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I am looking into how light is being bent by objects of large masses with the intention of creating a numerical simulation of this (basically a ray tracer taking this phenomenon into account). I was thinking about taking a ray as a photon with a specific energy and treating it as a regular particle with $m = \frac{h}{\lambda c}$ and $p = \frac{h}{\lambda}$. Then taking the gravitational force as $F_g = G\frac{Mm}{r^2} = \frac{dp}{dt}$ would give me an equation to numerically integrate. A nice side effect of this approach would be that gravitational blue/red shift would be taken care of automatically as I would be altering the momentum of the photon, changing its wavelength at the same time.

However, reading around this topic here, on Wikipedia, and some other sources, I came across the fact that the deflection predicted by Newtonian gravity is half the deflection given by general relativity. So my main question is if the Newtonian model is even usable here, in particular:

  1. To avoid the issue of half deflection, could I arbitrarily double the force exerted on the photon in my calculations? That seems to be a reasonable thing to do to me (ignoring the fact it is probably physically wrong), but I am not sure if it would not break up somewhere unexpected.
  2. Would this model correctly predict the blue/red shift? Or would it somehow break down and result in nonsense (even after some constant multiplication)?

I came across some blackhole ray tracers and they all used Newtonian style physics similar to what I described above, but at the same time there were usually some weird parts (which may be weird because I don't understand this concept well enough) like multiplying the force by $sin(\textrm{angle between particle velocity and direction from the particle to the black hole})^2$ etc., so I didn't get much clarification regarding my above questions from them.

Pandicon
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1 Answers1

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In GR, the trajectory of a photon can be determined by doing calculations on the schwarzschild metric, here it is in natural units where $c = 1$

$$ d\tau^2 = (1 - \frac {R_s}{r}) dt^2 - \frac {1}{(1 - \frac {R_s}{r})}dr^2 - r^2 d\theta^2 $$ Where $R_s$is the schwarzschild radius. We will not be taking into account of the $\phi$ coordinate axis, because it makes the calculatins messier, you can simply do a coordinate transformation and $\theta$ and everything would be completely alright. For a massless particle $\tau = 0$, therefore we have the following

$$ (1 - \frac {R_s}{r})dt^2 = \frac {dr^2}{(1 - \frac {R_s} {r})} + r^2d\theta^2.$$

Instead of writing $(1 - \frac {R_s}{r})$ again and again, we will label it $F(r)$, on dividing both sides by $dt^2$ we get

$$ F(r) = \frac{\dot{r}^2}{F(r)} + r^2\dot\theta^2$$

There is a conserved quantity, angular momentum which can make things a lot easier

$$ L = r^2\dot\theta \frac{E}{F(r)}$$

Where E is the energy of the photon. Now you can substitute this value in the earlier equation and get two equations one for $\dot r$ and one for $ \dot \theta$. Then you can use integrating methods like the runge-kutta method to determine the trajectory of the photon. Remember we used $c=1$, so you need to use natural units or retain $c$ by doing some dimensional analysis. Regarding the redshift, once you know the momentum of the photon, you can use this formula to determine the wavelength $$\lambda = \frac{h}{L}$$ Where $h$ is the plank's constant.

Ronny
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