changeTensorIndex

Description

The index of the tensors can be changed using raising and lowering operations.

Metric Tensor

The metric tensor can be in either covariant or contravariant form, this is transformed between each by doing a matrix inverse transformation.

Stress-energy Tensor

The stress-energy tensor can be either covariant, contravariant or a mixed-form. Transformations for covariant or contravariant are just raising/lowering operations given as:

$T^{\mu\nu} = g^{\mu\alpha} g^{\nu\beta} T_{\alpha \beta}$ or $T_{\mu\nu} = g_{\mu\alpha} g_{\nu\beta} T^{\alpha \beta}$

Likewise, the mixed form is found by:

$T^{\mu}_{\ \ \nu} = g^{\mu\alpha} T_{\alpha \nu}$

Method

The index-changing operations occur at each point in the spacetime doing the index summations for the stress-energy tensor or inverse operations on the metric tensor.

Syntax

[outputTensor] = changeTensorIndex(inputTensor, index, metricTensor)

Input Arguments

blue are required inputs.

orange are optional inputs with native default values.

Inputs	Format	Type	Description
inputTensor	struct	object	Input tensor, metric if transforming the metric or stress-energy tensor. If a stress-energy tensor object is an input, then the metricTensor is required as an input.
index	1x1 array	string	The selected target index. Can be either: "covariant", "contravariant" for both the metric or stress-energy tensor. If transforming a stress-energy tensor then "mixedupdown", "mixeddownup". index transformations can also be done, but only on the stress-energy tensor.
metricTensor	struct	object	Metric tensor object. This is only needed if transforming the stress-energy tensor index, which is input in inputTensor .

Output Arguments

Outputs	Format	Type	Description
outputTensor	struct	object	Returns tensor with new index, can be either metric or stress-energy depending on inputTensor .