# Correct Energy Mass equation

1. Jul 15, 2006

### ravisastry

Dear All,

whats 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.

2. Jul 15, 2006

### pervect

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

3. Jul 15, 2006

### dark88

http://www.karlscalculus.org/einstein.html [Broken]

Last edited by a moderator: May 2, 2017
4. Jul 15, 2006

### pmb_phy

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

5. Jul 16, 2006

### 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 ?

6. Jul 16, 2006

### pmb_phy

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

7. Jul 16, 2006

### pervect

Staff Emeritus
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.

Last edited: Jul 16, 2006
8. Jul 16, 2006

### MeJennifer

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!

9. Jul 16, 2006

### clj4

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

Last edited: Jul 16, 2006
10. Jul 16, 2006

### pmb_phy

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

Last edited: Jul 16, 2006
11. Jul 16, 2006

### clj4

"Why is c Lorentz invariant"

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.

12. Jul 17, 2006

### pmb_phy

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.
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

13. Jul 17, 2006

### clj4

14. Jul 17, 2006

### 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 lets hit the core... A ray of light consists of photos ? if yes, what the distance between each of the photos, whats their properties like .. do they attract each other ? or stuff like that..

15. Jul 17, 2006

### pmb_phy

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

16. Jul 17, 2006

### pmb_phy

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 refering 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

17. Jul 17, 2006

### Azael

Doesnt maxwells equations imply c=constant?

18. Jul 17, 2006

### pmb_phy

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

19. Jul 17, 2006

### Office_Shredder

Staff Emeritus

20. Jul 17, 2006

### pmb_phy

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

21. Jul 17, 2006

### pervect

Staff Emeritus
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'.

22. Jul 17, 2006

### clj4

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

23. Jul 17, 2006

### rbj

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, sorta 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.

24. Jul 18, 2006

### pmb_phy

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.

Last edited: Jul 18, 2006
25. Jul 18, 2006

### clj4

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.