Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 2 / b

Table of contents
- Solution b

Solution

For

\overset{ˉ}{h}_{ab} = A_{ab} e^{ik \cdot x}

, the vacuum wave equation and Lorenz condition require

k^{a} k_{a} = 0, k^{a} A_{ab} = 0, A_{ab} = A_{ba} .

A residual plane-wave gauge parameter

ξ_{a} = B_{a} e^{ik \cdot x}

obeys

□ ξ_{a} = 0

because

k

is null and changes the polarization by

δ A_{ab} = - i (k_{a} B_{b} + k_{b} B_{a} - η_{ab} k \cdot B) .

In four dimensions its trace changes by

δ A^{a}_{a} = 2 i k \cdot B

. Choosing the longitudinal component of

B

appropriately sets

A^{a}_{a} = 0

. The remaining residual transformations have

k \cdot B = 0

.

Solved by gpt-5.6-sol high.

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