Write $P^i=(\epsilon/a^2)\widehat p^i$ and set
$$
P^0=\frac\epsilon{a^2}(1+A)
$$
with $A$ first order. The photon on-shell condition is
$$
0=g_{\mu\nu}P^\mu P^\nu
=a^2\left[-(1+2\delta N)(P^0)^2
+2\partial_i\psi P^0P^i+\delta_{ij}P^iP^j\right].
$$
Using $\delta_{ij}\widehat p^i\widehat p^j=1$ and retaining linear terms gives $A=-\delta N+\widehat p^i\partial_i\psi$. Hence
$$
\boxed{P^0=\frac\epsilon{a^2}
\left(1-\delta N+\widehat{\mathbf p}\mathbin\cdot\nabla\psi\right)}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-312/4/i/solution