E=mc^2 proof. Why use Newtonian kinetic energy?

[SamRoss]

E=mc^2 proof. Why use Newtonian kinetic energy?

In "Does the Inertia of a Body Depend Upon its Energy-Content?" Einstein says that the difference in kinetic energy of a body before and after it releases some energy is...

K(0)-K(1)=E(gamma-1)

In his previous paper, he worked out the kinetic energy of masses as...

K=mc²(gamma-1)

Can't we immediately compare the two equations and conclude that E=mc^2? Why does Einstein bother converting the first equation above into

K(0)-K(1)=(1/2)(E/c²)v²

? It seems to me that this is a less precise method since he had to neglect magnitudes of fourth and higher order to get it.

**I know that me writing out gamma looks really dumb but I was having trouble with Latex.

 

[bcrowell]

In his previous paper, he worked out the kinetic energy of masses as...
K=mc²(gamma-1)

Wait, are you sure about this? I don't think he derives this until the "Does the inertia..." paper.

Why does Einstein bother converting the first equation above into

K(0)-K(1)=(1/2)(E/c²)v²

? It seems to me that this is a less precise method since he had to neglect magnitudes of fourth and higher order to get it.

This is a good question. I was just puzzling over the same thing myself. I think it's because he wants to make contact with his audience's knowledge of the Newtonian theory, and also because he's about to propose some experimental tests. For example, if you measure the inertia of some radium salts before and after decay, you're doing it under completely Newtonian conditions, i.e., the sample is not moving at speeds comparable to c, so any higher-order corrections aren't negligible.

**I know that me writing out gamma looks really dumb but I was having trouble with Latex.

Here is how to write that: K=mc^2(\gamma-1). To see how I did that, click on the Quote button on my post. Sometimes it shows up after you submit your post without rendering the math, but if you reload the page it will render correctly. Other people will always see it rendered correctly.

 

[SamRoss]

Wait, are you sure about this? I don't think he derives this until the "Does the inertia..." paper.

I often look at the book "The Principle of Relativity" from Dover Publications, which is just a reprinting of a lot of original papers on the subject. Assuming the reprintings are accurate, Einstein derives the kinetic energy formula in section 10 of "On the Electrodynamics of Moving Bodies".

I think it's because he wants to make contact with his audience's knowledge of the Newtonian theory, and also because he's about to propose some experimental tests.

Yeah, that makes a lot of sense. "Does the Inertia" came out only shortly after "Electrodynamics". Einstein might not have felt confident that his own kinetic energy formula would have been accepted by his audience. His audience might not have even seen the formula if they hadn't read his other paper. Thanks for the insight.

 

[bcrowell]

I often look at the book "The Principle of Relativity" from Dover Publications, which is just a reprinting of a lot of original papers on the subject. Assuming the reprintings are accurate, Einstein derives the kinetic energy formula in section 10 of "On the Electrodynamics of Moving Bodies".

I see. That's interesting. So in "On the electrodynamics..." he proves it by calculating work, and in "Does the inertia..." he derives it in a completely different way. There are some remarks about the definition of force in "On the electrodynamics..." (including footnote 9) that show that he feels the whole thing is problematic.