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

9.4 Full spatial coordinate-fixing choices 173
9.4 Full spatial coordinate-fixing choices
The spatial coordinate choices discussed in Sec. 9.3, namely vanishing shift, minimal distortion,
Gamma freezing, Gamma driver and related prescriptions, are relative to the propagation of the
coordinates (x i ) away from the initial hypersurface Σ0. They do not restrict at all the choice of
coordinates in Σ0. Here we discuss some coordinate choices that fix completely the coordinate
freedom, including in the initial hypersurface.
9.4.1 Spatial harmonic coordinates
The first full coordinate-fixing choice we shall discuss is that of spatial harmonic coordinates.
They are defined by
DjD j x i = 0 , (9.98)
in full analogy with the spacetime harmonic coordinates [cf. Eq. (9.23)]. The above condition
is equivalent to
1 ∂
√
γ ∂xj
√γγjk ∂xi
∂xk
=δi
= 0,
k
(9.99)
i.e.
∂
∂xj √ ij
γγ = 0. (9.100)
This relation restricts the coordinates to be of Cartesian type. Notably, it forbids the use of
spherical-type coordinates, even in flat space, for it is violated by γij = diag(1,r 2 ,r 2 sin 2 θ). To
allow for any type of coordinates, let us rewrite condition (9.100) in terms of a background flat
metric f (cf. discussion in Sec. 6.2.2), as
Dj
γ
f
1/2 γ ij
= 0 , (9.101)
where D is the connection associated with f and f := detfij is the determinant of f with
respect to the coordinates (xi ).
Spatial harmonic coordinates have been considered by Čadeˇz [72] for binary black holes
and by Andersson and Moncrief [16] in order to put the 3+1 Einstein system into an elliptichyperbolic
form and to show that the corresponding Cauchy problem is well posed.
Remark : The spatial harmonic coordinates discussed above should not be confused with spacetime
harmonic coordinates; the latter would be defined by gx i = 0 [spatial part of
Eq. (9.23)] instead of (9.98). Spacetime harmonic coordinates, as well as some generalizations,
are considered e.g. in Ref. [11].