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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.