Work in geometric units,

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Differential geometry of curved spacetime

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

Problem

c=1

. A rocket maintains constant proper acceleration

g_\star

along its own

\xi

-axis, so its worldline sits at

\xi=0

in the Rindler chart of the accelerated frame,

ds^2=-(1+g_\star\xi)^2d\eta^2+d\xi^2+dy^2+dz^2.

This chart is related to an inertial (Minkowski) chart

(t,x,y,z)

sharing the same

y,z

t=\left(\xi+\frac1{g_\star}\right)\sinh(g_\star\eta), \qquad x=\left(\xi+\frac1{g_\star}\right)\cosh(g_\star\eta)-\frac1{g_\star}.

((a)) State the signature of

g_{\mu\nu}

\xi=0

and say, using the equivalence-principle argument for a local inertial frame, why it has to come out this way at every point of the worldline. ((b)) At the event on the worldline where

g_\star\eta=1

(rapidity 1), the rocket's four-velocity in the Rindler chart is

u^\eta=1

u^\xi=u^y=u^z=0

(it is at rest in its own instantaneous chart). Use the vector component transformation law

u^{\mu'}=(\partial x^{\mu'}/\partial x^\mu)u^\mu

to find

u^t

and

u^x

in the Minkowski chart.

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