Universality of Doppler shift for energy of any massless field

[bcrowell]

In "Does the inertia of a body depend upon its energy content?," http://fourmilab.ch/etexts/einstein/E_mc2/www/ , Einstein invokes a result from his 1905 SR paper, which is that the Doppler shift of a light wave's frequency $D(v)=\sqrt{(1-v)/(1+v)}$ is also the factor by which its energy changes when one transforms from one frame to another. (He points out how remarkable the result is, and reading between the lines, clearly he cares about it because if it were invalid, then E=hf would have to be abandoned.) Making use of this fact, derived from Maxwell's equations, he proves the famous $E=mc^2$.

Now clearly from a modern point of view there is nothing so special about the electromagnetic field as opposed to any other fundamental field, so it's a little unsatisfactory to have the foundations of SR depend on this obscure fact about EM waves. If any other massless field or particle had an energy that *didn't* scale by the same D(v) under Lorentz boosts, then you could use it to prove that E equaled something other than $mc^2$. Therefore I'm sure that the D(v) factor *does* apply to all other massless phenomena. For example, it must also apply to low-amplitude gravitational waves. However, it seems kind of silly to be obligated to prove this individually for every massless phenomenon, just to make sure that the new phenomenon doesn't invalidate the foundations of relativity.

It seems to me that there must be some more generic argument that D(v) is a universal energy-scaling factor for all massless classical phenomena, and that this argument must carry through without having to use $E=mc^2$ as a prior assumption. Is there such an argument?

I'm not interested in arguments that the frequency scales by f for any massless field. That's straightforward. I'm interested in the scaling factor of energy.

This is all about classical physics, so I'm not interested in quantum-mechanical arguments. (Of course if you already believe in E=hf, then it's trivially obvious that E scales by the same factor as f.)

 

[atyy]

Perhaps one could try starting from the (traceless) stress-energy tensor of a conformal field?

 

[bcrowell]

Perhaps one could try starting from the (traceless) stress-energy tensor of a conformal field?

Sounds interesting -- can you say more? I don't know anything about conformal field theory.

 

[atyy]

Neither do I. I am reading from http://arxiv.org/abs/0908.0333 , chapter 4.

I think first I would start from the EM stress tensor, which is traceless, and integrate over some appropriate volume. The stress tensor transformation is known, and by the choice of volume, the transformation for the volume is also known, and so we should know the transformation for the integral (who knows, maybe we won't even need to use conformality, maybe all we need are the properties of an arbitrary stress-energy tensor). That's all the vague idea I have now.

The basic idea is that energy density should transform a certain way and volume in certain way, and together they should transform like frequency. One might may have to assume that the energy density is zero outside of the volume.

 

[atyy]

The de Broglie relation E=hf is supposed to hold for massive and massless particles. This seems to indicate that the transformation property should be obtainable from integrating the energy density of any stress-energy tensor, not just that of a massless particle.

The Lorentz invariance of charge Q comes from Integral(J₀dV), if the charge density has the proper boundary conditions (which is one of those things I need to think about before getting straight, so I can't tell you what the boundary conditions are). So I expect the Lorentz covariance of E to come from a similar calculation.

 

[bcrowell]

The de Broglie relation E=hf is supposed to hold for massive and massless particles. This seems to indicate that the transformation property should be obtainable from integrating the energy density of any stress-energy tensor, not just that of a massless particle.

Hmm...but for a massive particle, the energy *doesn't* transform by a factor of D under a boost. For example, suppose that in frame A a massive particle is moving to the right at 0.1c. Now I make a Lorentz boost into a frame B that's moving to the left at 0.9999c relative to frame A. We have D=7x10^-3, but the particle's energy in frame B isn't smaller by a factor of 7x10^-3 than its energy in frame A.

I think what happens with a massive particle is that neither E nor f transforms by a factor D. The derivation of D as a frequency Doppler shift specifically assumes that the wave is moving at c. As a special case, consider the case where the wave is nearly at rest in frame A, while frame B is moving at relativistic speed relative to A. Then the frequency transforms by a factor of gamma, not D.

 

[bcrowell]

I spent some time thinking about this while hiking yesterday, and I think I made some progress. This is still kind of rough, but I think it's starting to make sense. My purpose here is to see if I can get a good, freshman-level argument for E=mc^2 that can be used with students who don't know any E&M yet, and who also don't know conservation of momentum (but do know Newton's third law).

We know that a rocket that expels exhaust at a constant rate has a fixed dv over a fixed time interval dt in its own frame (neglecting the decrease in its mass due to fuel consumption), and this means that its v will approach c in an inertial frame. Work can be done indefinitely on it, but it will never surpass c. Therefore we know that relativistic kinetic energy can't just be (1/2)mv^2. It must approach infinity as v approaches c.

The most conservative thing we can do is to suppose that mass and energy are *not* equivalent: the mass of a system is just the sum of the masses of its constituent particles. We hope that the only new thing that happens in relativity is that the factor of (1/2)v^2 in Newtonian kinetic energy gets replaced with some other function. We would then want energy to be conserved in all frames, and this requirement is more conveniently imposed by expressing the unknown function in terms of D, since D values simply combine by multiplication in relative motion. So now we're looking for an unkonwn function f(D) that gives the velocity-dependent factor for relativistic kinetic energy.

Let a set of particles with unit mass all be moving in the positive direction in one dimension. Some collisions happen, and in the final state they have some other state of motion, but for convenience let's say that they are all still moving in the positive direction. Energy conservation is $f(D_1)+...=f(D_1')+...$ If energy conservation is to apply in all other frames, then we must have $f(kD_1)+...=f(kD_1')+...$. The simplest way to satisfy this is f(x)=x, although we could also have f(x)=x^p, or even f(x)=ln x if the particles are never created or destroyed.

Assuming the x^p form, the fact that k and D_i are both positive definite tells us that we can never have exactly zero energy, and not only that, but there is no frame in which a particle's energy is at a minimum. This shows that our assumptions don't work unless the particles being discussed are all moving at c. For example, our "particles" could be rays of light. (Since light rays can be created or destroyed, we can rule out f(x)=ln x.) The assumption that a particle's mass is fixed must therefore be invalid, because it gives results that only make sense for massless phenomena such as light rays.

I can now proceed with Einstein's original argument from "Does the inertia of a body depend upon its energy content?," except that I haven't established p=1. For example, p=3 would just seem to give E=3mc^2. Maybe I could show that this violates the work-KE theorem, but that would require establishing how force transforms, which would be difficult to do at a stage where they don't even know about Newtonian momentum.

 

[DrGreg]

For what it's worth, I once posted my own derivation of momentum & energy formulas, but I'm not sure that's really what you are looking for here.

 

[bcrowell]

Thanks, DrGreg -- that's extremely helpful! For one thing, it's purely mechanical, which is along the lines I'm looking for. Also, it doesn't use four-vectors, which is similar to what I'm trying to do. It's interesting how similar our approaches are. We both write down these unknown functions in terms of the rapidity (or in my case the thing I'm calling D, which is the exponential of minus the rapidity). Your argument intertwines the energy and momentum, whereas I was trying to find something to be used with students who don't know momentum (but who do know Newton's third law). It seems like I'm stuck on pinning down the value of the exponent p, and it may be that this is impossible without doing something momentum-ish. Maybe I should take my result, with unknown p, and substitute it into your expressions to see where $p \ne 1$ violates your axioms.

-Ben

 

[bcrowell]

OK, I think I have it now. DrGreg's $\alpha$ is the same as my integer exponent p. Having $p\ne 1$ violates DrGreg's equation F15, which expresses the requirement that a certain inelastic collision conserve energy in all frames. Thanks, DrGreg -- that was exactly what I needed!

I am confident of the logic that leads from my assumptions to my conclusions. The reason I am posting this is to ask whether the assumptions that I make are reasonable in this context. Could I simplify my argument still further with even better assumptions?

One place where I could have used some more convincing was at A7, where you assume that the form of the work-KE theorem is frame-invariant. A6 doesn't bother me because it can be taken as a definition of F (and I believe it's a pretty standard way of defining the force three-vector in the context of SR). But having used up that freedom to define F, it's not clear to me how we know a priori whether the work-KE theorem is frame-invariant.

You justify A4 and A5, proportionality of E and p to m, by considering "the limit as any two particles collide and coalesce at a relative speed approaching zero." Both of the statements and their justification are phrased in a way that restricts the discussion to particles with nonzero mass. This leaves open the question of how to generalize to zero-mass particles, although I guess you can handle it using limits and show that the energies and momenta of ultrarelativistic particles transform by the Doppler shift factor D.

One of your stated goals was to avoid the "rather ugly and tedious mathematics" of treatments like Einstein's. To me, the ugly and tedious part of Einstein's treatment is where he establishes, in "On the electrodynamics...," that the energy of a light wave transforms by the factor D. The remainder of the derivation, in "Does the inertia...," is actually pretty short and elegant. This was basically my motivation for looking for a derivation that would avoid grotting around with Maxwell's equations in order to derive D as the transformation for energy, which clearly has to be something that holds for more fundamental reasons for all massless phenomena, not just because of Maxwell's equations for light.

-Ben

 

[robphy]

Does this answer your question?
The Doppler Factor (the Bondi k-factor) and its reciprocal are eigenvalues of a Lorentz boost.

 

[DrGreg]

One of your stated goals was to avoid the "rather ugly and tedious mathematics" of treatments like Einstein's. To me, the ugly and tedious part of Einstein's treatment is where he establishes, in "On the electrodynamics...," that the energy of a light wave transforms by the factor D. The remainder of the derivation, in "Does the inertia...," is actually pretty short and elegant.

When I wrote that, two years ago, I was thinking of a different publication by Einstein, "Elementary derivation of the equivalence of mass and energy", Bull. Amer. Math. Soc. 41 (1935), 223-230.

In the context of the derivation you were discussing, I agree, the second part is elegant, it's only the first part that isn't. The first part can be avoided if you're prepared to accept $E = h \nu$, but it seems to be "cheating" to use quantum theory in a classical theory like relativity.

 

[atyy]

Hmm...but for a massive particle, the energy *doesn't* transform by a factor of D under a boost. For example, suppose that in frame A a massive particle is moving to the right at 0.1c. Now I make a Lorentz boost into a frame B that's moving to the left at 0.9999c relative to frame A. We have D=7x10^-3, but the particle's energy in frame B isn't smaller by a factor of 7x10^-3 than its energy in frame A.

I think what happens with a massive particle is that neither E nor f transforms by a factor D. The derivation of D as a frequency Doppler shift specifically assumes that the wave is moving at c. As a special case, consider the case where the wave is nearly at rest in frame A, while frame B is moving at relativistic speed relative to A. Then the frequency transforms by a factor of gamma, not D.

Boy, am I confused. http://en.wikipedia.org/wiki/Four-vector says the four wavevector is a four vector. But frequency doesn't transform as the first component of a four vector?

Edit: OK, I think I see what's happening. (E,p) and (w,k) transform the same way. For a massive particle, E is a function of p and m, and E' will be a function of E and p, not E alone. For a massless particle, E is a function of p alone, and we can eliminate p from E' as a function of E and p, to get E' as a function of E alone. So the general question is whether this is true for all conformal fields (should be, since I expect conformal fields to have k proportional to w, unlike massive fields).

 

[bcrowell]

When I wrote that, two years ago, I was thinking of a different publication by Einstein, "Elementary derivation of the equivalence of mass and energy", Bull. Amer. Math. Soc. 41 (1935), 223-230.

In the context of the derivation you were discussing, I agree, the second part is elegant, it's only the first part that isn't. The first part can be avoided if you're prepared to accept $E = h \nu$, but it seems to be "cheating" to use quantum theory in a classical theory like relativity.

Right, I agree that it's lame to appeal to quantum mechanics. Thanks for posting the link to the Einstein paper -- I hadn't been aware of it. Your version certainly cuts down on the cruft by using rapidity rather than velocity.

I've written up my own argument in section 12.4 of this book: http://www.lightandmatter.com/mechanics/ When it comes to proving that the exponent (what you call $\alpha$) is $\pm 1$, I copped out and gave a link to your version.

Since your derivation helped me on a crucial point, I'd like to acknowledge you, preferably by real-world name, in a footnote in that section of my book. Would this be OK? Possible concerns I can imagine would be that (a) you might want to remain anonymous on PF, or (b) you might not like my derivation and might not want your approval of it to be implied.

In any case, thanks!

-Ben

[EDIT] Oops, after writing up the argument in my book I realized there's a flaw in the logic having to do with the way that I tried to surgically separate the energy argument from the momentum one. The fact that momentum is conserved relaxes the constraint on the form of my function f(s). I'll have to see if it can be modified, or maybe I should just call it a heuristic rather than a rigorous proof.

 

[DrGreg]

Does this answer your question?
The Doppler Factor (the Bondi k-factor) and its reciprocal are eigenvalues of a Lorentz boost.

I am aware of that fact, but it's not obvious to me how that can help here.

I have wondered whether there's a k-calculus argument to establish formulas for energy and momentum. The light-cone-coordinate approach would suggest you should make use of $(E + pc)$ and $(E - pc)$, but I couldn't get any further than that.

 

[bcrowell]

I have a new version of my argument online now: section 12.4, http://www.lightandmatter.com/mechanics/ . I think it fixes the logical flaw in my earlier version.

 

[atyy]

Apart from your specific paedagogical purpose, do you think it would be too kludgey to say a massless particle is one in which E~p and a massless field is defined to be one which has a wave equation whose solutions obey w~k (so that dw/dk=c) then since (E,p) and (w,k) are 4-vectors, we get the required result that E'=DE and w'=Dw?

 

[bcrowell]

Apart from your specific paedagogical purpose, do you think it would be too kludgey to say a massless particle is one in which E~p and a massless field is defined to be one which has a wave equation whose solutions obey w~k (so that dw/dk=c) then since (E,p) and (w,k) are 4-vectors, we get the required result that E'=DE and w'=Dw?

That seems fine to me. Both pedagogically and logically, there's a variety of ways to arrange the material. You can pick and choose which things you feel should be more logically fundamental and which should be derived.

 

[atyy]

That seems fine to me. Both pedagogically and logically, there's a variety of ways to arrange the material. You can pick and choose which things you feel should be more logically fundamental and which should be derived.

OK, thanks. I still don't know how to get directly the energy transformation from the stress-energy tensor of an arbitrary massless field.

 

[PhilDSP]

Apart from your specific paedagogical purpose, do you think it would be too kludgey to say a massless particle is one in which E~p and a massless field is defined to be one which has a wave equation whose solutions obey w~k (so that dw/dk=c) then since (E,p) and (w,k) are 4-vectors, we get the required result that E'=DE and w'=Dw?

That seems fine to me also. But, isn't it the case then that the math becomes Euclidean? And the need for the 4-vectors drops away?