Question

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if is symmetric, show that 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.

$t^{\mu \nu }=t^{\nu \mu }$

Write the property of an anti-symmetric tensor.

role="math" localid="1650574388909" $t^{\mu \nu }=-t^{\nu \mu }$

Here, a negative sign for anti-symmetric tensor.

Determine the symmetry or antisymmetry of a tensor:

Consider

$t^{-\kappa \lambda }=A_{\mu }^{\kappa }{A}_{\nu }^{\lambda }{t}^{\mu \nu }$

Here, A 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,

$$\begin{gathered} t^{-\kappa \lambda }=A_{\mu }^{\kappa }{A}_{\nu }^{\lambda }{t}^{\mu \nu } \\ {t}^{-\kappa \lambda }=A_{\nu }^{\lambda }{A}_{\mu }^{\kappa }{t}^{\nu \mu } \\ {t}^{-\kappa \lambda }=A_{\mu }^{\kappa }{A}_{\mu }^{\lambda }(-t^{\mu \nu }) \\ {t}^{-\kappa \lambda }=\pm t^{-\kappa \lambda } \end{gathered}$$

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