general relativity - Does gravity slow the speed that light travels?

Does gravity slow the speed that light travels? Can we actual measure the time it takes light from the sun to reach us? Is that light delayed as it climbs out of the sun's gravity well?

Answer

This is one of those questions that is more subtle than it seems. In GR the velocity of light is only locally equal to $c$, and we (approximately) Schwarzschild observers do see the speed of light change as light moves to or away from a black hole (or any gravity well). Famously, the speed that radially moving light travels falls to zero at the event horizon. So the answer to your first question is that yes gravity does slow the light reaching us from the Sun.

To be more precise about this, we can measure the Schwarzschild radius $r$ by measuring the circumference of a circular orbit round the Sun and dividing by 2$\pi$. We can also measure the circumference of the Sun and calculate its radius, and from these values calculate the distance from our position to the Sun's surface. If we do this we'll find the average speed of light over this distance is less than $c$.

However suppose we measured the distance to the Sun's surface with a (long) tape measure. We'd get a value bigger than the one calculated in the paragraph above, and if we use this distance to calculate the speed of the light from the Sun we'd get an average speed of $c$.

So I suppose the only accurate answer to your question is: it depends.

Re your other question, assuming the spacetime around the Sun is described by the Schwarzschild metric, the time dilation at the surface of the Sun is given by:

$$ \text{time dilation factor} = \frac{1}{\sqrt{1 - r_s/r}} $$

where $r_s$ is the radius of a black hole with the mass of the Sun and $r$ is the radius of the Sun. The former is about 3,000m and the latter about 700,000,000m so I calculate the time dilation factor to be around 1.000002 and this is too small to measure directly.

However you can interpret gravitational lensing to be due to changes in the speed of light, and since we can measure the gravitational lensing due to the Sun you can argue we have measured its effect on the speed of light. This isn't really true as what gravitational lensing measures is the spacetime curvature. However the change in the speed of light (measured by a Schwarzschild observer) is an aspect of this.