  We construct a new Evans function for quasi-periodic solutions to the
linearisation of the sine-Gordon equation about a periodic travelling wave.
This Evans function is written in terms of fundamental solutions to a Hill's
equation. Applying Evans-Krein function theory to our Evans function, we
provide a new method for computing the Krein signatures of simple
characteristic values of the linearised sine-Gordon equation. By varying the
Floquet exponent parametrising the quasi-periodic solutions, we compute the
linearised spectra of periodic travelling wave solutions of the sine-Gordon
equation and and track dynamical Hamiltonian-Hopf bifurcations via the Krein
signature. Finally, we show that our new Evans function can be readily applied
to the general case of the nonlinear Klein-Gordon equation with a non-periodic
potential.
