Gravitational lensing thought experiment

[stoomart]

Hello all,

I'm starting to learn the math in physics and have the following thought experiment I want to use to work through something I own.

The setup: There is a wall 10m from the center of a 1m diameter sphere. There is a 1m diameter light source casting a 1m diameter shadow on the wall (unsure if luminosity and distance from the sphere of the light source matter).

Question 1: As the mass of the sphere increases, would the light bend and cause the shadow to shrink and eventually disappear or would the light scatter and fill in the shadow with lower energy light?

Question 2a: If the shadow shrinks and disappears, what equation would calculate the mass required to make the shadow disappear?

Question 2b: If the shadow fills in with lower energy light, does its luminosity increase in proportion to the sphere's mass?

Question 2bi: If the answer to 2b is yes, what equation would show the relationship between the luminosity of the light in the shadow and the mass of the sphere?

Question 2bii: If the answer to 2b is no, what equation would show the relationship between the luminosity of the light in the shadow and to its original luminosity?

 

[stoomart]

My first thought test had unexpected results. The answer to the main question was neither happened. The shadow actually grew larger as the sphere affected spacetime around it due to its close proximity to the wall. As the wall moved further away from the sphere, the shadow eventually returned to and maintained its original size.

Was there a flaw in my test or in my initial assumptions?

 

[Ibix]

What tests are you doing? I don't think the proximity of the wall to the sphere should have anything to do with it.

Edit: except for the degree of lensing necessary and the extent of the penumbra.

 

[stoomart]

What tests are you doing? I don't think the proximity of the wall to the sphere should have anything to do with it.

Edit: except for the degree of lensing necessary and the extent of the penumbra.

No real tests, just mental analysis based on my layman understanding of physics.
I assume no point in the original area of the umbra ever experiences the antumbra or penumbra due to the light source and sphere having the same diameter, and the sphere has no atmosphere to scatter the light like the Earth does with the sun.

 

[Charles Kottler]

I am by no means an expert, but I would expect the shadow to shrink as the mass increases. If the mass was a black hole the curvature would be such that multiple images of light sources behind it would reach the wall.

 

[stoomart]

I am by no means an expert, but I would expect the shadow to shrink as the mass increases. If the mass was a black hole the curvature would be such that multiple images of light sources behind it would reach the wall.

I would expect the same thing.

 

[Ibix]

Qualitatively, light behaves like anything else. As you replace the sphere with increasingly massive ones the light paths curve towards the mass. Thus the umbra will be smaller for a very massive sphere than for a light one.

Note that a 1m diameter sphere made of any normal matter will not appreciably curve light paths, so your scenario is unrealistic in the extreme.

As to what you are doing wrong, basically you can't expect your intuition to give you correct answers unless you have a large library of exact solutions to learn from. In other words, learn the maths and then develop intuition. The other way round doesn't really work for anything more sophisticated than what I've written here.

@m4r35n357 may have simulations to show you.

 

[stoomart]

Thanks Ibix, that helps a lot. I will work on learning the maths so I can answer 2a of this extremely unrealistic problem. I tend to learn better when working on problems I've created. If anyone else has the answer, I'm all for learning through reverse engineering.

 

[Ibix]

Carroll's lecture notes on GR definitely cover geodesics in the Schwarzschild metric, which is what you need to learn about. They are freely available online, but you can't avoid some fairly complex maths.

 

[stoomart]

Carroll's lecture notes on GR definitely cover geodesics in the Schwarzschild metric, which is what you need to learn about. They are freely available online, but you can't avoid some fairly complex maths.

I don't understand most of what I'm reading, but Carrol's notes seem a bit scant on light deflection. I found the following lecture on light deflection I'm going to work on understanding the maths in.

If I understand the basics though, my second thought test results are this: The size of the umbra decreases in proportion to increasing the distance to the wall and the mass of the sphere. If the wall didn't move, I suspect the umbra would increase until the light could no longer escape the sphere's gravitational field, forming an accretion disk.

http://www.reed.edu/physics/courses/Physics411/html/page2/files/Lecture.30.pdf

 

[m4r35n357]

Thanks Ibix, that helps a lot. I will work on learning the maths so I can answer 2a of this extremely unrealistic problem. I tend to learn better when working on problems I've created. If anyone else has the answer, I'm all for learning through reverse engineering.

This guy has videos and a C program that you can use to reproduce them or to reverse-engineer. I should warn you it's not an easy task, you will need to learn about geodesics in the Kerr spacetime to understand it.

I do have my own geodesic generator (it does full 4D Kerr-deSitter geodesics for light and matter using the constants of motion), but I don't know enough about relativistic ray-tracing to do the rest myself (I just use it to drive the program in the link I gave).

 

[Jonathan Scott]

I think we have to assume parallel incoming light (e.g. from a very distant source), as the idea of penumbra or similar makes it too complicated, and assume that the sphere is not rotating (so Schwarzschild solution rather than Kerr).

For weak fields, I think that as a good approximation a light beam which passes at closest distance $r$ from spherical mass $M$ is deflected inwards by an angle $2GM/rc^2$ radians, so for a given mass the amount of deflection varies as $1/r$. There are obviously corrections for strong fields, but that should give a reasonable idea of what happens.

Basically, if you have parallel light arriving from one side, the light closest to the surface is bent a little more than light further out, so the shadow shrinks slightly, and if the mass is just right, it will shrink to a point. For a sufficiently large mass, approaching a black hole (where the orbit radius in Schwarzschild coordinates is close to $3GM/c^2$, the "photon sphere" radius) light very close to it can in theory complete one or more orbits before escaping in some direction.

 

[Jonathan Scott]

For weak fields, I think that as a good approximation a light beam which passes at closest distance $r$ from spherical mass $M$ is deflected inwards by an angle $2GM/rc^2$ radians, so for a given mass the amount of deflection varies as $1/r$. There are obviously corrections for strong fields, but that should give a reasonable idea of what happens.

I think that should have been $4GM/rc^2$. But if it's important, look it up.