4.3 The BSSN formulation
The BSSN formulation is based on the 3+1 decomposition of Einstein’s field equations.
Unlike the harmonic formulation, which has been motivated by the mathematical structure
of the equations and the understanding of the Cauchy formulation in general relativity, this
system has been mainly developed and improved based on its capability of numerically evolving
spacetimes containing compact objects in a stable way. Interestingly, it turns out that in spite of the
fact that the BSSN formulation is based on an entirely different motivation, mathematical
questions like the well-posedness of its Cauchy problem can be answered, at least for most gauge
conditions.
In the BSSN formulation, the three metric
and the extrinsic curvature
are decomposed
according to
Here,
and
are the trace and the trace-less part, respectively, of the conformally-rescaled
extrinsic curvature. The conformal factor
is determined by the requirement for the conformal metric
to have unit determinant. Aside from these variables one also evolves the lapse (
), the shift (
)
and its time derivative (
), and the variable
In terms of the operator
the BSSN evolution equations are
Here, quantities with a tilde refer to the conformal three metric
, which is also used in order to raise
and lower indices. In particular,
and
denote the covariant derivative and the Christoffel
symbols, respectively, with respect to
. Expressions with a superscript
refer to their trace-less
part with respect to the conformal metric. Next, the sum
represents the Ricci tensor associated
with the physical three metric
, where
The term
in Eq. (4.55) is set equal to the right-hand side of Eq. (4.59). The parameter
in the
latter equation modifies the evolution flow off the constraint surface by adding the momentum constraint to
the evolution equation for the variable
. This parameter was first introduced in [10] in order
to compare the stability properties of the BSSN evolution equations with those of the ADM
formulation.
The gauge conditions, which are imposed on the lapse and shift in Eqs. (4.52, 4.54, 4.55), were
introduced in [52
] and generalize the Bona–Massó condition [62
] and the hyperbolic Gamma driver
condition [11]. It is assumed that the functions
,
and
are strictly
positive and smooth in their arguments, and that
and
are smooth functions of
their arguments. The choice
with
a positive constant, corresponds to the evolution system used in many black-hole simulations
based on
slicing and the moving puncture technique (see, for instance, [423] and references
therein). Finally, the source terms
,
and
are defined in the following way: denoting by
and
the Ricci tensors belonging to the three-metric
and the spacetime metric, respectively, and
introducing the constraint variables
the source terms are defined as
For vacuum evolutions one sets
,
and
. When matter fields are present, the
Einstein field equations are equivalent to the evolution equations (4.52 – 4.59) setting
,
,
and the constraints
,
and
.
When comparing Cauchy evolutions in different spatial coordinates, it is very convenient to reformulate
the BSSN system such that it is covariant with respect to spatial coordinate transformations. This is indeed
possible; see [77, 82
]. One way of achieving this is to fix a smooth background three-metric
,
similarly as in Section 4.1, and to replace the fields
and
by the scalar and vector fields
where
and
denote the determinants of
and
, and
is the covariant derivative associated to the latter. If
is flat
and time-independent, the corresponding BSSN equations are obtained by replacing
and
in Eqs. (4.52 – 4.59, 4.60, 4.61, 4.64 – 4.66).
4.3.1 The hyperbolicity of the BSSN evolution equations
In fact, the ADM formulation in the spatial harmonic gauge described in Section 4.2.3 and the BSSN
formulation are based on some common ideas. In the covariant reformulation of BSSN just mentioned, the
variable
is just the quantity
defined in Eq. (4.40), where
is replaced by the conformal
metric
. Instead of requiring
to vanish, which would convert the operator on the right-hand side
of Eq. (4.60) into a quasilinear elliptic operator, one promotes this quantity to an independent field
satisfying the evolution equation (4.59) (see also the discussion below Equation (2.18) in [390]). In this
way, the
-block of the evolution equations forms a wave system. However, this system is coupled
through its principal terms to the evolution equations of the remaining variables, and so one needs to
analyze the complete system. As follows from the discussion below, it is crucial to add the
momentum constraint to Eq. (4.59) with an appropriate factor
in order to obtain a hyperbolic
system.
The hyperbolicity of the BSSN evolution equations was first analyzed in a systematic way in [373
],
where it was established that for fixed shift and densitized lapse,
the evolution system (4.53, 4.56 – 4.59) is strongly hyperbolic for
and
and symmetric
hyperbolic for
and
. This was shown by introducing new variables and enlarging the
system to a strongly or symmetric hyperbolic first-order one. In fact, similar first-order reductions were
already obtained in [196, 188
]. However, in [373
] it was shown that the first-order enlargements are
equivalent to the original system if the extra constraints associated to the definition of the new variables are
satisfied, and that these extra constraints propagate independently of the BSSN constraints
,
and
. This establishes the well-posedness of the Cauchy problem
for the system (4.69, 4.53, 4.56 – 4.59) under the aforementioned conditions on
and
.
Based on the same method, a symmetric hyperbolic first-order enlargement of the evolution
equations (4.52, 4.53, 4.56 – 4.59) and fixed shift was obtained in [52
] under the conditions
and
and used to construct boundary conditions for BSSN. First-order
strongly-hyperbolic reductions for the full system (4.52 – 4.59) have also been recently analyzed
in [82
].
An alternative and efficient method for analyzing the system consists in reducing it to a first-order
pseudodifferential system, as described in Section 3.1.5. This method has been applied in [308
] to derive a
strongly hyperbolic system very similar to BSSN with fixed, densitized lapse and fixed shift. This system is
then shown to yield a well-posed Cauchy problem. In [52
] the same method was applied to the evolution
system (4.52 – 4.59). Linearizing and localizing, one obtains a first-order system of the form
. The eigenvalues of
are
,
,
,
,
,
,
, where we have defined
and
. The system is
weakly hyperbolic provided that
and it is strongly hyperbolic if, in addition, the parameter
and the functions
,
, and
can be
chosen such that the functions
are bounded and smooth. In particular, this requires that the nominators converge to zero at least as fast as
the denominators when
,
or
, respectively. Since
, the boundedness of
requires that
. For the standard choice
, the conditions on the gauge
parameters leading to strong hyperbolicity are, therefore,
,
and
. Unfortunately,
for the choice (4.62, 4.63) used in binary black-hole simulations these conditions reduce to
which is typically violated at some two-surface, since asymptotically,
and
while near black
holes
is small and
positive. It is currently not known whether or not the Cauchy problem is well
posed if the system is strongly hyperbolic everywhere except at points belonging to a set of
zero measure, such as a two-surface. Although numerical simulations based on finite-difference
discretizations with the standard choice (4.62, 4.63) show no apparent sign of instabilities near
such surfaces, the well-posedness for the Cauchy problem for the BSSN system (4.52 – 4.59)
with the choice (4.62, 4.63) for the gauge source functions remains an open problem when the
condition (4.72) is violated. However, a well-posed problem could be formulated by modifying the
choice for the functions
and
such that
and
are guaranteed to hold
everywhere.
Yet a different approach to analyzing the hyperbolicity of BSSN has been given in [219
, 220
] based on a
new definition of strongly and symmetric hyperbolicity for evolution systems, which are first order in time
and second order in space. Based on this definition, it has been verified that the BSSN system (4.69, 4.53,
4.56 – 4.59) is strongly hyperbolic for
and
and symmetric hyperbolic for
. (Note that this generalizes the original result in [373] where, in addition,
was
required.) The results in [220
] also discuss more general 3+1 formulations, including the one in [308] and
construct constraint-preserving boundary conditions. The relation between the different approaches to
analyzing hyperbolicity of evolution systems, which are first order in time and second order in space, has
been analyzed in [221].
Strong hyperbolicity for different versions of the gauge evolution equations (4.52, 4.54, 4.55), where the
normal operator
is sometimes replaced by
, has been analyzed in [222
]. See Table I in that
reference for a comparison between the different versions and the conditions they are subject to in order to
satisfy strong hyperbolicity. It should be noted that when
and
is replaced by
,
additional conditions restricting the magnitude of the shift appear in addition to
and
.
4.3.2 Constraint propagation
As mentioned above, the BSSN evolution equations (4.52 – 4.59) are only equivalent to Einstein’s field
equation if the constraints
are satisfied. Using the twice contracted Bianchi identities in their 3+1 decomposed form, Eqs. (4.45, 4.46),
and assuming that the stress-energy tensor is divergence free, it is not difficult to show that the
equations (4.52 – 4.59) imply the following evolution system for the constraint fields [52
, 220
]:
This is the constraint propagation system for BSSN, which describes the propagation of constraint
violations, which are usually present in numerical simulations due to truncation and roundoff errors. There
are at least three reasons for establishing the well-posedness of its Cauchy problem. The first reason is to
show that the unique solution of the system (4.74 – 4.76) with zero initial data is the trivial solution. This
implies that it is sufficient to solve the constraints at the initial time
. Then, any smooth enough
solution of the BSSN evolution equations with such data satisfies the constraint propagation system with
,
and
, and it follows from the uniqueness property of this system that the
constraints must hold everywhere and at each time. In this way, one obtains a solution to Einstein’s field
equations. However, in numerical calculations, the initial constraints are not exactly satisfied due to
numerical errors. This brings us to the second reason for having a well-posed problem at the level of
the constraint propagation system; namely, the continuous dependence on the initial data.
Indeed, the initial constraint violations give rise to constraint violating solutions; but, if these
violations are governed by a well-posed evolution system, the norm of the constraint violations
is controlled by those of the initial violations for each fixed time
. In particular, the
constraint violations must converge to zero if the initial constraint violations do. Since the
initial constraint errors go to zero when resolution is increased (provided a stable numerical
scheme is used to solve the constraints), this guarantees convergence to a constraint-satisfying
solution.
Finally, the third reason for establishing well-posedness for the constraint propagation system is the
construction of constraint-preserving boundary conditions, which will be explained in detail in
Section 6.
The hyperbolicity of the constraint propagation system (4.74 – 4.76) has been analyzed
in [220
, 52
, 81
, 80
], and [315
] and shown to be reducible to a symmetric hyperbolic first-order system for
. Furthermore, there are no superluminal characteristic fields if
. Because of finite
speed of propagation, this means that BSSN with
(which includes the standard
choice
) does not possess superluminal constraint-violating modes. This is an important
property, for it shows that constraint violations that originate inside black hole regions (which
usually dominate the constraint errors due to high gradients at the punctures or stuffing of the
black-hole singularities in the turducken approach [156, 81, 80]) cannot propagate to the exterior
region.
In [353
] a general result is derived, showing that under a mild assumption on the form of the
constraints, strong hyperbolicity of the main evolution system implies strong hyperbolicity of the constraint
propagation system, with the characteristic speeds of the latter being a subset of those of the former. The
result does not hold in general if “strong” is replaced by “symmetric”, since there are known examples for
which the main evolution system is symmetric hyperbolic, while the constraint propagation system is only
strongly hyperbolic [108
].