# A simple example and explanation of gravitational lensing

I've recently been looking into gravitational lensing for the purpose of creating interesting visual effects through cgi. However, the wikipedia page doesn't go into much detail about the mathematical process, only briefly touching on the Schwarzschild radius. It doesn't explain how to apply this.

So, for the sake of example, consider this picture: I've kept the numbers relatively small for the sake of simplicity. In this diagram we have an object which has a Schwarzschild radius of $1m$ (thus, a mass of about $6.7 \times 10^{26}kg$), a flat object $8m$ away which is emitting light, and a third object $13m$ away in the opposite direction, representing the observer. Again, for the sake of simplicity, assume the object creating this strong gravity well doesn't directly interact with light. For instance, assume it's dark matter.

Given this diagram, how would the light rays be affected by the dense object in the center? Can you explain the math behind this in a relatively simple manner?

• Depends on what "simple" means. Do you know how to numerically solve a differential equation and plot the trajectory? Commented May 3, 2017 at 15:01
• @Javier I passed calc 3 and still remember most of it, so I can solve most differential equations, yes. I'd like for it to be fairly simple, not for me, but for the average user who might be trying to do something similar. So, maybe simple enough for a high school grad to understand. Personally, though, I'm more worried about the physics terminology and such, as I know almost nothing about any form of physics beyond Newtonian. Commented May 3, 2017 at 15:07
• Sadly, general relativity is hard. You'd have to be a pretty advanced physics or math student to have the necessary background for the actual theory. If you just want to know how to find the trajectories of light rays, you only need the differential equation. I don't think I could explain the details of the math behind either of those things to a high schooler. Commented May 3, 2017 at 15:11
• @Javier Fair enough. I guess in that case, I'd rather know how to find the trajectories than nothing at all. Commented May 3, 2017 at 15:14

For weak lensing, lensing that is within the Newtonian limit, you have $$\frac{d^2u}{d\theta^2}~+~u~=~\frac{GM}{c^2b^2}~=~\Phi,~for~u~=~\frac{1}{r}$$ where $b$ is the impact parameter. We think of the photon as a classical particle moving with $v~~=~c$ at infinity. The inputs then are the boundary condition for the $u~\rightarrow~\infty$ and $u/sin\theta~\rightarrow~b$ for the general solution $u~=~u_0cos(\theta~-~\theta_0)~-~\Phi$. We have then set up the standard problem in classical mechanics for scattering. I will leave the rest of this to the reader to look up in a test on classical mechanics. This is also the attractive version of what was done to calculate Rutherford scattering.