## Riemann Curvature Tensor and working out independent components of a tensor generally

1. The problem statement, all variables and given/known data

(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?

2. Relevant 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) $$R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu} = R_{\sigma\rho\nu\mu}$$

3. 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!
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 1st: I don’t think all the symmetry constrains equations are independent. As: lower the upper indices in (1) using the metric tensor. Now apply 2nd equality of (4) to RHS and one gets (3). Hence I think you should only use (1), (2) & (4) or the set (2), (3) & (4) to derive the # of independent tensor elements. 2nd: I think (2) should be interpreted like a supermarket deal, ie “buy 2 get one free”. ie pick any two of the tensors, and then the third tensor is independent (or “free”) if you know (or have “bought”) the other two. Hope this helps, Mischa

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