Question 12.39: Define proper acceleration in the obvious way: α^μ ≡ dη^μ/dτ...

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)

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\text { (a) } \alpha^{0}=\frac{d \eta_{0}}{d \tau}=\frac{d \eta_{0}}{d t} \frac{d t}{d \tau}=\left[\frac{d}{d t}\left(\frac{c}{\sqrt{1-u^{2} / c^{2}}}\right)\right] \frac{1}{\sqrt{1-u^{2} / c^{2}}}

 

=\frac{c}{\sqrt{1-u^{2} / c^{2}}}\left(-\frac{1}{2}\right) \frac{\left(-\frac{1}{c^{2}}\right) 2 u \cdot a }{\left(1-u^{2} / c^{2}\right)^{3 / 2}}=\frac{1}{c} \frac{ u \cdot a }{\left(1-u^{2} / c^{2}\right)^{2}}.

 

\alpha =\frac{d \eta }{d \tau}=\frac{d t}{d \tau} \frac{d \eta }{d t}=\frac{1}{\sqrt{1-u^{2} / c^{2}}} \frac{d}{d t}\left(\frac{ u }{\sqrt{1-u^{2} / c^{2}}}\right)=\frac{1}{\sqrt{1-u^{2} / c^{2}}}\left\{\frac{ a }{\sqrt{1-u^{2} / c^{2}}}+ u (-t) \frac{-\frac{1}{c^{2}} 2 u \cdot a }{\left(1-u^{2} / c^{2}\right)^{3 / 2}}\right\}

 

=\frac{1}{\left(1-u^{2} / c^{2}\right)}\left[ a +\frac{ u ( u \cdot a )}{\left(c^{2}-u^{2}\right)}\right].

\text { (b) } \quad \alpha_{\mu} \alpha^{\mu}=-\left(\alpha^{0}\right)^{2}+ \alpha \cdot \alpha =-\frac{1}{c^{2}} \frac{( u \cdot a )^{2}}{\left(1-u^{2} / c^{2}\right)^{4}}+\frac{1}{\left(1-u^{2} / c^{2}\right)^{4}}\left[ a \left(1-\frac{u^{2}}{c^{2}}\right)+\frac{1}{c^{2}} u ( u \cdot a )\right]^{2}

 

=\frac{1}{\left(1-u^{2} / c^{2}\right)^{4}}\left\{-\frac{1}{c^{2}}( u \cdot a )^{2}+a^{2}\left(1-\frac{u^{2}}{c^{2}}\right)^{2}+\frac{2}{c^{2}}\left(1-\frac{u^{2}}{c^{2}}\right)( u \cdot a )^{2}+\frac{1}{c^{4}} u^{2}( u \cdot a )^{2}\right\}

 

=\frac{1}{\left(1-u^{2} / c^{2}\right)^{4}}\left\{a^{2}\left(1-\frac{u^{2}}{c^{2}}\right)^{2}+\frac{( u \cdot a )^{2}}{c^{2}}(\underbrace{-1+2-2 \frac{u^{2}}{c^{2}}+\frac{u^{2}}{c^{2}}}_{\left(1-\frac{u^{2}}{c^{2}}\right)}\right\}

 

=\frac{1}{\left(1-u^{2} / c^{2}\right)^{2}}\left[a^{2}+\frac{( u \cdot a )^{2}}{\left(c^{2}-u^{2}\right)}\right]

 

\text { (c) } \eta^{\mu} \eta_{\mu}=-c^{2} \text {, so } \frac{d}{d \tau}\left(\eta^{\mu} \eta_{\mu}\right)=\alpha^{\mu} \eta_{\mu}+\eta^{\mu} \alpha_{\mu}=2 \alpha^{\mu} \eta_{\mu}=0, \text { so } \alpha^{\mu} \eta_{\mu}=0 \text {. }

 

\text { (d) } K^{\mu}=\frac{d \rho^{\mu}}{d \tau}=\frac{d}{d \tau}\left(m \eta^{\mu}\right)=m \alpha^{\mu} . \quad K^{\mu} \eta_{\mu}=m \alpha^{\mu} \eta_{\mu}=0.

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