Question

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if ${\mathit{t}}^{\mathbf{\mu }\mathbf{v}}$is symmetric, show that${\overline{\mathbf{t}}}^{\mathbf{\mu }\mathbf{v}}$ is also symmetric, and likewise for antisymmetric).

Short answer

The symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation.

Step-by-step solution

Expression for the property of symmetric and antisymmetric tensor:

Write the property of the symmetric tensor.

${\mathbf{t}}^{\mathbf{\mu v}}\mathbf{=}{\mathbf{t}}^{\mathbf{v\mu }}$

Write the property of an anti-symmetric tensor.

${\mathbf{t}}^{\mathbf{\mu v}}\mathbf{=}\mathbf{-}{\mathbf{t}}^{\mathbf{v\mu }}$

Here, a negative sign for anti-symmetric tensor.

Determine the symmetry or antisymmetry of a tensor:

Consider$t^{-K\lambda }={\Lambda }_{\mu }^K{\Lambda }_v^{\lambda }{t}^{\mu v}$

Here,$\Lambda$ is the Lorentz transformation matrix.

Since$\mu$ and v both are summed from$0$ to $3$, both the values can be interchanged.

Hence, the above equation becomes,

$\begin{array}{c}{t}^{-\lambda K}={\Lambda }_{\mu }^K{\Lambda }_{\mu }^K{t}^{\mu v}\\ t^{-\lambda K}={\Lambda }_v^{\lambda }{\Lambda }_{\mu }^K{t}^{v\mu }\\ t^{-\lambda K}={\Lambda }_{\mu }^K{\Lambda }_{\mu }^{\lambda }(-t^{\mu v})\\ t^{-\lambda K}=\pm t^{-K\lambda }\end{array}$

Therefore, the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation.