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This is actually for a graduate course but it's a basic special relativity problem, i.e. undergraduate-level material, so I'm posting it here...
[tex]U^{\mu} = (1, 0, 0, 0)[/tex] (in the observer's rest frame)
[tex]V^{\mu} = (\gamma, \gamma \vec{v})[/tex]
[tex]P^{\mu} = (E, \vec{p})[/tex]
I got parts (a) and (b) easily enough by evaluating 4-vector products in the observer's rest frame,
(a) [tex]\gamma = -U^{\mu} V_{\mu} \gg 1[/tex]
(b) [tex]E = U^{\mu}P_{\mu}[/tex]
The problem is with part (c). We're supposed to express P' in terms of U, V, and P, but it doesn't seem to be possible without knowing the angle at which the scattered photon exits (or the angle at which the charged particle exits). Am I missing something, or is this actually impossible?
Homework Statement
Inverse Compton scattering describes the process whereby a photon scatters off a charged particle moving with a speed very nearly that of light. In this problem we analyze an inverse Compton scattering event "geometrically".
(a) An observer, moving with four-velocity U, observes a charged particle traveling with four-velocity V and rest mass m. Describe, in terms of U and V, the condition that the charged particle is moving with a speed very nearly that of light.
(b) The charged particle encounters a photon with four-momentum P. Express, in terms of the appropriate four-vectors, the energy of the photon incident on the charge particle as seen by the observer.
(c) Express, in terms of the appropriate four-vectors, the photon's four-momentum P' following the scattering event.
Homework Equations
[tex]U^{\mu} = (1, 0, 0, 0)[/tex] (in the observer's rest frame)
[tex]V^{\mu} = (\gamma, \gamma \vec{v})[/tex]
[tex]P^{\mu} = (E, \vec{p})[/tex]
The Attempt at a Solution
I got parts (a) and (b) easily enough by evaluating 4-vector products in the observer's rest frame,
(a) [tex]\gamma = -U^{\mu} V_{\mu} \gg 1[/tex]
(b) [tex]E = U^{\mu}P_{\mu}[/tex]
The problem is with part (c). We're supposed to express P' in terms of U, V, and P, but it doesn't seem to be possible without knowing the angle at which the scattered photon exits (or the angle at which the charged particle exits). Am I missing something, or is this actually impossible?