Argthtjtr

This question is about how to explain gravitational lensing to a layman, not about exact theoretical correctness.

I am accustomed to explaining optical refraction in terms of wavefronts and the fact that light moves at different speeds in different media. For example, I explain an optical lens by saying

A plane wave that is incident normal to the flat surface of a
plano-convex lens will propagate slower inside the lens. This means
that the portions of the wavefront that exit the lens first, near the
edges, will end up ahead of the portions that exit near the middle of
the lens. This results in the wavefronts downstream from the lens
having a concave curvature. Because the wave propagates perpendicular
to the wavefront, it converges to a point.

I would like to use an analogous explanation to describe the effects of a gravitational lens. Although the local speed of light is invariant in a vacuum, gravity causes the local frequency and wavelength of light in an initially flat, monochromatic, wavefront to vary with radial distance from a gravitating mass. When the light wave has passed far enough downstream from the gravitating mass, the frequency and wavelength return to their original values, but the wavefront is curved because of the changes experienced by different parts of the wavefront while passing through the gravitational field of the mass: the parts of the wavefront near the middle are delayed relative to those near the edges.

I've been trying to refine this explanation, but am running into a complication that, thus far, is beyond my skills. I understand time dilation in terms of gravitational redshift and blueshift. The complication is that both the wavelength and frequency of light (from the perspective of a distant observer) would seem to be affected by the gravitational field. It's not obvious to me how wavelength is affected by gravity. Locally, the wavelength change should balance the frequency change in such a way that the speed of light is $c$. This suggests that the apparent wavelength as judged by a distant observer should also change.

I don't know how to explain this in an intuitively satisfying way, probably because I don't understand well enough how a distant observer can remotely measure the length of something that is deep in a gravitational well. Any help will be greatly appreciated.

I think the distinction that you are making, with concerns about the intermediate blueshifts, redshifts, and time dilation, is too technical for someone whose education is outside of physics altogether. I would start with something simpler and add complexity as the responses of your audience suggest appropriate directions to go.

When I explain about gravitational lensing, I use an approach like this:

1. 
   

   The equivalence principle predicts that light should also fall in a gravitational field. (And experiments on Earth confirm this, but the details are tricky.) So if a distant star/galaxy and a nearby star/galaxy are along the same line of sight, light from the distant object that would have missed us can get bent back towards us by the nearby star's gravity, similar the way that a magnifying lens makes parallel light rays converge. I usually end up drawing some variant of the below:

   
   
   

   
   
   [source]

   
   
   

   Beware that most people's experience with magnifying lenses is to look through the lens at a virtual image. Looking through a magnifier with enough distance to see the inverted real image is a surprise for many adults, and putting your eye at the focus is just plain strange-looking. So you may have some intuition failures to cope with if you use this approach.

   
   


2. 
   

   If they get that gravity can bend light and make images of things appear in the "wrong" place, I ask them to remember a time they walked through a parking lot and noticed a dent on a shiny car. You don't actually see the dent. What you see is the reflection in the car of the car's surroundings. The reflection in the body of the car is distorted a little, because a car isn't flat like a mirror. And where there is a dent, there is more and different distortion. Sometimes you'll see the same object reflected more than once, from different sides of the dent. Most people have seen this enough times that they have a pretty good idea of the size and shape of the dent just from this reflected-image information: walking up to the car and touching the dent is not usually a surprise. And in the same way, an image like this one

   
   
   

   
   
   [source]

   
   
   

   lets you say things about the distribution of matter in the foreground galaxy cluster that's doing the lensing.

   
   

This little spiel has served me well.

Although these papers (1), (2), (3), (4) explain mostly from a ray perspective, they do derive an effective refractive index that depends on gravitational potential. It is easy to explain gravitational lensing from either the wave perspective or the ray perspective, based on the effective refractive index $n$:
$$n=1− 2Φ/c^2, $$ where $Φ$ is the Newtonian gravitational potential. This is a first-order approximation; some of the papers provide a more exact formula.