativistic fluid are given by the five conservation laws,

where $μ,ν = 0,...,3$ , and where $;μ$ denotes the covariant derivative with respect to coordinate $μ x$ . Furthermore, $ρ$ is the proper rest mass density of the fluid, $μ u$ its four-velocity, and $μν T$ is the stress-energy tensor, which for a perfect fluid can be written as

Here, $gμν$ is the metric tensor, $p$ the fluid pressure, and $h$ the specific enthalpy of the fluid defined by

where $𝜀$ is the specific internal energy. Note that we use natural units (i.e., the speed of light c = 1) throughout this review.

In Minkowski spacetime and Cartesian coordinates $(t,x1,x2,x3)$ , the conservation equations (1 , 2 ) can be written in vector form as

where i = 1 ,2 ,3. The state vector $u$ is defined by

and the flux vectors $Fi$ are given by

The five conserved quantities $D$ , $S1$ , $S2$ , $S3$ , and $τ$ are the rest mass density, the three components of the momentum density, and the energy density (measured relative to the rest mass energy density), respectively. They are all measured in the laboratory frame, and are related to quantities in the local rest frame of the fluid (primitive variables) through

D = ρW, (8 ) Si = ρhW 2vi, i = 1, 2,3, (9 ) 2 τ = ρhW − p − D, (10 )

where $vi$ are the components of the three-velocity of the fluid

and $W$ is the Lorentz factor,

The system of Equations (5

) with Definitions (6

, 7

, 8

, 9

, 10

, 11

, 12

) is closed by means of an equation of state (EOS), which we shall assume to be given in the form

In the non-relativistic limit (i.e., $v ≪ 1$ , $h → 1$ ) $D$ , $Si$ , and $τ$ approach their Newtonian counterparts $ρ$ , $ρvi$ , and $ρE = ρ𝜀 + ρv2∕2$ , and Equations (5 ) reduce to the classical ones. In the relativistic case the equations of system (5 ) are strongly coupled via the Lorentz factor and the specific enthalpy, which gives rise to numerical complications (see Section 2.3).

In classical numerical hydrodynamics it is very easy to obtain $i v$ from the conserved quantities (i.e., $ρ$ and $ρvi$ ). In the relativistic case, however, the task to recover $(ρ,vi,p)$ from $(D, Si,τ)$ is much more complicated. Moreover, as state-of-the-art SRHD codes are based on conservative schemes where the conserved quantities are advanced in time, it is necessary to compute the primitive variables from the conserved ones one (or even several) times per numerical cell and time step making this procedure a crucial ingredient of any algorithm (see Section 9.2).