The question is:
For a spacetime the Riemann tensor is given below:
$$R_{\mu \nu \rho \sigma} = \frac{R}{6} (g_{\mu \rho} g_{\nu \sigma} - g_{\mu \sigma} g_{ \nu \rho} )$$
What is the dimension of this spacetime?
My thoughts (what I think is the correct method but not 100% sure how to do):
- Contract the indexes on the left hand side until you obtain the Ricci scalar which in turn will automatically give a constraint on the number of dimensions.
Ricci tensor $ R_{ \nu \sigma} = g^{\mu \rho} R_{\mu \nu \rho \sigma}$
Ricci scalar $ R = g^{\nu\sigma} R_{\nu \sigma}$
The Ricci scalar itself should be a number, this number I think should be related to the number of dimensions.
So for us we have, $$R = g^{\nu\sigma} g^{\mu \rho} R_{\mu \nu \rho \sigma}$$
Not sure if plugging in our spacetime into this and working through would yield the correct answer. $$R = \frac{R}{6} g^{\nu\sigma} g^{\mu \rho}(g_{\mu \rho} g_{\nu \sigma} - g_{\mu \sigma} g_{ \nu \rho} )$$
Perhaps something like this? Not 100% sure though?