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Mass and Energy

In the hand-waving spirit of the preceding section, let's explore the consequences of Eq. (6). The BINOMIAL EXPANSION of $\gamma$ is

$$\gamma \; = \; (1 - \beta^2)^{-\onehalf} \; = \; 1 \; + \; \onehalf \, \beta^2 \; - \; \cdots$$ (24.7)

For small $\beta$,  we can take only the first two terms (later terms have still higher powers of $\beta \ll 1$ and can be neglected) to give the approximation

$$m' \; \approx \; (1 + \onehalf \beta^2) \; m \qquad \hbox{\ . . . . . . r} \qquad m' c^2 \; \approx \; m c^2 \; + \; \onehalf m u^2$$ (24.8)

The last term on the right-hand side is what we ordinarily think of as the KINETIC ENERGY T.  So we can write the equation (in the limit of small velocities) as

$$T \; = \; \gamma \, m c^2 \; - \; m c^2$$ (24.9)

It turns out that Eq. (9) is the exact formula for the kinetic energy at all velocities, despite the ``handwaving'' character of the derivation shown here.

We can stop right there, if we like; but the two terms on the right-hand side of Eq. (9) look so simple and similar that it is hard to resist the urge to give them names and start thinking in terms of them.^24.3 It is conventional to call $\gamma \, m c^2$  the TOTAL RELATIVISTIC ENERGY and m c² the REST MASS ENERGY. What do these names mean? The suggestion is that there is an irreducible energy E₀ = m c² associated with any object of mass m, even when it is sitting still! When it speeds up, its total energy changes by a multiplicative factor $\gamma$; the difference between the total energy $E = \gamma \, m c^2$ and E₀ is the energy due to its motion, namely the kinetic energy T.