Define proper acceleration in the obvious way:
\alpha^{\mu} \equiv \frac{d \eta^{\mu}}{d \tau}=\frac{d^{2} x^{\mu}}{d \tau^{2}}. (12.75)
(a) Find \alpha^{0} \text { and } \alpha in terms of u and a (the ordinary acceleration).
(b) Express \alpha_{\mu} \alpha^{\mu} in terms of u and a.
(c) Show that \eta^{\mu} \alpha_{\mu}=0.
(d) Write the Minkowski version of Newton’s second law, Eq. 12.68, in terms of \alpha^{\mu}. Evaluate the invariant product K^{\mu} \eta_{\mu}.
K^{\mu} \equiv \frac{d p^{\mu}}{d \tau} (12.68)