Light Travel Delay in Kerr Render Engine me and a friend are working on a render engine which can visualise Kerr Black Holes with a volumetric accretion disk and astrophysical jet. So far, this is what we got;

As you can see, we have the following working;

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*Accurat numerical integration of the Kerr Metric to draw the image of the BH (i.e trace the rays in curve spacetime)

*Single Scatter model for Volumes (i.e the Volumetrics use  a, for this situation, accurat model. We did tests with Multiple Scattering (Monte Carlo) and the results look basically identical but take 100s of times longer

*Volume integration along the spacelike geodesic´s (i.e the Volume is sampled while the ray marches through the scene, meaning the effects of curved spacetime are represented in the image of the disk)

This is all fine but a few things are missing. Such as Redshift, Doppler Beaming, the whole Astrophysical jet, accurat colors for the disk and much more. However, i am not smart enough to tackel any of those entirly on my own. I am much more familiar with rendering the party and doing that side of the work.
However, one effect i may just be able to do is Light travel delay. As far as i understand it, because some rays take their sweet as time to really interact with anything the image of the disk would have noticable distortions. I.e parts of the disk should look fairly normal where as others appear to be "behind" in terms of time.
Now, fairly naive approach to this would be to just messure the length of the geodesic each time the Volume is sampled and use this length together with some scale factor and c to determain how long it would have taken the ray to get there.
But, i was told this is indeed a naive approach because it just kind of ignores the whole black hole chillin nearby. Which would obviously have effects due to Time dilation.
But that is the thing, i am not sure how to do this. Since i am a bit bad at explaining stuff i dont understand, in the following i want to breakdown how the rendering works so that there is no miscommunication. So that yall smart people can point me to the right path on how to impliment light travel delay.
How the rendering works
As is so often, here the rays are traced backwards. So rays (photons if you will) are emitted from the camera location and wander about, Pathtracing. The general way it works goes like this;

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*Compute stepsize, this depends on where the ray currently is. The closer it is the the EH, the smaller the step becomes. This happens in stages. Let d be the distance to the center, then the stepSize equals; d > 10 = 20, d < 10 = 2, d < 4 = exponential function. This function lowers the step size exponentially down to a threshold of 0.075. Though the step Size may get as small as 0.0001 to illiminate discontinuations on the Y-Axis.

*Once the step Size is computed, we check break conditions. So if the ray is outside the draw distance it gets shot, if it is at the Event Horizon it gets shot.

*Assuming the ray is still alive, we then check if the current ray position is inside the bounding volume of the disk. If so we perform the local volume integration. If not;

*using a 4th order integration scheme the ray is marched "forward" according to the equations of motion of the Kerr Metric. This happens in 3+1 space, so

*The 4D spacetime coordinates are converted to a vec3 to move the ray in cartisian space because thats how computers work. And well then it is back to 1.


here you can see how this looks in action. These are 6 Geodesics of the render from above moving around from the cameras origin.
Now, what i would like to be able to do is to look at a ray´s position at step 3 when the Volume is sampled, look at its past and say "with all effects taken into consideration, this particular ray´s light would have reached the camera in 5 seconds". And then use these 5 seconds or whatever to rotate the disk accordingly. So that each ray sees a slightly differnt rotation due to time dilation and the travel delay.
What this would visually mean, i think, is that if you for example move the camera backwards faster than light the disk should rotate backwards.
As such, the core question i have is how i can calculate this idk call it "local time" for a given ray position knowing its "past", which technically is the future because it is backwards traced but whatever.
I really hope this is understandable. Thank you for reading, any help, helps :D
 A: In order to include the effect of time delay, you will need to keep track of the global time "position" of each of the rays. It seems you are already calculating this in step 4, but then throwing it away in step 5. So just keep track of it.
Having kept track of the global time of the ray, in step 3 you integrate against the disk at the that position at that particular time.
In order to do this, you will need some sort of model for the dynamics of the disk. In the question you suggest just using a rigid rotation of the disk. As a first approximation this is okay, but not very realistic. In practice there will be differential rotation with the inner edge of the disk rotating faster then outer edge.
A: Since you are ray tracing in four dimensions, you already have the emission time of the light: it's one of the four coordinates of the intersection point of the null ray with the accretion disc.
You should not think of the travel time as something extra to calculate. Instead, you should imagine that you are ray tracing a static four-dimensional scene (because you are). If you place the camera at the correct position in 4D, and fill the world-tube of the accretion disc with some appropriately helical pattern, then you'll automatically get the right light-travel-time effects. There's nothing more that you need to do.
