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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-4/10d/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 10D / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Take velocities relative to a comoving observer at

O

. Spatial homogeneity means that changing the observer to

A

gives the same relative velocity law, so

v (r_{B} - r_{A}) = v (r_{B} - r_{O}) - v (r_{A} - r_{O}) .

With

O

as origin this implies

v (0) = 0

and

v (a + b) = v (a) + v (b)

. A physical velocity field is continuous; additivity first gives rational homogeneity and then, by continuity, real homogeneity. Hence

v_{i} (r) = j \sum H_{ij} r_{j}, H_{ij} = [v (e_{j})]_{i},

with coefficients independent of position. Without a regularity assumption, the additive functional equation alone admits non-linear pathological solutions; continuity is the usual implicit velocity-field assumption.

Isotropy requires

v (R r) = R v (r)

for every spatial rotation, or

H R = R H

. Rotations about

e_{i}

H e_{i}

to be parallel to

e_{i}

, and rotations exchanging coordinate directions force those diagonal coefficients to agree. Therefore

H_{ij} = H δ_{ij}

and

v = Hr

, the Hubble law.

Allow

H = H (t)

. Solving

\overset{r}{˙} = H (t) r

gives

r (t) = a (t) x

for a constant comoving coordinate

x

, where

a (t) = a (t_{0}) exp (\int_{t_{0}}^{t} H (s) d s)

. Thus

H = \overset{a}{˙} / a

, with

a

the scale factor.

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