StaticConformal

Tom Goodale et al

Date

Abstract

Base thorn to provide the variables for the static conformal factor

1 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_{i j}^{physical} = ψ^{4} g_{i j}^{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 $1 ∕ ψ$ normalisation since this is the most common use of the derivative. I.e., the grid functions are

\begin{array}{rcl} p s i & = & ψ, \\ p s i x & = & ψ_{x} ∕ ψ, etc \\ p s i x x & = & ψ_{i j} ∕ ψ etc \end{array}

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.

2 Utilities

StaticConformal provides aliased functions to convert between physical and conformal 3-metric values. It is very important to understand that these functions apply the conversion in place. That is, if gxx contains the conformal metric value, when the routine is exited it will now contain the physical metric value. These functions do not change the value of conformal_state and should be used with due care. (These functions are for example used by some analysis thorns who work only with the physical metric, they apply the transformation on entry to the analysis routine and switch it back on exit).

Convert from conformal to physical:: subroutine ConfToPhysInPlace (nx, ny, nz,
                              psi,
                              gxx, gxy, gxz, gyy, gyz, gzz)
   implicit none
   CCTK_INT,                         intent(in)    :: nx, ny, nz
   CCTK_REAL, dimension(nx, ny, nz), intent(in)    :: psi
   CCTK_REAL, dimension(nx, ny, nz), intent(inout) :: gxx, gxy, gxz, gyy, gyz, gzz
end subroutine ConfToPhysInPlace
Convert from physical to conformal: