Einstein’s Relativity Used to Weigh a Star

The mechanism is completely different, but gravitational lensing also involves changing the angle of a ray of light.

The point is to remind the reader that light doesn't always "travel in straight lines". Obviously refraction and gravity aren't the same thing, but nonetheless you can measure the index of refraction of a material by looking at the deflection of light through it in (sort of) the same way you can measure the gravity well of a star by looking at similar deflections.

Probably, if you think of refraction variationally in terms of Fermat's principle.

There is a way of looking at this phenomenon in terms of an optimal path problem, Fermat's Principal of Least Time.

Suppose you have the following situation

    x1  |  n2
        |
        |
    n1  |  x2

We have two different materials, with index of refraction n1 and n2. Now suppose that light has traveled between points x1 and x2, we can ask the question, what path did it take? (Note here, we are asking about classical rays of light, so we're thinking in terms of geometrical optics, not quantum :) )

When light travels through the left hand side, it's traveling at a velocity, c/n1, but in the right, it will travel at c/n2.

It turns out that we can answer the question of which path the light takes by asking "Which path takes the shortest amount of time?". From that assumption you can actually re-derive Snell's Law.

An example when n2 is very large (and thus moves slower on the right hand side):

    x1  |  n2
      \ |
       \|
    n1  |--x2

the light will travel more distance on the left hand side because it can move faster there, and thus get close as possible to the destination. It will then make the last bit of distance in the right hand side.

Now Snell's law says

  n1 sin θ_input = n2 sin θ_output

Since n2 is very large, the only way to make this work is to make sin θ_output very small. So we see in this case at least, both approaches agree as to what will happen.

So it turns out this principal works even in the case of gravitational spacetime distortion, and can be applied there as well.

This is actually what inspires me as a physicist. So back in the day people were looking for an encapsulated theory of why the hell things happened. The Principle of Least Time was the original theory, but that didn't make sense (look at a spring) so then there was The Principle of Least Energy, but that also wasn't good enough. Until Hamilton came along and gave us the Principle of Least Action. This is the basis for all physics. All of our laws incorporate this theory. Einstein took it one direction and Maxwell took it another and Planck another. So all of these phenomena are basically the same. Just different manifestations.

Physics isn't 10,000 problems it's 10,000 manifestations of like 5.