Introductio

Ad hunc locum, diligenter examinavimus modum quo transformationes Lorentzianae efficiuntur, id est, rationem qua coordinatae in spatio-temporis Minkowski cuiusdam eventus particularis mutantur cum observantur ex diversis systematibus inertialibus. Quod nunc faciemus erit perscrutari prospectum diversum pro his evolutionibus, eas visualizantes tamquam transformationes rotationum spatio-temporis. Mox reperiemus hunc accessum utilitates afferre ad gradum algebraicum, quae in genere calculos simpliciorem reddunt, praesertim cum plures transformationes Lorentzianas continuas componere necesse est.

Rotationes in Spatio-Temporis Minkowski

Incipiamus perpendentes quomodo variae rotationes spatiales in spatio-temporis Minkowski efficiantur. Cum hoc sit spatium quattuor dimensionum, commodissimum est ad rotationem constituendam id facere respectu cuiusdam plani specifici. Hoc modo possumus definire rotationes super planis xy, xz et yz, itemque super planis xt, yt et zt. Rotationes peractae in planis ex axibus spatialibus compositis sunt rotationes pure spatiales, dum illae in planis ex axibus spatii et temporis constantibus sunt rotationes spatio-temporales. Interim intendemus ad intellegendum accurate rotationes pure spatiales, ut postea hoc scientiam ad rotationes spatio-temporales extendamus.

Rotationes pure spatiales

Incipiamus studium rotationum spatialium recognoscendo quomodo rotationes in plano xy perficiantur. Ad hoc, ponamus nos habere punctum cum coordinatis (a,b) respectu systematis definiti per axes \hat{x} et \hat{y}. Deinde perpendamus rationem quae has coordinatas conectit cum illis quas observaret systema referentiae rotatum. Hoc systema definitur per axes \hat{x}^\prime et \hat{y}^\prime, qui rotati sunt angulo \theta respectu systematis originalis, ut ostenditur in imagine sequenti:

Ad obtinendas rationes inter coordinatas (a,b) et (a^\prime,b^\prime) mensas ex unoquoque systemate, uti possumus sequentibus lineis directricibus:

Ita nunc simplex est obtinere aequationes transformationis

\begin{array}{rcl} a^\prime & = & \phantom{-}a\cos(\theta) + b\sin(\theta) \\ b^\prime & = & -a \sin(\theta) + b \cos(\theta) \end{array}

Generalizatio Matricialis pro Rotationibus Tridimensionalibus

Hoc systema aequationum commodius repraesentari potest in forma matriciali.

\left(\begin{array}{r} a^\prime \\ b^\prime \end{array}\right) = \left(\begin{array}{cc} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta)\end{array}\right) \left(\begin{array}{r} a \\ b \end{array}\right)

Hoc est commodum, quia hinc facile est generalizare ad dimensiones maiores. Exempli gratia, punctum cum coordinatis (a,b,c) in systemate formato per axes \hat{x}, \hat{y} et \hat{z}, observatum ex alio systemate formato per axes \hat{x}^\prime, \hat{y}^\prime et \hat{z}^\prime, quod distinguitur a systemate originali per rotationem in angulo \theta respectu plani \hat{x}\hat{y}, esset:

\left(\begin{array}{r} a^\prime \\ b^\prime \\ c^\prime \end{array}\right) = \left(\begin{array}{ccc} \cos(\theta) & \sin(\theta) & 0 \\ -\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1\end{array}\right) \left(\begin{array}{r} a \\ b \\ c\end{array}\right)

Ex hoc consequimur diversas matrices transformationis rotationum pro singulis planis spatialibus.

\begin{array}{rll} R_{xy}(\theta)= & \left(\begin{array}{ccc} \cos(\theta) & \sin(\theta) & 0 \\ -\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1\end{array}\right) & \begin{array}{l} \text{Rotatio in angulo }\theta\\ \text{super planum }xy \end{array} \\ \\ R_{yz}(\theta)= & \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\theta) & \sin(\theta) \\ 0 & -\sin(\theta) & \cos(\theta)\end{array}\right) & \begin{array}{l} \text{Rotatio in angulo }\theta\\ \text{super planum }yz \end{array} \\ \\ R_{xz}(\theta)= & \left(\begin{array}{ccc} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta)\end{array}\right) & \begin{array}{l} \text{Rotatio in angulo }\theta\\ \text{super planum }xz \end{array} \end{array}

Ad computandam transformationem inversam harum transformationum rotationis, satis est substituere \theta per -\theta.

Rotationes Spatiales pro Eventibus cum Coordinatis Spatio-Temporis

Similiter ac generalizavimus ex duabus ad tres dimensiones, hoc ad quattuor dimensiones extendere possumus. Ut cohaerentiam cum sermone relativitatis specialis teneamus, interest intellegere significationem uniuscuiusque coordinatae. Generaliter, coordinatae spatio-temporis sic exprimuntur:

x^\mu = (x^0, x^1, x^2, x^3) = (ct, x, y, z)

Hic, superindices non denotant potentias, sed indicant proprietates uniuscuiusque coordinatae. Coordinata cum superindice 0 repraesentat dimensionem temporalem, dum coordinatae cum superindicibus 1, 2 et 3 respondent dimensionibus spatialibus. Hoc intellecto, rotationes pure spatiales in spatio-temporis Minkowski describuntur per has relationes:

Rotatio respectu plani xy: \underbrace{\left(\begin{array}{r} x^{\prime 0} \\ x^{\prime 1} \\ x^{\prime 2} \\ x^{\prime 3} \end{array}\right)}_{\large{x^{\prime \mu}}} = \underbrace{\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & \sin(\theta) & 0 \\ 0 & -\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)}_{\large{{R_{xy}(\theta)^\mu}_\nu}} \underbrace{\left(\begin{array}{c} x^0 \\ x^1 \\ x^2 \\ x^3 \end{array}\right)}_{\large{x^{\nu}}}

Rotatio respectu plani yz: \underbrace{\left(\begin{array}{c} x^{\prime 0} \\ x^{\prime 1} \\ x^{\prime 2} \\ x^{\prime 3} \end{array}\right)}_{\large{x^{\prime \mu}}} = \underbrace{\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ {} 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\theta) & \sin(\theta) \\ 0 & 0 & -\sin(\theta) & \cos(\theta) \end{array}\right)}_{\large{{R_{yz}(\theta)^\mu}_\nu}} \underbrace{\left(\begin{array}{r} x^0 \\ x^1 \\ x^2 \\ x^3 \end{array}\right)}_{\large{x^{\nu}}}

Rotatio respectu plani xz: \underbrace{\left(\begin{array}{c} x^{\prime 0} \\ {}x^{\prime 1} \\ x^{\prime 2} \\ x^{\prime 3} \end{array}\right)}_{\large{x^{\prime \mu}}} = \underbrace{\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & 0 & \sin(\theta) \\ 0 & 0 & 1 & 0 \\ 0 & -\sin(\theta) & 0 & \cos(\theta) \end{array}\right)}_{\large{{R_{xz}(\theta)^\mu}_\nu}} \underbrace{\left(\begin{array}{r} x^0 \\ {} x^1 \\ x^2 \\ x^3 \end{array}\right)}_{\large{x^{\nu}}}

Hae transformationes eadem prorsus servant proprietates ac earum homologae in tribus dimensionibus.

Rotationes hyperbolicae spatii-temporis

Introducens parametron velocitatis

Similitudo inter transformationes Lorentzianas et rotationem spatialem obtineri potest introducendo quod vocamus parametron velocitatis

\psi_{ss^\prime_x}= \text{argtanh}(\beta_{ss^\prime_x}).

Cum \beta_{ss^\prime_x}\in]-1,1[, habetur \psi_{ss^\prime_x}\in\mathbb{R}. Praeterea animadvertamus quod ex hoc sequetur \gamma_{ss^\prime_x}=\cosh(\psi_{ss^\prime_x}) et \gamma_{ss^\prime_x} \beta_{ss^\prime_x} = \sinh(\psi_{ss^\prime_x}). Hoc obtinetur ex sequentibus calculis:

Manifestum est quod \psi_{ss^\prime_x}= \text{argtanh}(\beta_{ss^\prime_x}) aequivalet dicere \beta_{ss^\prime_x} =\tanh(\psi_{ss^\prime_x}); atque ideo:

\begin{array}{rl} \gamma^2_{ss^\prime_x} &= \dfrac{1}{1-\beta^2_{ss^\prime_x}} \\ \\ & = \dfrac{1}{1-\tanh^2(\psi_{ss^\prime_x})} \\ \\ {} & = \dfrac{\cosh^2(\psi_{ss^\prime_x})}{\cosh^2(\psi_{ss^\prime_x}) - \sinh^2(\psi_{ss^\prime_x})} \\ \\ & = \cosh^2(\psi_{ss^\prime_x}) \end{array}

Cum tam factor gamma quam cosinus hyperbolicus semper sint maiores vel pares 1, tandem demonstratur \gamma_{ss^\prime_x} = \cosh(\psi_{ss^\prime_x}).

Similiter, continuando calculos antea factos, habetur:

\gamma^2_{ss^\prime_x} \beta^2_{ss^\prime_x} = \cosh^2(\psi_{ss^\prime_x}) \tanh^2(\psi_{ss^\prime_x})= \sinh^2(\psi_{ss^\prime_x}).

Atque ideo \gamma_{ss^\prime_x} \beta_{ss^\prime_x} = \sinh(\psi_{ss^\prime_x}).

Formulatio Rotationum Spatio-Temporalium ut Rotationes Hyperbolicae

His ad hunc locum perventis, nunc possumus rescribere factorum ad impulsum velocitatis pertinentium atque factoris gamma usum parametro velocitatis in transformationibus Lorentzianis. Consideratis duobus systematibus inertialibus S et S^\prime in configuratione normali, ubi secundum impellitur super axem x, \beta_{ss^\prime_x}, habetur:

\begin{array}{rl} ct^\prime &= \gamma_{ss^\prime_x}(ct - \beta_{ss^\prime_x} x) \\ &= \gamma_{ss^\prime_x} ct - \gamma_{ss^\prime_x}\beta_{ss^\prime_x} x \\ &= ct\cosh(\psi_{ss^\prime_x}) - x\sinh(\psi_{ss^\prime_x}), \\ \\ x^\prime &= \gamma_{ss^\prime_x}(x - \beta_{ss^\prime_x} ct) \\ &= -\gamma_{ss^\prime_x}\beta_{ss^\prime_x} ct + \gamma_{ss^\prime_x}x \\ &= -ct \sinh(\psi_{ss^\prime_x}) + x\cosh(\psi_{ss^\prime_x}), \\ \\ y^\prime &= y, \\ \\ z^\prime &= z. \end{array}

Hoc systema aequationum hanc repraesentationem matricialem admittit:

Rotatio Hyperbolica Spatii-Temporis super Planum tx:

\begin{array}{rl} \underbrace{\left( \begin{array}{c} ct^\prime \\ x^\prime \\ y^\prime \\ z^\prime \end{array} \right)}_{\large{x^{\prime \mu}}} &= \underbrace{\left( \begin{array}{cccc} \cosh(\psi_{ss^\prime_x}) & -\sinh(\psi_{ss^\prime_x}) & 0 & 0 \\ - \sinh(\psi_{ss^\prime_x}) & \cosh(\psi_{ss^\prime_x}) & 0 & 0 \\ {} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)}_{\large{{R_{tx}(\psi_{ss^\prime_x})^\mu}_\nu}} \underbrace{\left( \begin{array}{c} ct \\ x \\ y \\ z \end{array} \right)}_{\large{x^{\nu}}} \end{array}

Similiter, habemus rotationes hyperbolicas super unumquodque planum spatio-temporis:

Rotatio Hyperbolica Spatii-Temporis super Planum ty:

\begin{array}{rl} \underbrace{\left( \begin{array}{c} ct^\prime \\ x^\prime \\ y^\prime \\ z^\prime \end{array} \right)}_{\large{x^{\prime \mu}}} &= \underbrace{\left(\begin{array}{cccc} \cosh(\psi_{ss^\prime_y}) & 0 & -\sinh(\psi_{ss^\prime_y}) & 0 \\ 0 & 1 & 0 & 0 \\ {} - \sinh(\psi_{ss^\prime_y}) & 0 & \cosh(\psi_{ss^\prime_y}) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)}_{\large{{R_{ty}(\psi_{ss^\prime_y})^\mu}_\nu}} \underbrace{\left( \begin{array}{c} ct \\ x \\ y \\ z \end{array} \right)}_{\large{x^{\nu}}} \end{array}

Rotatio Hyperbolica Spatii-Temporis super Planum tz:

\begin{array}{rl} \underbrace{\left( \begin{array}{c} ct^\prime \\ x^\prime \\ y^\prime \\ z^\prime \end{array} \right)}_{\large{x^{\prime \mu}}} &= \underbrace{\left( \begin{array}{cccc} \cosh(\psi_{ss^\prime_z}) & 0 & 0 & -\sinh(\psi_{ss^\prime_z}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {} - \sinh(\psi_{ss^\prime_z}) & 0 & 0 & \cosh(\psi_{ss^\prime_z}) \end{array}\right)}_{\large{{R_{tz}(\psi_{ss^\prime_z})^\mu}_\nu}} \underbrace{\left( \begin{array}{c} ct \\ x \\ y \\ z \end{array} \right)}_{\large{x^{\nu}}} \end{array}

Propter formam et proprietates algebraicas, hae transformationes simillimae sunt rotationi spatiali, nisi quod loco functionum trigonometricarum functiones hyperbolicae adhibentur. Quamquam non sunt rotationes in sensu stricto, quandam analogiam servant cum rotationibus initio recognitis. Exempli gratia, similiter ac fit in rotationibus, transformatio inversa obtinetur substituendo parametron velocitatis correspondentem \psi per -\psi. Hae transformationes interdum appellantur rotationes hyperbolicae, et parametron velocitatis etiam notum est ut angulus hyperbolicus.