3+1 formalism and bases of numerical relativity - LUTh ...

172 Choice of foliation and spatial coordinates
Recently, van Meter et al. [264] and Brügmann et al. [69] have considered a modified version of
Eq. (9.91), by replacing all the derivatives ∂/∂t by ∂/∂t − β j ∂/∂x j , i.e. writing
⎧
⎪⎨
⎪⎩
∂βi ∂t
∂Bi ∂t
∂βi
− βj = kBi
∂xj − βj ∂Bi
∂x j = ∂˜ Γ i
∂t − βj ∂˜ Γ i
∂x j − ηBi .
(9.93)
In particular, Brügmann et al. [69, 184] have computed binary black hole mergers using (9.93)
with k = 3/4 and η ranging from 0 to 3.5/M, whereas Herrmann et al. [158] have used (9.93)
with k = 3/4 and η = 2/M.
9.3.6 Other dynamical shift gauges
Shibata (2003) [228] has introduced a spatial gauge that is closely related to the hyperbolic
Gamma driver: it is defined by the requirement
∂βi
= ˜γij Fj + δt
∂t ∂Fj
, (9.94)
∂t
where δt is the time step used in the numerical computation and 4
Fi := D j ˜γij. (9.95)
From the definition of the inverse metric ˜γ ij , namely the identity ˜γ ik˜γkj = δi j , and relation
(9.80), it is easy to show that Fi is related to ˜ Γ i by
Fi = ˜γij ˜ Γ j −
˜γ jk − f jk
Dk˜γij. (9.96)
Notice that in the weak field region, i.e. where ˜γ ij = f ij + h ij with fikfjlh kl h ij ≪ 1, the second
term in Eq. (9.96) is of second order in h, so that at first order in h, Eq. (9.96) reduces to
Fi ≃ ˜γij ˜ Γ j . Accordingly Shibata’s prescription (9.94) becomes
∂β i
∂t ≃ ˜ Γ i + ˜γ ij δt ∂Fj
∂t
. (9.97)
If we disregard the δt term in the right-hand side and take the time derivative of this equation,
we obtain the Gamma-driver condition (9.89) with k = 1 and η = 0. The term in δt has been
introduced by Shibata [228] in order to stabilize the numerical code.
The spatial gauge (9.94) has been used by Shibata (2003) [228] and Sekiguchi and Shibata
(2005) [221] to compute axisymmetric gravitational collapse of rapidly rotating neutron stars
to black holes, as well as by Shibata and Sekiguchi (2005) [234] to compute 3D gravitational
collapses, allowing for the development of nonaxisymmetric instabilities. It has also been used
by Shibata, Taniguchi and Uryu (2003-2006) [237, 238, 236] to compute the merger of binary
neutron stars, while their preceding computations [239, 240] rely on the approximate minimal
distortion gauge (Sec. 9.3.3).
4 let us recall that D i := f ij Dj