Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 38D / c

Table of contents
- Solution c

Solution

Introduce

T = t cosh χ, R = t sinh χ .

Then

- d T^{2} + d R^{2} + R^{2} d Ω^{2} = - d t^{2} + t^{2} (d χ^{2} + sinh^{2} χ d Ω^{2}),

which is the Minkowski metric. Since

t > 0

,

T^{2} - R^{2} = t^{2} > 0, T > 0,

so these coordinates cover exactly the interior of the future light cone of its vertex

p

. The

t = constant

slices are future hyperboloids. Their spatial isometries are precisely the proper orthochronous Lorentz transformations of Minkowski spacetime that fix

p

.

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