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I couldn't find this question on the suggested "similar questions". If this has been asked before please direct me to answer. My question is "why isn't the wavelength of light,which is in the direction of motion, going at the speed of light Lorentz contracted to zero instead of its value?"

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I mean basically, it's because light is weird. Light is simply not intelligible as a classical thing which is moving and keeps an internal clock that is oscillating in time and has a wavelength by virtue of having some spatial extent. There is no classical thing that a photon easily corresponds to. Our best guess about what a photon "sees" involves roughly speaking the entire rest of the universe being "ahead" of it except for the event which emitted it, which is "behind" it, in some sort of one-dimensional timeless existence.

Quantum mechanics specifies that energies become frequencies and momentums become wavelengths, and the photon certainly has both of these, $E=p~c$, but we can change how these properties look together by simply moving relative to the photon and thus inducing a relativistic Doppler effect—so neither one is deeply intrinsic. Those features of wavelength and frequency correspond to something about how we interact with the photon, not to anything intrinsic about the photon itself.

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    $\begingroup$ The question is a classical question with a classical answer. The quantum stuff is totally irrelevant. $\endgroup$ – Ben Crowell Mar 20 at 18:32
  • $\begingroup$ @BenCrowell you are correct in that you can view this purely as a question about what classical plane waves travelling at $c$ look like in relativity. Nevertheless I disagree in pedagogical practice: assuming that they're an undergraduate they are learning about photons at around the same time that they are learning about relativity, and one trend I have noticed in folks learning about photons is that the very language we use to talk about them suggests that these properties like energy and momentum are somehow intrinsic rather than relational. $\endgroup$ – CR Drost Mar 20 at 18:58
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The Lorentz contraction factor $\gamma$ describes the length of an object in relation to the object's length in its own rest frame. Light doesn't have a rest frame, so the same analysis doesn't apply.

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The wavelength of light in the lights own rest frame (if there were such a frame!) is infinite. What do you get if you divide infinity with infinity? It's not well-defined and you can say that the wave length you see is the infinitely contracted infinite wavelength.

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The short answer:

  • Length Contraction involves
    the apparent spatial separation of two parallel [timelike] worldlines, marking the ends of a stick.
    The contraction depends only on the relative-speed (but not direction) along the x-axis.

  • The Doppler Effect involves
    the apparent spatial separation of two parallel lightlike lines, marking the wavelength... the successive wavefronts of a light wave.
    The scaled wavelength depends on the relative-velocity,
    where the approaching case (where the wavelength appears decreased)
    is different from the receding case (where the wavelength appears increased) .

The long answer:

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