Gravity effect on observed speed of light

[Dale]

What is his estimation of the distance between the start line and the finish line?

The problem is that this question is ambiguous in curved spacetime. There are many different ways you could determine the distance, and in GR they can lead to different results. For example, you could simply subtract the coordinate positions, but since the coordinate positions are arbitrary so is this procedure. You could do a radar ranging experiment or you could use a tape measure, but those only work in a static spacetime. You could connect them by a spacelike geodesic and determine the spacetime interval, but there may be more than one geodesic. Etc.

In any case, you do not need to answer this question to answer the OP. It is clear that the pulse through the gravitational field will arrive later. If you prefer to attribute that to a changed (radar) distance or a changed (coordinate) speed of light is up to you. My preference is the former.

 

[atyy]

Again, seems inevitable. Red light has different energy than blue, per photon. If turnaround twin is sending pulses of constant energy from his/her point of view, inertial twin will receive red pulses with less energy and blue pulses with more energy. How could it be otherwise?

To make this better defined, assume the emitted pulses are directed, not spherical. Then, the turnaround twin changes momentum and KE a tiny bit after each pulse. Further, note that the angle of receiver needed to capture a complete directed pulse will change from the red shift case to the blue shift case. Assume receiver is big enough to capture whole pulse in both cases (relativistic beaming effects eliminated). Then, clearly the receiver will, as I said above, receive red pulses of lower total energy compared to the blue pulses. Small momentum and KE changes to the emitting, turnaround twin, preserve conservation of energy and momentum.

How is the equivalence of energy and frequency justified in classical physics?

 

[harrylin]

How is the equivalence of energy and frequency justified in classical physics?

Here we are discussing special relativity, according to which:

f'/f = E'/E

"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."

- http://www.fourmilab.ch/etexts/einstein/specrel/www/ section 8 (Einstein, 1905).

 

[atyy]

Here we are discussing special relativity, according to which:

f'/f = E'/E

"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."

- http://www.fourmilab.ch/etexts/einstein/specrel/www/ section 8 (Einstein, 1905).

But for anyone observer, there is no change in energy.

 

[PAllen]

But for anyone observer, there is no change in energy.

For any inertial observer there is no change in energy after the light is emitted. But two different observers disagree on energy of emitted light. Emitter thinks they emitted white light, total energy E, receding observer thinks they emitted red light of energy less than E.

On the other hand, for any observer whose state of motion is changing, they see a change in the energy of light they are passing through, as their state of motion changes.

I am really confused - I don't think any of this is controversial. If we just replace light with bullets, nobody disagrees; light is not fundamentally different. And for bullets you can certainly make an analogy between an accelerated observer seeing bullets lose KE, and bullets fired up gravity well lose KE. Specifically imagine a long rocket accelerating, bullets fired from its back to front. Someone at the front of the rocket will find tham less energetic than when they were fired at the bottom. You can explain this as accelerating toward the bullets or a fictitious gravity field. Now place same rocket on the surface of a planet. Bullets fired from the bottem will be less energetic at the top. You can say the lost energy going up a gravity well or that the rocket is accelerating upwards (to maintain static position in gravity well). Principle of equivalence says the situations are equivalent.

 

[atyy]

For any inertial observer there is no change in energy after the light is emitted. But two different observers disagree on energy of emitted light. Emitter thinks they emitted white light, total energy E, receding observer thinks they emitted red light of energy less than E.

On the other hand, for any observer whose state of motion is changing, they see a change in the energy of light they are passing through, as their state of motion changes.

I am really confused - I don't think any of this is controversial. If we just replace light with bullets, nobody disagrees; light is not fundamentally different. And for bullets you can certainly make an analogy between an accelerated observer seeing bullets lose KE, and bullets fired up gravity well lose KE. Specifically imagine a long rocket accelerating, bullets fired from its back to front. Someone at the front of the rocket will find tham less energetic than when they were fired at the bottom. You can explain this as accelerating toward the bullets or a fictitious gravity field. Now place same rocket on the surface of a planet. Bullets fired from the bottem will be less energetic at the top. You can say the lost energy going up a gravity well or that the rocket is accelerating upwards (to maintain static position in gravity well). Principle of equivalence says the situations are equivalent.

No, I am confused. What's amazing to me is that the energy argument is not an equivalence principle argument. To me it is an argument from conservation of energy and semiclassical "old" quantum physics argument. From QM we have E=hf. From classical gravity, we have dE=g.dh, which implies df. No relativity even (all in one Galilean frame)! So somehow classical gravity knows about QM? (Yes, but how?)

The point of view from GR is just red shifts are due to light going along geodesics, and the local acceleration of the observer, and works even in cases where there is no conserved energy.

 

[PAllen]

No, I am confused. What's amazing to me is that the energy argument is not an equivalence principle argument. To me it is an argument from conservation of energy and semiclassical "old" quantum physics argument. From QM we have E=hf. From classical gravity, we have dE=g.dh, which implies df. No relativity even (all in one Galilean frame)! Some somehow classical gravity knows about QM? (Yes, but how?)

The point of view from GR is just red shifts are due to light going along geodesics, and the local acceleration of the observer, and works even in cases where there is no conserved energy.

Harrylin's link earlier show's how Einstein derived that different frames (similarly changing frames) see light energy and frequency proportionally changed, purely from SR and Maxwell. To be successful, QED had to incorporated SR + Maxwell in the classical limit; so it did, as a result of which E=hf can be said to follow.

My meta point is that there were features 'hidden' in classical theories that became clearer in the carry over to quanum theories. The classical theories didn't 'know about' quantum theories, instead the quantum theory clarified the classical theory.

An exampel is radiation in SR + Maxwell can be a very subtle issue, as radiation is non-local in this framework. In QED many of these tricky cases are very straightforward because radiation in QED is locally defined.

 

[harrylin]

No, I am confused. What's amazing to me is that the energy argument is not an equivalence principle argument. To me it is an argument from conservation of energy and semiclassical "old" quantum physics argument. From QM we have E=hf. From classical gravity, we have dE=g.dh, which implies df. No relativity even (all in one Galilean frame)! Some somehow classical gravity knows about QM?
[..]

E/E0=f/f0 => E ~ f.
Thus I would say, QM knows SR!

Edit: Oops I was delayed in replying, meanwhile PAllen said it all.

 

[atyy]

But the classical theories are complete in themselves. If we want to bring in QM, then we could just as well say QED is a low energy approximation to a non-relativistic theory.

Anyway, going back to Harrylin's point that purely clasically E and f transform the same way, does that necessarily mean that E~f, purely classically?

 

[PAllen]

But the classical theories are complete in themselves. If we want to bring in QM, then we could just as well say QED is a low energy approximation to a non-relativistic theory.

Anyway, going back to Harrylin's point that purely clasically E and f transform the same way, does that necessarily mean that E~f, purely classically?

It seems to. Assume E=g(f) for some arbitrary g, and that by SR, E0=g(f0). We also have E/E0 = f/ f0 = expression of v and c = constant relative to E and f. Call it k. Then E0=kE, f0=kf, then:

kE=g(k f)

k g(f) = g (k f)

This implies g'(f) = g'(kf) for all f (g' as derivative). Then g'(f)=g'(f/k)=g'(f/k^2)... If g continuous, the g'(f) = g'(0). Thus g' constant, thus g(f) = c f. QED (not quantum electrodynamics).

 

[atyy]

It seems to. Assume E=g(f) for some arbitrary g, and that by SR, E0=g(f0). We also have E/E0 = f/ f0 = expression of v and c = constant relative to E and f. Call it k. Then E0=kE, f0=kf, then:

kE=g(k f)

k g(f) = g (k f)

This implies g'(f) = g'(kf) for all f (g' as derivative). Then g'(f)=g'(f/k)=g'(f/k^2)... If g continuous, the g'(f) = g'(0). Thus g' constant, thus g(f) = c f. QED (not quantum electrodynamics).

Hmmm, very interesting. A universal constant. By dimensional analysis, the constant cannot be formed from G and c, so SR implies a new constant of nature?

 

[PAllen]

Hmmm, very interesting. A universal constant. By dimensional analysis, the constant cannot be formed from G and c, so SR implies a new constant of nature?

Maybe this doesn't really say quite that much. I think all it says is that fixing a volume of spacetime, holding field amplitudes constant, the EM energy content of waves of contained waves is proportional to frequency. It doesn't say there is some smallest unit energy. To make the factor more fundamental, you would have to state is (energy * time)/( field amplitude), which would be a different sort of animal than plank's constant.

On further thought, I wouldn't be suprised if this result follows in a fairly direct way from Maxwell's equations, though I am not able to do this myself. An intuitive argument is simply that a wave of given amplitude has given energy per wave front area. Then, it would follow immediately that the shorter the wavelength, the more energy per unit volume, thus E~f.

 

[atyy]

The surprise to me is not that this implies a quantum of energy. The surprise is that it seems to imply a relationship between energy and frequency, purely classically. I had always thought that energy was related to amplitude in classical terms (eg. the formula for the Poynting vector says nothing about frequency), and that E~f is a purely quantum mechanical relation. So if E~f purely classically, and the constant of proportoinality is a universal constant, then by dimensional analysis, it seems to predict a new universal constant with the same units as Planck's constant - so classical physics would seem to know about quantum physics. It's because of this implication that I'm skeptical that classical physics implies E~f, though I cannot find any mistake in your reasoning.

 

[harrylin]

The surprise to me is not that this implies a quantum of energy. The surprise is that it seems to imply a relationship between energy and frequency, purely classically. I had always thought that energy was related to amplitude in classical terms (eg. the formula for the Poynting vector says nothing about frequency), and that E~f is a purely quantum mechanical relation. So if E~f purely classically, and the constant of proportoinality is a universal constant, then by dimensional analysis, it seems to predict a new universal constant with the same units as Planck's constant - so classical physics would seem to know about quantum physics. It's because of this implication that I'm skeptical that classical physics implies E~f, though I cannot find any mistake in your reasoning.

Perhaps you overlooked the number of photons? I think that the energy is of a light wave is not hf but nhf. There's also an old thread about this, here:
https://www.physicsforums.com/archive/index.php/t-63023.html

Harald

 

[PAllen]

I went back and reviewed derivations of energy density and flux of plane waves, and also read the prior sections of Einstein's 1905 paper. I believe all the mysteries can be resolved.

1) The mistake in my argument that E~f could be derived classically is the very assumption that E=g(f) for some unknown g. Classically, E is simply independent of f, and depends only on amplitude.

2) The classical derivation of the change in E for a traveling wave for change in motion of an observer involves a change in amplitude. Since the amplitude, energy, and frequency change in tandem, then when looked at from a quantum point of view, number of photons is preserved, each having different energy for different observers. (Same number of lower energy photons is represented classically as lower amplitude).

3) I believe all my prior statements about the effect of motion, acceleration, and gravity on the the energy of light, and the analogies with bullets, are essentially correct, if not always precisely worded. I was never setting out to reason strictly classically. Obviously, any statements about classical justification for E~f are incorrect.

 

[harrylin]

I went back and reviewed derivations of energy density and flux of plane waves, and also read the prior sections of Einstein's 1905 paper. I believe all the mysteries can be resolved.

1) The mistake in my argument that E~f could be derived classically is the very assumption that E=g(f) for some unknown g. Classically, E is simply independent of f, and depends only on amplitude.

2) The classical derivation of the change in E for a traveling wave for change in motion of an observer involves a change in amplitude. Since the amplitude, energy, and frequency change in tandem, then when looked at from a quantum point of view, number of photons is preserved, each having different energy for different observers. (Same number of lower energy photons is represented classically as lower amplitude).

3) I believe all my prior statements about the effect of motion, acceleration, and gravity on the the energy of light, and the analogies with bullets, are essentially correct, if not always precisely worded. I was never setting out to reason strictly classically. Obviously, any statements about classical justification for E~f are incorrect.

I'm afraid that I don't follow you here; I think that the discussion is simply a different topic, although related.

In a follow-up paper of the same year[1], Einstein elaborated further on the passage that I cited earlier. He started with stressing that the energy of an emitted wave as measured in a system that is moving relative to the source, is different from the energy as measured in the "rest" system.

He thus elaborated on the fact that E'/E = f'/f.

I thought that this was the discussion topic following your post #38 (elaborating on gravitation effects on light with PoE), but I must admit that I did not follow the discussion carefully.

[1] http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

 

[PAllen]

I'm afraid that I don't follow you here; I think that the discussion is simply a different topic, although related.

In a follow-up paper of the same year[1], Einstein elaborated further on the passage that I cited earlier. He started with stressing that the energy of an emitted wave as measured in a system that is moving relative to the source, is different from the energy as measured in the "rest" system.

He thus elaborated on the fact that E'/E = f'/f.

I thought that this was the discussion topic following your post #38 (elaborating on gravitation effects on light with PoE), but I must admit that I did not follow the discussion carefully.

[1] http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

But it surprisingly does *not* follow from E'/E = f'/f over Lorentz transform, that E~f, and Einstein never makes this latter claim. My derivation of this is all fine *if* E is determined by some initially unknown g(f). However, if E is independent of f, my derivation amounts to a proof based on aerodynamics, that "If elephants could fly, then pigs certainly could too".

The key is that in section 7 of Einstein's paper he also derived the way amplitude transformed, and the way E transformed *follows* from the way amplitude transforms. As a result, the fact that f transforms the same as E is giving you no additional information. So we have the classically known fact that E is determined by amplitude and independent of f; amplitude transform under Lorentz determines E transform under Lorentz. It happens that E'/E=f'/f, but this says nothing about dependence of E on f in general.

[Edit: For emphasis: certainly one cannot say that amplitude depends on f! One can say that amplitude depends on number of photons and f, once you quantize.

The interesting consequence of the fact that E'/E=f'/f combined with the amplitude transform law, results that when light is quantized, a Lorentz transform preserves the number of photons.]

 

[PAllen]

To make the classical vs. quantum relation between energy, frequency, and amplitude concrete, consider a concrete example:

Imagine in one frame we have 3 pulses of light, each with energy E0, amplitude A0, and frequency f1, f2, and f3. Classically, there is no relation between E and f. Now the quantum situation is simply that pulse 1 has E0/hf1 photons, pulse 2 has E0/hf2 photons, etc.

Now apply a Lorentz transform. We have the following relations:

E0' is related to A0' the same as E0 is to A0.
E0'/E0 = f1'/f1 = f2'/f2 = f3'/f3 (1)

As a result of (1), E0'/hf1' = E0/hf1, so the number of photons is preserved. However, from a classical point of view, there remains no relation between E and f.

 

[atyy]

Do you think section 8 of http://www.fourmilab.ch/etexts/einstein/specrel/www/ is correct? eg. is the R² in the same in primed and unprimed coordinates?

(Yes, maybe it's just that E is not a function of f at all, but why doesn't that "fall out" of the mathematics? I would have thought maybe E=g(f), Eo=h(g(fo), or that v(f) implicitly rather than v=constant)

 

[PAllen]

Do you think section 8 of http://www.fourmilab.ch/etexts/einstein/specrel/www/ is correct? eg. is the R² in the same in primed and unprimed coordinates?

(Yes, maybe it's just that E is not a function of f at all, but why doesn't that "fall out" of the mathematics? I would have thought maybe E=g(f), Eo=h(g(fo), or that v(f) implicitly rather than v=constant)

I didn't see any problem with section 8, which was based on the amplitude transform derived in section 7.

It seems to me that what my earlier derivation shows is the IF E=g(f), g must be linear, but that if E is independent of f, the argument shows nothing.

 

[atyy]

I didn't see any problem with section 8, which was based on the amplitude transform derived in section 7.

It seems to me that what my earlier derivation shows is the IF E=g(f), g must be linear, but that if E is independent of f, the argument shows nothing.

In Einstein's section 8 is the volume transformed correctly? Is it the volume of a light sphere?

The Energy E in section 8 is not my naive understanding of energy. I would expect E ~ A². But he defined E ~ AS.

I've actually never read this section

 

[harrylin]

To make the classical vs. quantum relation between energy, frequency, and amplitude concrete, consider a concrete example:

Imagine in one frame we have 3 pulses of light, each with energy E0, amplitude A0, and frequency f1, f2, and f3. Classically, there is no relation between E and f. Now the quantum situation is simply that pulse 1 has E0/hf1 photons, pulse 2 has E0/hf2 photons, etc.

Now apply a Lorentz transform. We have the following relations:

E0' is related to A0' the same as E0 is to A0.
E0'/E0 = f1'/f1 = f2'/f2 = f3'/f3 (1)

As a result of (1), E0'/hf1' = E0/hf1, so the number of photons is preserved. However, from a classical point of view, there remains no relation between E and f.

Earlier you wrote: "light energy and frequency [are] proportionally changed, purely from SR and Maxwell".

I still think that that is correct!

In shorthand, for the change of speed, dE~df. And as E=0 when f=0, also E~f for the case under consideration.
This tells me that if we move away at such a speed that the received frequency is half of that in rest, the energy that we can absorb is also half of that in rest.

Proportional does not necessarily mean causal. That is a mistake that is (was) notoriously made in medical studies.

Regards,
Harald

 

[harrylin]

In Einstein's section 8 is the volume transformed correctly? Is it the volume of a light sphere?

The Energy E in section 8 is not my naive understanding of energy. I would expect E ~ A². But he defined E ~ AS.

I've actually never read this section

You overlooked a square, right?
He has that the [energy] ~ [amplitude per volume]² * [volume].
Looks good to me!

 

[PAllen]

Earlier you wrote: "light energy and frequency [are] proportionally changed, purely from SR and Maxwell".

I still think that that is correct!

Yes, I agree.

In shorthand, for the change of speed, dE~df. And as E=0 when f=0, also E~f for the case under consideration.

Here there is an issue. f=0 is no longer a wave, and power radiation arguments completely break down. Classically, any f > 0 can be associated with any E > 0. Quantum mechanics comes along and says for any E there is a maximum f =(E/h), and for any f, there is a minimum E = hf. But within the broad range where quantization is insignificant (hf << E), then the classical picture is essentially correct: for a give f, the energy of a light pulse can be whatever you want.

This tells me that if we move away at such a speed that the received frequency is half of that in rest, the energy that we can absorb is also half of that in rest.

correct, despite the disagreement above.

Proportional does not necessarily mean causal. That is a mistake that is (was) notoriously made in medical studies.

Regards,
Harald

In this case, proportional is wrong as regards light pulses or corresponding sections of traveling waves. All that is true is for a given light pulse, Energy and frequency change in proportion, *not* that the energy is proportional to the frequency.

[Edit: I guess we can agree that *for a given body of light*, E is proportional to f *under the operations of change of motion of an observer or measuring device*. But it is not proportional in any other sense. A different body of light can have the same f and any other E desired (classically).]

 

[harrylin]

[..]
In this case, proportional is wrong as regards light pulses or corresponding sections of traveling waves. All that is true is for a given light pulse, Energy and frequency change in proportion, *not* that the energy is proportional to the frequency.

[Edit: I guess we can agree that *for a given body of light*, E is proportional to f *under the operations of change of motion of an observer or measuring device*. But it is not proportional in any other sense. A different body of light can have the same f and any other E desired (classically).]

Yes, nothing else is implied than that of a given light pulse, the observed E~f.

 

[harrylin]

E/E0=f/f0 => E ~ f.
Thus I would say, QM knows SR! [..]

Correction: I should not have written that, as the similarity is just in the equations and not in their physical meanings as we now elaborated: SR's E~f has little to do with QM's E=hf.

But also, I had forgotten that E=hf was proposed in 1901 already - so SR knew about QM!

 

[atyy]

You overlooked a square, right?
He has that the [energy] ~ [amplitude per volume]² * [volume].
Looks good to me!

Yes, the volume factor is necessary. What I didn't understand was why we are allowed to have the volume factor, eg. for a plane wave solution. Apparently, some approproate volume can be used if you have an approximately monochromatic wavepacket. It's limited enough in space that its volume is objective, but unlimited enough that its wavelength is well defined. So we really need a bullet of light, exactly what the semiclassical photon is. At least this is what I gathered from comments in Brau's "Modern problems in classical electrodynamics", but I haven't seen a detailed calculation.

 

[harrylin]

Yes, the volume factor is necessary. What I didn't understand was why we are allowed to have the volume factor, eg. for a plane wave solution. Apparently, some approproate volume can be used if you have an approximately monochromatic wavepacket. It's limited enough in space that its volume is objective, but unlimited enough that its wavelength is well defined. So we really need a bullet of light, exactly what the semiclassical photon is. At least this is what I gathered from comments in Brau's "Modern problems in classical electrodynamics", but I haven't seen a detailed calculation.

Oops I had put the square at the wrong place, it should have been:
[energy] ~ [(amplitude)² per volume] * [volume].

But I wonder if you actually read that chapter, as Einstein took there the example of a sphere that encloses a wave:

"We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k."