Generalise the order-by-order counting of coordinate freedoms against metric…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

The Geometry of Gravity, Ch Redshift, local inertial frames and the failure of flat spacetime

Problem

Generalise the order-by-order counting of coordinate freedoms against metric derivatives used for four-dimensional spacetime to a spacetime of

n

dimensions (one time and

n-1

space dimensions). ((a)) Using the same Taylor-series argument (first, second and third derivatives of the coordinate change matched against

g_{\mu\nu}(P)

, its first derivatives and its second derivatives), show that the number of second derivatives of the metric at

P

that cannot be set to zero by any choice of coordinates is

N(n)=\frac{n^2(n^2-1)}{12}.

Check that

N(4)=20

, as claimed for ordinary spacetime, and evaluate

N(6)

. ((b)) On the Moon's surface,

g_{\rm Moon}=1.62\ \mathrm{m\,s^{-2}}

. A ball is thrown vertically upward from the surface with speed

v_0=3.00\ \mathrm{m\,s^{-1}}

and caught again at the same height a flight time

T=2v_0/g_{\rm Moon}

later. Using

\Phi(\ell)=g_{\rm Moon}\ell

for height

\ell

above the surface,

g_{00}=-(1+2\Phi/c^2)

d\tau/dt\approx1+\Phi/c^2-v^2/2c^2

for slow motion, and

\delta\int d\tau=0

, confirm that the freely falling worldline satisfies

\ddot\ell=-g_{\rm Moon}

. Then compute, to leading order in

1/c^2

, the height term and the speed term whose difference is the ball's net proper-time gain over the flight relative to a clock held at the launch point, and check that the height term is exactly twice the speed term.

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