## Light speed measured from distance

Clocks run slower, deeper they are in the gravity field, or faster they are receding from observer.

Question: If we could measure speed of light near the event horizon, or at far away galaxy, by means of measuring from distance, would we measure light going slower there?

I am not asking about measuring speed of light there, or speed of light coming from gravity well, or distant galaxy, and passing us. I am asking what we would measure from here as speed of light there.
 I prefer to define the following quantities (example for diagonal metric, and 1+1 dim): ds^2 = g_tt*dt*dt - g_xx*dx*dx physical time differensial: dtau := sqrt(g_tt)*dt physical length differential: dl := sqrt(g_xx)*dx physical speed: v := dl/dtau Since ds^2=0 for light, you will always have v = 1 for light. However, if you want to calculate |dx/dt| for light, that could be anything depending on your coordinate system. E.g. in Schwarzchild coordinates, radial movement of light towards the origin will have |dr/dt| > 1, since this is just a "coordinate velocity", not physical velocity. If you define "speed of light" to be my "physical velocity" then you would measure always 1, and this is the most sensible thing to do IMO. If you define it as my "coordinate velocity", then you could get anything you want, depending on your coordinate system. Torquil