def from_metric(cls, metric):
"""
Get Schouten tensor calculated from a metric tensor
Parameters
----------
metric : ~einsteinpy.symbolic.metric.MetricTensor
Space-time Metric from which Christoffel Symbols are to be calculated
Raises
------
ValueError
Raised when the dimension of the tensor is less than 3
"""
if metric.dims >= 3:
t_ricci = RicciTensor.from_metric(metric)
r_scalar = RicciScalar.from_riccitensor(t_ricci,
parent_metric=None)
dims = metric.dims
t_schouten = (t_ricci.tensor() -
(r_scalar.expr * metric.lower_config().tensor() /
(2 * (dims - 1)))) / (dims - 2)
return cls(t_schouten,
metric.syms,
config="ll",
parent_metric=metric)
raise ValueError(
"Dimension of the space/space-time should be 3 or more")
def from_metric(cls, metric):
"""
Get Weyl tensor calculated from a metric tensor
Parameters
----------
metric : ~einsteinpy.symbolic.metric.MetricTensor
Space-time Metric from which Christoffel Symbols are to be calculated
Raises
------
ValueError
Raised when the dimension of the tensor is less than 3
"""
if metric.dims > 3:
metric_cov = metric.lower_config()
t_riemann = RiemannCurvatureTensor.from_metric(metric)
# Riemann Tensor with covariant indices is needed
t_riemann_cov = t_riemann.change_config("llll", metric=None)
t_ricci = RicciTensor.from_riemann(t_riemann, parent_metric=None)
r_scalar = RicciScalar.from_riccitensor(t_ricci, parent_metric=None)
g = metric_cov
dims = g.dims
# Indexing for resultant Weyl Tensor is iklm
C = np.zeros(shape=(dims, dims, dims, dims), dtype=int).tolist()
for t in range(dims ** 4):
i, k, l, m = (
t % dims,
(int(t / dims)) % (dims),
(int(t / (dims ** 2))) % (dims),
(int(t / (dims ** 3))) % (dims),
)
C[i][k][l][m] = t_riemann_cov[i, k, l, m] + (
(
(
t_ricci[i, m] * g[k, l]
- t_ricci[i, l] * g[k, m]
+ t_ricci[k, l] * g[i, m]
- t_ricci[k, m] * g[i, l]
)
/ (dims - 2)
)
+ (
r_scalar.expr
* (g[i, l] * g[k, m] - g[i, m] * g[k, l])
/ ((dims - 1) * (dims - 2))
)
)
C = sympy.simplify(sympy.Array(C))
return cls(C, metric.syms, config="llll", parent_metric=metric)
if metric.dims == 3:
return cls(
sympy.Array(np.zeros((3, 3), dtype=int)),
metric.syms,
config="llll",
parent_metric=metric,
)
raise ValueError("Dimension of the space/space-time should be 3 or more")
def from_metric(cls, metric):
t_ricci = RicciTensor.from_metric(metric)
r_scalar = RicciScalar.from_riccitensor(t_ricci, t_ricci.parent_metric)
einstein_tensor = t_ricci.tensor() - (1 / 2) * metric.tensor() * r_scalar.expr
return cls(
einstein_tensor,
metric.syms,
config="ll",
parent_metric=t_ricci.parent_metric,
)