{"id":34859,"date":"2024-01-07T13:00:47","date_gmt":"2024-01-07T13:00:47","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34859"},"modified":"2025-09-21T04:38:46","modified_gmt":"2025-09-21T04:38:46","slug":"rotationes-hyperbolicae-spatii-temporis","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/rotationes-hyperbolicae-spatii-temporis\/","title":{"rendered":"Rotationes Hyperbolicae Spatii-Temporis"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Rotationes Hyperbolicae Spatii-Temporis<\/h1>\n<p class=\"eq\"><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione recognoscemus quomodo transformationes Lorentzianae possint reinterpretari tamquam transformationes rotationum spatii-temporis. Incipiemus examinando rotationes in spatio quattuor dimensionum Minkowski, distinguentes inter rotationes pure spatiales et eas quae axes spatio-temporales implicant.<\/br><\/em><\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nPost hanc lectionem studiosus poterit:<\/p>\n<ol>\n<li><strong>Intellegere<\/strong> transformationes rotationis in spatio-temporis Minkowski.<\/li>\n<li><strong>Intellegere<\/strong> transformationes Lorentzianas ut rotationes spatio-temporales.<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>INDEX<\/strong><br \/>\n<a href=\"#0\"><strong>Introductio<\/strong><\/a><br \/>\n<a href=\"#1\"><strong>Rotationes in Spatio-Temporis Minkowski<\/strong><\/a><br \/>\n<a href=\"#2\">Rotationes pure spatiales<\/a><br \/>\n<a href=\"#3\">Generalizatio Matricialis pro Rotationibus Tridimensionalibus<\/a><br \/>\n<a href=\"#4\">Rotationes Spatiales pro Eventibus cum Coordinatis Spatio-Temporis<\/a><br \/>\n<a href=\"#5\"><strong>Rotationes hyperbolicae spatii-temporis<\/strong><\/a><br \/>\n<a href=\"#6\">Introducens parametron velocitatis<\/a><br \/>\n<a href=\"#7\">Formulatio Rotationum Spatio-Temporalium ut Rotationes Hyperbolicae<\/a><br \/>\n<a href=\"#8\"><strong>Conclusiones<\/strong><\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/PCB-XC3XwQE?si=rBjMJhQEZ8O2wBLg\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><br \/>\n<\/center>\n<\/div>\n<p><a name=\"0\"><\/a><\/p>\n<h2>Introductio<\/h2>\n<p style=\"text-align:justify;\">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.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Rotationes in Spatio-Temporis Minkowski<\/h2>\n<p style=\"text-align:justify;\">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 <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xy<\/span><\/bdi>, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xz<\/span><\/bdi> et <bdi><span class=\"katex-eq\" data-katex-display=\"false\">yz<\/span><\/bdi>, itemque super planis <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xt<\/span><\/bdi>, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">yt<\/span><\/bdi> et <bdi><span class=\"katex-eq\" data-katex-display=\"false\">zt<\/span><\/bdi>. 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.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Rotationes pure spatiales<\/h3>\n<p style=\"text-align:justify;\">\nIncipiamus studium rotationum spatialium recognoscendo quomodo rotationes in plano <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xy<\/span><\/bdi> perficiantur. Ad hoc, ponamus nos habere punctum cum coordinatis <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b)<\/span><\/bdi> respectu systematis definiti per axes <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/bdi> et <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}<\/span><\/bdi>. Deinde perpendamus rationem quae has coordinatas conectit cum illis quas observaret systema referentiae rotatum. Hoc systema definitur per axes <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}^\\prime<\/span><\/bdi> et <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}^\\prime<\/span><\/bdi>, qui rotati sunt angulo <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> respectu systematis originalis, ut ostenditur in imagine sequenti:\n<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotaciontheta.jpg\" alt=\"Rotatio in angulo theta plani xy\" width=\"623\" height=\"495\" class=\"aligncenter size-full wp-image-25994 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotaciontheta.jpg\" alt=\"Rotatio in angulo theta plani xy\" width=\"623\" height=\"495\" class=\"aligncenter size-full wp-image-25994 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotaciontheta.jpg 623w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotaciontheta-300x238.jpg 300w\" sizes=\"(max-width: 623px) 100vw, 623px\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align:justify;\">\nAd obtinendas rationes inter coordinatas <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b)<\/span><\/bdi> et <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a^\\prime,b^\\prime)<\/span><\/bdi> mensas ex unoquoque systemate, uti possumus sequentibus lineis directricibus:\n<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia.png\" alt=\"Lineae directrices ad obtinendam rationem inter systemata rotata\" width=\"827\" height=\"620\" class=\"aligncenter size-full wp-image-25998 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia.png\" alt=\"Lineae directrices ad obtinendam rationem inter systemata rotata\" width=\"827\" height=\"620\" class=\"aligncenter size-full wp-image-25998 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia.png 827w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia-300x225.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia-768x576.png 768w\" sizes=\"(max-width: 827px) 100vw, 827px\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align:justify;\">\nIta nunc simplex est obtinere aequationes transformationis\n<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcl} a^\\prime &amp; = &amp; \\phantom{-}a\\cos(\\theta) + b\\sin(\\theta) \\\\ b^\\prime &amp; = &amp; -a \\sin(\\theta) + b \\cos(\\theta)\n\n\\end{array} <\/span>\n<p><\/bdi><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Generalizatio Matricialis pro Rotationibus Tridimensionalibus<\/h3>\n<p style=\"text-align:justify;\">\nHoc systema aequationum commodius repraesentari potest in forma matriciali.\n<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\begin{array}{r} a^\\prime \\\\ b^\\prime \\end{array}\\right) = \\left(\\begin{array}{cc} \\cos(\\theta) &amp; \\sin(\\theta) \\\\ -\\sin(\\theta) &amp; \\cos(\\theta)\\end{array}\\right) \\left(\\begin{array}{r} a \\\\ b \\end{array}\\right) <\/span>\n<p><\/bdi><\/p>\n<p style=\"text-align:justify;\">Hoc est commodum, quia hinc facile est generalizare ad dimensiones maiores. Exempli gratia, punctum cum coordinatis <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c)<\/span><\/bdi> in systemate formato per axes <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{z}<\/span>, observatum ex alio systemate formato per axes <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}^\\prime<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}^\\prime<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{z}^\\prime<\/span>, quod distinguitur a systemate originali per rotationem in angulo <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> respectu plani <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}\\hat{y}<\/span><\/bdi>, esset:<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\begin{array}{r} a^\\prime \\\\ b^\\prime \\\\ c^\\prime \\end{array}\\right) = \\left(\\begin{array}{ccc} \\cos(\\theta) &amp; \\sin(\\theta) &amp; 0 \\\\ -\\sin(\\theta) &amp; \\cos(\\theta) &amp; 0 \\\\ 0 &amp; 0 &amp; 1\\end{array}\\right) \\left(\\begin{array}{r} a \\\\ b \\\\ c\\end{array}\\right) <\/span>\n<p><\/bdi><\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/sistemarotadoxy3d.png\" alt=\"Rotationes Spatiales\" width=\"725\" height=\"597\" class=\"aligncenter size-full wp-image-26014 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/sistemarotadoxy3d.png\" alt=\"Rotationes Spatiales\" width=\"725\" height=\"597\" class=\"aligncenter size-full wp-image-26014 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/sistemarotadoxy3d.png 725w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/sistemarotadoxy3d-300x247.png 300w\" sizes=\"(max-width: 725px) 100vw, 725px\" \/><\/noscript><\/p>\n<p style=\"text-align:justify;\">Ex hoc consequimur diversas matrices transformationis rotationum pro singulis planis spatialibus.<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rll} R_{xy}(\\theta)= &amp; \\left(\\begin{array}{ccc} \\cos(\\theta) &amp; \\sin(\\theta) &amp; 0 \\\\ -\\sin(\\theta) &amp; \\cos(\\theta) &amp; 0 \\\\ 0 &amp; 0 &amp; 1\\end{array}\\right) &amp; \\begin{array}{l} \\text{Rotatio in angulo }\\theta\\\\ \\text{super planum }xy \\end{array} \\\\ \\\\ R_{yz}(\\theta)= &amp; \\left(\\begin{array}{ccc} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; \\cos(\\theta) &amp; \\sin(\\theta) \\\\ 0 &amp; -\\sin(\\theta) &amp; \\cos(\\theta)\\end{array}\\right) &amp; \\begin{array}{l} \\text{Rotatio in angulo }\\theta\\\\ \\text{super planum }yz \\end{array} \\\\ \\\\ R_{xz}(\\theta)= &amp; \\left(\\begin{array}{ccc} \\cos(\\theta) &amp; 0 &amp; \\sin(\\theta) \\\\ 0 &amp; 1 &amp; 0 \\\\ -\\sin(\\theta) &amp; 0 &amp; \\cos(\\theta)\\end{array}\\right) &amp; \\begin{array}{l} \\text{Rotatio in angulo }\\theta\\\\ \\text{super planum }xz \\end{array} \\end{array} <\/span>\n<p><\/bdi><\/p>\n<p style=\"text-align:justify;\">Ad computandam transformationem inversam harum transformationum rotationis, satis est substituere <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">-\\theta<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Rotationes Spatiales pro Eventibus cum Coordinatis Spatio-Temporis<\/h3>\n<p style=\"text-align:justify;\">\nSimiliter 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:\n<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x^\\mu = (x^0, x^1, x^2, x^3) = (ct, x, y, z)<\/span>\n<p><\/bdi><\/p>\n<p style=\"text-align:justify;\">\nHic, 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:<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><strong>Rotatio respectu plani xy:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\n\\underbrace{\\left(\\begin{array}{r}\n\nx^{\\prime 0} \\\\ x^{\\prime 1} \\\\ x^{\\prime 2} \\\\ x^{\\prime 3} \\end{array}\\right)}_{\\large{x^{\\prime \\mu}}} = \\underbrace{\\left(\\begin{array}{cccc}\n\n1 &amp; 0 &amp; 0 &amp; 0 \\\\ 0 &amp; \\cos(\\theta) &amp; \\sin(\\theta) &amp; 0 \\\\ 0 &amp; -\\sin(\\theta) &amp; \\cos(\\theta) &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 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}}} <\/span>\n<p><\/bdi><br \/>\n<bdi><\/p>\n<p style=\"text-align:center;\"><strong>Rotatio respectu plani yz:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\n\\underbrace{\\left(\\begin{array}{c}\n\nx^{\\prime 0} \\\\ x^{\\prime 1} \\\\ x^{\\prime 2} \\\\ x^{\\prime 3} \\end{array}\\right)}_{\\large{x^{\\prime \\mu}}} = \\underbrace{\\left(\\begin{array}{cccc} 1 &amp; 0 &amp; 0 &amp; 0 \\\\\n\n{} 0 &amp; 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; \\cos(\\theta) &amp; \\sin(\\theta) \\\\ 0 &amp; 0 &amp; -\\sin(\\theta) &amp; \\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}}} <\/span>\n<p><\/bdi><\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><strong>Rotatio respectu plani xz:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\n\\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 &amp; 0 &amp; 0 &amp; 0 \\\\ 0 &amp; \\cos(\\theta) &amp; 0 &amp; \\sin(\\theta) \\\\ 0 &amp; 0 &amp; 1 &amp; 0 \\\\ 0 &amp; -\\sin(\\theta) &amp; 0 &amp; \\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}}} <\/span>\n<p><\/bdi><\/p>\n<p style=\"text-align:justify;\">Hae transformationes eadem prorsus servant proprietates ac earum homologae in tribus dimensionibus.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Rotationes hyperbolicae spatii-temporis<\/h2>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Introducens parametron velocitatis<\/h3>\n<p style=\"text-align:justify;\">Similitudo inter transformationes Lorentzianas et rotationem spatialem obtineri potest introducendo quod vocamus <strong>parametron velocitatis<\/strong><\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}= \\text{argtanh}(\\beta_{ss^\\prime_x}).<\/span><\/bdi><\/p>\n<p style=\"text-align:justify;\">Cum <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}\\in]-1,1[<\/span>, habetur <span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}\\in\\mathbb{R}<\/span>. Praeterea animadvertamus quod ex hoc sequetur <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x}=\\cosh(\\psi_{ss^\\prime_x})<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} \\beta_{ss^\\prime_x} = \\sinh(\\psi_{ss^\\prime_x})<\/span>. Hoc obtinetur ex sequentibus calculis:<\/p>\n<p style=\"text-align:justify;\">Manifestum est quod <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}= \\text{argtanh}(\\beta_{ss^\\prime_x})<\/span><\/bdi> aequivalet dicere <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x} =\\tanh(\\psi_{ss^\\prime_x})<\/span><\/bdi>; atque ideo: <\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl} \\gamma^2_{ss^\\prime_x} &amp;= \\dfrac{1}{1-\\beta^2_{ss^\\prime_x}} \\\\ \\\\ &amp; = \\dfrac{1}{1-\\tanh^2(\\psi_{ss^\\prime_x})} \\\\ \\\\ {} &amp; = \\dfrac{\\cosh^2(\\psi_{ss^\\prime_x})}{\\cosh^2(\\psi_{ss^\\prime_x}) - \\sinh^2(\\psi_{ss^\\prime_x})} \\\\ \\\\ &amp; = \\cosh^2(\\psi_{ss^\\prime_x}) \\end{array}<\/span><\/bdi><\/p>\n<p style=\"text-align:justify;\">Cum tam factor gamma quam cosinus hyperbolicus semper sint maiores vel pares 1, tandem demonstratur <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} = \\cosh(\\psi_{ss^\\prime_x})<\/span><\/bdi>.<\/p>\n<p style=\"text-align:justify;\">Similiter, continuando calculos antea factos, habetur:<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\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})<\/span>.<\/bdi><\/p>\n<p style=\"text-align:justify;\">Atque ideo <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} \\beta_{ss^\\prime_x} = \\sinh(\\psi_{ss^\\prime_x})<\/span><\/bdi>. <\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Formulatio Rotationum Spatio-Temporalium ut Rotationes Hyperbolicae<\/h3>\n<p style=\"text-align:justify;\">\nHis 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 <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> in configuratione normali, ubi secundum impellitur super axem <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}<\/span>, habetur:\n<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl} ct^\\prime &amp;= \\gamma_{ss^\\prime_x}(ct - \\beta_{ss^\\prime_x} x) \\\\ &amp;= \\gamma_{ss^\\prime_x} ct - \\gamma_{ss^\\prime_x}\\beta_{ss^\\prime_x} x \\\\ &amp;= ct\\cosh(\\psi_{ss^\\prime_x}) - x\\sinh(\\psi_{ss^\\prime_x}), \\\\ \\\\ x^\\prime &amp;= \\gamma_{ss^\\prime_x}(x - \\beta_{ss^\\prime_x} ct) \\\\ &amp;= -\\gamma_{ss^\\prime_x}\\beta_{ss^\\prime_x} ct + \\gamma_{ss^\\prime_x}x \\\\ &amp;= -ct \\sinh(\\psi_{ss^\\prime_x}) + x\\cosh(\\psi_{ss^\\prime_x}), \\\\ \\\\ y^\\prime &amp;= y, \\\\ \\\\\n\nz^\\prime &amp;= z. \\end{array} <\/span><\/bdi><\/p>\n<p style=\"text-align:justify;\">Hoc systema aequationum hanc repraesentationem matricialem admittit:<\/p>\n<p style=\"text-align:center;\"><strong>Rotatio Hyperbolica Spatii-Temporis super Planum tx: <\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl} \\underbrace{\\left( \\begin{array}{c} ct^\\prime \\\\ x^\\prime \\\\ y^\\prime \\\\ z^\\prime \\end{array} \\right)}_{\\large{x^{\\prime \\mu}}} &amp;= \\underbrace{\\left( \\begin{array}{cccc} \\cosh(\\psi_{ss^\\prime_x}) &amp; -\\sinh(\\psi_{ss^\\prime_x}) &amp; 0 &amp; 0 \\\\ - \\sinh(\\psi_{ss^\\prime_x}) &amp; \\cosh(\\psi_{ss^\\prime_x}) &amp; 0 &amp; 0 \\\\\n\n{} 0 &amp; 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 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} <\/span>\n<p style=\"text-align:justify;\">Similiter, habemus rotationes hyperbolicas super unumquodque planum spatio-temporis:<\/p>\n<p style=\"text-align:center;\"><strong>Rotatio Hyperbolica Spatii-Temporis super Planum ty: <\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl} \\underbrace{\\left( \\begin{array}{c} ct^\\prime \\\\ x^\\prime \\\\ y^\\prime \\\\ z^\\prime \\end{array} \\right)}_{\\large{x^{\\prime \\mu}}} &amp;= \\underbrace{\\left(\\begin{array}{cccc} \\cosh(\\psi_{ss^\\prime_y}) &amp; 0 &amp; -\\sinh(\\psi_{ss^\\prime_y}) &amp; 0 \\\\ 0 &amp; 1 &amp; 0 &amp; 0 \\\\ {} - \\sinh(\\psi_{ss^\\prime_y}) &amp; 0 &amp; \\cosh(\\psi_{ss^\\prime_y}) &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 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} <\/span>\n<p style=\"text-align:center;\"><strong>Rotatio Hyperbolica Spatii-Temporis super Planum tz: <\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl} \\underbrace{\\left( \\begin{array}{c} ct^\\prime \\\\ x^\\prime \\\\ y^\\prime \\\\ z^\\prime \\end{array} \\right)}_{\\large{x^{\\prime \\mu}}} &amp;= \\underbrace{\\left( \\begin{array}{cccc} \\cosh(\\psi_{ss^\\prime_z}) &amp; 0 &amp; 0 &amp; -\\sinh(\\psi_{ss^\\prime_z}) \\\\ 0 &amp; 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 &amp; 0 \\\\ {} - \\sinh(\\psi_{ss^\\prime_z}) &amp; 0 &amp; 0 &amp; \\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} <\/span>\n<p style=\"text-align:justify;\">\nPropter 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 <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">-\\psi<\/span>. Hae transformationes interdum appellantur <strong>rotationes hyperbolicae<\/strong>, et parametron velocitatis etiam notum est ut <strong>angulus hyperbolicus<\/strong>.\n<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclusiones<\/h2>\n<p style=\"text-align:justify;\">\n        Adhuc modo accurato tractavimus notionem rotationum in spatio-temporis Minkowski, quod nobis sinit altiorem comprehensionem transformationum Lorentzianarum adipisci. Per hoc studium, hos nexus principales consecuti sumus:\n    <\/p>\n<ul>\n<li><strong>Reinterpretatio Transformationum Lorentzianarum<\/strong>: Didicimus visualizare et intellegere transformationes Lorentzianas non solum tamquam mutationes in coordinatis ob diversos systematum referentiae, sed etiam ut rotationes in spatio-temporis.<\/li>\n<li><strong>Comprehensio Rotationum in Spatio-Temporis Minkowski<\/strong>: Diligenter examinavimus rotationes intra spatium quattuor dimensionum Minkowski.<\/li>\n<li><strong>Exploratio Rotationum Hyperbolicarum Spatii-Temporis<\/strong>: Tandem introduximus notionem rotationum hyperbolicarum spatii-temporis, investigantes earum similitudinem cum rotationibus spatialibus consuetis.<\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Rotationes Hyperbolicae Spatii-Temporis Summarium: In hac lectione recognoscemus quomodo transformationes Lorentzianae possint reinterpretari tamquam transformationes rotationum spatii-temporis. Incipiemus examinando rotationes in spatio quattuor dimensionum Minkowski, distinguentes inter rotationes pure spatiales et eas quae axes spatio-temporales implicant. PROPOSITA DISCENDI: Post hanc lectionem studiosus poterit: Intellegere transformationes rotationis in spatio-temporis Minkowski. Intellegere transformationes Lorentzianas ut rotationes spatio-temporales. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26205,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1250,1286],"tags":[],"class_list":["post-34859","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physica","category-relativitas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Rotationes Hyperbolicae Spatii-Temporis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quomodo transformationes Lorentzianae possint reinterpretari ut rotationes hyperbolicae spatii-temporis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/rotationes-hyperbolicae-spatii-temporis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Rotationes Hyperbolicae Spatii-Temporis\" \/>\n<meta property=\"og:description\" content=\"Disce quomodo transformationes Lorentzianae possint reinterpretari ut rotationes hyperbolicae spatii-temporis.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/rotationes-hyperbolicae-spatii-temporis\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-01-07T13:00:47+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-21T04:38:46+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacioneshiperbolicas-1-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Rotationes Hyperbolicae Spatii-Temporis\" \/>\n<meta name=\"twitter:description\" content=\"Disce quomodo transformationes Lorentzianae possint reinterpretari ut rotationes hyperbolicae spatii-temporis.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacioneshiperbolicas-1.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Rotationes Hyperbolicae Spatii-Temporis\",\"datePublished\":\"2024-01-07T13:00:47+00:00\",\"dateModified\":\"2025-09-21T04:38:46+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/\"},\"wordCount\":1825,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/01\\\/rotacioneshiperbolicas-1.jpg\",\"articleSection\":[\"Physica\",\"Relativitas\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/rotationes-hyperbolicae-spatii-temporis\\\/\",\"name\":\"Rotationes Hyperbolicae Spatii-Temporis - 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