A Field evolution equations
In the BSSN evolution system, we define the following variables in terms of the standard ADM 4-metric
, 3-metric
, and extrinsic curvature
:
The evolution system consists of 15 equations for the various field terms,
where the matter source terms contain various projections of the stress-energy tensor, defined through the
relations
We have introduced the notation
. All quantities with a tilde involve the conformal 3-metric
, which is used to raise and lower indices. In particular,
and
refer to the covariant derivative
and the Christoffel symbols with respect to
. Parentheses indicate symmetrization of indices, and the
expression
denotes the trace-free part of the expression inside the brackets. In the BSSN
approach, the Ricci tensor is typically split into two pieces, whose respective contributions are given by
These equations must be supplemented with gauge conditions that determine the evolution of the lapse
function
and shift vector
. Noting that some groups introduce slight variants of these, the moving
puncture gauge conditions that have become popular for all GR merger calculations involving BHs and NSs
typically take the form
where
is an intermediate quantity used to convert the second-order “Gamma-driver” shift condition
into a pair of first-order equations, and
is a user-prescribed term used to control dissipation
in the simulation. Note that it is possible to replace the three instances of
in the shift
evolution equations 51 and 52 by the shift-advected time derivative
without
changing the stability or hyperbolicity properties of the evolution scheme; in both cases moving
punctures translate smoothly across a grid over long time periods [322] and both systems are
strongly hyperbolic so long as the shift vector does not grow too large within the simulation
domain [129].
The generalized harmonic formulation involves recasting the Einstein field equations, Eq. 9 in the form
and, after some tensor algebra, rewriting the Ricci tensor in the form
The Christoffel coefficients are calculated from the full 4-metric,
and the gauge source terms
are defined in Eq. 25.
Given well-posed initial data for the metric and its first time derivative (since the system is
second-order in time according to Eq. 54), the evolution of the system may be treated by a
first-order reduction that specifies the evolution of the four functions
along with the
spacetime metric
, its projected time derivatives
, and its spatial derivatives
subject to a constraint specifying that the derivative terms
remain consistent with the metric
in time. In practice, one typically introduces a constraint for the source functions, defining
and then modifies the evolution equation by appending a constraint damping term to the RHS of the stress
energy-term (following [130, 231
]
where
is the unit normal vector to the hypersurface (see Eq. 15). The gauge conditions used in the
first successful simulations of merging BH binaries [231] consisted of the set
with
constant parameters used to tune the evolution. The first one drives the coordinates toward
the ADM form and the latter provides dissipation. The binary NS-NS merger work of [6] chose harmonic
coordinates with
.