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c The metric given here corresponds to that of flat spacetime
c but with non-trivial slicing and a shift vector such that
c the resulting metric is still time independent. I take
c the flat metric in spherical coordinates and define a new
c time coordinate as:
c
c t = t - a f(r)
c new
c
c where f(r) is a gaussian centered at r=0 with amplitude 1.
c Finally, I transform back to cartesian coordinates.
c For -1 < fp < 1, the transformation above results in spatial
c slices.
C
C Author: unknown
C Copyright/License: unknown
C
c $Header$
#include "cctk.h"
#include "cctk_Parameters.h"
subroutine Exact__Minkowski_shift(
$ x, y, z, t,
$ gdtt, gdtx, gdty, gdtz,
$ gdxx, gdyy, gdzz, gdxy, gdyz, gdzx,
$ gutt, gutx, guty, gutz,
$ guxx, guyy, guzz, guxy, guyz, guzx,
$ psi, Tmunu_flag)
implicit none
DECLARE_CCTK_PARAMETERS
c input arguments
CCTK_REAL x,y,z,t
c output arguments
CCTK_REAL gdtt, gdtx, gdty, gdtz,
$ gdxx, gdyy, gdzz, gdxy, gdyz, gdzx,
$ gutt, gutx, guty, gutz,
$ guxx, guyy, guzz, guxy, guyz, guzx
CCTK_REAL psi
LOGICAL Tmunu_flag
c local variables
CCTK_REAL a,s
CCTK_REAL r,r2,x2,y2,z2
CCTK_REAL f,fp,fpr,fpr2
c constants
CCTK_REAL zero,one,two
parameter (zero=0.0d0, one=1.0d0, two=2.0d0)
c This is a vacuum spacetime with no cosmological constant
Tmunu_flag = .false.
c Read parameters
a = Minkowski_shift__amplitude
s = Minkowski_shift__sigma
c Find transformation function.
x2 = x**2
y2 = y**2
z2 = z**2
r2 = x2 + y2 + z2
r = sqrt(r2)
f = a*exp(-r2/s**2)
fp = - two*f*r/s**2
fpr = fp/r
fpr2 = fpr**2
c Give metric components.
gdtt = - one
gdtx = - x*fpr
gdty = - y*fpr
gdtz = - z*fpr
gdxx = one - x2*fpr2
gdyy = one - y2*fpr2
gdzz = one - z2*fpr2
gdxy = - x*y*fpr2
gdzx = - x*z*fpr2
gdyz = - y*z*fpr2
c Inverse metric. And yes, it is this simple, simpler
c than the metric itself. I tripled checked!
gutt = - one + fp**2
gutx = - x*fpr
guty = - y*fpr
gutz = - z*fpr
guxx = one
guyy = one
guzz = one
guxy = zero
guzx = zero
guyz = zero
return
end
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