This post was initially intended to be a one-liner:
One might find it amusing that this embarrassing realization is keeping me up at this unearthly hour (it's 4.30am now); but just imagine how Einstein managed catch any sleep at all after deducing E=mc^2 and its corollary -- the atomic bomb. After all, humans have been known to go to war, pre-emptively, over nuclear weapons.
Anyway, I suspect that he might have lost even more sleep upon realizing that "...henceforth space by itself, and time by itself, are doomed to fade away into mere shadows..."
But, really, why is E=mc^2 ?!
I spent a long time on google trying to find the answer, but mostly found unsatisfactory circular arguments. I also discovered that the meaning of E=mc^2 is not at all clear. For example, it is somewhat of a myth that mass can be converted into energy --- indeed the popular explanation for the mysterious "source" of atomic or nuclear energy. Another popular saying is that mass and energy are two forms of the "same thing". What does that even mean anyway?! Mass and energy are the same thing......!
Perhaps one should first recognize the distinction between matter and mass. These are more or less the same thing for Newton: matter has mass, stuff that has mass is matter. But in Einstein's relativity, mass, and even "rest mass" does not really behave like matter. (That's not to say that it is clear what "matter" is, in the first place; suffice to say, matter is "stuff" and mass/energy is some property of "stuff".) The rest mass of a system of two objects need not be equal to the sum of the individual rest masses! Anyway, the point is that it is more accurate to say that binding energy, rather than mass, is converted to radiant energy in an atomic bomb.
The upshot of all this babble is rather shocking. Assuming Einstein's own postulate about relativity (i.e. the speed of light c is a constant), E=mc^2 is nothing more than a definition of (relativistic) mass in terms of the energy E. One then asks what energy is, and this is truly a deep question. My own understanding is that it appears, via Noether's theorem, as a conserved quantity corresponding to the symmetry of physics under time translations.
So, is the most famous equation in Physics nothing more than a smoke-bomb?
I can't give a definitive answer, since I'm rather confused myself, and also quite groggy. Einstein himself derived the formula without recourse to his relativity. He simply imagined a box with a light and realized that mass (understood as resistance to motion) had to be equivalent to energy (multiplied by c^2)! In this context, the speed of light c is not assumed to be a constant, so that E=mc^2 is not sufficient for special relativity. This is also the reason why my one-liner didn't make complete sense when I thought a bit harder, leading to a much longer post than I had originally planned.
Relativity is a bitch.
Do we really need another atomic bomb to appreciate the relativity of space and time?Everyone knows that the most famous equation in physics is E=mc^2. Most people know that it is related to nuclear-ish things and the inexplicable thing called "relativity". Some people know that electromagnetism led to Einstein's special relativity, when he pondered what light would look like, if he was chasing after it. I discovered, to my greatest horror, that I don't quite know where E=mc^2 comes from!
One might find it amusing that this embarrassing realization is keeping me up at this unearthly hour (it's 4.30am now); but just imagine how Einstein managed catch any sleep at all after deducing E=mc^2 and its corollary -- the atomic bomb. After all, humans have been known to go to war, pre-emptively, over nuclear weapons.
Anyway, I suspect that he might have lost even more sleep upon realizing that "...henceforth space by itself, and time by itself, are doomed to fade away into mere shadows..."
But, really, why is E=mc^2 ?!
I spent a long time on google trying to find the answer, but mostly found unsatisfactory circular arguments. I also discovered that the meaning of E=mc^2 is not at all clear. For example, it is somewhat of a myth that mass can be converted into energy --- indeed the popular explanation for the mysterious "source" of atomic or nuclear energy. Another popular saying is that mass and energy are two forms of the "same thing". What does that even mean anyway?! Mass and energy are the same thing......!
Perhaps one should first recognize the distinction between matter and mass. These are more or less the same thing for Newton: matter has mass, stuff that has mass is matter. But in Einstein's relativity, mass, and even "rest mass" does not really behave like matter. (That's not to say that it is clear what "matter" is, in the first place; suffice to say, matter is "stuff" and mass/energy is some property of "stuff".) The rest mass of a system of two objects need not be equal to the sum of the individual rest masses! Anyway, the point is that it is more accurate to say that binding energy, rather than mass, is converted to radiant energy in an atomic bomb.
The upshot of all this babble is rather shocking. Assuming Einstein's own postulate about relativity (i.e. the speed of light c is a constant), E=mc^2 is nothing more than a definition of (relativistic) mass in terms of the energy E. One then asks what energy is, and this is truly a deep question. My own understanding is that it appears, via Noether's theorem, as a conserved quantity corresponding to the symmetry of physics under time translations.
So, is the most famous equation in Physics nothing more than a smoke-bomb?
I can't give a definitive answer, since I'm rather confused myself, and also quite groggy. Einstein himself derived the formula without recourse to his relativity. He simply imagined a box with a light and realized that mass (understood as resistance to motion) had to be equivalent to energy (multiplied by c^2)! In this context, the speed of light c is not assumed to be a constant, so that E=mc^2 is not sufficient for special relativity. This is also the reason why my one-liner didn't make complete sense when I thought a bit harder, leading to a much longer post than I had originally planned.
Relativity is a bitch.
