Purpose

This thorn provides the variables defining a static conformal factor which is used to transform the physical metric. If this thorn is active and the ADMBase::metric_type parameter is set to static conformal, then the ADMBase::g... variables are the conformal values as opposed to the physical values.

The transformation is

$$g_{ij}^{\mbox{physical}} = \psi^4 g_{ij}^{\mbox{conformal}}$$

The extrinsic curvature is not transformed.

Memory is provided for the conformal factor psi, its first derivatives psix, psiy, psiz, and its second derivatives psixx, psixy, psixz, psiyy, psiyz, and psizz depending on the setting of the conformal_storage parameter.

Note that the first and second ``derivative'' grid functions have an additional factor of $1 / \psi$ normalisation since this is the most common use of the derivative. I.e., the grid functions are

$$\begin{eqnarray*} {\tt psi} &=& \psi, \\ {\tt psix} &=& \psi_x/\psi, \qquad \mbox{etc}\\ {\tt psixx} &=& \psi_{ij}/\psi \qquad \mbox{etc} \end{eqnarray*}$$

Thorns need to check the value of the grid scalar conformal_state to determine how many levels of these variables have actually been calculated before using the conformal factor:

conformal_state=0

No conformal factor has been calculated -- thorns may assume the conformal factor is 1 at all points. (I.e., the metric is physical.)

conformal_state=1

The conformal factor has been calulated, but no derivatives.

conformal_state=2

The conformal factor and its first derivatives have been calculated.

conformal_state=3

The conformal factor and its first and second derivatives have been calculated.

Note that this means that if you only want to know whether psi contains the values for the conformal factor you can check for conformal_state > 0.