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giokara
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I have to solve the following problem and I am also given the solution. I do not find the same answer as given and I do not understand why.
Consider a centrifuge with radius r, spinning at a constant angular velocity ω. Two atoms E and A on the edge of the centrifuge are situated in the rotation plane an form a angle α. Atom E emits light with wavelength λ(E). A photon is absorbed by atom A (in a new position A'). Is there a Doppler shift? If yes, give the functional relationship in α,r and ω. Calculate therefore the quantity λ(A)/λ(E) in the lab system, given E = [itex]u^{\mu}[/itex][itex]p_{\mu}[/itex] with E the energy of the photon, p the four-momentum of the photon and u the 4-velocity of the atom.
E = [itex]u^{\mu}[/itex][itex]p_{\mu}[/itex]
As the receiving atom A' sees the emitting atom moving with a certain velocity, there will be a Doppler shift. The angle between E and A' in the lab system is found by demanding that the photon propagates at the speed of light. When all the angles are known, it is easy to find the four-velocity of A' and E in the lab system. By choosing an appropriate coordinate system for the lab system, the classical Lorentztransformation can be used to transform the 4-velocity of E to the instantaneous rest frame in A'. Finally, the Doppler shift can be found in the rest frame of A'.
The given solution is that there will be no Doppler shift. Why is that? Is it because that in the problem statement, the Doppler shift has to be determined in the lab system?
Many thanks in advance!
Homework Statement
Consider a centrifuge with radius r, spinning at a constant angular velocity ω. Two atoms E and A on the edge of the centrifuge are situated in the rotation plane an form a angle α. Atom E emits light with wavelength λ(E). A photon is absorbed by atom A (in a new position A'). Is there a Doppler shift? If yes, give the functional relationship in α,r and ω. Calculate therefore the quantity λ(A)/λ(E) in the lab system, given E = [itex]u^{\mu}[/itex][itex]p_{\mu}[/itex] with E the energy of the photon, p the four-momentum of the photon and u the 4-velocity of the atom.
Homework Equations
E = [itex]u^{\mu}[/itex][itex]p_{\mu}[/itex]
The Attempt at a Solution
As the receiving atom A' sees the emitting atom moving with a certain velocity, there will be a Doppler shift. The angle between E and A' in the lab system is found by demanding that the photon propagates at the speed of light. When all the angles are known, it is easy to find the four-velocity of A' and E in the lab system. By choosing an appropriate coordinate system for the lab system, the classical Lorentztransformation can be used to transform the 4-velocity of E to the instantaneous rest frame in A'. Finally, the Doppler shift can be found in the rest frame of A'.
The given solution is that there will be no Doppler shift. Why is that? Is it because that in the problem statement, the Doppler shift has to be determined in the lab system?
Many thanks in advance!