# Experiment design: can one actually measure the speed of non-local light in curved spacetime

The equivalence principle tells us that in some local neighborhood, every free-falling observer in a general relativistic spacetime will measure the speed of light to be $$c$$; this literally means at a given tangent space, a light beam's 4-velocity is a null vector. Following the geodesic postulate, it will go on its merry way following a null geodesic.

But as we know, light beams once sufficiently far away in a curved spacetime will appear to travel at coordinate speeds not necessarily $$c$$. The canonical example is the speeds $$\frac{dr}{dt}$$ of radial light beams approaching zero as they near the event horizon in the Schwarzschild spacetime from the perspective of an observer at asymptotic infinity (and so when using the standard Schwarzschild coordinates).

But what practical experiment could give a value for $$\frac{dr}{dt}$$? One can't bounce radar signals off the beam, for example. And even then, how would $$r$$ values be found for the beam of light at a given $$t$$?

Ingredients: We have physical meaning for the $$t$$ coordinate as proper time for the asymptotic observer (call their worldline $$\mathcal{I}$$). And the $$r$$ coordinate gives the physical circumference for circle $$t = 0, \theta = 0, r = R, \phi \in [0,2\pi)$$.

• Is this what you're looking for? en.wikipedia.org/wiki/Shapiro_time_delay Mar 22, 2023 at 23:02
• Yes and no. The Shapiro time delay is fundamentally demonstrating the curvature of spacetime. It's still in some sense giving global information: about total travel time on a null geodesic. Never do we see at an instant the light "changing speed." So I guess the answer to my question is then that it's not possible, due to this classic issue of only being able to measure the roundtrip/global travel times of light. Mar 22, 2023 at 23:08

Answer for timelike geodesics: The observer at infinity takes a flashlight that strobes at 1 Hz, points it towards them, and lets it free fall towards the black hole. Using the gravitational redshift, the $$r$$ coordinate of the flashlight can be deduced at many times $$t_n$$ where $$n \in \mathbb{N}$$, thus letting the far-off observer calculate a discrete approximation to $$\frac{dr}{dt}$$, which in this set-up is a meaningful quantity thanks to redshift.