Is E=MC2 the Complete Equation for Energy-Mass Relation?

[ravisastry]

Dear All,

what's the correct relation between Energy & Mass. I've read somewhere that E=MC2 is not complete and the nuclear labs use a similar, modified equation ? pls guide me and if someone can provide links to the proper derivation of this equation, it'll be great.

 

[pervect]

Are you perhaps looking for (m c^2)^2 = E^2 - (pc)^2?

 

[dark88]

try this link
http://www.karlscalculus.org/einstein.html

 

[pmb_phy]

Dear All,

what's the correct relation between Energy & Mass. I've read somewhere that E=MC2 is not complete and the nuclear labs use a similar, modified equation ? pls guide me and if someone can provide links to the proper derivation of this equation, it'll be great.

The only thing that I'm aware of is that there is a difference of opinion of what the m is in that equation. Some use it to refer to relativistic mass while others use it to refer to proper mass. That expression does fail when the system is not a closed system (e.g. a rod with forces acting to compress the rod).

Pete

 

[ravisastry]

somehow.. I'm having a strange feeling, thought this equation may be correct, the correct explanation for the assumptions behind it is lacking. Why is the speed of light constant, irrespective of the observers motion ?

 

[pmb_phy]

somehow.. I'm having a strange feeling, thought this equation may be correct, the correct explanation for the assumptions behind it is lacking. Why is the speed of light constant, irrespective of the observers motion ?

Nobody knows why the speed of light is Lorentz invariant (i.e. remains unchanged by a Lorentz transformation, i.e. change from one inertial frame to another.

A derivation of E = mc² is found at

http://www.geocities.com/physics_world/sr/mass_energy_equiv.htm
http://www.geocities.com/physics_world/sr/einsteins_box.htm

http://www.geocities.com/pmb_phy/ref/sachs_1973.pdf
http://www.geocities.com/pmb_phy/ref/warren_1976.pdf

Pete

 

[pervect]

somehow.. I'm having a strange feeling, thought this equation may be correct, the correct explanation for the assumptions behind it is lacking. Why is the speed of light constant, irrespective of the observers motion ?

The constancy of the speed of light came from experiment (starting with the Michelson-Morley experiment, which eventually won the experimenters the Nobel prize), not from any theoretical assumptions. In fact, the result was quite a surprise to the experimenters.

The constancy of the speed of light has since been confirmed by numerous different experiments, the latest of which are much more precise than the original experiments.

Theories have to be made to fit the facts, not the other way around.

 

[MeJennifer]

Nobody knows why the speed of light is Lorentz invariant (i.e. remains unchanged by a Lorentz transformation, i.e. change from one inertial frame to another.

Well one way to look at it (as Minkowski did almost 100 years ago!) is that lightspeed is not a velocity at all but simply the number of meters in one second!

 

[clj4]

Nobody knows why the speed of light is Lorentz invariant (i.e. remains unchanged by a Lorentz transformation, i.e. change from one inertial frame to another.

The answer is simple : because the Lorentz transform derivation is predicated on c=const. Look at Einstein's 1905 derivation of the Lorentz transforms.

 

[pmb_phy]

The answer is simple : because the Lorentz transfor derivation is predicated on c=const. Look at Einstein's 1905 derivation of the Lorentz transforms.

c = invariant cannot be proven. It is one of the postulates/laws of special relativity, i.e. nobody knjows why c = invariant.

What you've stated here makes no sense to me. Why you refer to "the answer" above, what is the question to which you are answering? Is it "why is c invariant?" If your answer is "because the Lorentz transformation is predicated on c=invariant." then that is no answer at all. The question was "why" c = invariant. That the Lorentz transformation is predicated on c = invariant it does imply prove that c = invariant. It only assumes it.

Look at Einstein's 1905 derivation of the Lorentz transforms.

I've read it several times over years, thanks. I'd recommend that you yourself take another look at it. In the beginning section Einstein states the invariance of the speed of light as a postulate just as I explained above. To be exact Einstein wrote

...also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.

Nobody in the history of physics has ever derived the constancy of c and thus we don't know why it is constant. Postulates are those assertions which cannot be derived from other, more basic, postulates.

Pete

 

[clj4]

c = invariant cannot be proven. It is one of the postulates/laws of special relativity, i.e. nobody knjows why c = invariant.

What you've stated here makes no sense to me. Why you refer to "the answer" above, what is the question to which you are answering? Is it "why is c invariant?" If your answer is "because the Lorentz transformation is predicated on c=invariant." then that is no answer at all. The question was "why" c = invariant. That the Lorentz transformation is predicated on c = invariant it does imply prove that c = invariant. It only assumes it.
I've read it several times over years, thanks. I'd recommend that you yourself take another look at it. In the beginning section Einstein states the invariance of the speed of light as a postulate just as I explained above. To be exact Einstein wrote
Nobody in the history of physics has ever derived the constancy of c and thus we don't know why it is constant. Postulates are those assertions which cannot be derived from other, more basic, postulates.

Pete

Your question was (textually):

"Why is c Lorentz invariant"
My answer stands, what I answered is that your question made no sense because the Lorentz transforms are predicated on c=const.

There is also ample experimental proof that light speed is invariant:
-wrt the speed of the light source
-wrt the speed of the observer

I can give you the list of those experiments.

 

[pmb_phy]

Your question was (textually):

"Why is c Lorentz invariant"

Sorry clj4 but I never asked that question. I used the term "Lorentz invariant" to refer to those quantities (scalars) which do not change as you change from one set of spacetime coordinates S to another S' where both S and S' correspond to inertial frames of reference.
My answer stands, what I answered is that your question made no sense because the Lorentz transforms are predicated on c=const.

There is also ample experimental proof that light speed is invariant:
-wrt the speed of the light source
-wrt the speed of the observer

I can give you the list of those experiments.

You can list them if you'd like to but I don't claim that the c=const. postulate is incorrect. In fact I hold it to be correct. One can't prove the second postulate by observing nature. One can only confirm predictions made by the postulate. I.e. you're speaking about experimental confirmation, not of proof.

Pete

 

[clj4]

Nobody knows why the speed of light is Lorentz invariant (i.e. remains unchanged by a Lorentz transformation, i.e. change from one inertial frame to another.

Your exact post, right? The answer I gave you stands.

 

[ravisastry]

guys, pervect mentions that constancy of c was derived experimentally. well, then y did einstein mention it as a postulate while deriving E=MC2 ?
and how far the photon theory of light is consistent.. I agree that the photo electric effect won the nobel.. but let's hit the core... A ray of light consists of photos ? if yes, what the distance between each of the photos, what's their properties like .. do they attract each other ? or stuff like that..

 

[pmb_phy]

guys, pervect mentions that constancy of c was derived experimentally. well, then y did einstein mention it as a postulate while deriving E=MC2 ?

Things like the constancy of light are never derived] by any type of experiment. They are merely observed or experimentally confirmed.

and how far the photon theory of light is consistent.. I agree that the photo electric effect won the nobel.. but let's hit the core... A ray of light consists of photos ?

The notion of a beam of light consisting of photons is consistent with observation. Observation is also consisent with light being a wave. That's where the particle-wave duality came into being. Whether photons really exist or not is hard to say. I read an article once by one of those Nobel Laureates (Willis Lamb) in which he asserted that photons don't exist.

Pete

 

[pmb_phy]

Your exact post, right? The answer I gave you stands.

Your responded to

Nobody knows why the speed of light is Lorentz invariant (i.e. remains unchanged by a Lorentz transformation, i.e. change from one inertial frame to another.

with "The answer is simple : because the Lorentz transform derivation is predicated on c=const." You claimed that I asked "Why is c Lorentz invariant" when I never asked this question to anyone at anytime. I only asked you if Is it "why is c invariant?" was the question you were anwsering. You never responded to this question. You seemed to be objecting to my using "c is Lorentz invariant" as a statement of the second postulare of relativity. It appears to me that you think this is circular logic. It is not. The expression for the Lorentz transformation is derivced with the second postulate. When I used the term "Lorentz transformation" I was referring to "that change in coordinate such that c=constant."

Long story short - The only thing that I was asking you was what was the question to which you posted an answer when you said "The answer is simple:" Complaints/objections about my use of the term "Lorentz transformation" detract from the users question. His question was "why is c=constant?" The answer is "Nobody knows why." I was trying to avoid the use of the phrase "c=constant" since I've never liked that phrase. It seems to me that some people would misread this law as "c does not change in time."

Pete

 

[Azael]

Doesnt maxwells equations imply c=constant?

 

[pmb_phy]

Doesnt maxwells equations imply c=constant?

There is a more general form of Maxwell's equations which has the value of the photon's proper mass explicitly in Maxwell's equations. If this value is different than zero then the speed of light is not constant. The corresponding Lagrangian is called the Proca Lagrangian.

See

http://en.wikipedia.org/wiki/Variable_speed_of_light

Pete

 

[Office_Shredder]

Doesnt maxwells equations imply c=constant?

Short answer: yes.

 

[pmb_phy]

Short answer: yes.

That is incorrect. The most general form of Maxwell's equation accounts for the photon's proper mass. The constancy of the speed of light is tied to the proper mass of the photon. Postulating c=constant is equivalent to postulating that the photon's proper mass is zero. However it has never been proved that the photon's proper mass is zero. All that has been done was to reduce the photon proper mass to a small quantity. The expermimental evidence of photon proper mass does not allow us yet to assume that the value is zero. There is, however, an upper bound to it.

Otherwise if you believe that the "short answer" is yes then please provide proof. So far in relativity today it is a postulate and thus cannot be proven.

Pete

 

[pervect]

I like the short answer, and I would not call it incorrect, either.

I would agree that it's possible that photons might have a very small rest mass and that Maxwell's equations would have to be revised if photons did have a rest mass.

The key point being that a revision to the equations would be necessary. Thus Maxwell's equations (as you would look them up in Wiki) do imply that light moves at 'c'.

 

[clj4]

However it has never been proved that the photon's proper mass is zero. All that has been done was to reduce the photon proper mass to a small quantity. The expermimental evidence of photon proper mass does not allow us yet to assume that the value is zero. There is, however, an upper bound to it.

It is not possible to measure a "zero" . This is due to the limitations of our measuring devices. Nevertheless the mass of the photon has been measured repeatedly and the measurements have been converging rapidly towards zero thru a set of incredibly small quantities.

A limit on the photon mass can be obtained through satellite
measurements of planetary magnetic fields. The Charge Composition
Explorer spacecraft was used to derive a limit of 6x10^-16 eV with high
certainty. This was slightly improved in 1998 by Roderic Lakes in a
laborartory experiment which looked for anomalous forces on a Cavendish
balance. The new limit is 6x10^-17 eV. See here:

http://pdg.lbl.gov/2005/tables/gxxx.pdf

Studies of galactic magnetic fields suggest a much better limit of less
than 3x10^-27 eV but there is some doubt about the validity of this
method.
References:

[1]E. Fischbach et al., Physical Review Letters, 73,
514-517 25 July 1994.


[2]http://pdg.lbl.gov/2005/tables/gxxx.pdf

 

[rbj]

Doesnt maxwells equations imply c=constant?

Short answer: yes.

That is incorrect. The most general form of Maxwell's equation accounts for the photon's proper mass. The constancy of the speed of light is tied to the proper mass of the photon. Postulating c=constant is equivalent to postulating that the photon's proper mass is zero.

Pete, i also have to disagree. Maxwell's equations are about the wave-like property of light (or, more generally, E&M radiation). Maxwell's equations exist without any reference to particle-like properties of light. no photon mass, no photons at all.

the reason that it was reasonable (but extremely insightful) for Einstein to postulate that $c$ is constant for all intertial observers is because there was nothing in Maxwell's equations to suggest that there was a medium for the E&M wave to propagate in (and this was, of course, sort of confirmed with Michaelson-Morley). then if you cannot tell the difference between a vacuum moving past your face at high velocity and a "stationary" vacuum, then there is no reason for $\epsilon_0$, $\mu_0$, or $c^2 = 1/(\epsilon_0 \mu_0)$ to be different for different inertial observers. while it seemed radical at the time, in fact, it was the most reasonable conclusion to come to. otherwise (since $\mu_0$ was defined by the definition of unit current) you would have to explain why one inertial observer in a vacuum experienced a different $\epsilon_0$ than another and there is no good reason for such a difference.

the postulates that no inertial frame is qualitative different (or "better") than any other inertial frame of reference and that we can't tell the difference between a "stationary" vacuum and a vacuum "moving" past our faces at a high velocity, that there is no difference and that Maxwell's Equations should work the same for any and all inertial frames so then the speed of E&M must be measured to be the same in all inertial frames, even if it is the same beam of light viewed by two observers moving relative to each other.

 

[pmb_phy]

Pete, i also have to disagree. Maxwell's equations are about the wave-like property of light (or, more generally, E&M radiation). Maxwell's equations exist without any reference to particle-like properties of light. no photon mass, no photons at all.

That is only true if one assumes that the photon's proper mass is zero, i.e. add the photon mass zero postulate of you define the term Maxwell's equations to be those equations of electrodynamics for which the proper mass is zero. Do you have the text Classical Electrodynamics - 3rd Ed. J. D. Jackson, Wiley Press (1999)? If so then please see Section 12.8 Proca Lagrangian; Photon Mass Effects page 600. It reads

The conventional Maxwell equations and the Lagrangian (12.85) are based on the hypothesis that the photon has zero mass. As discussed in the Introduction, it can always be asked what evidence there is for the masslessness of the photon or equivalently for the inverse suqare law of the Coulomb force and what consequences would result from a nonvanishing mass. We modify the Lagrangian density (12.85) by adding a "mass" term. The resulting equation is known as the Proca Lagrangian, Proca having been the first to consider it (1930, 1936). The Proca Lagrangian is

$$L_{Proca} = -\frac{1}{16\pi} F_{\alpha\beta}F^{\alpha\beta} + \frac{\mu^2}{8\pi}A_{\alpha}A^{\alpha} - \frac{1}{c}-J_{\alpha}A^{\alpha}$$

The parameter $\mu$ has dimensions of inverse length and is the reciprocal Compton wavelength of the photon ($\mu = m_{\lambda}/{\hbar}$. Instead of (12.89) the Proca equations of motion are

$$\partial^{\alpha}F_{\alpha\beta} + {\mu}^2A_{\alpha} = \frac{4\pi}{c}J_{\alpha}$$

The most general equations of electrodynamics therefore include a mass term. So what you say is true if and only if one starts with the hypothesis that the photon's proper mass is zero and one understands the term Maxwell's equations to be those equations of electrodynamics for which the proper mass is zero. The experimental lower bound for the photon mass is

$m_{\gamma} < 4x10^{-51}$kg

Otherwise it appears to me that you're postulating that the photon proper mass is zero. What experimental evidence do you know for this? For the reason why this is not experimentally found to be zero please see page 5-9 in Jackson in the section labeled 1.2 Inverse Square Law or the Mass of the Photon.

Pete

ps - I too once thought as you do. I even wrote a letter to the editor of the American Journal of Physics on this topic. The letter was rejected by the editor for publication because the experimetal lower limit on the photon's proper mass is not zero.

 

[clj4]

That is only true if one assumes that the photon's proper mass is zero, i.e. add the photon mass zero postulate of you define the term Maxwell's equations to be those equations of electrodynamics for which the proper mass is zero. Do you have the text Classical Electrodynamics - 3rd Ed. J. D. Jackson, Wiley Press (1999)? If so then please see Section 12.8 Proca Lagrangian; Photon Mass Effects page 600. It reads


The most general equations of electrodynamics therefore include a mass term. So what you say is true if and only if one starts with the hypothesis that the photon's proper mass is zero and one understands the term Maxwell's equations to be those equations of electrodynamics for which the proper mass is zero. The experimental lower bound for the photon mass is

$m_{\gamma} < 4x10^{-51}$kg

Otherwise it appears to me that you're postulating that the photon proper mass is zero. What experimental evidence do you know for this? For the reason why this is not experimentally found to be zero please see page 5-9 in Jackson in the section labeled 1.2 Inverse Square Law or the Mass of the Photon.

Pete

ps - I too once thought as you do. I even wrote a letter to the editor of the American Journal of Physics on this topic. The letter was rejected by the editor for publication because the experimetal lower limit on the photon's proper mass is not zero.

The editor of AJP is an idiot, some of us know that. The mere fact that he stated an impossibility : "the experimetal lower limit on the photon's proper mass is not zero." proves it. See my previous post, we can never measure a "zero", we can only approach it asymptotically. If the editor of AJP (and we know who he is) had any knoewledge of experimental physics on one hand and of QED on the other hand he would not have written back to you such an idiocy. You shouldn't allow such situations influence you.

 

[pmb_phy]

The editor of AJP is an idiot, some of us know that. The mere fact that he stated an impossibility : "the experimetal lower limit on the photon's proper mass is not zero." proves it.

Actually it proves nothing. Though I don't care much for the editor's personality in my opinion he was very correct as Proca demonstrated and for which Jackson clarifies and details.The fact that we can never measure something to be exactly zero only goes to prove that we can never measure the photon's proper mass to be exactly zero. A similar situation occurred for neutrinos. Why do you think that it took so long for physicists to show that a nuetrino has non-zero proper mass? Many physicists before the experimental work was done assume, incorrectly, that the proper mass of the neutrino was zero, only later to be proven wrong.

If the editor of AJP (and we know who he is) had any knowledge of experimental physics on one hand and of QED on the other hand he would not have written back to you such an idiocy. You shouldn't allow such situations influence you.

It is on experimental physics on which the non-zero lower bound exists.

It appears that you believe that I changed my mind by taking the editor's comment on pure faith. You are dead wrong. What influences me is what I undertake to study.

The statement regarding the editor was in a "ps" and not part of my post. It was used strictly as mere side note, hence the ps. Never in my entire life have I ever taken anything on mere assertions made by others. After I read the editor's comments I looked into the Proca Lagrangian and realized that he is correct if Jackson's text is correct, which I believe it to be. In any case in this thread I'm not going by the editor. I'm going by Jackson. Exactly why do you think people speak of the Proca Lagrangian in modern physics? Are you familiar with this Lagrangian?

Recall a post by robphy who once said (https://www.physicsforums.com/archive/index.php/t-76872.html)

The "speed of light" axiom would then need to be dropped and only the principle of relativity would remain.


The "speed of light" axiom would probably be replaced by something equivalent to a "maximum signal speed" axiom.

Bravo robphy. Nicely stated.

Regarding QFT; I'm not fluent in that theory so I have no comment on it. But I do suggest that you read the wikedia link that I gave above.

See also http://www.iop.org/EJ/abstract/0034-4885/68/1/R02/

In any case this is getting off point. Azael 's question regarded whether Maxwell's equations prove that c = constant.

The following is what I stand by

The generalization of Maxwell's equations (i.e. the Proca Euations) implies otherwise that c = constant if and only if $m_{\gamma}$ = 0.[/itex]. If and only if one postulates a zero proper mass for the photon can one make such an assertion and be correct. I.e. the only way to drop one postulate is to assert another. I.e. one can drop the second postulate (c=const.) if and only if one assumes another.

And it is on that I rest and thus bow out of this thread since any other contribution I make will simply be repeating what I've already stated. I really hate those never ending debates. :)

Pete

ps - I believe that Rielly Atkinson will chime in on the QFT and its relationship to nonzero photon mass.

Rielly - Does experimental evidence exist which comfirms that the photon mass is exactly zero?

 

[jtbell]

Doesnt maxwells equations imply c=constant?

Not by themselves. I think you also have to assume that Maxwell's equations hold in all inertial reference frames, in order to conclude that c is constant.

Before Michelson-Morley and similar experiments, I think most physicists assumed that Maxwell's equations are valid only in the reference frame in which the luminiferous ether (the hypothetical medium that carries light waves) is at rest. In other reference frames, one would have to modify Maxwell's equations to include terms that contain the relative velocity of the ether, and of course the speed of light in those frames would not equal c.

 

[Meir Achuz]

Not by themselves. I think you also have to assume that Maxwell's equations hold in all inertial reference frames, in order to conclude that c is constant.

Before Michelson-Morley and similar experiments, I think most physicists assumed that Maxwell's equations are valid only in the reference frame in which the luminiferous ether (the hypothetical medium that carries light waves) is at rest. In other reference frames, one would have to modify Maxwell's equations to include terms that contain the relative velocity of the ether, and of course the speed of light in those frames would not equal c.

It would even be worse. c would have to be a 2nd rank tensor, varying with the seasons.

 

[CarlB]

It would even be worse. c would have to be a 2nd rank tensor, varying with the seasons.

The standard view is that relativity is a principle of the universe that is perfect and exact. Of course this has not been tested to much accuracy, human limitations being what they are. But in that view, there is no question that c has to be a scalar.

But if you take the view of Poincare, that there is an ether, just very difficult to ascertain (or look at the flat space gauge gravity theory which makes the assumption of a flat ether quite natural), then it also makes sense to generalize c.

The Dirac equation uses c. This equation can be used for any spin-1/2 particle, for example electrons, or neutrinos. The equation covers a single particle of indeterminant type. Now if you want to generalize the Dirac equation so that it covers a single particle of a more general indeterminant type, then it is natural to have to expand those 4x1 spinors out to something larger. An obvious choice is 4x4 spinors, or matrices. That is, each of the four columns makes an independent Dirac equation with no interference, voila, it looks to Mr. experimenter to be four separate particles that each just happen to coincidentally use the same Dirac equation.

But in doing this, you also have an opportunity to generalize c. In this context, c is a scalar, or what is the same thing, a scalar multiple of the unit 4x4 matrix. Just so long as you preserve the Dirac equation, there is no particular reason to suppose that c is scalar. In this case, making c more general means that your four particles get mixed together.

Now where this gets interesting is when you algebraically determine exactly what choices you can make for c as a 4x4 matrix, and still get the Dirac equation. The result looks a lot like electroweak symmetry breaking. I typed up the results (but written in the language of the Geometric Algebra) here:
http://brannenworks.com/PPANIC05.pdf

So is c a constant? Yes, if you limit yourself to just E&M, but when you look at more general forces, it makes sense to generalize c. Einstein's relativity is all about light, and that is only a very limited part of what kinds of things go moving around in spacetime. Light is very simple, matter is a lot more complicated. So why should the use of c in describing matter be the same as the c used in describing light? For more general particles, you may need a more general c. Now I'm not saying that you can't do this without generalizing c. Of course you can do the same thing by simply assuming another variable to break the symmetry. But since you can do it with c, why not keep the number of arbitrary stuff down and use c. As soon as you agree that spontaneous symmetry breaking is needed, you already admit that you're going to have to break something.

Carl

 

[reilly]

The answer is simple : because the Lorentz transform derivation is predicated on c=const. Look at Einstein's 1905 derivation of the Lorentz transforms.

Einstein's or Lorentz's derivations, or Maxwell's Eq.'s in vacuum are all elegant and profound -- but if they did not do well when exposed to Nature, we would not know much about them. And they are all based on the vacuum speed of light being constant for all inertial frames, so that they do well when confronted with experiments and observations

Pete is right -- we don't have a clue why c=constant for all intertial observers -- one could consider this equality a gift from Nature, but that doesn't say much.
Regards,
Reilly Atkinson