Historical origin of energy momentum equation?

[SamRoss]

Does anyone know who first posed the energy relation E²=m²c⁴+p²c² and where its original appearance can be found?

 

[Drakkith]

Maybe this can help you?
http://en.wikipedia.org/wiki/E=mc²#History

 

[tom.stoer]

You should check

http://www-itp.particle.uni-karlsruhe.de/~schreck/general_relativity_seminar/Zur_Elektrodynamik_bewegter_Koerper.pdf
http://www-itp.particle.uni-karlsruhe.de/~schreck/general_relativity_seminar/Ist_die_Traegheit_eines_Koerpers_von_seinem_Energieinhalt_abhaengig.pdf

Of course this relation follows immediately from the relativistiv energy and momentum - and these have been discussed by Einstein in his first papers - but I don't know whether he explicitly wrote down the relation you are asking for.

 

[SamRoss]

You should check

http://www-itp.particle.uni-karlsruhe.de/~schreck/general_relativity_seminar/Zur_Elektrodynamik_bewegter_Koerper.pdf
http://www-itp.particle.uni-karlsruhe.de/~schreck/general_relativity_seminar/Ist_die_Traegheit_eines_Koerpers_von_seinem_Energieinhalt_abhaengig.pdf

Of course this relation follows immediately from the relativistiv energy and momentum - and these have been discussed by Einstein in his first papers - but I don't know whether he explicitly wrote down the relation you are asking for.

I've read those papers (in English). Relativistic energy is in there although I don't think relativistic momentum is. In any case, do you know a proof that utilizes the two quantities? Most of the proofs that I see depend on the ambiguous relativistic mass.

 

[tom.stoer]

The modern view is that m² = E² - p² corresponds to the first Casimir of the Poincare group - but Einstein would hate that! So I guess the only (historically correct) way is to deal with the "relativistic mass" which Einstein didn't like, too, as you can see from some remarks like the following one from a 1948 letter to Lincoln Barnett:

"It is not good to introduce the concept of the mass M = m/(1-v²/c²)^½ of a body for which no clear definition can be given. It is better to introduce no other mass than 'the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion."

 

[SamRoss]

The modern view is that m² = E² - p² corresponds to the first Casimir of the Poincare group - but Einstein would hate that! So I guess the only (historically correct) way is to deal with the "relativistic mass" which Einstein didn't like, too, as you can see from some remarks like the following one from a 1948 letter to Lincoln Barnett:

"It is not good to introduce the concept of the mass M = m/(1-v²/c²)^½ of a body for which no clear definition can be given. It is better to introduce no other mass than 'the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion."

What do you think Einstein would have said the correct proof was?

 

[tom.stoer]

Einstein observed that introducing velocity dependend xyz-masses always leads to confusion. Just look at p(v) = mv /(1-v²/c²)^½; introducing M(v) = m /(1-v²/c²)^½ one can write p(v) = M(v)*v which looks rather familiar; it seems that one can save the Newtonian equation for the momentum p.

But what about E(v); neither the total nor the kinetic energy can be re-written using the same trick. E(v) = M(v)*v^2 / 2 is nonsense, and E = M(v)*c² is correct but does not correspond to any Newtonian formula. So all equations Einstein derived are correct, but using new terms for new, velocity dependend entities does not help.

 

[jtbell]

In any case, do you know a proof that utilizes the two quantities? Most of the proofs that I see depend on the ambiguous relativistic mass.

$$E = \frac{m_0 c^2} {\sqrt{1 - v^2/c^2}}$$

$$p = \frac{m_0 v} {\sqrt{1 - v^2/c^2}}$$

Use algebra to eliminate v between these two equations. Sheldon Cooper would say, "Easy-peasy."