import numpy as np
import sympy
from einsteinpy.symbolic.helpers import _change_name
from einsteinpy.symbolic.tensor import BaseRelativityTensor, _change_config
[docs]
class ChristoffelSymbols(BaseRelativityTensor):
"""
Class for defining christoffel symbols.
"""
def __init__(
self, arr, syms, config="ull", parent_metric=None, name="ChristoffelSymbols"
):
"""
Constructor and Initializer
Parameters
----------
arr : ~sympy.tensor.array.dense_ndim_array.ImmutableDenseNDimArray or list
Sympy Array or multi-dimensional list containing Sympy Expressions
syms : tuple or list
Tuple of crucial symbols denoting time-axis, 1st, 2nd, and 3rd axis (t,x1,x2,x3)
config : str
Configuration of contravariant and covariant indices in tensor. 'u' for upper and 'l' for lower indices. Defaults to 'ull'.
parent_metric : ~einsteinpy.symbolic.metric.MetricTensor
Metric Tensor from which Christoffel symbol is calculated. Defaults to None.
name : str
Name of the Christoffel Symbols Tensor. Defaults to "ChristoffelSymbols".
Raises
------
TypeError
Raised when arr is not a list or sympy Array
TypeError
syms is not a list or tuple
ValueError
config has more or less than 3 indices
"""
super(ChristoffelSymbols, self).__init__(
arr=arr, syms=syms, config=config, parent_metric=parent_metric, name=name
)
self._order = 3
if not len(self.config) == self._order:
raise ValueError("config should be of length {}".format(self._order))
[docs]
@classmethod
def from_metric(cls, metric):
"""
Get Christoffel symbols calculated from a metric tensor
Parameters
----------
metric : ~einsteinpy.symbolic.metric.MetricTensor
Space-time Metric from which Christoffel Symbols are to be calculated
"""
dims = metric.dims
tmplist = np.zeros((dims, dims, dims), dtype=int).tolist()
mat, syms = metric.lower_config().tensor(), metric.symbols()
matinv = sympy.Matrix(mat.tolist()).inv()
for t in range(dims**3):
# i,j,k each goes from 0 to (dims-1)
# hack for codeclimate. Could be done with 3 nested for loops
k = t % dims
j = (int(t / dims)) % (dims)
i = (int(t / (dims**2))) % (dims)
if k <= j:
tmpvar = 0
for n in range(dims):
tmpvar += (matinv[i, n] / 2) * (
sympy.diff(mat[n, j], syms[k])
+ sympy.diff(mat[n, k], syms[j])
- sympy.diff(mat[j, k], syms[n])
)
tmplist[i][j][k] = tmplist[i][k][j] = tmpvar
return cls(tmplist, syms, config="ull", parent_metric=metric)
[docs]
def change_config(self, newconfig="lll", metric=None):
"""
Changes the index configuration(contravariant/covariant)
Parameters
----------
newconfig : str
Specify the new configuration. Defaults to 'lll'
metric : ~einsteinpy.symbolic.metric.MetricTensor or None
Parent metric tensor for changing indices.
Already assumes the value of the metric tensor from which it was initialized if passed with None.
Compulsory if not initialized with 'from_metric'. Defaults to None.
Returns
-------
~einsteinpy.symbolic.christoffel.ChristoffelSymbols
New tensor with new configuration. Defaults to 'lll'
Raises
------
Exception
Raised when a parent metric could not be found.
"""
if metric is None:
metric = self._parent_metric
if metric is None:
raise Exception("Parent Metric not found, can't do configuration change")
new_tensor = _change_config(self, metric, newconfig)
new_obj = ChristoffelSymbols(
new_tensor,
self.syms,
config=newconfig,
parent_metric=metric,
name=_change_name(self.name, context="__" + newconfig),
)
return new_obj
