Looking for more accurate energy-momentum transformations for photons

[jtbell]

I do not believe that Einstein would have used the word "photon" at all. A search of the paper you cite didn't find the word anywhere in the paper.

In fact, the word "photon" wasn't invented until long after Einstein's 1905 papers.

Photon: New light on an old name (ArXiv)

 

[pervect]

In fact, the word "photon" wasn't invented until long after Einstein's 1905 papers.

Photon: New light on an old name (ArXiv)

I suspected as much, but I thought I'd be more careful in my answer since I wasn't entirely clear on the details of the history of the term.

Nowadays, since we know what a photon is, we can write the exact momentum-energy relationship for a photon as E = |p|c , that the energy is equal to the magnitude of the momentum multiplied by c. We take the magnitude, because momentum is a vector.

 

[tade]

It's worth pointing out that this thread has segued from the question in the title (energy-momentum transformations for photons) to the transformations for light. This is a good thing.

To ensure no misunderstanding, the line I quoted comes from the paragraph immediately before he introduces the frequency transform.

The connecting formula between particle dynamics and wave models is the Planck-Einstein relation; it can be stated that the relation applies at all distances right?

 

[tade]

Here's one possible setup, involving "standard configuration" of a primed and unprimed frame:

In the primed frame, a spectrometer remains at rest at the origin while a light-source moving parallel to the $x^\prime$-axis [edit: in the negative $x^\prime$-direction, as per standard configuration] briefly sends a monochromatic light wave directly toward the spectrometer. The unprimed frame is the light-source's rest frame.

The goal is to find an expression for the wavelength (and ultimately the frequency) of the light as measured in the primed frame. Steps:

• Use time dilation to express the wave's primed period in terms of its unprimed wavelength.
• Express the (primed) distance the light travels during the primed period in terms of the unprimed wavelength.
• Express the (primed) radial component of the light-source's displacement during the primed period in terms of the unprimed wavelength. (This will involve the cosine of the primed angle between the light's velocity vector and the positive $x^\prime$-direction.)
• Sum the previous two results—that's your primed wavelength in terms of the unprimed wavelength and the aforementioned primed angle. You may also Lorentz-transform the primed angle to the equivalent corresponding [edited] unprimed angle (aberration).

The way @SiennaTheGr8 explained it is what I had in mind. There's a bit more complexity than I alluded to because of the necessity of transforming the angles, which I had forgotten - apologies.

thanks guys, so, I am supposed to select a pair of non-simultaneous events in the source's rest frame, which are simultaneous in the moving frame?

 

[tade]

Right, but Einstein's 1905 paper showed that amplitude (per wave solutios of Maxwell's equations) transforms the same way as frequency. Lest one think this implies a classical derivation of the frequency/energy relations, this is not the case. Classically, you could associated any amplitude to a chosen frequency. It is just that having specified them in one frame, they transform the same to a different frame.

hi, after using Ibix/Sienna's method, we get $f'=γ(1-βcosθ)f$, which is supposed to match the frequency/energy relations right?

 

[SiennaTheGr8]

hi, after using Ibix/Sienna's method, we get $f'=γ(1-βcosθ)f$, which is supposed to match the frequency/energy relations right?

Well, that is the frequency-transformation formula you obtain, but if you want to use it to show that light's energy transforms the same way then you need to invoke the Planck–Einstein relation for photons. It's "cheating" in a sense, and not entirely satisfying, but it's not "wrong," and it's not entirely ahistorical actually (Einstein's paper on the photoelectric effect was published a bit before his paper on special relativity, though it would take several years before the Planck–Einstein relation was widely accepted).

It's a bit trickier to show that a classical light wave's energy transforms in that way. You need to transform the appropriate electromagnetic energy density ("appropriate" means that $\mathbf{E} \perp \mathbf{B}$ and $E = B$), and then you need to multiply the result by the transformation of the appropriate volume (what "appropriate" means here is rather subtle, and I'm still grappling with what Einstein's doing there at the beginning of §8 of his SR paper).