Suppose I have a pulsar at some radius $r_p$ from a black hole and an observer at a radius $r_o$ where $r_o \gg r_p$.
Assume that both the pulsar and observer are stationary. The pulsar 'pulses' at $\tau_1$ and emits a photon which travels along a null geodesic to an observer. At time $\tau_2$, a period $\Delta \tau_p$ later, it pulses again, emitting a second photon which travels along the same geodesic.
We can integrate the photon path from $r_p \rightarrow r_o$ and so determine the total coordinate flight time $t$.
Photons 1 and 2 then arrive at the observer at time $\tau_1 +t$ and $\tau_2 +t$
My question is: What period does the observer say that the pulsar has? The pulsar is in a region of stronger gravity, so its clock should run slower compared to the observer's clock, but it seems to me that since the photons both travel along the same geodesic, the observer will say that the period is also $\Delta \tau_p$?