A Tumblr/blog by an astronomy PhD student, documenting her work, recording her life and explaining the wider craft. Because science is tasty!
Dougal’s Paradox.
CALLING ALL PHYSICISTS, COSMOLOGISTS AND GENERALLY SMART PEOPLE.
Okay. So I’m asking you guys for some help. Help to solve a problem I’ve devised and can’t come up with the correct answer on my own (hence why I named it after myself, go me). So here goes…
On the event horizon of a black hole you should expect electromagnetic radiation to be stuck there and not moving. Therefore you can conclude that its velocity is equal to zero. However, the relationship of v=fλ must equal zero. This means that either the waves frequency (f) or wavelength (λ) must equal zero. The interesting thing about this is the energy involved (calculated by E=hf, where h is Planck’s constant) has two values. If f=0 then E=0, if λ=0 then the wave has an infinite amount of energy (due to rearranging v=fλ to give v/λ=f and substituting to give E=h(v/ λ)). Both these answers are of course paradoxes.
So smartypants, what is going on here?
You’re confusing two frames of reference here I think, so getting the wrong conclusion when there isn’t an issue at all. You have two people, one with a flashing torch/flashlight that’s falling towards the black hole and is now nearing the event horizon, and his friend who’s watching him fall in. As torch-dude gets closer to the event horizon and passes it, he doesn’t slow down or feel any different, his flashing torch is still flashing at one flash a second or whatever. He, and the light from the torch, do not stop moving on the event horizon.
However, his friend watching would see something different. The light from the torch has to climb out of a huge gravitational potential well, and, like anything climbing a hill of sorts, it loses energy. The speed of light in a vacuum, as always, is c*, so using your equation E=h(c/λ), the wavelength will get longer and redder, i.e it will undergo redshift. If his friend sat around long enough, it would look like torch-dude was flashing his torch slower and slower, taking forever to reach the event horizon and that last torch flash as he crosses the event horizon would redshift away into the radio, never finishing. It would look like torch-dude is still outside the event horizon and has ‘stopped’ there, but he really hasn’t.
It’s best to think of it like the light is a spring being stretched from the event horizon to the observer as the spring’s being reeled in at speed c, tighter and tighter as time goes on, rather than individual ‘pellets’ of light (photons), because then you end up getting photons stuck on the event horizon and that’s not quite right. Any light emitted from the torch after the event horizon will just end up in the black hole.
*v(light) is defined as v=1/sqrt(εµ), where ε and µ are properties (permittivity and permeability) of the material the light is travelling in and have nothing to do with energy of the light. The specific value of c is from the ε and µ of a vacuum.
