
#20
Nov2009, 02:54 PM

P: 24

Could anyone expand on this please? I would appreciate the help.
I have the the effective potential in the schwarzschild metric as (L being angular momentum) [tex]V_{eff} = ( 1  \frac{r_{s}}{r})(mc^{2} + \frac{L^{2}}{mr^{2}})[/tex] Would this be enough information to solve for r and [tex]\Phi[/tex] so that I could plot the trajectories. Also could I use the same equation for plotting the course of an observer descending into the black hole? 



#21
Nov2009, 03:09 PM

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P: 5,500

Here is some python code I wrote that does pretty much what you're talking about. It's meant to be short and easy to understand, so it uses a pretty crude method of doing the numerical integration. If you want to do accurate numerical calculations of geodesics, you'd want to substitute a better integration method. There are various generalpurpose subroutines out there, e.g., in the book Numerical Recipes in C. What my code does is to calculate the deflection of a light ray that grazes the sun. Actually, it calculates half he deflection for a ray that grazes the sun, with the mass of the sun scaled up by a factor of 1000 in order to keep the result from being overwhelmed by rounding errors in my elcheapo integration method.




#22
Nov2009, 03:15 PM

P: 24

I actually have the numerical recipes book next to me although I may avoid solving the elliptic integral I mentioned earlier, and just use the approximation for light deflection, i.e. 4GM/bc[tex]^{2}[/tex], to get the einstein ring effect, and focus on the trajectory of the observer descending into the black hole.




#23
Nov2009, 04:44 PM

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#24
Nov2109, 04:16 AM

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P: 6,038

[tex]\frac{d}{d\lambda} \left[ \left( \frac{dr}{d \lambda} \right)^2 \right] ?[/tex] 



#25
Nov2109, 02:49 PM

P: 24

[tex]\left( \frac{dr}{d \lambda} \right)^2 &= E^2  V^2(r) \right)[/tex] and [tex]V^2(r) = \left(1  \frac{2M}{r} \right)\frac{L^2}{r^2}[/tex] I have [tex] \frac{d^2r}{d\lambda^2} = \frac{1}{2}\frac{d}{dr}V^2(r)[/tex] 



#26
Nov2209, 03:01 PM

P: 24

[tex]\frac{d}{dr}V^2(r) = \frac{2L^2M}{r^4}  \frac{2L^2(\left 1  \frac{2M}{r} \right)}{r^3}[/tex] 



#27
Nov2309, 07:16 AM

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After simplifying the right side of




#28
Nov2309, 11:47 AM

P: 24

That would be
[tex]\frac{d^2r}{d\lambda^2} =  \frac{L^2(3Mr)}{r^4}[/tex] Which gives me the closest radius for a stable orbit of a photon as 3M 



#29
Nov2309, 08:52 PM

P: 24





#30
Nov2309, 09:07 PM

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P: 6,038

We now have a system of three firstorder coupled differential equations that determine the worldlines for "photons". In order to solve these equations, we need three initial conditions. Imagine firing a photon from a laser. One possibility for the three initial conditions: the [itex]\left( r , \phi \right)[/itex] position from which the photon is fired and the direction in which the photon is fired. These initial conditions determine [itex]L[/itex]. Alternatively, [itex]L[/itex] could be used as one of the initial conditions.
Much more on all of this later. 



#31
Nov2409, 08:44 AM

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P: 6,038

[tex] \begin{equation*} \begin{split} \frac{d \phi}{d \lambda} &= \frac{L}{r^2} \\ \frac{dr}{d\lambda} &= p \\ \frac{dp}{d \lambda} &= \frac{L^2(r  3M)}{r^4}.\\ \end{split} \end{equation*} [/tex] Assuming that all required values are given, write a few lines of (pseudo)code that uses the simplest, most intuitive method (Euler's method) to solve these equations. 



#32
Nov2609, 06:37 PM

P: 24

Having a bit of trouble with this but I have p as
p = [tex]\frac{L^2(2Mr)}{2r^3}[/tex] 



#33
Mar910, 09:14 AM

P: 24

wow it's been a while since I looked at this project.
Thought i'd come back to it :) A few questions. Since I lack a good maths background, I would appreciate some advice on how I would implement these 3 equations into a numerical solver like Euler's or RungeKutta to get back some results that I could plot. e.g. Do I predefine L?, how do I know which direction the photon is travelling? currently trying to use this : http://www.ee.ucl.ac.uk/~mflanaga/java/RungeKutta.html 


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