2 The­o­ret­i­cal Back­ground

The Mi­ata-light in­ter­ac­tion is ob­vi­ously de­scribed by the two pil­lars of mod­ern physics, spe­cial rel­a­tiv­ity and quan­tum me­chan­ics. Since the reader is surely well aware of quan­tum elec­tro­dy­nam­ics, ei­ther in Feyn­man’s path in­te­gral ap­proach or in canon­i­cal for­mu­la­tion, it needs no in­tro­duc­tion. How­ever, a few key re­sults will be sum­ma­rized in sub­sec­tions 2.1 and 2.2 for ready ref­er­ence. For more in­for­ma­tion see [2, pp. 1-152]. A more eas­ily ac­ces­si­ble source may be [8].

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

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

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.

2.2 Quan­tum me­chan­ics

Schrödinger, in his fa­mous equa­tion, as­so­ci­ated en­ergy with the par­tial time dif­fer­en­ti­a­tion op­er­a­tor, and lin­ear mo­men­tum with the par­tial space dif­fer­en­ti­a­tion op­er­a­tor in a given di­rec­tion:

$$E\quad\Longleftrightarrow\quad i\hbar\frac{\partial}{\pa... ...ongleftrightarrow\quad -i\hbar\frac{\partial}{\partial x} %$$ (3)

Here $\hbar$ is the scaled Planck’s con­stant and $i$ is $\sqrt{-1}$.

The above re­sults are of crit­i­cal im­por­tance for this pa­per, be­cause the Mi­ata-light in­ter­ac­tion is due to ex­change of the en­ergy and mo­men­tum of pho­tons of light. There­fore, it is help­ful to make the above re­la­tions spe­cific for pho­tons:

$$E = \hbar\omega \qquad p = \hbar \frac{\omega}{c} %$$ (4)

These ex­pres­sions are known as the Planck-Ein­stein and de Broglie re­la­tions. They may be de­rived by ap­ply­ing Schrödinger’s as­so­ci­a­tions on a com­plex, prop­a­gat­ing mono­chro­matic light wave. As is well known, the first ex­pres­sion is con­sis­tent with Ein­stein’s re­la­tion (1) in view of the fact that pho­tons have zero rest mass.

2.3 Aero­dy­nam­ics

The pri­mary fac­tor lim­it­ing the max­i­mum speed of a Mi­ata is aero­dy­namic re­sis­tance. This re­sis­tance is re­lated to bound­ary lay­ers along the sur­face of the Mi­ata in which the air is be­ing dragged along. Go­ing down­stream, this air forms a wake be­hind the Mi­ata. The wake is the pri­mary cause of re­sis­tance, [see 3, pp. 570-571].

The wake is al­ways rel­a­tively wide be­cause the bound­ary layer sep­a­rates from the sur­face at some point, [5,6,7]. This greatly in­creases the aero­dy­namic re­sis­tance. Con­trol­ling sep­a­ra­tion re­mains a dif­fi­cult prob­lem, [4].

There is also the prob­lem that the bound­ary layer is nor­mally tur­bu­lent. By it­self, tur­bu­lence will in­crease drag due to its ther­mo­dy­nam­i­cally ir­re­versible me­chan­ics, [9]. How­ever, of­ten tran­si­tion to tur­bu­lence de­creases drag in­stead be­cause it also tends to de­lay sep­a­ra­tion.

The high speed of, in par­tic­u­lar, white Mi­atas, will also bring in com­press­ibil­ity ef­fects, [1]. Note that such ef­fects are largest in el­e­vated speed ar­eas such as near the top of the wind­shield header.