- #1
Barnak
- 63
- 0
I need to build a tensor from the product of the metric components, like this (using three factors, not less, not more) :
[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.
[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.