HPS 0410 Einstein for Everyone

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The Calculus Behind
The World's Quickest Derivation of E = mc2

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

Consider a body of mass $$m$$ that moves at a speed $$v$$ very close to the speed of light. A force $$F$$ acts on it and, as a result, the energy $$E$$ of the body increases. The speed of the body $$v$$ cannot exceed c and, as the force continues to act, the speed $$v$$ approaches c asymptotically. We have:

$$\dfrac{dE}{dt} = Fv = \dfrac{d(mv)}{dt}v = v^2 \dfrac{dm}{dt} + mv\dfrac{dv}{dt}$$

In this last equation, t is the independent variable. So the equation tracks how energy $$E(t)$$, mass $$m(t)$$ and speed $$v(t)$$ grown as a function of time t.

Our concern, however, is to see how mass grows as we increase the energy of the body. To track that, we need to make energy $$E$$ the independent variable; and to take time $$t(E)$$, mass $$m(E)$$ and speed $$v(E)$$ all to be functions of $$E$$. Multiplying the last equation by $$\dfrac{dt(E)}{dE}$$and using the chain rule, we recover:

$$1 = v^2 \dfrac{dm}{dt}\dfrac{dt}{dE} + mv \dfrac{dv}{dt}\dfrac{dt}{dE} = v^2 \dfrac{dm}{dE} + mv \dfrac{dv}{dE}$$ Rearranging we recover $$\dfrac{dm}{dE} = \dfrac{1}{v^2} - \dfrac{m}{v}\dfrac{dv}{dE}$$

We now take the limit in which the energy $$E$$ grows large. In that limit, $$v$$ approaches c asymptotically and $$\dfrac{dv}{dE}$$ approaches zero. In this limit we have: $$\lim_{E \to \infty} \dfrac{dm}{dE} = \dfrac{1}{c^2}$$

This equation is the result. It tells us that mass $$m$$ grows incrementally by $$\dfrac{1}{c^2}$$ for each unit of energy $$E$$ added. While this result holds generally, this simple demonstration returns the result only in the special case in which the energy $$E$$ of the body has grown so large that its speed is close to c.

Thanks to Christian Seberino for suggesting this calculation.

Copyright John D. Norton. May 8, 2015.