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Homework Statement
(My first post on this forum)
Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition).
Problem: Dirac lists several symmetry properties of the curvature tensor from its definition in terms of the Christoffel Symbols and then claims without proof that from these symmetry properties, the curvature tensor has 20 independent components (reduced from 256).
a) Any hints on how he got that?
b) How do you work out the number of independent tensor components in general, given whatever symmetry constraints on the indices?
Homework Equations
Definition: R^{\beta}_{\nu\rho\sigma} = \Gamma^{\beta}_{\nu\sigma,\rho} - \Gamma^{\beta}_{\nu\rho,\sigma} + \Gamma^{\alpha}_{\nu\sigma}\Gamma^{\beta}_{\alpha\rho} - \Gamma^{\alpha}_{\nu\rho}\Gamma^{\beta}_{\alpha\sigma}
Dirac's symmetry constraints:
(1) R^{\beta}_{\nu\rho\sigma} = -R^{\beta}_{\nu\sigma\rho}
(2) R^{\beta}_{\nu\rho\sigma} + R^{\beta}_{\rho\sigma\nu} + R^{\beta}_{\sigma\nu\rho} = 0
(3) R_{\mu\nu\rho\sigma} = -R_{\nu\mu\rho\sigma}
(4) R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu} = R_{\sigma\rho\nu\mu}
The Attempt at a Solution
I tried to work out the general case first. A tensor of 2 indices in N dimensions has N^{2} components. If it is a symmetric tensor then it has \frac{1}{2}N(N+1) independent components and hence symmetry constrains \frac{1}{2}N(N-1) components. Similarly, if the tensor is antisymmetric then that constrains \frac{1}{2}N(N+1) components.
N = 4 here. Assuming what I said above is correct equations (1) and (3) will constrain 20 components in total.
Now I'm stuck because:
a) Should I treat equation (4) as three conditions?
b) If I do treat equation (4) as three conditions, am I correct in saying that R_{\rho\sigma\mu\nu} = R_{\sigma\rho\nu\mu} constrains 12 components because there are two pairs of symmetric indices?
c) How do I interpret things like R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu} in the above context?
d) How should I interpret the cyclic sum of equation (2) in terms of what I said above?
Thanks!