Given a black hole of some size, say $10^8$ solar masses, how can the size of its sphere of influence of light be calculated?
To clarify, ultimately I'd like to be able to calculate the apparent angle of a black hole's lensing effects, as viewed from some distance.
(I'm no physicist, so I might be making incorrect assumptions about gravitational lensing. If this is the case, sorry!)
Answer
The calculation of light bending isn't hard if you make the assumption that you're well away from the event horizon, but it's probably a bit much for a non-physicist so I'll just quote the result. If you stay well away from the event horizon the approximate equation for the angle that the light is bent is:
$$ \Delta \phi = \frac{4GM}{c^2r} $$
where $M$ is the mass of your object and $r$ is the distance of closest approach. So for example for a light ray just grazing the Sun, mass $1.99 \times 10^{30}$kg at a distance of $6.96 \times 10^8$m the equation tells us the light is bent by about $8.5 \times 10^{-6}$ radians or just under 2 seconds of arc. This was the bending that Eddington measured in his famous (if now disputed) 1919 experiment to test general relativity.
You can use this to work out how much light is bent by your $10^8$ solar mass black hole. However note that there is no cut-off i.e. no distance at which light stops being bent. It just gets bent less and less as you get farther away. You would have to decide for yourself what angle you consider negligable and work out the corresponding distance.
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