My physics teacher claimed to derive $E=mc^2$ by manipulating the equation for the speed of a standing wave on a string. A commonly known fact about the string wave equation is that speed can be calculated by $c^2=\frac{T}{\mu}$ where $\mu$ is the mass per unit length of the string. "Since Fds is energy we can work out $E = mc^2$". I feel like this is cheating. In fact, I know this is cheating but still, why does it work so well in terms of the units for energy? Isn't this just a funny coincidence?
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2$\begingroup$ related: Why is there an emergent quasi Lorentz symmetry in classical string lagrangian?. $\endgroup$– AccidentalFourierTransformCommented Feb 10, 2017 at 14:16
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$\begingroup$ These aren't really related. We already know that kinetic energy is generically quadratic in velocity, just by dimensional analysis. But $E = mc^2$ says there's an extra contribution to the energy even when objects aren't moving at all! $\endgroup$– knzhouCommented Feb 10, 2017 at 16:36
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2 Answers
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Here c is not the speed of light so the equation obtained is not the same as Einstein's equation. The fact that energy has dimensions of mass times some velocity squared is trivial and you can get it by dimensional analysis. Or just look at definition of kinetic energy.
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$\begingroup$ ok i'll accept this. but isn't it then strange that c^2 is simply a constant in the context of einstein's relativity? (E = m), yet it is also a speed squared. That's the essential 'coincidence' that must be explained then isn't it? $\endgroup$– user86425Commented Feb 10, 2017 at 15:41
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1$\begingroup$ @rpfphysics I'm not sure I understand what coincidence you're talking about. If you're relating mass to energy then you need to multiply by a speed squared to make the units work. Another example would be the expression for kinetic energy: you have energy on one side and mass on the other, and you're multiplying the mass by a speed squared ($v^2/2$). $\endgroup$– JoldCommented Feb 10, 2017 at 17:20
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$\begingroup$ c^2 is dimensionless though $\endgroup$– user86425Commented Feb 12, 2017 at 11:45
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If the relation is correct then it will be dimensionally correct too u can manipulate it to get any dimensionally correct relation
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$\begingroup$ And in above equation c is speed of standing wave not speed of light so the equation just looks like E =mc^2 $\endgroup$ Commented Feb 10, 2017 at 16:35