Tom Goodale et al



Basic analysis of the metric and extrinsic curvature tensors

1 Purpose

This thorn provides analysis routines to calculate the following quantities:

2 Trace of Extrinsic Curvature

The trace of the extrinsic curvature at each point on the grid is placed in the grid function trK. The algorithm for calculating the trace uses the physical metric, that is it includes any conformal factor.

trK trK = 1 ψ4gijK ij (1)

3 Determinant of 3-Metric

The determinant of the 3-metric at each point on the grid is placed in the grid function detg. This is always the determinant of the conformal metric, that is it does not include any conformal factor.

detg detg = g132 g 22 + 2 g12 g13 g23 g11 g232 g 122 g 33 + g11 g22 g33 (2)

4 Transformation to Spherical Cooordinates

The values of the metric and/or extrinsic curvature in a spherical polar coordinate system (r,𝜃,ϕ) evaluated at each point on the computational grid are placed in the grid functions (grr, grt, grp, gtt, gtp, gpp) and (krr, krt, krp, ktt, ktp, kpp). In the spherical transformation, the 𝜃 coordinate is referred to as q and the ϕ as p.

The general transformation from Cartesian to Spherical for such tensors is

Arr = sin2𝜃cos2ϕA xx + sin2𝜃sin2ϕA yy + cos2𝜃A zz + 2sin2𝜃cosϕsinϕA xy +2sin𝜃cos𝜃cosϕAxz + 2sin𝜃cos𝜃sinϕAyz Ar𝜃 = r(sin𝜃cos𝜃cos2ϕA xx + 2 sin𝜃cos𝜃sinϕcosϕAxy + (cos2𝜃 sin2𝜃)cosϕA xz +sin𝜃cos𝜃sin2ϕA yy + (cos2𝜃 sin2𝜃)sinϕA yz sin𝜃cos𝜃Azz) Arϕ = rsin𝜃(sin𝜃sinϕcosϕAxx sin𝜃(sin2ϕ cos2ϕ)A xy cos𝜃sinϕAxz +sin𝜃sinϕcosϕAyy + cos𝜃cosϕAyz) A𝜃𝜃 = r2(cos2𝜃cos2ϕA xx + 2cos2𝜃sinϕcosϕA xy 2sin𝜃cos𝜃cosϕAxz + cos2𝜃sin2ϕA yy 2sin𝜃cos𝜃sinϕAyz + sin2𝜃A zz) A𝜃ϕ = r2 sin𝜃(cos𝜃sinϕcosϕA xx cos𝜃(sin2ϕ cos2ϕ)A xy + sin𝜃sinϕAxz +cos𝜃sinϕcosϕAyy sin𝜃cosϕAyz) Aϕϕ = r2 sin2𝜃(sin2ϕA xx 2sinϕcosϕAxy + cos2ϕA yy)

If the parameter normalize_dtheta_dphi is set to yes, the angular components are projected onto the vectors (rd𝜃,rsin𝜃dϕ) instead of the default vector (d𝜃,dϕ). That is,