6.2
Sources of matter
6.2.1
Cosmological constant
A cosmological constant is implemented in the
3 + 1 framework simply by introducing the quantity
as an effective isotropic pressure in the stress-energy
tensor
The matter source terms can then be written as
with
.
6.2.2
Scalar field
The dynamics of scalar fields is governed by
the Lagrangian density
where
is the scalar Riemann curvature,
is the interaction potential,
is typically assumed to be
, and
is the field-curvature coupling constant (
for minimally coupled fields and
for conformally coupled fields). Varying the action yields the
Klein-Gordon equation
for the scalar field and
for the stress-energy tensor, where
.
For a massive, minimally coupled scalar
field
[46],
and
where
, and
is a general potential which, for example, can be set to
in the chaotic inflation model. The covariant form of the scalar
field equation (25) can be expanded as in
[107]
to yield
which, when coupled to Equation (31), determines the evolution of the scalar field.
6.2.3
Collisionless dust
The stress-energy tensor for a fluid composed
of collisionless particles (or dark matter) can be written simply
as the sum of the stress-energy tensors for each particle
[161],
where
is the rest mass of the particles,
is the number density in the comoving frame, and
is the 4-velocity of each particle. The matter source terms are
There are two conservation laws: the conservation of particles
, and the conservation of energy-momentum
, where
is the covariant derivative in the full 4-dimensional spacetime.
Together these conservation laws lead to
, the geodesic equations of motion for the particles, which can
be written out more explicitly in the computationally convenient
form
where
is the coordinate position of each particle,
is determined by the normalization
,
is the Lagrangian derivative, and
is the “transport” velocity of the particles as measured by
observers at rest with respect to the coordinate grid.
6.2.4
Ideal gas
The stress-energy tensor for a perfect fluid
is
where
is the 4-metric,
is the relativistic enthalpy, and
,
,
and
are the specific internal energy (per unit mass), pressure, rest
mass density and four-velocity of the fluid. Defining
and
as the generalization of the special relativistic boost factor,
the matter source terms become
The hydrodynamics equations are derived from the normalization of
the 4-velocity,
, the conservation of baryon number,
, and the conservation of energy-momentum,
. The resulting equations can be written in flux conservative
form as
[162
]
where
,
,
,
, and
is the determinant of the 4-metric satisfying
. A prescription for specifying an equation of state (e.g.,
for an ideal gas) completes the above equations.
Update
When solving Equations (45,
46,
47), an artificial viscosity method is needed to handle the
formation and propagation of shock fronts
[162, 85, 84]
. These methods are computationally cheap, easy to implement, and
easily adaptable to multi-physics applications. However, it has
been demonstrated that problems involving very high Lorentz
factors are somewhat sensitive to different implementations of
the viscosity terms, and can result in substantial numerical
errors if solved using time explicit methods
[126]
.
On the other hand, a number of different
formulations
[75]
of these equations have been developed to take advantage of the
hyperbolic and conservative nature of the equations in using high
resolution and non-oscillatory shock capturing schemes (although
strict conservation is only possible in flat spacetimes - curved
spacetimes exhibit source terms due to geometry). These
techniques potentially provide more accurate and stable
treatments in highly relativistic regimes. A particular
formulation used together with high resolution Godunov techniques
and approximate Riemann solvers is the following
[139, 26]
:
where
and
,
,
,
,
,
, and
.
Update
Although Godunov-type schemes are accepted as more accurate
alternatives to AV methods, especially in the limit of high
Lorentz factors, they are not immune to problems and should
generally be used with caution. They may produce unexpected
results in certain cases that can be overcome only with
problem-specific fixes or by adding additional dissipation. A few
known examples include the admittance of expansion shocks,
negative internal energies in kinematically dominated flows, the
‘carbuncle’ effect in high Mach number bow shocks, kinked Mach
stems, and odd/even decoupling in mesh-aligned shocks
[135]
. Godunov methods, whether they solve the Riemann problem exactly
or approximately, are also computationally much more expensive
than their simpler AV counterparts, and it is more difficult to
incorporate additional physics.
A third class of computational fluid dynamics
methods reviewed here is also based on a conservative hyperbolic
formulation of the hydrodynamics equations. However, in this case
the equations are derived directly from the conservation of
stress-energy,
to give
with curvature source terms
. The variables
and
are the same as those defined in the internal energy
formulation, but now
are different expressions for energy and momenta. An alternative
approach of using high resolution, non-oscillatory, central
difference (NOCD) methods
[99, 100]
has been applied by Anninos and Fragile
[12
]
to solve the relativistic hydrodynamics equations in the above
form. These schemes combine the speed, efficiency, and
flexibility of AV methods with the advantages of the potentially
more accurate conservative formulation approach of Godunov
methods, but without the cost and complication of Riemann solvers
and flux splitting.
NOCD and artificial viscosity methods have been
discussed at length in
[12]
and compared also with other published Godunov methods on their
abilities to model shock tube, wall shock and black hole
accretion problems. They find that for shock tube problems at
moderate to high boost factors, with velocities up to
, internal energy formulations using artificial viscosity methods
compare quite favorably with total energy schemes, including NOCD
methods and Godunov methods using either approximate or exact
Riemann solvers. However, AV methods can be somewhat sensitive to
parameters (e.g., viscosity coefficients, Courant factor, etc.)
and generally suspect in wall shock problems at high boost
factors (
). On the other hand, NOCD methods can easily be extended to
ultra-relativistic velocities (
) for the same wall shock tests, and are comparable in accuracy
to the more standard but complicated Riemann solver codes. NOCD
schemes thus provide a reasonable alternative for relativistic
hydrodynamics, though it should be noted that low order versions
of these methods can be significantly more diffusive than either
the AV or Godunov methods.
6.2.5
Imperfect fluid
Update
The perfect fluid equations discussed in Section
6.2.4
can be generalized to include viscous stress in the
stress-energy tensor,
where
and
are the dynamic shear and bulk viscosity coefficients,
respectively. Also,
is the expansion of fluid world lines,
is the trace-free spatial shear tensor with
and
is the projection tensor.
The corresponding energy and momentum
conservation equations for the internal energy formulation of
Section
6.2.4
become
For the NOCD formulation discussed in Section
6.2.4
it is sufficient to replace the source terms in the energy and
momentum equations (53,
53,
54) by