Gravity Deflecting Light and Gravity Lensing

[haytil]

I'm interested in the deflection angles of light rays passing by extremely dense gravitational objects, specifically black holes. First, I'd like to find the formula for the deflection angle of an incoming light ray.

My googling (as well as a textbook from college, Hartle's "Gravity") states that the deflection angle = 4GM/(b * c^2) = 2 R/b

where b is the impact parameter of the undeflected light ray and R is the Schwarzschild radius of the object in question.

-However, in Hartle (p. 259), it states that this is true for b >> M.

-Furthermore, this gives a maximum deflection angle of 2 radians (when b = R, as a lesser value of b would mean the light ray enters the object's event horizon). This doesn't jive with my conceptual understanding of light rays being deflected by gravity (it seems as if maximum deflection should equal 2*pi. If the impact parameter is greater, the deflection is less. But if the impact parameter is less than R, than the light ray should spiral inwards - essentially, a "deflection" or more than 2*pi, as the light ray spirals inwards.) Otherwise, why would 2 radians "magically" be the critical angle, which seems a random value?

These two facts lead me to believe that the simple formula above is an approximation and is very wrong in the vicinity of a black hole. Thus, I wonder if anyone has the formula for the deflection angle of light where b ~ R.

Secondly, I'm also interested in the "lensing" effect a black hole can have with an object, particularly when the black hole lies directly between the object and the viewer. I wonder if there is a formula for the magnification an apparent image has, given:

R (Schwarzschild radius, and thus M of the black hole)
dO (distance between object and black hole)
dV (distance between black hole and viewer)
w (width of object being viewed, not sure if this is important).
d (distance between the

The simpler the formulae, the better, but accuracy is more important.

The application is for real-time calculation of optical effects for a game I'm proto-typing. If I need to, I can simplify the formula myself (for quicker processing), but would rather do so given the accurate original formula, rather than an approximation (as above).

Thank you.

 

[Ich]

http://arxiv.org/abs/gr-qc/0611086"
http://arxiv.org/abs/gr-qc/0102068"

 

[Entropia]

Hello,

Thank you for your interest in the deflection angles of light rays passing by extremely dense gravitational objects, specifically black holes. The formula you have mentioned, 4GM/(b * c^2) = 2 R/b, is known as the Einstein deflection formula and it is an approximation that is valid for small angles and large impact parameters (b >> M). As you have correctly pointed out, this formula is not accurate for larger angles and smaller impact parameters, which are more relevant for the study of black holes.

In the case of a black hole, the maximum deflection angle is indeed 2 radians, as you have mentioned. This is because the light ray can only be deflected by a maximum of 180 degrees before it crosses the black hole's event horizon. This maximum deflection angle is also known as the critical angle. However, for smaller impact parameters, the light ray will indeed spiral inwards, as you have mentioned. This is known as gravitational lensing and it can produce multiple images of the same object. The exact formula for the deflection angle in this case is more complex and it depends on the specific properties of the black hole, such as its mass and spin.

In terms of the "lensing" effect of a black hole, there is indeed a formula for the magnification of an apparent image. This formula takes into account the same properties as the deflection angle formula, such as the mass and distance of the black hole. However, it also depends on the size and distance of the object being viewed, as well as the wavelength of the light being observed. The formula is quite complex and it is known as the Schwarzschild lens equation. You can find more information about it in research papers or textbooks on black holes and gravitational lensing.

In terms of simplifying the formula for real-time calculation, I would recommend using the Einstein deflection formula as an approximation for small angles and large impact parameters. However, for more accurate results, you would need to use the more complex formulas mentioned above. I hope this helps answer your questions and good luck with your game prototype!