Transforming Energy and Momentum to a New Frame - from which it is easy to show that Similarly we can show that These are the Lorentz transformations

Transforming Energy and Momentum to a New Frame We have shown Notice we can write this last equation in the form . That is to say, depends only on the rest mass of the particle and the speed of light. It does not depend on the velocity of the particle, so it must be the same—for a particular particle —in all inertial frames. This is reminiscent of the invariance of , the interval squared between two events, under the Lorentz transformations. One might guess from this that the laws governing the transformation from E , p in one Lorentz frame to E ′, p ′ in another are similar to those for x , t . We can actually derive the laws for E , p to check this out. As usual, we consider all velocities to be parallel to the x -axis. We take the frame S' to be moving in the x -direction at speed v relative to S . Consider a particle of mass m 0 (rest mass) moving at u ′ in the x ′ direction in frame S' , and hence at u along x in S , where . The energy and momentum in S' are and in S :

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Thus giving

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Unformatted text preview: from which it is easy to show that Similarly, we can show that These are the Lorentz transformations for energy and momentum of a particle—it is easy to check that . Photon Energies in Different Frames For a zero rest mass particle, such as a photon, E = cp , E 2 – c 2 p 2 = 0 in all frames. Thus . Since E = cp , E ′ = cp ′ we also have . Notice that the ratios of photon energies in the two frames coincides with the ratio of photon frequencies found in the Doppler shift. As we shall see when we cover quantum mechanics, the photon energy is proportional to the frequency, so these two must of course transform in identical fashion. But it’s interesting to see it come about this way. Needless to say, relativity gives us no clue on what the constant of proportionality (Planck’s constant) is: it must be measured experimentally. But the same constant plays a role in all quantum phenomena, not just those concerned with photons....
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