In the BSSN formulation, the three metric and the extrinsic curvature are decomposed according to 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  and generalize the Bona–Massó condition  and the hyperbolic Gamma driver condition . 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 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
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 fields16 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).
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 ). 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 , where it was established that for fixed shift and densitized lapse,[196, 188]. However, in  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  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 .
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  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  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
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  where, in addition, was required.) The results in  also discuss more general 3+1 formulations, including the one in  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 .
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 . 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 .
As mentioned above, the BSSN evolution equations (4.52 – 4.59) are only equivalent to Einstein’s field equation if the constraints[52, 220]: 17 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  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  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 .
Living Rev. Relativity 15, (2012), 9
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