Can Rest Energy be Derived from the Stress-Energy Tensor?

[Count Iblis]

Einstein's 1905 paper didn't demonstrate that $$E = mc^2$$ did it? It rather demonstrated that $$\delta(E) = \delta(mc^2)$$

But it then directly follows that adding an energy of E to an object in any form will increase its mass by E/c^2. Because after the radiation is absorbed, the energy can be converted in any other form (such a transformation is irrelevant in the argument). The whole point of Einstein's reasoning is to pinpoint the reason why an object has an inertia. He shows that the intertia depends on energy and nothing else than energy. Therefore we don't need to postulate an independent physical quantity for inertia (i.e. mass) anymore. What we call mass is the same as the energy content of an object.

 

[bcrowell]

I think he wrote that the fact that a conserved four momentum exists at all should be derived (e.g. by assuming Lorentz invariance of the Lagrangian and then using Noether's theorem, but he doesn't elaborate, I think he only mentions that it requires a knowledge of field theory) and that simply assuming that there exists a conserved four momentum and then deriving the expression is not a rigorous argument.

I guess this is a matter of taste, then, because I see it the other way around. Writing down a Lagrangian is most likely going to boil down to making a guess based on aesthetics, looking for the simplest expression that will make sense physically. IMO the other version is much more rigorous.

Of course we don't have anything in physics that is totally equivalent to the mathematician's concept of a proof -- at least not in the kind of context we're talking about here, where we're trying to extend an old theory to be consistent with newly imposed principles. This is probably why there's so much room for disagreement, with, e.g., Einstein and Ohanian disagreeing on whether Einstein's 1905 derivation is correct.

 

[PhilDSP]

Einstein's 1905 paper didn't demonstrate that $$E = mc^2$$ did it? It rather demonstrated that $$\delta(E) = \delta(mc^2)$$

Sorry about the strange looking equation by the way. Thanks to bcrowell's post I see now that the tex Delta argument must be capitalized.

$$\Delta(E) = \Delta(mc^2)$$

 

[snoopies622]

Well, thanks to everyone for helping me out here. I'm still trying to understand dx's posts but he and I have started discussing them via visitor messages so this thread won't become burdened with my ignorance of the calculus of variations. I'm also still thinking about bcrowell's post #16 and I like the reasoning behind it.