Physlib.Relativity.Tensors.RealTensor.Units.Basic

Unit tensors for real Lorentz tensors

i. Overview

This file computes the unit tensors of `realLorentzTensor` in the standard covariant and contravariant bases.

ii. Key results

- `realLorentzTensor.unitTensor_repr_apply` identifies the standard-basis components of either real Lorentz unit tensor with the Kronecker delta.

iii. Table of contents

A. Basis components

iv. References

A. Basis components

1 declaration

theorem

Components of the real Lorentz unit tensor equal the Kronecker delta $\delta$

#unitTensor_repr_apply

For any spatial dimension $d$ and index color $c$ (representing a representation of the Lorentz group, such as contravariant or covariant indices), the components of the real Lorentz unit tensor in the standard basis are given by the Kronecker delta. Specifically, let $\phi = (\phi_0, \phi_1)$ be a multi-index for a tensor of type $(\tau(c), c)$ , where $\tau(c)$ is the dual color of $c$ . The $\phi$ -th component of the unit tensor is $1$ if $\phi_0 = \phi_1$ and $0$ otherwise.