2.1 Spe­cial rel­a­tiv­ity

The speed of any Mi­ata is small, but not van­ish­ingly small, com­pared to the speed of light. (There are some math­e­mat­i­cal is­sues as­so­ci­ated with the pre­vi­ous state­ment that will be ad­dressed in a planned sec­ond vol­ume of this book.) There­fore rel­a­tivis­tic me­chan­ics must be used.

Ein­stein’s fa­mous re­la­tion

$$E = m c^{2} %$$ (2.1)

im­plies that a mov­ing Mi­ata picks up ad­di­tional mass.

How­ever, the prin­ci­ple of rel­a­tiv­ity, as first for­mu­lated by Poin­caré, al­lows the view­point of a dri­ver in­side the Mi­ata. Physics is the same re­gard­less of the rel­a­tive mo­tion of the ob­servers. The dri­ver view­point will fre­quently be used in the cur­rent pa­per to sim­plify the ar­gu­ments.

The most im­por­tant re­la­tion for the pur­pose of this pa­per is the rel­a­tivis­tic Doppler shift. The equa­tion that gov­erns the dif­fer­ence in ob­served wave­length $\lambda$ of light, and the cor­re­spond­ing dif­fer­ence in ob­served fre­quency $\omega$, be­tween mov­ing ob­servers is

$$\lambda_v = \lambda_0 \sqrt{\frac{c + v}{c - v}}$$

$$\omega_v = \omega_0 \sqrt{\frac{c - v}{c + v}} %$$ (2.2)

Here the sub­script 0 stands for the emit­ter of the light, and sub­script $v$ for an ob­server mov­ing with speed $v$ away from the emit­ter. If the ob­server moves to­wards the emit­ter, $v$ is neg­a­tive. (To be true, the for­mu­lae above ap­ply whether the ob­server 0 is emit­ting the light or not. But in most prac­ti­cal ap­pli­ca­tions, ob­server 0 is in­deed the emit­ter.)

Of course, Mi­atas do not drive in vac­uum but in the at­mos­phere. For­tu­nately, this ef­fect may be ig­nored as sec­ondary on light prop­a­ga­tion as long as no sig­nif­i­cant H$_{2}$O in liq­uid form is present. (In any case, Mi­atas are known to dis­agree with these so-called rain con­di­tions.) How­ever, the at­mos­phere is very im­por­tant be­cause of aero­dy­namic drag. These is­sues will be ad­dressed fur­ther in sub­sec­tion 2.3.