import numpy as np
import sympy
from einsteinpy.symbolic.christoffel import ChristoffelSymbols
from einsteinpy.symbolic.helpers import _change_name
from einsteinpy.symbolic.tensor import BaseRelativityTensor, _change_config
[docs]
class RiemannCurvatureTensor(BaseRelativityTensor):
"""
Class for defining Riemann Curvature Tensor
"""
def __init__(
self,
arr,
syms,
config="ulll",
parent_metric=None,
name="RiemannCurvatureTensor",
):
"""
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 'ulll'.
parent_metric : ~einsteinpy.symbolic.metric.MetricTensor
Metric Tensor related to this Riemann Curvature Tensor.
name : str
Name of the Tensor. Defaults to "RiemannCurvatureTensor".
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 4 indices
"""
super(RiemannCurvatureTensor, self).__init__(
arr=arr, syms=syms, config=config, parent_metric=parent_metric, name=name
)
self._order = 4
if not len(config) == self._order:
raise ValueError("config should be of length {}".format(self._order))
[docs]
@classmethod
def from_christoffels(cls, chris, parent_metric=None):
"""
Get Riemann Tensor calculated from Christoffel Symbols.
Reimann Tensor is given as:
.. math::
R^{t}{}_{s r n}=\\Gamma^{t}{}_{s n, r} - \\Gamma^{t }{}_{s r, n } +
\\Gamma^{p}{}_{s n}\\Gamma^{t}{}_{p r} - \\Gamma^{p}{}_{s r}\\Gamma^{t}{}_{p n}
Parameters
----------
chris : ~einsteinpy.symbolic.christoffel.ChristoffelSymbols
Christoffel Symbols from which Riemann Curvature Tensor to be calculated
parent_metric : ~einsteinpy.symbolic.metric.MetricTensor or None
Corresponding Metric for the Riemann Tensor.
None if it should inherit the Parent Metric of Christoffel Symbols.
Defaults to None.
"""
if not chris.config == "ull":
chris = chris.change_config(newconfig="ull", metric=parent_metric)
arr, syms = chris.tensor(), chris.symbols()
dims = len(syms)
riemann_list = (np.zeros(shape=(dims, dims, dims, dims), dtype=int)).tolist()
for i in range(dims**4):
# t,s,r,n each goes from 0 to (dims-1)
# hack for codeclimate. Could be done with 4 nested for loops
n = i % dims
r = (int(i / dims)) % (dims)
s = (int(i / (dims**2))) % (dims)
t = (int(i / (dims**3))) % (dims)
temp = sympy.diff(arr[t, s, n], syms[r]) - sympy.diff(arr[t, s, r], syms[n])
for p in range(dims):
temp += arr[p, s, n] * arr[t, p, r] - arr[p, s, r] * arr[t, p, n]
riemann_list[t][s][r][n] = sympy.simplify(temp)
if parent_metric is None:
parent_metric = chris.parent_metric
return cls(riemann_list, syms, config="ulll", parent_metric=parent_metric)
[docs]
@classmethod
def from_metric(cls, metric):
"""
Get Riemann Tensor calculated from a Metric Tensor
Parameters
----------
metric : ~einsteinpy.symbolic.metric.MetricTensor
Metric Tensor from which Riemann Curvature Tensor to be calculated
"""
ch = ChristoffelSymbols.from_metric(metric)
return cls.from_christoffels(ch, parent_metric=None)
[docs]
def change_config(self, newconfig="llll", metric=None):
"""
Changes the index configuration(contravariant/covariant)
Parameters
----------
newconfig : str
Specify the new configuration. Defaults to 'llll'
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.riemann.RiemannCurvatureTensor
New tensor with new configuration. Configuration defaults to 'llll'
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 = RiemannCurvatureTensor(
new_tensor,
self.syms,
config=newconfig,
parent_metric=metric,
name=_change_name(self.name, context="__" + newconfig),
)
return new_obj
