Light deviation in Schwarzschild-metric

[timo]

PROBLEM:

The deviation angle of a photon passing by a simple black hole as described by the Schwarzschild-metric is given by

[math] \delta = \frac{4M}{R} [/math] where R is the minimum r-coordinate of the trajectory´s points.

This equation is derived as an approximation and I doubt it´s validity for small R a bit for reasons given below.

 

 

STORY:

I´m currently working on a numeric simulation about black holes to visualize some effects of GR. Within this scope I decided to test my results for light deviation against the predicted values. I discovered a huge difference between my values and the prediction:

At distances R>100 (M=1 in my case, btw) the values look fine and the relative error <5% is absolutely acceptable. But for smaller R the relative error climbs up to >40%. What makes me especially surprised it that the relative error is a nice 1/R - type plot. Hence, the cause of this difference is almost certainly systematically.

 

My first guess of course was that the cause of this error is my integration routine or the fact that I can´t start the ray at infinite deistance for obvious reasons. But neither reducing the stepsize in my integration by a factor 0.01 nor increasing the original distance by a factor of 10 (1000->10000) made any noticeable difference.

Insertion: If anyone here has experience with Python: Do the timesteps I give to the "odeint"-routine have any impact on the result or was the reduction of the stepsize useless because the routine does not care about them when integrating?

 

While slowly running out of easily verificable options where the problem with my code lies I came up with the idea that maybe above approximation which is said to have been verified at a relative error <1% on the sun according to one of my books is completely wrong for small R.

 

My thoughts about that:

- Due to the radius of the sun these measurement have been made at R>10^5 so there is no contradition between these experiments and my numeric results.

- Above equation is derived by an expansion in 1/r which is cut off (approximation, as I allready said). It seems only reasonable the the higher powers uf 1/r cause a significant difference for small R.

- From pure intuition I miss a pi in the equation. I tested 2pi/R which in fact gave good results for small distances but -of course- completely failed for bigger ones.

 

 

QUESTION:

I wrote quite some text for a simple question: Does someone here have experience with GR and can either verify my assumption that 4M/R gives wrong results for small R (preferrably with a better formula + derivation) or convince me that 4M/R must be correct for all R ?

 

 

Well, I might be able to figure out the question myself within the next week but it´s late in the night and I can´t get some sleep so I can also post my question here as well. Maybe someone is interested in this or has a usefull answer for me.

 

EDIT: A simple test with [math] \delta = \frac{4}{R} + \frac{8}{R^{2}} + \frac{16}{R^{3}} + \frac{32}{R^{4}} [/math] gave great results and pushed the errors way below 1% except for extremely small R (close to the limit of my integration routine). This seems to back up my assumption. Yet, an internet resource containing a better approximation than 4M/R would be still interesting for me.

[jana]

Hi Atheist,

 

What you've called R is actually the impact parameter, not the point of closest approach, and is usually written as b. If b < 3^3/2M the photon is captured. Note that the approximation you begin with is valid only for M<<1.

[timo]

Do you have any sources on that? According to two of my books and my notes of the lecture on GR I heard R is the smallest r of the trajectory´s points and not the impact parameter.

I might try pluging the impact parameter in there as 4M/b and see what I get but since b>R the prediction values will be even worse (predicted values are too small in case I didn´t explicitely metion it before).

 

Validity for M<<1 is essentially the same as validity for R>>1, here, so your 2nd statement is just what I said. What I need (better: Could use) is a source that explicitely shows me I´m right or wrong - with an explanation why so!

 

Thx for the answer, anyways. Maybe you can tell me why the approx is only valid for M<<1 or what better approx you´d suggest ?

 

EDIT: No need to further think about it if you don´t know a good answer. As i already expected in my 1st post I found a sufficient answer myself

[jana]

Do you have any sources on that?

 

Any decent text on GR. I use "General Relativity" by Wald.

 

 

Validity for M<<1 is essentially the same as validity for R>>1

 

If the physical quantities you're after depend on M and R only through their ratio M/R, then yes, it doesn't matter.

 

 

why the approx is only valid for M<<1

 

The functional dependence on M of the analytic formula (which I don't know off hand) for deflection differs from that on R such that the approximations in terms of limits of M or of R (with other variables held fixed) will be inequivalent.

[[Tycho?]]

Swoosh

 

 

 

 

Thats the sound of this thread going over my head.