How does relativity show that energy is proportional to frequency?

[lightarrow]

In the book: "Introduction to Special Relativity" by Robert Resnick, Appendix C3: Gravitational red shift (I have the italian version), talking about photons, he says:

"...Actually, it's not necessary to use the Quantistic Theory. We can show from relativity itself that E is proportional to nu, since from relativistic transforms of energy and momentum, it comes that the energy of an electromagnetic pulse varies with the same factor of which varies its frequency when it's observed in a different frame of reference (see question 3)".

Is someone able to show me this? I am very interested in.

 

[Meir Achuz]

E and nu are each the zeroth component of 4-vectors.
But they are different, unrelated 4-vectors, unless QM is introduced.
E and nu transform in the same way, but there is no reason in SR to assume they are proportional. RR is just making an illogical extension.

 

[lightarrow]

So, what do you think he wanted to mean, that could easily be solved from a first year undergraduate student?

 

[pmb_phy]

In the book: "Introduction to Special Relativity" by Robert Resnick, Appendix C3: Gravitational red shift (I have the italian version), talking about photons, he says:

"...Actually, it's not necessary to use the Quantistic Theory. We can show from relativity itself that E is proportional to nu, since from relativistic transforms of energy and momentum, it comes that the energy of an electromagnetic pulse varies with the same factor of which varies its frequency when it's observed in a different frame of reference (see question 3)".

Is someone able to show me this? I am very interested in.

I don't buy it. The constant of proportionality, h = Planck's constant (all of which means that E is proportional to nu) is a quantum concept.

So, what do you think he wanted to mean, that could easily be solved from a first year undergraduate student?

Seems to me that he just plain screwed up.

Best wishes

Pete

 

[Meir Achuz]

Could there be a slight mistranslation from English to Italian to English?
Hello...Ciao... goodbye.

 

[lightarrow]

Could there be a slight mistranslation from English to Italian to English?
Hello...Ciao... goodbye.

I don't know from the original english version to the italian version, but surely not in my (even if not perfect) translation from italian into english. I asked in the thread because I couldn't believe it too: it seemed a Nobel Prize discovery to me!
I wish I could read the original english version of that book.

 

[selfAdjoint]

I don't buy it. The constant of proportionality, h = Planck's constant (all of which means that E is proportional to nu) is a quantum concept.
Seems to me that he just plain screwed up.

Best wishes

Pete

The value of the constant of proportionality, and the fact that it comes in minimal chunks, belongs to quantum mechanics. It is not obvious to me that the proportionality itself needs to. And so, without the peculiarities of h, the discovery seems interesting but not Earth'shaking.

 

[lightarrow]

The value of the constant of proportionality, and the fact that it comes in minimal chunks, belongs to quantum mechanics. It is not obvious to me that the proportionality itself needs to. And so, without the peculiarities of h, the discovery seems interesting but not Earth'shaking.

Ok, but, being able to establish that proportionality, it would be possible to determine the proportionality constant in a different way than from quantum mechanics, isnt'it?

 

[Ich]

In the original paper, you find:

It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.

So a light complex has energy E=hf, with h being a constant that is proportional to "the amount of light in the light complex".
But you need another hypothesis outside SR: that the "amount of light" comes in quanta - photons, with a corresponding constant value of h. Maybe it's not a coincidence that Einstein postulated this hypothesis in the same year.

 

[lightarrow]

...So a light complex has energy E=hf, with h being a constant that is proportional to "the amount of light in the light complex".
But you need another hypothesis outside SR: that the "amount of light" comes in quanta - photons, with a corresponding constant value of h. Maybe it's not a coincidence that Einstein postulated this hypothesis in the same year.

I'm sorry, what is meant here as "light complex"?

 

[Ich]

A certain "amount of light". Could be a photon, a light pulse of finite duration, or a defined region of a planar wave.

 

[lightarrow]

A certain "amount of light". Could be a photon, a light pulse of finite duration, or a defined region of a planar wave.

So, from the phrase: "It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law" it seems that it's possible to compute the energy of "a light complex" in another ref. frame in a different way than from doppler effect. Is this possible?

 

[actionintegral]

I am thinking about trying this sometime. I would calculate the energy of an harmonic oscillator in the rest frame, and then switching to another frame relativistically.

(But I am secretly hoping someone else does this first...)

 

[Meir Achuz]

So, from the phrase: "It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law" it seems that it's possible to compute the energy of "a light complex" in another ref. frame in a different way than from doppler effect. Is this possible?

The relativistic Doppler effect (derived in many textbooks) just follows from the Lorentz transformation of the four vector (nu;k) of an EM wave.

 

[actionintegral]

If I am looking at an harmonic oscillator at rest with me, it takes a time T to complete a period. A moving guy will see a longer period than I will. He will see a lower frequency than I will. So a moving harmonic oscillator has a lower energy than a rest one.

 

[lightarrow]

The relativistic Doppler effect (derived in many textbooks) just follows from the Lorentz transformation of the four vector (nu;k) of an EM wave.

It is exactly for this reason that I wonder how the energy of an EM wave in different ref. frames can be determined in a way different from this.

What I mean is that, from what others have wrote, including R. Resnick, it WOULD SEEM that:

1. We know how an EM wave's frequency trasforms from one ref. frame to another (relativistic doppler effect).
2. We know how an EM wave's energy trasforms from one ref. frame to another (How?!)
3. They transforms in the same way --> Energy is proportional to frequency.

I don't think this is possible at all, this is my concern.

 

[Ich]

2. We know how an EM wave's energy trasforms from one ref. frame to another (How?!)

Just read Einstein's paper. When you transform the electromagnetic field of a wave in some finite region, you find that the Amplitude increases with speed. The region itself becomes smaller (but contains the same number of wave fronts), and A²*Size (of the region) - the energy of the light complex - increases in the same way as the frequency does. Remarkable.

 

[lightarrow]

Just read Einstein's paper. When you transform the electromagnetic field of a wave in some finite region, you find that the Amplitude increases with speed. The region itself becomes smaller (but contains the same number of wave fronts), and A²*Size (of the region) - the energy of the light complex - increases in the same way as the frequency does. Remarkable.

That's very interesting, Ich. Do you know where can I find that paper ?

 

[Ich]

http://www.fourmilab.ch/etexts/einstein/specrel/www/"
§§ 7,8.

 

[lightarrow]

Thank you Ich.

 

[Meir Achuz]

1. We know how an EM wave's frequency trasforms from one ref. frame to another (relativistic doppler effect).
2. We know how an EM wave's energy trasforms from one ref. frame to another (How?!)
3. They transforms in the same way --> Energy is proportional to frequency.

3 does not follow from 1 and 2.