Conceptual help with relativistic transformations for the energy of light waves

[Square47]

I'm in a math class reading Einstein's original paper "On the Electrodynamics of Moving Bodies," from 1905.

I'm stuck in Section 8, "transformation of the energy of light rays." We're basically trying to show that that Placnk's constant is Lorentz invariant- if anyone has an easy way of deriving this, that would help too.

So far, we have the equation for the relativistic doppler effect: When a moving system k moves parallel to a wave of light relative to a stationary system K, the equation for frequency is

nu'/nu = squrt[(1-v/c)/(1+vc)]

where nu is the frequency of light from stationary system K and nu' is the frequency of light from moving system k, and v is the speed of the moving system k relative to the stationary system K. When

I take this equation to mean that the frequency increases when k moves with -v (taking the velocity of light in the positive direction) and decreases when v is positive. The first case is blue shift, the second red.

This is the first time that the direction of the velocity of system k has mattered for relativistic effects. This makes sense for the doppler effect, because one direction is red shift and the other is blue shift. But I don't see why the direction would matter for energy transformations.

More specifically, the energy in a volume containing light is proportional to the light's amplitude times the volume. If E is energy, A is amplitude, and S is volume (and variable's with an apostrophe are the measured magnitudes in the moving k system), then E'/E = A'^2(S')/(A^2(S)).

Einstein finds that A'^2/A^2 = (1-v/c)/(1+v/c). It appears therefore that, like the frequency, the amplitude increases when system k moves in the opposite direction of light. Why is that?

What's worse, the volume enclosing the light undergoes Lorentz contraction. Einstein assumes that the volume is a sphere moving along with the light as viewed in system K, and applies the Lorentz transformations to find the volume as viewed from system k. I follow his derivation, and it leads to a ratio of the volumes S'/S = sqrt ((1+v/c)/(1-v/c)), the reciprocal of the frequency transformation. Shouldn't this mean that the volume expands when k moves in the same direction as the light, and contracts when k moves in the opposite direction of the light (relative to K)? But I thought Lorentz contraction doesn't depend on direction, only on the velocity.

If anyone has an easy way to think about this, that would be immensely helpful. I hope the equations are readable.

 

[mark726]

Thank you for sharing your thoughts and questions regarding Einstein's paper "On the Electrodynamics of Moving Bodies." I can understand how daunting it can be to work through complex equations and theories, but I assure you that with persistence and a clear understanding of the concepts, you will be able to tackle Section 8 and beyond.

Firstly, let's address your question on the direction of the velocity in the relativistic doppler effect. You are correct in stating that the direction of the velocity does matter for this effect, as it determines whether the frequency of light will be blue shifted or red shifted. This is because the velocity of the system k relative to the stationary system K affects the perceived wavelength of the light waves. When the velocity is in the same direction as the light, the perceived wavelength decreases, resulting in a higher frequency (blue shift). On the other hand, when the velocity is in the opposite direction of the light, the perceived wavelength increases, leading to a lower frequency (red shift). Therefore, the direction of the velocity is an important factor in determining the frequency of light in the relativistic doppler effect.

Moving on to the transformation of energy of light rays, you correctly pointed out that the amplitude of the light waves also changes when viewed from the moving system k. This is because the amplitude is directly related to the energy of the light waves. As the frequency and amplitude are interrelated, with the frequency increasing with the decrease in wavelength, the amplitude also increases. This is why the amplitude is affected by the direction of the velocity, as the frequency is.

Regarding the Lorentz contraction of the volume enclosing the light, it is important to note that this contraction is a result of the Lorentz transformations applied to the spatial coordinates. The contraction is not dependent on the direction of the velocity, but rather on the relative velocity between the two systems. This is why the ratio of the volumes S'/S is the reciprocal of the frequency transformation, as the frequency and volume are inversely related.

I understand that these concepts may be difficult to grasp at first, but I encourage you to continue working through the equations and seeking help when needed. Perhaps discussing with your classmates or seeking guidance from your professor can also aid in your understanding. Rest assured that with persistence and a clear understanding of the concepts, you will be able to make sense of Section 8 and beyond. Keep up the good work and don't hesitate to ask