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# -*- coding: utf-8 -*-
from . import curvature
from . import diff
from . import utils
import numpy as np
def calc_expansion(x, z, metric, curv, surf, direction = 1):
"""
Calculate expansion of null geodesics on a sequence of surfaces [1]. The
surfaces are specified as level surfaces of F(r, θ) = r - h(θ).
[1] Alcubierre (2008): Introduction to 3+1 Numerical Relativity, chapter
6.7, specifically equation (6.7.13).
:param array_like x: 1D array of x coordinates.
:param array_like z: 1D array of z-coordinates.
:param array_like metric: 4D array of spatial metric values at the grid
formed by x and z. metric[i, j, k, l] is the ijth
component of the metric at the point (X=x[l],
Z=z[k]).
:param array_like curv: values of the extrinsic curvature, otherwise same as
metric.
:param callable surf: A callable that specifies the surfaces. Accepts an
array of θ and returns the array of correponding h.
:param int direction: Values of 1/-1 specify that the expansion of outgoing
or ingoing geodesics is to be computed.
:rtype: array_like, shape (z.shape[0], x.shape[0])
:return: Expansion values at the grid formed from x and z.
"""
dX = [x[1] - x[0], 0, z[1] - z[0]]
X, Z = np.meshgrid(x, z)
R = np.sqrt(X ** 2 + Z ** 2)
Theta = np.where(R > 1e-8, np.arccos(Z / R), 0.0)
metric_u = utils.matrix_invert(metric)
Gamma = curvature.calc_christoffel(x, z, metric)
trK = np.einsum('ij...,ij...', metric_u, curv)
F = R[:]
for i in range(Theta.shape[0]):
F[i] -= surf.eval(Theta[i])
dF = np.empty((3,) + F.shape)
dF[0] = diff.fd4(F, 1, dX[0])
dF[1] = 0.0
dF[2] = diff.fd4(F, 0, dX[2])
s_l = direction * dF[:]
s_u = np.einsum('ij...,j...->i...', metric_u, s_l)
s_u /= np.sqrt(np.einsum('ij...,i...,j...', metric, s_u, s_u))
ds_u = np.zeros((3,) + s_u.shape)
for i in range(3):
for j in range(3):
if i == 1 or j == 1:
continue
diff_dir = 1 if (i == 0) else 0
ds_u[i, j] = diff.fd4(s_u[j], diff_dir, dX[i])
ds_u[1, 1] = np.where(np.abs(X) > 1e-8, s_u[0] / X, ds_u[0, 0])
Div_s_u = np.einsum('ii...', ds_u) + np.einsum('iki...,k...', Gamma, s_u)
H = Div_s_u - trK + np.einsum('ij...,i...,j...', curv, s_u, s_u)
return H
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