A <strictly stationary spacetime> has an everywhere timelike Killing field $K$. Choose a complete orthonormal set of <positive-frequency solutions> satisfying
$$
i\mathcal L_Ku_j=\omega_ju_j,
\qquad \omega_j>0,
$$
with positive <Klein-Gordon inner product>. Expand
$$
\widehat\Phi=\sum_j(a_ju_j+a_j^\dagger u_j^*)
$$
with the sum replaced by an integral for continuous labels. Equivalently, the coefficients are projections using the Klein-Gordon product: $a_j=(u_j,\widehat\Phi)_{KG}$ and $a_j^\dagger=-(u_j^*,\widehat\Phi)_{KG}$.

The stationary vacuum is uniquely selected, up to degeneracies and unitary changes of positive-frequency basis, by
$$
a_j|0\rangle=0
$$
for every $j$. Acting with the $a_j^\dagger$ constructs the <bosonic Fock space>, the symmetric direct sum of all particle-number sectors.

In a nonstationary spacetime no preferred timelike Killing flow exists, so there is no canonical split into positive and negative frequencies. Different splits mix creation and annihilation operators by <Bogoliubov transformations>[Bogoliubov transformation] and lead to the <vacuum ambiguity in a nonstationary spacetime>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 3 / c / ii / Solution

A strictly stationary spacetime has a globally defined Killing vector that is timelike everywhere.


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