The general-relativistic hydrodynamics equations consist of the local conservation laws of the stress-energy
tensor, (the Bianchi identities) and of the matter current density,
(the continuity
equation):
The stress-energy tensor for a nonperfect (unmagnetized) fluid is defined as (see, e.g., [264])
whereIn the following, we will neglect nonadiabatic effects, such as viscosity and heat transfer, assuming the stress-energy tensor to be that of a perfect fluid
where we have introduced the relativistic specific enthalpy, Introducing an explicit coordinate chart , the previous conservation equations
read
In order to close the system, the equations of motion (1) and the continuity equation (2
) must be
supplemented with an equation of state (EOS) relating some fundamental thermodynamic quantities. In
general, the EOS takes the form
The available EOSs have become sophisticated enough to take into account physical and chemical
processes such as molecular interactions, quantization, relativistic effects, nuclear physics, etc. Nevertheless,
due to their simplicity, the most widely employed EOSs in numerical simulations in astrophysics are the
ideal fluid EOS, , where
is the adiabatic index, and the polytropic EOS (e.g., to build
equilibrium stellar models),
,
being the polytropic constant and
the
polytropic index. State-of-the-art, microphysical EOSs that describe the interior of compact stars at
nuclear matter densities have also been developed. While they are being increasingly used in the
numerical modelling of relativistic stars, the true EOS of neutron-star interiors remains still largely
unknown, as reproducing the associated densities and temperatures is not amenable to laboratory
experimentation.
In the “test-fluid” approximation, where the fluid self-gravity is neglected, the dynamics of the system is
completely governed by Equations (1) and (2
), together with the EOS (9
). In those situations in which such
an approximation does not hold, the previous equations must be solved in conjunction with the Einstein
gravitational-field equations,
The Arnowitt–Deser–Misner (ADM) 3+1 metric equations have been shown over the years to be
intrinsically unstable for long-term numerical simulations, especially for those dealing with black-hole
spacetimes. Recently, there have been diverse attempts to reformulate those equations into forms better
suited for numerical investigations (see [363, 39
, 5
] and references therein). Among the various
approaches proposed, the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) reformulation of the
ADM system [363
, 39
] is very appropriate for long-term stable numerical work. BSSN makes
use of a conformal decomposition of the 3-metric,
and the trace-free part of
the extrinsic curvature,
, with the conformal factor
chosen to satisfy
. In this formulation, as shown by [39
], in addition to the evolution equations
for the conformal three-metric
and the conformal-traceless extrinsic-curvature variables
, there are evolution equations for the conformal factor
, the trace of the extrinsic
curvature
and the “conformal connection functions”
. Indeed, BSSN (or slight
modifications thereof) is currently the standard 3+1 formulation for most numerical relativity
groups [5, 158
]. Long-term stable applications include strongly-gravitating systems such as
neutron stars (both, isolated and in binary systems) and single and binary black holes. Such
binary–black-hole evolutions, possibly the grandest challenge of numerical relativity ever, since the
beginning of the field, have only been possible in the last few years (see, e.g., [323
] and references
therein).
Alternatively, a characteristic initial-value–problem formulation of the Einstein equations was developed
in the 1960s by Bondi, van der Burg, and Metzner [59], and by Sachs [344]. This approach has gradually
advanced to a state where long-term stable evolutions of caustic-free spacetimes in multiple dimensions are
possible, even including matter fields (see [217] and references therein). A comprehensive review of the
characteristic formulation is presented in a Living Reviews article by Winicour [419]. Examples of this
formulation in general-relativistic hydrodynamics are discussed in various sections of the current
article.
Traditionally, most of the approaches for numerical integrations of the general-relativistic hydrodynamics equations have adopted spacelike foliations of the spacetime, within the 3+1 formulation. More recently, however, covariant forms of these equations, well suited for advanced numerical methods, have also been developed. This is reviewed next (Section 2.1) in a chronological way.
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