- #1

BiGyElLoWhAt

Gold Member

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I think this is the appropriate subforum.

I'm curious as to what approaches have been taken. I know this prediction isn't correct. I can think of at least a couple ways that I could go about this. They may or may not give the same prediction.

One approach would be to simply use kinematics, and force a rescale after every interval dt. I haven't tried it. This might be a computational thing.

What I mean is start with an initial velocity and position, use ##a = GM/r^2## and integrate. However, we need to maintain that the velocity is always c, so we would need to include a factor of something like ##\frac{c}{|c + \int adt|}##.

I think I'm going to end up with something like:

##\frac{c}{|c + \int adt|} (\int (adt) + v_0)## for the velocity function, which I could then integrate again to get a position function. I could write ##v_0## as ##c<cos(\theta),sin(\theta)>## and now I have a parameter to vary to get different paths.

Another approach I found on stack exchange is to use a test mass and take the limit as m-> 0.

I'm interested in thoughts on these as well as other possible ways you might try to tackle this.

Edit:

I think my rescale equation isn't quite right. That should work computationally, not analytically (assuming there is an analytical solution). The issue is that this isn't a rescale every dt, it's only a rescale at the end. I'm currently thinking about how I should put it in so that it also works analytically. Will post back when I think of it.

I'm curious as to what approaches have been taken. I know this prediction isn't correct. I can think of at least a couple ways that I could go about this. They may or may not give the same prediction.

One approach would be to simply use kinematics, and force a rescale after every interval dt. I haven't tried it. This might be a computational thing.

What I mean is start with an initial velocity and position, use ##a = GM/r^2## and integrate. However, we need to maintain that the velocity is always c, so we would need to include a factor of something like ##\frac{c}{|c + \int adt|}##.

I think I'm going to end up with something like:

##\frac{c}{|c + \int adt|} (\int (adt) + v_0)## for the velocity function, which I could then integrate again to get a position function. I could write ##v_0## as ##c<cos(\theta),sin(\theta)>## and now I have a parameter to vary to get different paths.

Another approach I found on stack exchange is to use a test mass and take the limit as m-> 0.

I'm interested in thoughts on these as well as other possible ways you might try to tackle this.

Edit:

I think my rescale equation isn't quite right. That should work computationally, not analytically (assuming there is an analytical solution). The issue is that this isn't a rescale every dt, it's only a rescale at the end. I'm currently thinking about how I should put it in so that it also works analytically. Will post back when I think of it.

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