General Relativity: An Introduction for Physicists · Problem 2.11

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · independent-components-metric-derivatives

Problem

Prove the following results used in Section 2.11: ((a))

(\partial x^a/\partial x'^b)_P

has

N^2

independent values; ((b))

(\partial^2 x^a/\partial x'^b \partial x'^c)_P

has

\frac{1}{2}N^2(N+1)

independent values; ((c))

(\partial^3 x^a/\partial x'^b \partial x'^c \partial x'^d)_P

has

\frac{1}{6}N^2(N+1)(N+2)

independent values; ((d))

g'_{ab}(P)

has

\frac{1}{2}N(N+1)

independent values; ((e))

(\partial g'_{ab}/\partial x'^c)_P

has

\frac{1}{2}N^2(N+1)

independent values; ((f))

\left(\partial^2 g'_{ab}/\partial x'^c \partial x'^d\right)_P

has

\frac{1}{4}N^2(N+1)^2

independent values. Hence show that, in a general Riemannian manifold, at least

N^2(N^2-1)/12

of the second derivatives

\left(\partial^2 g'_{ab}/\partial x'^c \partial x'^d\right)_P

will not vanish in any coordinate system.