= Solution
Use the metric $\eta=\operatorname{diag}(-1,1,1,1)$ and <four-velocity> $u^\mu=\gamma(c,\mathbf v)$. Its derivative with respect to <proper time> is the <four-acceleration>. Since
$$
\frac{d\gamma}{dt}=\frac{\gamma^3}{c^2}\mathbf v\cdot\mathbf a,
$$
a purely transverse laboratory acceleration has $d\gamma/dt=0$. With $d/d\tau=\gamma d/dt$,
$$
A^\mu=\frac{du^\mu}{d\tau}=(0,\gamma^2\mathbf a).
$$
The invariant <proper acceleration> is the magnitude measured in the <instantaneous rest frame>, $a_0^2=A^\mu A_\mu$. Consequently,
$$
\boxed{a_0=\gamma^2a\qquad(\mathbf a\perp\mathbf v).}
$$
For longitudinal acceleration the factor is instead $\gamma^3$ multiplying the laboratory acceleration. The distinction matters for circular storage-ring motion, where the acceleration is centripetal and transverse even though its magnitude is nonzero.
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