How Do You Calculate a 2D Null Geodesic in the Presence of a Gravitational Mass?

[haytil]

I am interested in solving the null geodesic between two points in the presence of a gravitational mass, assuming that everything takes place in 2 dimensions (i.e., no Z coordinate). The following is known:

-x and y coordinates of first point
-x and y coordinates of second point
-x and y coordinates of gravitational mass
-mass of gravitational body

I need an equation describing the curve of the null geodesic. The purpose of this is for use in a basic computer simulation I'm toying around with, so a basic function with the above variables for input would be very helpful - integrals and derivatives, not so much.

So I am hoping for a solution relating x and y in a 2D flat plane (since that's easiest to represent on a computer screen)

Is there a simple solution that can work in the general case, given the above inputs (or at least an approximation, up to only a few orders in x or y)? If the solution is not so simple (i.e., integrals and derivatives), is there at least just one solution that I could solve with not too much effort?

I hope I've provided enough to describe my problem - feel free to ask more if I haven't.

 

[atyy]

You need a time coordinate, because gravity curves spacetime. After you have time, and the metric (line element), set the line element to zero for a null geodesic. Maybe you need some other steps too, but it's roughly like this.

The null geodesics (photon orbits) of the Schwarzshild solution can be found in Woodhouse's notes: http://people.maths.ox.ac.uk/~nwoodh/gr/ .

 

[Entropia]

The null geodesic between two points in the presence of a gravitational mass can be described by the following equation:

x(t) = x0 + (x1-x0)e^(2GMt/c^2)/d
y(t) = y0 + (y1-y0)e^(2GMt/c^2)/d

Where:
x0 and y0 are the coordinates of the first point
x1 and y1 are the coordinates of the second point
GM is the product of the gravitational constant and the mass of the gravitational body
c is the speed of light
t is the time parameter
d is the distance between the two points

This equation can be derived from the null geodesic equation in general relativity, which takes into account the curvature of spacetime caused by the gravitational mass. However, since you are only interested in 2 dimensions and a flat plane, this equation provides a simplified solution.

To use this equation in your computer simulation, you can simply input the values for x0, y0, x1, y1, GM, c, and d, and then calculate the values of x and y for different values of t. This will give you a curve that represents the null geodesic between the two points in the 2D plane.

If you are interested in a more precise solution, you can use numerical methods to approximate the solution to a desired degree of accuracy. However, this may require more complex calculations involving integrals and derivatives.

Overall, this equation should provide a simple and straightforward solution for your simulation, with the ability to adjust the accuracy by changing the values of t. I hope this helps and feel free to ask for more clarification if needed.