From a recent post I’ve had the scripts lying around to calculate the trajectories of particles around black holes. The paths were pretty crazy, which got me wondering what such a system would look like. For that I need to calculate how photons move around black holes, which is a slightly different problem, and doesn’t offer up much in the way of puns for post titles…
Black holes of course don’t emit any light, and don’t reflect any light. These are the two mechanisms by which we see most objects, which won’t work for a black hole. Instead we’ll have to consider how the gravitational field of the black hole distorts the path of light travelling nearby. For the sake of simplicity, we’ll only consider one black hole which distorts the image of a scene far off into the background.
To think about how to approach this problem, consider the fancy 3D drawing below. Without the black hole present, an image is formed on a camera when light rays from the background scene hit the lens. This background is emitting light in all directions, so the most efficient way of calculating what the image looks like is to go backwards – send light rays out of the camera and see where they end up on the background. Do this for each pixel in the camera and you end up with an image. As the object is ‘infinitely far away’, different pixels just correspond to different light ray angles.
The presence of the black hole distorts these light rays – in the example above a ray which might normally sample a part of the background way out to the top will actually sample a region near the middle. Other rays will end up inside the black hole, and so the corresponding pixel will be black. The task is then, for every pixel in the camera sensor, calculate the trajectory of a photon leaving the camera and passing round the black hole and see where it ends up. For a decent image this might require rays, which is a lot of computational effort. Instead we can do something simpler – due to the symmetry of the problem we just need a map of angles ‘in’ to angles ‘out’ – plotted below.
We send a ray out of the camera at angle , calculate its path (blue) to a point where it’s travelling in mostly a straight line, and measure the new angle . Do this for enough s and we can find the function relating input and output angles . This function is then easily applied to transform the background image in a single step, and is much quicker than calculating the path for each pixel. I will neglect gravitational redshift in this approximation.
With a procedure outlined then, onto the first step: calculating photon trajectories. A point mass induces a Schwarzchild metric on its local spacetime, where the metric defines the length of ‘straight lines’ over coordinates : (assume Einstein summation convention).
The equations of motion are then
where the affine connection is related to the metric by
The parameter simply measures position along the trajectory. For massive particles this is most naturally the proper time of the particle, but for photons (where proper time doesn’t elapse) such an interpretation isn’t possible.
Crunching through the algebra then, the relevant equations of motion are
where is the position of the photon as indicated in the diagram above, is the mass of the black hole and is the gravitational constant. is a constant of motion, identified with the angular momentum of the photon. Making the substitution and tidying up a bit, the surprisingly simple result is the 2nd order ODE
where we’ve eliminated such that we’re only interested in the shape of the trajectory. The initial conditions we need to supply are , and , which can be derived from a bit of geometry. For simplicity I’ll use units where .
Plotted below are a few results for , and . The behaviour of the photons is dictated by their distance of closest approach. Too close and the photons spiral into the black hole, plotted red. A bit farther and the photons swing round the black hole severely. If you were to look in this direction then you’d be able to see round corners, or even see yourself staring back. Farther still and the photon paths become curved, but not excessively.
The relation between input and output angle is plotted below, and reflects what we see above. For large the deviation is small, around . At some point drops below zero, which is when we start seeing round corners, and below the photons are just absorbed.
What does this look like? Well I can apply the above function to an image to produce the following, assuming a field of view of about . The black hole is a bit farther away in order to make the image clearer.
Pretty nice! From the above we can see that some photons get bent right back around, for which I need a ‘background’ image in all directions. For simplicity I just tiled the background, which is why right in the middle, where the ‘blue’ rays might be going, you can see a few different bright spots corresponding to the bright patch on the left.
That isn’t to say that multiple images don’t occur, as the second image of the bright cloud on the right hand side of the black hole is indeed a gravitationally-induced second image. The light has bent around the back of the black hole and is shot towards the camera.
The obvious ring that you can see is an Einstein ring. This is a position where , satisfied for a ring of points around the camera. A single object in the background is then mapped to a ring in the image, and these have quite amazingly been observed in astronomy, see the linked Wikipedia page for a lovely example.
Or in a higher-quality Youtube version here:
Of course we don’t just have to use space scenes, what if one really did appear at the LHC?
It probably wouldn’t do the beam pipes much good…
There are a few more things which would be interesting to explore with this procedure. For example, using high-quality environment maps to get a proper 360-degree view rather than tiling the same image. Also, for small angles, the light rays wrap around the black hole an increasing number of times. This gives rise to multiple Einstein rings at closer and closer intervals. I could also make an animation of what the approach to a black hole looks like. All fun things to do, but they’ll have to wait until next time as my typing fingers are starting to give out.