In dimensions the fully lowered Riemann tensor is antisymmetric in its first pair of…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor

Problem

In nn dimensions the fully lowered Riemann tensor RρσμνR_{\rho\sigma\mu\nu} is antisymmetric in its first pair of indices and in its last pair, unchanged when the two pairs are exchanged as blocks, and obeys the cyclic identity Rρ[σμν]=0R_{\rho[\sigma\mu\nu]} = 0. The cyclic identity adds one independent condition for each set of four different index values, (n4)\binom{n}{4} conditions in all. Take n=5n = 5. ((a)) How many independent values NN can an antisymmetric pair of indices, such as [ρσ][\rho\sigma], take? ((b)) Treat RρσμνR_{\rho\sigma\mu\nu} as a symmetric N×NN\times N array indexed by the two pairs, count its independent entries, and subtract the cyclic conditions. Check the result against n2(n2−1)/12n^2(n^2-1)/12. ((c)) The Ricci tensor is symmetric, with n(n+1)/2n(n+1)/2 components. How many independent Riemann components are left over once the Ricci tensor is fixed?

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