Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 2 / c

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- Solution c

Solution

Applying

ϵ = b_{i} (t) x^{i}

to

A_{μ} = 0

gives

A_{0} = \dot{b}_{i} (t) x^{i}, A_{i} = b_{i} (t) .

Its field strength vanishes, so it is a large gauge transformation of the background. To arise as the zero-momentum limit of a physical transverse perturbation,

b_{i} (t)

must obey the zero-momentum vector equation of motion. In the variables used in the question this is

\partial_{t} (a^{3} \dot{b}_{i}) = 0,

with a constant growing solution and a decaying solution proportional to

\int^{t} d t^{'} / a^{3} (t^{'})

. This condition makes the large gauge mode an adiabatic mode that can be continued to small nonzero momentum.

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