{"id":27314,"date":"2024-01-07T13:00:09","date_gmt":"2024-01-07T13:00:09","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27314"},"modified":"2024-07-01T01:36:52","modified_gmt":"2024-07-01T01:36:52","slug":"rotacoes-hiperbolicas-de-espaco-tempo","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/rotacoes-hiperbolicas-de-espaco-tempo\/","title":{"rendered":"Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Rota\u00e7\u00e3o Hiperb\u00f3licas de Espa\u00e7o-Tempo<\/h1>\n<p class=\"eq\"><em><strong>Resumo:<\/strong><br \/>\nNesta aula, revisaremos como as transforma\u00e7\u00f5es de Lorentz podem ser reinterpretadas como transforma\u00e7\u00f5es de rota\u00e7\u00f5es de espa\u00e7o-tempo. Come\u00e7aremos examinando as rota\u00e7\u00f5es no espa\u00e7o de quatro dimens\u00f5es de Minkowski, distinguindo entre rota\u00e7\u00f5es puramente espaciais e aquelas que envolvem eixos espa\u00e7o-temporais.<\/br><\/em><\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\nAo finalizar esta aula o aluno ser\u00e1 capaz de:<\/p>\n<ol>\n<li><strong>Compreender<\/strong> as transforma\u00e7\u00f5es de rota\u00e7\u00e3o no espa\u00e7o-tempo de Minkowski.<\/li>\n<li><strong>Compreender<\/strong> as transforma\u00e7\u00f5es de Lorentz como rota\u00e7\u00f5es espa\u00e7o-temporais.<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#0\"><strong>Introdu\u00e7\u00e3o<\/strong><\/a><br \/>\n<a href=\"#1\"><strong>Rota\u00e7\u00f5es no Espa\u00e7o-Tempo de Minkowski<\/strong><\/a><br \/>\n<a href=\"#2\">Rota\u00e7\u00f5es espaciais puras<\/a><br \/>\n<a href=\"#3\">Generaliza\u00e7\u00e3o Matricial para Rota\u00e7\u00f5es Tridimensionais<\/a><br \/>\n<a href=\"#4\">Rota\u00e7\u00f5es Espaciais para Eventos com Coordenadas de Espa\u00e7o-Tempo<\/a><br \/>\n<a href=\"#5\"><strong>Rota\u00e7\u00f5es hiperb\u00f3licas de espa\u00e7o-tempo<\/strong><\/a><br \/>\n<a href=\"#6\">Introduzindo o par\u00e2metro de velocidade<\/a><br \/>\n<a href=\"#7\">Formulando as Rota\u00e7\u00f5es Espa\u00e7o-Temporais como Rota\u00e7\u00f5es Hiperb\u00f3licas<\/a><br \/>\n<a href=\"#8\"><strong>Conclus\u00f5es<\/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>Introdu\u00e7\u00e3o<\/h2>\n<p style=\"text-align:justify;\">At\u00e9 este ponto, examinamos detalhadamente a forma como s\u00e3o efetuadas as transforma\u00e7\u00f5es de Lorentz, isto \u00e9, a maneira como as coordenadas no espa\u00e7o-tempo de Minkowski de um evento espec\u00edfico se alteram quando s\u00e3o observadas de diferentes referenciais inerciais. O que faremos a seguir ser\u00e1 revisar uma perspectiva diferente para estes desenvolvimentos, visualizando-as como transforma\u00e7\u00f5es de rota\u00e7\u00f5es de espa\u00e7o-tempo. Logo descobriremos que esta abordagem traz vantagens ao n\u00edvel alg\u00e9brico, que simplificam em geral os c\u00e1lculos, especialmente ao combinar v\u00e1rias transforma\u00e7\u00f5es de Lorentz consecutivas.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Rota\u00e7\u00f5es no Espa\u00e7o-Tempo de Minkowski<\/h2>\n<p style=\"text-align:justify;\">Comecemos analisando como s\u00e3o efetuadas as diversas rota\u00e7\u00f5es espaciais no espa\u00e7o-tempo de Minkowski. Dado que este \u00e9 um espa\u00e7o de quatro dimens\u00f5es, o mais pr\u00e1tico para estabelecer uma rota\u00e7\u00e3o \u00e9 faz\u00ea-lo em rela\u00e7\u00e3o a um plano espec\u00edfico. Desta forma, podemos definir rota\u00e7\u00f5es sobre os planos <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xy<\/span><\/bdi>, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xz<\/span><\/bdi> e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">yz<\/span><\/bdi>, assim como tamb\u00e9m sobre os planos <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xt<\/span><\/bdi>, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">yt<\/span><\/bdi> e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">zt<\/span><\/bdi>. As rota\u00e7\u00f5es realizadas em planos formados por eixos espaciais s\u00e3o rota\u00e7\u00f5es puramente espaciais, enquanto que as realizadas em planos compostos por eixos de espa\u00e7o e tempo s\u00e3o rota\u00e7\u00f5es espa\u00e7o-temporais. Por ora, focaremos em entender em detalhe as rota\u00e7\u00f5es espaciais puras para depois ampliar este conhecimento \u00e0s rota\u00e7\u00f5es espa\u00e7o-temporais.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Rota\u00e7\u00f5es espaciais puras<\/h3>\n<p style=\"text-align:justify;\">\nComecemos nosso estudo das rota\u00e7\u00f5es espaciais revisando como se realizam as rota\u00e7\u00f5es no plano <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xy<\/span><\/bdi>. Para isso, suponhamos que temos um ponto com coordenadas <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b)<\/span><\/bdi> em rela\u00e7\u00e3o ao sistema definido pelos eixos <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/bdi> e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}<\/span><\/bdi>. Em seguida, analisemos a rela\u00e7\u00e3o que conecta estas coordenadas com as que observaria um sistema de refer\u00eancia rotado. Este sistema \u00e9 definido pelos eixos <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}^\\prime<\/span><\/bdi> e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}^\\prime<\/span><\/bdi>, que est\u00e3o rotados um \u00e2ngulo <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> em rela\u00e7\u00e3o ao sistema original, como mostra a figura a seguir:\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=\"Rota\u00e7\u00e3o em um \u00e2ngulo theta do plano 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=\"Rota\u00e7\u00e3o em um \u00e2ngulo theta do plano xy\" width=\"623\" height=\"495\" class=\"aligncenter size-full wp-image-25994 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotaciontheta.jpg 623w, http:\/\/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;\">\nPara obter as rela\u00e7\u00f5es entre as coordenadas <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b)<\/span><\/bdi> e <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a^\\prime,b^\\prime)<\/span><\/bdi> medidas de cada sistema, podemos utilizar as seguintes linhas guia:\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=\"Linhas guia para a obten\u00e7\u00e3o da rela\u00e7\u00e3o entre sistemas rotados\" 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=\"Linhas guia para a obten\u00e7\u00e3o da rela\u00e7\u00e3o entre sistemas rotados\" width=\"827\" height=\"620\" class=\"aligncenter size-full wp-image-25998 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia.png 827w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/rotacion-conlineasguia-300x225.png 300w, http:\/\/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;\">\nDe modo que agora \u00e9 simples obter as equa\u00e7\u00f5es de transforma\u00e7\u00e3o\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>Generaliza\u00e7\u00e3o Matricial para Rota\u00e7\u00f5es Tridimensionais<\/h3>\n<p style=\"text-align:justify;\">\nEste sistema de equa\u00e7\u00f5es pode ser representado de maneira mais conveniente em sua forma matricial.\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;\">Isso \u00e9 conveniente, pois a partir daqui \u00e9 f\u00e1cil generalizar para dimens\u00f5es maiores. Por exemplo, um ponto com coordenadas <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c)<\/span><\/bdi> no sistema formado pelos eixos <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{z}<\/span>, observado de outro sistema formado pelos eixos <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}^\\prime<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{y}^\\prime<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{z}^\\prime<\/span>, que se distingue do sistema original por uma rota\u00e7\u00e3o em um \u00e2ngulo <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> em rela\u00e7\u00e3o ao plano <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}\\hat{y}<\/span><\/bdi>, seria:<\/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=\"Rota\u00e7\u00f5es Espaciais\" 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=\"Rota\u00e7\u00f5es Espaciais\" width=\"725\" height=\"597\" class=\"aligncenter size-full wp-image-26014 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/01\/sistemarotadoxy3d.png 725w, http:\/\/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;\">A partir disso, obtemos as diferentes matrizes de transforma\u00e7\u00e3o de rota\u00e7\u00f5es para cada um dos planos espaciais.<\/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{Rota\u00e7\u00e3o em um \u00e2ngulo }\\theta\\\\ \\text{sobre o plano }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{Rota\u00e7\u00e3o em um \u00e2ngulo }\\theta\\\\ \\text{sobre o plano }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{Rota\u00e7\u00e3o em um \u00e2ngulo }\\theta\\\\ \\text{sobre o plano }xz \\end{array} \\end{array} <\/span>\n<p><\/bdi><\/p>\n<p style=\"text-align:justify;\">Para calcular a transforma\u00e7\u00e3o inversa destas transforma\u00e7\u00f5es de rota\u00e7\u00e3o, basta substituir <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">-\\theta<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Rota\u00e7\u00f5es Espaciais para Eventos com Coordenadas de Espa\u00e7o-Tempo<\/h3>\n<p style=\"text-align:justify;\">\nDe maneira similar a como generalizamos de duas a tr\u00eas dimens\u00f5es, podemos estender isso a quatro dimens\u00f5es. Para manter a coer\u00eancia com a linguagem da relatividade especial, \u00e9 importante entender o significado de cada coordenada. Geralmente, as coordenadas de espa\u00e7o-tempo s\u00e3o expressas da seguinte maneira:\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;\">\nAqui, os super\u00edndices n\u00e3o denotam pot\u00eancias, mas indicam as caracter\u00edsticas de cada coordenada. A coordenada com super\u00edndice 0 representa a dimens\u00e3o temporal, enquanto que as coordenadas com super\u00edndices 1, 2 e 3 correspondem \u00e0s dimens\u00f5es espaciais. Com isso em mente, as rota\u00e7\u00f5es puramente espaciais no espa\u00e7o-tempo de Minkowski s\u00e3o descritas pelas seguintes rela\u00e7\u00f5es:<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><strong>Rota\u00e7\u00e3o em rela\u00e7\u00e3o ao plano 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><\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><strong>Rota\u00e7\u00e3o em rela\u00e7\u00e3o ao plano 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>Rota\u00e7\u00e3o em rela\u00e7\u00e3o ao plano 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;\">Estas transforma\u00e7\u00f5es mant\u00eam exatamente as mesmas propriedades que suas hom\u00f3logas em tr\u00eas dimens\u00f5es.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Rota\u00e7\u00f5es hiperb\u00f3licas de espa\u00e7o-tempo<\/h2>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Introduzindo o par\u00e2metro de velocidade<\/h3>\n<p style=\"text-align:justify;\">A semelhan\u00e7a entre as transforma\u00e7\u00f5es de Lorentz e uma rota\u00e7\u00e3o espacial pode ser obtida introduzindo o que chamamos de <strong>par\u00e2metro de velocidade<\/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;\">Dado que <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}\\in]-1,1[<\/span>, tem-se que <span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}\\in\\mathbb{R}<\/span>. Al\u00e9m disso, notemos que, a partir disso se ter\u00e1 que <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x}=\\cosh(\\psi_{ss^\\prime_x})<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} \\beta_{ss^\\prime_x} = \\sinh(\\psi_{ss^\\prime_x})<\/span>. Isso se obt\u00e9m a partir dos seguintes c\u00e1lculos:<\/p>\n<p style=\"text-align:justify;\">\u00c9 claro que <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}= \\text{argtanh}(\\beta_{ss^\\prime_x})<\/span><\/bdi> \u00e9 equivalente a dizer <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x} =\\tanh(\\psi_{ss^\\prime_x})<\/span><\/bdi>; e portanto: <\/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;\">Como tanto o fator gama como o cosseno hiperb\u00f3lico s\u00e3o sempre maiores ou iguais a 1, fica demonstrado finalmente que <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;\">De forma semelhante continuando os c\u00e1lculos realizados anteriormente, tem-se:<\/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;\">E portanto <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>Formulando as Rota\u00e7\u00f5es Espa\u00e7o-Temporais como Rota\u00e7\u00f5es Hiperb\u00f3licas<\/h3>\n<p style=\"text-align:justify;\">\nTendo chegado a este ponto, agora podemos reescrever o fator associado ao boost de velocidade e o fator gama utilizando o par\u00e2metro de velocidade nas transforma\u00e7\u00f5es de Lorentz. Considerando dois sistemas inerciais <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> em configura\u00e7\u00e3o padr\u00e3o, onde ao segundo se aplica um boost sobre o eixo <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}<\/span>, tem-se:\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;\">Este sistema de equa\u00e7\u00f5es admite a seguinte representa\u00e7\u00e3o matricial:<\/p>\n<p style=\"text-align:center;\"><strong>Rota\u00e7\u00e3o Hiperb\u00f3lica de Espa\u00e7o-Tempo sobre o Plano 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;\">De maneira an\u00e1loga, temos rota\u00e7\u00f5es hiperb\u00f3licas sobre cada um dos planos de espa\u00e7o-tempo:<\/p>\n<p style=\"text-align:center;\"><strong>Rota\u00e7\u00e3o Hiperb\u00f3lica de Espa\u00e7o-Tempo sobre o Plano 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>Rota\u00e7\u00e3o Hiperb\u00f3lica de Espa\u00e7o-Tempo sobre o Plano 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;\">\nPor sua forma e propriedades alg\u00e9bricas, estas transforma\u00e7\u00f5es s\u00e3o muito similares a uma rota\u00e7\u00e3o espacial, apenas que em vez de usar fun\u00e7\u00f5es trigonom\u00e9tricas, utilizam fun\u00e7\u00f5es hiperb\u00f3licas. Embora n\u00e3o sejam rota\u00e7\u00f5es no sentido estrito, mant\u00eam certa analogia com as rota\u00e7\u00f5es revisadas no in\u00edcio. Por exemplo, de maneira similar a como ocorre com as rota\u00e7\u00f5es, a transforma\u00e7\u00e3o inversa se obt\u00e9m substituindo o par\u00e2metro de velocidade correspondente <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">-\\psi<\/span>. Estas transforma\u00e7\u00f5es s\u00e3o \u00e0s vezes denominadas como <strong>rota\u00e7\u00f5es hiperb\u00f3licas<\/strong>, e o par\u00e2metro de velocidade \u00e9 conhecido tamb\u00e9m como <strong>\u00e2ngulo hiperb\u00f3lico<\/strong>.\n<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclus\u00f5es<\/h2>\n<p style=\"text-align:justify;\">\n        At\u00e9 agora abordamos de maneira exaustiva o conceito das rota\u00e7\u00f5es no espa\u00e7o-tempo de Minkowski, o que nos permite ter uma compreens\u00e3o mais profunda das transforma\u00e7\u00f5es de Lorentz. Atrav\u00e9s deste estudo, alcan\u00e7amos os seguintes pontos chave:\n    <\/p>\n<ul>\n<li><strong>Reinterpreta\u00e7\u00e3o das Transforma\u00e7\u00f5es de Lorentz<\/strong>: Aprendemos a visualizar e compreender as transforma\u00e7\u00f5es de Lorentz n\u00e3o apenas como mudan\u00e7as nas coordenadas devido a diferentes referenciais, mas tamb\u00e9m como rota\u00e7\u00f5es no espa\u00e7o-tempo.<\/li>\n<li><strong>Compreens\u00e3o das Rota\u00e7\u00f5es no Espa\u00e7o-Tempo de Minkowski<\/strong>: Examinamos detalhadamente as rota\u00e7\u00f5es dentro do espa\u00e7o de quatro dimens\u00f5es de Minkowski.<\/li>\n<li><strong>Explora\u00e7\u00e3o das Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo<\/strong>: Finalmente, introduzimos o conceito de rota\u00e7\u00f5es hiperb\u00f3licas de espa\u00e7o-tempo, examinando suas semelhan\u00e7as com as rota\u00e7\u00f5es espaciais usuais.<\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Rota\u00e7\u00e3o Hiperb\u00f3licas de Espa\u00e7o-Tempo Resumo: Nesta aula, revisaremos como as transforma\u00e7\u00f5es de Lorentz podem ser reinterpretadas como transforma\u00e7\u00f5es de rota\u00e7\u00f5es de espa\u00e7o-tempo. Come\u00e7aremos examinando as rota\u00e7\u00f5es no espa\u00e7o de quatro dimens\u00f5es de Minkowski, distinguindo entre rota\u00e7\u00f5es puramente espaciais e aquelas que envolvem eixos espa\u00e7o-temporais. OBJETIVOS DE APRENDIZAGEM: Ao finalizar esta aula o aluno ser\u00e1 capaz [&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":12,"footnotes":""},"categories":[637,693],"tags":[],"class_list":["post-27314","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fisica-pt","category-relatividade"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubra como as transforma\u00e7\u00f5es de Lorentz podem ser reinterpretadas como rota\u00e7\u00f5es hiperb\u00f3licas de espa\u00e7o-tempo.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/pt\/rotacoes-hiperbolicas-de-espaco-tempo\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo\" \/>\n<meta property=\"og:description\" content=\"Descubra como as transforma\u00e7\u00f5es de Lorentz podem ser reinterpretadas como rota\u00e7\u00f5es hiperb\u00f3licas de espa\u00e7o-tempo.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/rotacoes-hiperbolicas-de-espaco-tempo\/\" \/>\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=\"2024-01-07T13:00:09+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-07-01T01:36:52+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=\"Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo\" \/>\n<meta name=\"twitter:description\" content=\"Descubra como as transforma\u00e7\u00f5es de Lorentz podem ser reinterpretadas como rota\u00e7\u00f5es hiperb\u00f3licas de espa\u00e7o-tempo.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/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\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo\",\"datePublished\":\"2024-01-07T13:00:09+00:00\",\"dateModified\":\"2024-07-01T01:36:52+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/\"},\"wordCount\":2289,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/01\\\/rotacioneshiperbolicas-1.jpg\",\"articleSection\":[\"F\u00edsica\",\"Relatividade\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/rotacoes-hiperbolicas-de-espaco-tempo\\\/\",\"name\":\"Rota\u00e7\u00f5es Hiperb\u00f3licas de Espa\u00e7o-Tempo - 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