In two dimensions the chapter found that second derivatives of the metric at a point…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

The Geometry of Gravity, Ch Manifolds, vectors, tensor fields and the metric

Problem

In two dimensions the chapter found that 99 second derivatives of the metric at a point are opposed by 88 free coefficients of a cubic change of coordinates, leaving exactly 11 combination that no coordinate choice removes; in four dimensions the same count gives 100100 against 8080, leaving 2020. Both counts rest on two general facts: a symmetric metric in nn dimensions has M(n)=n(n+1)/2M(n)=n(n+1)/2 independent components, and a coefficient symmetric in three lower indices has (n+23)\binom{n+2}{3} independent choices for each of its nn upper-index values. ((a)) Write, in terms of M(n)M(n), the number of independent second derivatives ∂γ∂δgμν(P)\partial_\gamma\partial_\delta g_{\mu\nu}(P) at a point, and check that your formula reproduces 99 at n=2n=2 and 100100 at n=4n=4. ((b)) Write the number of free coefficients of a symmetric cubic term in the coordinate change, in terms of nn and (n+23)\binom{n+2}{3}, and check that it reproduces 88 at n=2n=2 and 8080 at n=4n=4. ((c)) Apply both formulas to n=3n=3 dimensions, and state how many independent second derivatives of the metric survive at a point in three dimensions, whatever coordinates are chosen.

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