differential_geometry.symbolic.lower_index#

Lower a single index of a tensor using the provided metric.

This function supports:

A full metric \(g_{\mu\nu}\) (shape (n,n)).
A diagonal metric (1D array of length n) for orthogonal coordinates.

General Formula (when metric is a full matrix):

\[T_{\ldots\nu\ldots} \;=\; T^{\ldots\mu\ldots}\; g_{\mu\nu},\]

Orthogonal Diagonal Case (1D array):

\[T_{\ldots\mu\ldots} \;=\; T^{\ldots\mu\ldots} \;\times\; g_{\mu\mu}.\]

Parameters:

tensor (MutableDenseNDimArray) – A symbolic tensor of arbitrary rank.
metric (MutableDenseNDimArray) – The metric used to lower the index. Either a full (n x n) matrix or a 1D array of length n.
axis (int) – The index position to lower.

Returns:

A new tensor with the specified index lowered.

Return type:

MutableDenseNDimArray

See also

raise_index

Examples

Full matrix usage:

>>> import sympy as sp
>>> from pymetric.differential_geometry.symbolic import lower_index
>>>
>>> r, theta = sp.symbols('r theta', positive=True)
>>> # Metric for polar coords
>>> g = sp.Matrix([[1, 0], [0, r**2]])
>>> # Rank-2 tensor
>>> T = sp.Array([
...     [sp.Function("T0")(r, theta), sp.Function("T1")(r, theta)],
...     [sp.Function("T2")(r, theta), sp.Function("T3")(r, theta)]
... ])
>>>
>>> lower_index(T, g, axis=1)
[[T0(r, theta), r**2*T1(r, theta)], [T2(r, theta), r**2*T3(r, theta)]]

Orthogonal diagonal usage (just multiply each slice):

>>> g_diag = sp.Array([1, r**2])
>>> lower_index(T, g_diag, axis=1)
[[T0(r, theta), r**2*T1(r, theta)], [T2(r, theta), r**2*T3(r, theta)]]

differential_geometry.symbolic.lower_index#

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