So I know that $$R_{μν}:=R^λ_{μλν}$$ is the Ricci curvature tensor (where $R^λ_{μλν}$ is the Riemann Tensor). This is in Einstein's field equations: $$R_{μν}-\frac{1}{2}g_{μν}R=\frac{8πG}{c^4}Τ_{μν}$$ and hence I presume that μ and ν are identifies of the components of the tensors been looked at. But in this first equation what does the λ represent (I think it may just be another place identifier)? (and is may assumption about μ and ν correct?)
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3$\begingroup$ Do you know Einstein notation? $\endgroup$ – Qmechanic♦ Jul 17 '14 at 16:22
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$\begingroup$ @Qmechanic I have read this article, and others, on Einstein notation and get the general idea, but I can't find any the explain what to do with something like $R^λ_{μλν}$ please could you explain, I also presume we sum from 1 to 4 due to us using 4 dimensions? $\endgroup$ – user43487 Jul 17 '14 at 17:00
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The Ricci tensor is built adding up some of the components of the Riemann tensor, as the definition specifies:
$$ R_{\mu\nu} = R^{\lambda}_{\ \mu\lambda\nu} $$
The repetition of two indices above and below means that all these components must be summed:
$$ R^{\lambda}_{\ \mu\lambda\nu} = R^{0}_{\ \mu 0\nu} + \dots + R^{3}_{\ \mu 3\nu} $$
This operation is like taking the trace of a matrix. For example, if we have a 3x3 matrix A, its trace will be
$$ \mathop{\rm{tr}}A = \sum_{i=1}^3A_{ii} = A_{11} + A_{22} + A_{33} $$
It has not a particular meaning other than this one.
