I need to build a tensor from the product of the metric components, like this (using three factors, not less, not more) :(adsbygoogle = window.adsbygoogle || []).push({});

[itex]H^{\mu \nu \lambda \kappa \rho \sigma} = g^{\mu \nu} \, g^{\lambda \kappa} \, g^{\rho \sigma} + g^{\mu \lambda} \, g^{\nu \kappa} \, g^{\rho \sigma} + ...[/itex],

however, that [itex]H^{\mu \nu \lambda \kappa \rho \sigma}[/itex] tensor should be fully symmetric under pairs of indices :

[itex]H^{\mu \nu \lambda \kappa \rho \sigma} \equiv H^{(\mu \nu) \lambda \kappa \rho \sigma} \equiv H^{\mu \nu (\lambda \kappa) \rho \sigma} \equiv H^{\mu \nu \lambda \kappa (\rho \sigma)}[/itex]

How can I do that ? Someone know what should be that tensor, explicitely ?

With only two times the metric, it would be easy :

[itex]H^{\mu \nu \lambda \kappa} = g^{\mu \nu} \, g^{\lambda \kappa} + g^{\mu \lambda} \, g^{\nu \kappa} + g^{\mu \kappa} \, g^{\nu \lambda}[/itex]

but I don't know how to do it with three times the metric.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Symetrisation of products of the metric

**Physics Forums | Science Articles, Homework Help, Discussion**