Date

Abstract

Basic analysis of the metric and extrinsic curvature tensors

### 1 Purpose

This thorn provides analysis routines to calculate the following quantities:

• The trace of the extrinsic curvature ($trK$).
• The determinant of the 3-metric ($detg$).
• The components of the 3-metric in spherical coordinates
(${g}_{rr},{g}_{r𝜃},{g}_{r\varphi },{g}_{𝜃𝜃},{g}_{\varphi 𝜃},{g}_{\varphi \varphi }$).
• The components of the extrinsic curvature in spherical coordinates
(${K}_{rr},{K}_{r𝜃},{K}_{r\varphi },{K}_{𝜃𝜃},{K}_{𝜃\varphi },{K}_{\varphi \varphi }$).
• The components of the 3-Ricci tensor in cartesian coordinates
(${\mathsc{ℛ}}_{ij}$) for $i,j\in \left\{1,2,3\right\}$.
• The Ricci scalar ($\mathsc{ℛ}$).

### 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.

 $trK\equiv trK=\frac{1}{{\psi }^{4}}{g}^{ij}{K}_{ij}$ (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.

 $detg\equiv detg=-{g}_{13}^{2}\ast {g}_{22}+2\ast {g}_{12}\ast {g}_{13}\ast {g}_{23}-{g}_{11}\ast {g}_{23}^{2}-{g}_{12}^{2}\ast {g}_{33}+{g}_{11}\ast {g}_{22}\ast {g}_{33}$ (2)

### 4 Transformation to Spherical Cooordinates

The values of the metric and/or extrinsic curvature in a spherical polar coordinate system $\left(r,𝜃,\varphi \right)$ 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 $\varphi$ as p.

The general transformation from Cartesian to Spherical for such tensors is

$\begin{array}{rcll}{A}_{rr}& =& {sin}^{2}𝜃{cos}^{2}\varphi {A}_{xx}+{sin}^{2}𝜃{sin}^{2}\varphi {A}_{yy}+{cos}^{2}𝜃{A}_{zz}+2{sin}^{2}𝜃cos\varphi sin\varphi {A}_{xy}& \text{}\\ & & +2sin𝜃cos𝜃cos\varphi {A}_{xz}+2sin𝜃cos𝜃sin\varphi {A}_{yz}& \text{}\\ {A}_{r𝜃}& =& r\left(sin𝜃cos𝜃{cos}^{2}\varphi {A}_{xx}+2\ast sin𝜃cos𝜃sin\varphi cos\varphi {A}_{xy}+\left({cos}^{2}𝜃-{sin}^{2}𝜃\right)cos\varphi {A}_{xz}& \text{}\\ & & +sin𝜃cos𝜃{sin}^{2}\varphi {A}_{yy}+\left({cos}^{2}𝜃-{sin}^{2}𝜃\right)sin\varphi {A}_{yz}-sin𝜃cos𝜃{A}_{zz}\right)& \text{}\\ {A}_{r\varphi }& =& rsin𝜃\left(-sin𝜃sin\varphi cos\varphi {A}_{xx}-sin𝜃\left({sin}^{2}\varphi -{cos}^{2}\varphi \right){A}_{xy}-cos𝜃sin\varphi {A}_{xz}& \text{}\\ & & +sin𝜃sin\varphi cos\varphi {A}_{yy}+cos𝜃cos\varphi {A}_{yz}\right)& \text{}\\ {A}_{𝜃𝜃}& =& {r}^{2}\left({cos}^{2}𝜃{cos}^{2}\varphi {A}_{xx}+2{cos}^{2}𝜃sin\varphi cos\varphi {A}_{xy}-2sin𝜃cos𝜃cos\varphi {A}_{xz}+{cos}^{2}𝜃{sin}^{2}\varphi {A}_{yy}& \text{}\\ & & -2sin𝜃cos𝜃sin\varphi {A}_{yz}+{sin}^{2}𝜃{A}_{zz}\right)& \text{}\\ {A}_{𝜃\varphi }& =& {r}^{2}sin𝜃\left(-cos𝜃sin\varphi cos\varphi {A}_{xx}-cos𝜃\left({sin}^{2}\varphi -{cos}^{2}\varphi \right){A}_{xy}+sin𝜃sin\varphi {A}_{xz}& \text{}\\ & & +cos𝜃sin\varphi cos\varphi {A}_{yy}-sin𝜃cos\varphi {A}_{yz}\right)& \text{}\\ {A}_{\varphi \varphi }& =& {r}^{2}{sin}^{2}𝜃\left({sin}^{2}\varphi {A}_{xx}-2sin\varphi cos\varphi {A}_{xy}+{cos}^{2}\varphi {A}_{yy}\right)& \text{}\end{array}$

If the parameter normalize_dtheta_dphi is set to yes, the angular components are projected onto the vectors $\left(rd𝜃,rsin𝜃d\varphi \right)$ instead of the default vector $\left(d𝜃,d\varphi \right)$. That is,