Formulatio transformationum Galilei

Physica Newtoniana innititur principio relativitatis formato per transformationes Galilei, ubi tempus constituitur tamquam coordinata universalis omnibus observatoribus inercialibus; id est: t=t^\prime. Sub hac affirmatione ostensum est transformationem linearem quae observationes duorum systematum inercialium S et S^\prime refert, recognitorum in lectione de principio relativitatis, formam habere transformationis linearis:

\begin{array}{rl} t^\prime &= At + Bx,\\ x^\prime &= Dt + Ex,\\ y^\prime &= y, \\ z^\prime &=z, \end{array} [1]

habet hanc formam cum systemata inercialia S et S^\prime sunt in configuratione standardi et S^\prime movetur cum velocitate v_{ss^\prime_x}\hat{x} respectu S

\begin{array}{rlr} {}t^\prime &= t \\ x^\prime &= x - v_{ss^\prime_x}t \\ y^\prime &= y \\ z^\prime &= z \end{array} [2]

Transformatio inversa

Ex quadam quasi symmetria algebraica possumus scribere transformationem inversam:

\begin{array}{rl} t &= t^\prime \\ x &= x^\prime + v_{ss^\prime_x}t \\ y &= y^\prime \\ z &= z^\prime \end{array} [3]

Tempus absolutum et additio velocitatum

Ex prima aequatione transformationum Galilei (utraque, [2] vel [3]) habetur coordinata temporalis cuiusdam eventus non pendere ab systemate ex quo spectatur, dum secunda concedit obtinere quod vulgo intelligitur ut “sensus communis” coniunctus additioni velocitatum. Si particula movetur cum velocitate constanti v_{ss^\prime_x} super axem \hat{x} systematis S, tunc eius velocitas in S^\prime determinatur per

\displaystyle v^\prime_x = \frac{dx^\prime}{dt^\prime} = \frac{dx^\prime}{dt} = \frac{d}{dt}\left(x - v_{ss^\prime_x} t \right) = v_x - v_{ss^\prime_x}

Ex hac ultima expressione derivando ostenditur accelerationem cuiuslibet particulae eandem esse in S et in S^\prime, id est: dv^\prime_x/dt^\prime = dv_x/dt.

Geometria Galileana spatii et temporis

Si consideremus duo eventa A et B quae habent coordinatas (t_A,x_A,y_A,z_A) et (t_B,x_B,y_B,z_B), respective. Facile apparet quantitates \Delta t = t_B - t_A et \Delta r^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 separatim invariabiles esse sub transformationibus Galilei, quod nos ducit ad considerandum spatium et tempus tamquam entia separata. Porro \Delta r^2 suggerit hoc proprietatem geometricam ipsius spatii esse. Nos agnoscimus \Delta r^2 tamquam quadratum distantiae inter eventa in spatio euclideo. Hoc definit geometriam spatii et temporis in contextu mechanicae Newtonianae.

Relativitas Galilei et leges physicae

Applicata ad dynamicam Newtonianam

In sectione superiore vidimus quod, in contextu physicae Newtonianae, duo systemata inercialia diversa et quaecumque semper videbunt easdem accelerationes. Hoc, cum secunda lege Newtoni coniunctum, significat omnia systemata inercialia semper observatura esse eandem dynamicam. Id est:

\displaystyle F_x = m\frac{dv_x}{dt}= m\frac{dv^\prime_x}{dt^\prime} = F^\prime_x.

Haec ultima expressio nobis ostendit physicam non mutari cum fiunt transformationes Galilei, quod aequivalet dicere: physica eadem est omnibus observatoribus inercialibus.

Applicata ad propagationem undae

Quamquam haec constantia physicae sub mutationibus observatorum inercialium exspectanda est, primum quia hoc percipimus dum movemur, deinde quia hoc per calculos priores obtentum est, tamen non semper ita impletur. Causus maxime notabilis phaenomeni quod non servatur sub transformationibus Galilei est propagationis undarum; in genere, aequatio quae modulat propagationem undae \psi in spatio et tempore est huius formae

\displaystyle \nabla^2 \psi = \frac{1}{v_0^2}\frac{\partial^2 \psi}{\partial t^2} [4]

ubi v_0 est celeritas propagationis undae.

Quis effectus transformationum Galilei in propagationem undarum?

Ad hoc exstat responsio brevis et longa. Responsio brevis est quod “etiamsi idem phaenomenon observetur, diversi observatores inerciales videbunt ‘physicam’ diversam”. Responsio longa consistit in inspiciendo quomodo mutetur aequatio propagationis undae cum applicatur transformatio Galilei; ad hoc faciendum, primum sumimus aequationem [4] eamque expandimus super singulas coordinatas obtinentes:

\displaystyle \frac{\partial ^2 \psi}{\partial x^2} + \frac{\partial ^2 \psi}{\partial y^2} + \frac{\partial ^2 \psi}{\partial z^2} = \frac{1}{v_0^2} \frac{\partial ^2 \psi}{\partial t^2}. [5]

Cum hac aequatione prope, nunc oportet uti aequationibus [3], ad re-exprimendas derivatas in altero systemate inerziali.

Transformatio primarum derivatarum

Secundum expressiones [3] et derivando unamquamque variabilem respectu variabilium primarum obtinetur:

\displaystyle \frac{\partial x^\prime}{\partial x} = \frac{\partial y^\prime}{\partial y}= \frac{\partial z^\prime}{\partial z}= \frac{\partial t^\prime}{\partial t}= 1

\displaystyle \frac{\partial x^\prime}{\partial t} = - v_{x_0}

Dum omnes aliae evanescunt:

\displaystyle \frac{\partial t^\prime}{\partial x} = \frac{\partial t^\prime}{\partial y} = \frac{\partial t^\prime}{\partial z} = \frac{\partial x^\prime}{\partial y} = \frac{\partial x^\prime}{\partial z} = \frac{\partial y^\prime}{\partial x} = \frac{\partial y^\prime}{\partial z} = \frac{\partial y^\prime}{\partial t} = \frac{\partial z^\prime}{\partial x} = \frac{\partial z^\prime}{\partial y} = \frac{\partial z^\prime}{\partial t} = 0

His ad manum, nunc possumus derivatas \psi computare per regulam catenae:

\displaystyle\frac{\partial \psi}{\partial x} = \frac{\partial \psi}{\partial x^\prime} \underbrace{\frac{\partial x^\prime}{\partial x}}_{=1} + \frac{\partial \psi}{\partial y^\prime} \underbrace{\frac{\partial y^\prime}{\partial x}}_{=0} + \frac{\partial \psi}{\partial z^\prime} \underbrace{\frac{\partial z^\prime}{\partial x}}_{=0} + \frac{\partial \psi}{\partial t^\prime} \underbrace{\frac{\partial t^\prime}{\partial x}}_{=0} = \frac{\partial \psi}{\partial x^\prime}.

Et similiter habebitur pro aliis duabus variabilibus spatialibus:

\displaystyle\frac{\partial \psi}{\partial y} = \frac{\partial \psi}{\partial y^\prime}.

\displaystyle\frac{\partial \psi}{\partial z} = \frac{\partial \psi}{\partial z^\prime}.

Attamen derivata temporalis aliquas differentias exhibebit:

\begin{array}{rl} \displaystyle \frac{\partial \psi}{\partial t} &= \displaystyle\frac{\partial \psi}{\partial x^\prime}\underbrace{\frac{\partial x^\prime}{\partial t}}_{=-v_{x_0}} + \frac{\partial \psi}{\partial y^\prime}\underbrace{\frac{\partial y^\prime}{\partial t}}_{=0} + \frac{\partial \psi}{\partial z^\prime}\underbrace{\frac{\partial z^\prime}{\partial t}}_{=0} + \frac{\partial \psi}{\partial t^\prime}\underbrace{\frac{\partial t^\prime}{\partial t}}_{=1}\\ &=\displaystyle -v_{x_0} \frac{\partial \psi}{\partial x^\prime} + \frac{\partial \psi}{\partial t^\prime}, \end{array}

Transformatio secundarum derivatarum

Pro parte spaziali pergere poterimus sine magnis difficultatibus, eventus sunt:

\displaystyle \frac{\partial^2 \psi}{\partial x^2} = \frac{\partial^2 \psi}{\partial {x^\prime}^2}. [6]

\displaystyle \frac{\partial^2 \psi}{\partial y^2} = \frac{\partial^2 \psi}{\partial {y^\prime}^2} [7]

\displaystyle \frac{\partial^2 \psi}{\partial z^2} = \frac{\partial^2 \psi}{\partial {z^\prime}^2} [8]

At vero pars temporalis, sicut iam ex primis derivatis praevideri poterat, magnas differentias ostendit:

\begin{array}{rl} \displaystyle\frac{\partial^2 \psi}{\partial t^2} &=\displaystyle \frac{\partial}{\partial t}\left( -v_{x_0} \frac{\partial \psi}{\partial x^\prime} + \frac{\partial \psi}{\partial t^\prime} \right)\\ & \displaystyle = -v_{x_0} \frac{\partial }{\partial t} \left(\frac{\partial \psi}{\partial x^\prime} \right) + \frac{\partial }{\partial t} \left(\frac{\partial \psi}{\partial t^\prime} \right)\\ &\displaystyle = -v_{x_0} \frac{\partial }{\partial x^\prime} \left(\frac{\partial \psi}{\partial t} \right) + \frac{\partial }{\partial t^\prime} \left(\frac{\partial \psi}{\partial t} \right)\\ &\displaystyle = -v_{x_0} \frac{\partial }{\partial x^\prime} \left(-v_{x_0} \frac{\partial \psi}{\partial x^\prime} + \frac{\partial \psi}{\partial t^\prime} \right) + \frac{\partial }{\partial t^\prime} \left(-v_{x_0} \frac{\partial \psi}{\partial x^\prime} + \frac{\partial \psi}{\partial t^\prime} \right) \end{array}

\displaystyle\frac{\partial^2 \psi}{\partial t^2} = v_{x_0}^2 \frac{\partial^2 \psi}{\partial {x^\prime}^2} - 2v_{x_0}\frac{\partial^2 \psi}{\partial x^\prime \partial t^\prime} + \frac{\partial^2 \psi}{\partial {t^\prime}^2}. [9]

Applicatio transformationum Galilei super propagationem undarum

Hoc modo fieri potest transformatio Galilei super aequationem propagationis undae substituendo aequationes [6,7,8] et [9] in [5], unde provenit:

\displaystyle \frac{\partial^2 \psi}{\partial {x^\prime}^2} + \frac{\partial^2 \psi}{\partial {y^\prime}^2} + \frac{\partial^2 \psi}{\partial {z^\prime}^2} = \frac{1}{v_0^2} \left(\color{red}{ v_{x_0}^2 \frac{\partial^2 \psi}{\partial {x^\prime}^2} - 2v_{x_0}\frac{\partial^2 \psi}{\partial x^\prime \partial t^\prime}} + \frac{\partial^2 \psi}{\partial {t^\prime}^2} \right). [10]

Ubi animadvertitur formam propagationis undarum non servari sub transformationibus Galilei propter apparentiam terminorum additorum rubro notatorum. Quamquam hoc nunc magnas consequentias non habet, in lectionibus futuris videbimus hoc ipsum esse punctum quod, ut ita dicamus, “frangit” physicam classicam, viam aperiens relativitati speciali.