Why is energy-momentum symmetry important in special relativity?

[krab]

(I couldn't resist the title, because I am fed up with all the crackpot threads about why some aspect of modern physics is "incorrect".)

I work at an accelerator lab, dealing with particles whose velocities are not small compared with the speed of light. To me, saying that SR is incorrect is exactly like saying "the internal combustion engine is a fraud and does not work". It's a statement of such silliness as to leave one breathless. It is not as if I use some tiny piece of SR which is just small enough that I wiould get the same answer from another theory (which for example does not insist that simultaneity is relative). No, SR is fleshed out by me and others in excruciating mathematical detail, and if it is even a tiny bit wrong, our accelerator would not work!

Anyway, I thought I would share with everyone an aspect of the beauty of SR that I have never seen described anywhere else. Before SR, it was known that the pair of variables (t,E) acted very like the pairs (x,p_x), (y,p_y), (z,p_z). (These are called "canonical pairs".) It was known for example that in field-free regions,
[tex]E={p_x^2\over 2m}+{p_y^2\over 2m}+{p_z^2\over 2m}[/tex]
Hamilton showed that considering E to be a function of x,y,z,p_x,p_y,p_z, all the equations of motion could be derived from it. (It's called the "Hamiltonian".) For example, the equations could be derived from the "principal of least action", which involves the following integral
[tex]\int p_xdx+p_ydy+p_zdz-Edt[/tex]
Notice the symmetry (except for a sign change) between the afore-mentioned canonical pairs.

This symmetry obtains at a very deep level. For example, one could "pretend" that the independent variable is z instead of time t. Then all the equations would no longer answer the question, "Where is the particle at time t and what are its momentum components?", but rather, "I'm at z, so what are the x and y coordinates, the momenta p_x and p_y, and energy E and by the way, what time is it?" This can be obtained by solving the above E equation for p_z, and using p_z as if it were the new Hamiltonian. Amazingly, all the derived dynamics is exactly the same as if t were the independent variable.

There is truly a cyclic symmetry among x,y,z,t, and among p_x,p_y,p_z,E. All this was well-known before anyone ever dreamed of SR. So the question was: Why doesn't the equation for E display this symmetry explicitly? In fact the known dependence of E on momentum was very unsymmetric; for example, the momenta are squared, and the energy is not.

Then SR came along and everything made sense because:
[tex]E^2-p_x^2c^2-p_y^2c^2-p_z^2c^2=m^2c^4[/tex]
(or, in words, the norm of the 4-momentum is the rest energy). Notice the restoration of symmetry. Notice as well that this equation converges to the previous one (aside from a constant added to E) in the limit [itex]E<<mc^2[/itex]

Beauty, ain't it?

 

[chroot]

krab,

I moved your post here because I think no one will benefit from it in the TD section.

geistkiesel,

I deleted your post because I couldn't detect any substantive argument in it.

- Warren

 

[Doc Al]

(I couldn't resist the title, because I am fed up with all the crackpot threads about why some aspect of modern physics is "incorrect".)

I know that feeling.

Beauty, ain't it?

Indeed it is! Thanks for posting that, krab.

 

[russ_watters]

(I couldn't resist the title, because I am fed up with all the crackpot threads about why some aspect of modern physics is "incorrect".)

I work at an accelerator lab...

Its frustrating for me and I'm an engineer - I never work with it. I can't imagine how much it must annoy you.

 

[HallsofIvy]

And the logic of those who dispute relativity is always the same: "I don't understand it therefore it is wrong"!

 

[(Q)]

Krab

Excellent post! However, if HallsofIvy statement is correct, then it's not likely the crackpots will understand it anyway.

 

[turin]

krab,
I totally agree with your intent (my thesis was on GR, so I feel somewhat of a pang when someone tries to tell me it's bogus), but, I am not entirely convinced that it would be impossible to come up with:

E² - p.pc² = m²c⁴

without SR. The trivial counter-argument that comes to mind is to make the above equation an axiom. A not quite so trivial counter-argument (but almost as ad hoc) would be the generalization of the Schroedinger Equation (for the sake of symmetry) to the Dirac Equation that involves completing the square and requires the use of the Pauli Matrices. In fact, symmetry has seemed to me a powerful argument in itself, so one could just argue for:

E² + αp.p = constant

and then:

E² - p.pc² = m²c⁴

would likely flesh out naturally (i.e. α would be found to equal -c² and the constant would be m²c⁴).

Please comment on my admittedly ignorant statements.

 

[kurious]

KRAB:

"I'm at z, so what are the x and y coordinates, the momenta p_x and p_y, and energy E and by the way, what time is it?"

But if I bring the uncertainty principle into this argument ( momentum x position = h bar) does the symmetry still apply, given that, in general, the uncertainty in momentum is different from the uncertainty in position.

 

[quantumdude]

But if I bring the uncertainty principle into this argument ( momentum x position = h bar) does the symmetry still apply, given that, in general, the uncertainty in momentum is different from the uncertainty in position.

Yes, the symmetry still applies. Don't forget, the uncertainty principle does not hold betwee the components of the 4-momentum, it holds between conjugate pairs.

The uncertainty principle reads:

&Delta;x&Delta;p_x>=hbar
&Delta;y&Delta;p_y>=hbar
&Delta;z&Delta;p_z>=hbar
&Delta;t&Delta;E>=hbar

which is symmetrical with respect to p and E, as well as x and t.

 

[yogi]

Most of the questions that arise re SR do not involve the correctness of the interval transforms, but the propriety of the postulates - there are alternative ways to arrive at the Lorentz transforms (for example those of Lorentz). Before you condemn the "crackpot threads" you should take the time to read what is being questioned. Here is a quote from one of the doubters:

“There is no idea of which I would be sure that it would stand the
test of time, and I have doubts whether I am on the right way
In general ...feelings of dissatisfaction come from the inside.”

Albert Einstein

 

[kurious]

“There is no idea of which I would be sure that it would stand the
test of time, and I have doubts whether I am on the right way
In general ...feelings of dissatisfaction come from the inside.”

Albert Einstein

And Einstein could not at first believe he was right and Newton was not.
I don't think relativity is wrong - I just think that one day there will be a more fundamental explanation of why it is right.

 

[DrChinese]

“There is no idea of which I would be sure that it would stand the
test of time, and I have doubts whether I am on the right way
In general ...feelings of dissatisfaction come from the inside.”

Albert Einstein

And Einstein could not at first believe he was right and Newton was not.
I don't think relativity is wrong - I just think that one day there will be a more fundamental explanation of why it is right.

"More fundamental explanation" is in the eye of the beholder. "More fundamental description" is not likely, as the equations are already quite straightforward. The only real possibility is why c has the value it does.

And I want to throw my lot in with moving this thread from TD to SR/GR. 'Bout time...

 

[quantumdude]

Most of the questions that arise re SR do not involve the correctness of the interval transforms, but the propriety of the postulates -

In what way is the propriety of the postulates still questionable? Both have been directly tested experimentally.

there are alternative ways to arrive at the Lorentz transforms (for example those of Lorentz).

We all know that.

Before you condemn the "crackpot threads" you should take the time to read what is being questioned.

Most of the crackpots threads attack the interval transforms, and say that we should go back to Galileo.

Here is a quote from one of the doubters:

“There is no idea of which I would be sure that it would stand the
test of time, and I have doubts whether I am on the right way
In general ...feelings of dissatisfaction come from the inside.”

Albert Einstein

This doesn't carry much weight. You can find similar quotes from the founders of quantum mechanics, expressing doubts about their subject. Once the trauma of the paradigm shift wears off and the experimental confirmation starts raining down in buckets, the doubts get quenched pretty fast.

 

[yogi]

Kurious - I believe you have the time sequence reversed - the quote from Einstein was near the end of his life - after he had reflected upon these ideas for many years.

Dr Chinese - I would wager that a more fundamental description is on the horizon - SR is not really a physical theory as originally interpreted by Einstein himself - it is a mathematically elegant exemplification of symmetry without physics - but as early as 1915 Einstein begin leaning toward the notion of some sort of medium -- having properties necessary to explain gravity vis a vis G.R. - we will likely not find anything as beautiful and simple as SR - but it is likely that a better understanding of space will lead to a functional relationship between the transforms and the universe.

 

[yogi]

Tom - your comment, as with that posted by Kurious - would only be appropriate if the doubts were voiced in his early writings - Einstein reflected upon his theories for the better part of his life - after considerable experimental evidence had been amassed to substantiate time dilation (which can be used to derive all of the other SR relationships that have been experimentally confirmed).

When you say the postulates have been experimentally confirmed - you bypass the essence of the argument - the alternative theories such as Lorentz Ether, local G field modification of space, Inflow theory, etc, are also validated by the experiments - The Selleri tranforms lead to exactly the same results. I am not advocating anything personally - what I am about is to question dogmatic certainty.

 

[krab]

Most of the questions that arise re SR do not involve the correctness of the interval transforms, but the propriety of the postulates - there are alternative ways to arrive at the Lorentz transforms (for example those of Lorentz). Before you condemn the "crackpot threads" you should take the time to read what is being questioned.

You have not defined "crackpot threads". See below. Most of us here, and especially the mentors, "take the time to read what is being questioned".

Here is a quote from one of the doubters:
?There is no idea of which I would be sure that it would stand the test of time, and I have doubts whether I am on the right way In general ...feelings of dissatisfaction come from the inside.? --Einstein

I don't know why this is a remarkable quote. All the really great physicists had in common the humility to realize that they do not have a lock on the truth. But those I would call "crackpots" are exactly the opposite. Countless times, I've read threads by people who say "Einstein wrong...", or "Michelson-Morley misinterpreted". Here's a typical example:

It is a well known fact that QFT or QED (Quantum Electro dynamics ) postulates that electro-magnetic radiation of all wave-lengths and frequencies is propagated by means of quantum entangled pairs ( i.e electron/positron annihilation and formation ) . Yet a little cogitation will show that this theory of propagation is flawed.

It is always "a little cogitation" or some such. Yet it always develops that the crackpot knows far far less physics than the people who developed, worked on, work with the theory. In fact it is very striking how often such people say "I'm not much good with math...". The same people are always so certain of their own theories that they then misinterpret the humility of great physicists as being a sign that these theories were/are really shaky.

 

[yogi]

Krab - yes - I agree with all that you have said in your last post - there are many unfounded, ridiculous and arbitrary theories based upon nonsense... not worth audience or reply. What is of concern to me is the danger in asserting that SR (or any other theory for that matter) should be closed to further examination. SR is of particular interest because it can be interpreted in different ways, and has been by well reasoning intelligent persons. I wouldn't give a hoot for a theorist that is unable to derive the mathematical relationships that are required to evalute the merits of his musings

 

[DrChinese]

I am not advocating anything personally - what I am about is to question dogmatic certainty.

The dogmatic certainty is that previous experiments have been done which yield constraints as to theory development. The certainty is not that the existing theory is perfect, final, or somehow exempt from modification when new results are found.

This is freely acknowledged by mainstream science, as is the idea that we are ignorant in many areas and much more work remains. That too is a "dogmatic certainty". But not exactly a dark conspiracy of closed-minded hacks.

 

[yogi]

Dr Chinese - that is the way it should be - but history teaches us that authoritarians resist change - I don't have Planck's quote correct - but it was to the effect that we have to wait for the present generation to die off before the new ideas can take root.

 

[Nereid]

When you say the postulates have been experimentally confirmed - you bypass the essence of the argument - the alternative theories such as Lorentz Ether, local G field modification of space, Inflow theory, etc, are also validated by the experiments - The Selleri tranforms lead to exactly the same results. I am not advocating anything personally - what I am about is to question dogmatic certainty.

Excuse my ignorance, but do these alternative theories (or at least some of them) yield the same predictions (etc) as SR/GR, no matter what (in other words, they are, from the point of view of anything that could ever be done, experimentally or observationally, identical to SR/GR)?

If not, what specific predictions do these make that can, in principle, be tested (and which differ significantly from those of SR/GR)?

 

[quantumdude]

When you say the postulates have been experimentally confirmed - you bypass the essence of the argument -

How? If the postulates have been confirmed experimentally, then on what basis could anyone question their "propriety"? Aesthetic appeal?

 

[yogi]

Nereid. In the case of the inflow theory, some very definite test have peen proposed by Tom Martin - he has published these on the internet - you can probably come across them - search on google under his name and gravity.

Tom - what is at root with regard to further scruntity is not acedemic exercise - it is an attempt to understand space - SR gives no hint as to why the transforms work - the physics, if there is any, is put in by fiat in the second postulate. Lorentz Ether Theory arrives at the same result by a different postulate (based upon actual physical contraction) If space actually behaves that way - this could lead to just how and why masses condition space in GR - and perhaps a way to modify that affect for some useful purposes. Einstein's convention of one way isotrophy was a necessity --forced because of his conviction that it was impossible to detect absolute motion with respect to space - with the discovery of the CBR anisotropy, it appears that such a predispostion may be unfounded. It was not necessary to explain MMx since that is a round trip affair. Certainly Einstein's postulates have asthetic appeal - simple, elegant, symmetry. But are they correct - have they really been proved. Time dilation would appear to be validated - but even that has been questioned as being consequent to the fact that decay process are modified by high velocity - rather than being a confirmation of time dilation (I didn't say I believed this - I am just pointing out, that the so called proofs of SR are suscepible to alternative explanations). Not too long ago, on these boards, reference was made to an article that showed MMx could not be efficacious as a means for detecting etheral motion when Doppler was properly considered.

When the assertion is made that the hypothesis have been proven, the implication is that they are true - what has really been verified is that, at least in the Earth centered reference frame, light appears to be isotropic as measured by an observer in that frame. We have not done these experiments in free space, we do not know whether time dilations are real or apparent, and we have no good physics to explain why things should change when we shift from one inertial system to another.

 

[kurious]

YOGI

SR gives no hint as to why the transforms work - the physics, if there is any, is put in by fiat in the second postulate


Just about everything we experience has a cause and effect mechanism - one ato m bumps into another and transfers momentum etc. Why shouldn't relativity have some underlying causes like this - why can't particles in the vacuum cause an object moving through the vacuum to experience a force that physically makes it contract, or cause a force that slows clocks down? There is no proof that this can't happen and given the absolute reference frame of the cmbr we now have some hope that it can.

 

[selfAdjoint]

I have to tentatively support Yogi. We always say that special relativity has been amply proved to the point of engineering in the accelerators and particle experiments. But what has actually been validated is the Lorentz transformations. And we SAY that that validates the two postulates because we can derive the Lorentz transformations from them. But does anyone have actual proof of the postulates?

Note that Voigt essentially deduced the second postulate from an ether theory where all matter was excitations in the ether and the max speed of the excitations was independent of the speed of the matter, just as waves on the surface of a lake go at a particular speed that is independent of the speeds of the boats producing them.

Of course there are well known problems with ether theories, but that doesn't by itself rule out some alternative to special relativity. Or does it? At the very least, we should be prepared to defend SR at a deeper level than just checking the Lorentz transforms.

 

[Hurkyl]

Well, isn't the constancy of the speed of light easily derivable from the Lorentz transforms? Verifying the transforms should thus verify this postulate.


And furthermore, doesn't the verification of invariant physical laws (such as p = ymv) count as verification of the postulate of relativity?

 

[jdavel]

seflAdjoint said, "Note that Voigt essentially deduced the second postulate from an ether theory..."

I did a quick google search for this and didn't come up with anything. Do you know a link where this theory is explained.

Or can you explain it? Is Voigt's "ether" the medium for light waves? If so, doesnt' light have to travel at a constant speed wrt this medium? Then, if I'm moving wrt the medium wouldn't I measure a different value for light speed from someone who's stationary wrt the medium?

I just don't see how an ether can be consistent with the second posulate of SR. What am I missing?

 

[selfAdjoint]

Here is a http://psroc.phys.ntu.edu.tw/cjp/v39/211.pdf . His work was published, but it was only a sideline to his professional concentration; the determination of the physical constants of various materials.

I repeat that to get an idea of Voigt's concept, you should think of surface water waves in a harbor stirred up by boats. The waves do NOT add their speed to that of the boats; that's why boats have wakes!

Much later, after special relativity had been accepted, Voigt was honored at a meeting of German physicists. Acknowledging the applause the by then aged Voigt said "sometimes the old ways are still good" or words to the effect.

 

[quantumdude]

Tom - what is at root with regard to further scruntity is not acedemic exercise - it is an attempt to understand space - SR gives no hint as to why the transforms work

Sure it does. It states that the transforms are true because they follow from the postulates. Of course, SR gives no hint as to why the postulates are true, but experiment can tell us whether or not they are true.

- the physics, if there is any, is put in by fiat in the second postulate. Lorentz Ether Theory arrives at the same result by a different postulate (based upon actual physical contraction) If space actually behaves that way - this could lead to just how and why masses condition space in GR - and perhaps a way to modify that affect for some useful purposes.

What I'm not understanding is this:

How is it that the postulates of SR could be considered "improper" if they are experimentally validated? Even if some more fundamental physical mechanism were discovered, how would that invalidate the postulates? All the postulates say is that whatever inertial frame you decide to do an experiment in, you'll observe the same laws and the same speed of light. Surely a "physical explanation" would have to agree with that.

Certainly Einstein's postulates have asthetic appeal - simple, elegant, symmetry. But are they correct - have they really been proved.

They have been confirmed by Earth-bound experiments. Of course, we could always raise the objection that we have yet to do all those thought experiments with superfast spaceships, but I have no problem accepting the data collected from particle accelerators as evidence of SR.

Time dilation would appear to be validated - but even that has been questioned as being consequent to the fact that decay process are modified by high velocity - rather than being a confirmation of time dilation (I didn't say I believed this - I am just pointing out, that the so called proofs of SR are suscepible to alternative explanations).

No problem, since SR doesn't even give an "explanation" of the phenomenon. It just predicts what you will measure.

When the assertion is made that the hypothesis have been proven, the implication is that they are true - what has really been verified is that, at least in the Earth centered reference frame, light appears to be isotropic as measured by an observer in that frame. We have not done these experiments in free space,

Sure, there is more experimental work to do. That's true of every theory.

we do not know whether time dilations are real or apparent,

We don't? What would it take to convince you that they are real?

and we have no good physics to explain why things should change when we shift from one inertial system to another.

"What" changes when we shift from one system to another?

 

[jdavel]

selfAdjoint,

Thanks (I think!). I got a headache looking at that notation, but I'll give it a shot.

As to your suggestion, "to get an idea of Voigt's concept, you should think of surface water waves in a harbor stirred up by boats. The waves do NOT add their speed to that of the boats; that's why boats have wakes!" I don't see how this can be relevant.

Mechanical waves are never dependent on the velocity of their source. The wave equation says nothing about the source. All the motion it describes is wrt the medium through which the wave propagates. In the case of water waves the medium is, of course, the water. So water waves travel at a constant speed wrt the water. If an observer is stationary wrt the water, he will observe the waves moving at the speed given in the wave equation for water waves. If, on the other hand, the observer is moving wrt the water, he'll measure the waves moving at a different speed. This variation can be used by the observer to find his speed wrt the water.

But with light it's apparently different, because no variation in light speed has ever been measured. No matter how our speed changes (and it has to change, because we go around the sun) we always measure light speed as c. Doesn't this rule out the possibility of a medium through which light propagates at constant speed?

The first day that MM did their experiment it was remotely possible that the Earth happened to be stationary wrt an ether, unlikely, it seems to me, but possible. But when they did it 6 months later and the fringes didn't shift, the possibility of there being a stationary ether died! How all those brilliant guys (including Maxwell!) could have gone on believing in it is beyond me.

 

[quantumdude]

We always say that special relativity has been amply proved to the point of engineering in the accelerators and particle experiments. But what has actually been validated is the Lorentz transformations.

But that does verify the postulates. The relationship between the postulates and the transformations is of the "if and only if" type. That is, you can start from the transformations and get back to the postulates, if you want.

I think this is what Yogi is not understanding. If we accept the transformations as valid, then there are no "alternatives" to SR. There are only possible explanations as to why SR is true.

And we SAY that that validates the two postulates because we can derive the Lorentz transformations from them. But does anyone have actual proof of the postulates?

Sure, what about the Alvager experiment in which the speed of light of decaying pions was measured to be 'c', despite the fact that the pions themselves were moving at 99+% of the speed of light?

 

[quantumdude]

Well, isn't the constancy of the speed of light easily derivable from the Lorentz transforms? Verifying the transforms should thus verify this postulate.


And furthermore, doesn't the verification of invariant physical laws (such as p = ymv) count as verification of the postulate of relativity?

I think it does. You can take the transformation as fundamental, use it to figure out what a physical law will look like in another inertial frame, and deduce that they are all the same. You can similarly deduce the velocity addition law from the transforms and see that the invariance of the speed of light is a consequence of the transforms. Indeed, Halliday and Resnick takes this reversed approach, because it is mathematically simpler. Then in a subsequent course from, say, Taylor and Wheeler, the transformations are deduced from the postulates.

 

[yogi]

Hyrkyl - Tom - which transforms? - I do not see how you can recover c isotrophy from the interval transforms - they can be arrived at without assuming one-way light constancy. Same with respect to Self-adjoint's comment - the accelerators appear to validate time dilation which results from the invarience of the spacetime intervals in the two frames. What is consequent, is that the transform that results from the one way light constancy assumption has a term which drops out when one forms the interval transform. Again Tom - with respect to your comment that the transforms are true because they follow the postulate - the interval transforms are what is being tested in accelerators - and they can be arrived at by other means

SR is fundamentally conclusionary - maybe its correct, but If time dilation is real (which is how I am interpreting Tom's position) then there must be a physical reason - For example, it is not necessary to introduce the acceleration factor in the twin paradox to avoid the paradox because ---if the clocks actually run at different rates on the outbound journey they will also run at the same different rate on the inbound journey. Yet almost every textbook treatise will fall back upon Einstein's 1905 interpretation to explain the Twin Paradox (because the voyager feels the acceleration on turn around) - so what - if traveling twin's clock has a different reading than the Earth clock when the turnaround point is reached - the problem is resolved without resort to GR or whatever. But this means that time dilation (or time loss) is a real phenomena and it requires a physical explanation.

I am not at all sure that Einstein was true to his 1905 trajectory regarding the apparency of time dilation and the necessity of having to consider which twin underwent acceleration - several articles have been published questioning whether his later writings reveal a shift in his thinking on this subject.

 

[krab]

... But this means that time dilation (or time loss) is a real phenomena[sic] and it requires a physical explanation...

"Real" as opposed to what? How do you define real? I really don't understand the point you are making.

 

[quantumdude]

Hyrkyl - Tom - which transforms? - I do not see how you can recover c isotrophy from the interval transforms - they can be arrived at without assuming one-way light constancy.

It's not hard to see. Just take the transforms, and use them to predict what speed of light you would observe if you were moving with respect to the source. It's quite obvious that you will recover the invariance of 'c' if you take the transforms for granted.

Same with respect to Self-adjoint's comment - the accelerators appear to validate time dilation which results from the invarience of the spacetime
intervals in the two frames.

No, that is not right. The data from the accelerators has nothing to do with "the invariance of the spacetime intervals in the two frames". The time dilation is measured without any reference to any spacetime theory.

What is consequent, is that the transform that results from the one way light constancy assumption has a term which drops out when one forms the interval transform. Again Tom - with respect to your comment that the transforms are true because they follow the postulate - the interval transforms are what is being tested in accelerators - and they can be arrived at by other means

And again, with respect to my comment, the transforms bear an "if and only if" relationship to the postulates. Like it or not, the truth of the postulates implies the truth of the transforms and vice versa.

edit: fixed a spelling error

 

[yogi]

Krab - real vs apparent - in time dilation the loss is permanent because one clock is running faster than the other - so when they are compared by bringing them back together, there is a temporal difference. In contrast, Length contraction is an apparent observation - it is not real but merely consequent to observations made between moving reference frames while they are moving.

Tom - we seem to be on a different page - perhaps I am not expressing things well - but let me try again - if we take the example of the well worn light clock, we get the interval transforms directly because the result depends from two over and back beams as in MMx (in actuality a two light clock experiment). We are not measuring one way velocity, but only round trip velocity. The fact that the round trip velocity is constant does not lead to one way isotrophy. Round trip experiments always involve a (v/c)^2 (second order correction) whereas one way experiments involve (v/c) as in aberration.
(you of course already know this - but point I am attempting to make is that the extension of the over and back experiment(s) to one way isotrophy does not follow). The interval transforms are second order corrections - when you go in reverse you do not necessarily recover the vx/c^2 term unless you reintroduce Einstien's convention that the over and back time periods are equal.