cpf@lasspvax.UUCP (Courtenay Footman) (12/17/85)
In article <166@bambi.UUCP> mike@bambi.UUCP (Michael Caplinger) writes: >Can anybody tell me why a charged tachyon moving through free space would >generate Cherenkov radiation? I thought I understood the mechanism by which >"ordinary" Cherenkov radiation was generated (say, that caused by fast >neutrons in water) but that involves atomic transitions in surrounding atoms >in the medium, no? 1) Yes, a charged tachyon would generate radiation. 2) Here, yes and no. In the normal Cherenkov effect, if there were no interactions of electro-magnetic radiation (light) with surrounding atoms, then the index of refraction would be one, light would travel at c, and there would be no radiation, since the particle would be traveling less than the speed of light in the medium. The interactions serve to lower that speed to less than c, so that you can have a particle traveling faster than that speed, which is all you need for Cherenkov radiation. The Cherenkov effect is a classical one; one does not need QM to describe it. Here is a long, heuristic description of Cherenkov radiation. Things in braces, like this [1], refer to parenthetical remarks that are given at the end of the article. To begin with, I will describe the origin of ordinary EM radiation, caused by the acceleration of a charged particle. I will attempt to describe the beautiful pictures in Perkins[2] (Oh, for a way of sending graphics). First, consider a stationary charged particle. Its field lines are equally distributed and all point towards it. Accelerate the particle for a small time del t (*Dt) over a small distance. Wait a while, t, and then look at the field lines again. Outside a radius c(t+*Dt), (from the original position), the field lines are unchanged. Inside the radius ct, the field lines are straight and point to where the particle is now. Inside the shell from ct to c(t+*Dt), things are complicated, because all E field lines must be continuous and cannot cross, and so, inside the shell, the E field lines are almost transverse to the shell, so that the field lines inside the shell can connect to the field lines outside the shell. Also, because there dE/dt is non-zero, there is a B field inside that shell. Inside and outside the shell, the E field is proportional to 1/r^2 (2 different r's); inside, E, and thus B, is proportional to 1/r^2*(r/l) is proportional to 1/r. (l is the distance the particle was accelerated over.) Thus we have a spherical shell, propagating outward at c, with transverse E and B fields, whose field strengths go like 1/r, and whose energy goes like E^2 = 1/r^2. This, then is EM radiation. Now then, lets see where the Cherenkov effect comes from. Start with the same stationary particle, and accelerate it to faster than c [3]. (In the direction opposite the particle's final velocity, there will be a normal EM pulse; this is caused by boundary conditions, and will be ignored.) If we look at the field distribution at time t, we will see a configuration that looks very much like a the shock cone of a sonic boom \ \ field lines point \ field lines point toward original toward current position . position. / / / Again, on the cone, like the shell, the field lines are complicated, changing, proportional to 1/d. They are EM radiation. But, you might object [4], everyone knows that Cherenkov radiation is mostly in the direction of particle motion, while this cone seems to be going backward. However, this is a snapshot of where the light is now; if you take another snapshot a bit later, the hole cone has moved to the right, (plus out a bit), so that the direction of the radiation is mostly forward; furthermore, the faster the particle, the more collimated the light is. This analysis does not describe everything -- it omits the back reaction on the particle. That can be done in this picture, but is messy, and is not needed for an understanding of what is going on. Thus, any charged particle that is going faster than the speed of propagation of the EM field will emit radiation; the reason it is going faster (tachyon or particle in a medium) does not matter. I hope my version of physics without equations or pictures, or 'How to wave your hands when your hands are not visible', has not succeeded in creating complete confusion. Notes: [1] like this [2] Perkins is the second book in the Berkeley Physics series. It is an almost miraculously good introductory text to electro-magnetic theory. The only reason that it is not used in all introductory courses is that it uses some very elementary vector differential calculus, and for some reason most professors only want to introduce elementary vector integral calculus at that point. [3] The reason I do this is to introduce sensible boundary conditions- you can't create an electric charge, and I don't want to have the tachyon come in from minus infinity. If you don't want stationary tachyons, make it a tachyon going around in a circle; if that disturbs you, put it in a QM bound state. It doesn't make too much difference. [4] A very bad rhetorical device; always means "I have a statement to make and don't know how else to introduce it" -- Courtenay Footman arpa: cpf@lnsvax.tn.cornell.edu Newman Lab. of Nuclear Studies usenet (finally this will work): Cornell University {decvax,ihnp4,cmcl2,vax135}!cornell!lnsvax!cpf