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Contracting tensors

  1. Mar 13, 2013 #1
    Can anyone explain how to contract [itex] R_{\alpha \beta} [/itex] to [itex] R^{\rho}_{\alpha\beta\sigma} [/itex] without multiplying it by 16 i.e [itex] g^{\rho\xi}g_{\xi\sigma}[/itex] It is in a sum with other tensor products and so I obviusly can't just multiply one term by anything ither than 1.

    Should [itex] \eta [/itex]'s be used although are these valid in cosmological space-times i.e [itex] dt^2 -a^2 dx^2 [/itex]

    I apologise if any indices aren't in the correct order, I am self taught
     
  2. jcsd
  3. Mar 13, 2013 #2

    George Jones

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    [itex] g^{\rho\xi}g_{\xi\sigma} = \delta^\rho_\sigma[/itex] not 16. Consequently,

    [tex] g^{\sigma\xi}g_{\xi\rho} R^{\rho}{}_{\alpha\beta\sigma} = \delta^\sigma_\rho R^{\rho}{}_{\alpha\beta\sigma} = R^{\sigma}{}_{\alpha\beta\sigma}[/tex]
    No.
     
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