{"id":27328,"date":"2024-01-07T13:00:51","date_gmt":"2024-01-07T13:00:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27328"},"modified":"2024-08-15T05:56:56","modified_gmt":"2024-08-15T05:56:56","slug":"rotations-hyperboliques-de-lespace-temps","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/rotations-hyperboliques-de-lespace-temps\/","title":{"rendered":"Rotations Hyperboliques de l&#8217;Espace-Temps"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Rotations Hyperboliques de l&#8217;Espace-Temps<\/h1>\n<p class=\"eq\"><em><strong>R\u00e9sum\u00e9:<\/strong><br \/>\nDans ce cours, nous examinerons comment les transformations de Lorentz peuvent \u00eatre r\u00e9interpr\u00e9t\u00e9es comme des transformations de rotations de l&#8217;espace-temps. Nous commencerons par examiner les rotations dans l&#8217;espace de Minkowski \u00e0 quatre dimensions, en distinguant les rotations purement spatiales de celles impliquant des axes espace-temps.<\/br><\/em><\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE:<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de:<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> les transformations de rotation dans l&#8217;espace-temps de Minkowski.<\/li>\n<li><strong>Comprendre<\/strong> les transformations de Lorentz comme des rotations espace-temps.<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>INDEX<\/strong><br \/>\n<a href=\"#0\"><strong>Introduction<\/strong><\/a><br \/>\n<a href=\"#1\"><strong>Rotations dans l&#8217;Espace-Temps de Minkowski<\/strong><\/a><br \/>\n<a href=\"#2\">Rotations spatiales pures<\/a><br \/>\n<a href=\"#3\">G\u00e9n\u00e9ralisation Matricielle pour les Rotations Tridimensionnelles<\/a><br \/>\n<a href=\"#4\">Rotations Spatiales pour des \u00c9v\u00e9nements avec des Coordonn\u00e9es Espace-Temps<\/a><br \/>\n<a href=\"#5\"><strong>Rotations hyperboliques de l&#8217;espace-temps<\/strong><\/a><br \/>\n<a href=\"#6\">Introduction du param\u00e8tre de vitesse<\/a><br \/>\n<a href=\"#7\">Formuler les Rotations Espace-Temps comme des Rotations Hyperboliques<\/a><br \/>\n<a href=\"#8\"><strong>Conclusions<\/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>Introduction<\/h2>\n<p style=\"text-align:justify;\">Jusqu&#8217;\u00e0 pr\u00e9sent, nous avons examin\u00e9 en d\u00e9tail la fa\u00e7on dont se produisent les transformations de Lorentz, c&#8217;est-\u00e0-dire la mani\u00e8re dont les coordonn\u00e9es dans l&#8217;espace-temps de Minkowski d&#8217;un \u00e9v\u00e9nement sp\u00e9cifique se modifient lorsqu&#8217;elles sont observ\u00e9es depuis diff\u00e9rents r\u00e9f\u00e9rentiels inertiels. Ce que nous allons faire ensuite est d&#8217;examiner une perspective diff\u00e9rente pour ces d\u00e9veloppements, en les visualisant comme des transformations de rotations de l&#8217;espace-temps. Nous d\u00e9couvrirons bient\u00f4t que cette approche offre des avantages au niveau alg\u00e9brique, simplifiant en g\u00e9n\u00e9ral les calculs, notamment lorsqu&#8217;il s&#8217;agit de combiner plusieurs transformations de Lorentz cons\u00e9cutives.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Rotations dans l&#8217;Espace-Temps de Minkowski<\/h2>\n<p style=\"text-align:justify;\">Commen\u00e7ons par analyser comment se produisent les diff\u00e9rentes rotations spatiales dans l&#8217;espace-temps de Minkowski. \u00c9tant donn\u00e9 qu&#8217;il s&#8217;agit d&#8217;un espace \u00e0 quatre dimensions, la mani\u00e8re la plus pratique d&#8217;\u00e9tablir une rotation est de le faire par rapport \u00e0 un plan sp\u00e9cifique. Ainsi, nous pouvons d\u00e9finir des rotations sur les plans <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>, ainsi que sur les plans <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>. Les rotations effectu\u00e9es dans les plans form\u00e9s par des axes spatiaux sont des rotations purement spatiales, tandis que celles r\u00e9alis\u00e9es dans des plans compos\u00e9s d&#8217;axes espace et temps sont des rotations espace-temps. Pour l&#8217;instant, nous nous concentrerons sur la compr\u00e9hension des rotations spatiales pures avant d&#8217;\u00e9largir cette connaissance aux rotations espace-temps.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Rotations spatiales pures<\/h3>\n<p style=\"text-align:justify;\">\nCommen\u00e7ons notre \u00e9tude des rotations spatiales en examinant comment se r\u00e9alisent les rotations dans le plan <bdi><span class=\"katex-eq\" data-katex-display=\"false\">xy<\/span><\/bdi>. Pour cela, supposons que nous avons un point avec des coordonn\u00e9es <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b)<\/span><\/bdi> par rapport au syst\u00e8me d\u00e9fini par les 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>. Ensuite, analysons la relation qui relie ces coordonn\u00e9es \u00e0 celles qu&#8217;observerait un syst\u00e8me de r\u00e9f\u00e9rence rot\u00e9. Ce syst\u00e8me est d\u00e9fini par les 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 sont rot\u00e9s d&#8217;un angle <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> par rapport au syst\u00e8me original, comme le montre la figure suivante:\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=\"Rotation dans un angle theta du plan 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=\"Rotation dans un angle theta du plan 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;\">\nPour obtenir les relations entre les coordonn\u00e9es <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> mesur\u00e9es depuis chaque syst\u00e8me, nous pouvons utiliser les lignes de guide suivantes:\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=\"Lignes de guide pour obtenir la relation entre les syst\u00e8mes rot\u00e9s\" 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=\"Lignes de guide pour obtenir la relation entre les syst\u00e8mes rot\u00e9s\" 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;\">\nAinsi, il est maintenant facile d&#8217;obtenir les \u00e9quations de transformation\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>G\u00e9n\u00e9ralisation Matricielle pour les Rotations Tridimensionnelles<\/h3>\n<p style=\"text-align:justify;\">\nCe syst\u00e8me d&#8217;\u00e9quations peut \u00eatre repr\u00e9sent\u00e9 de mani\u00e8re plus pratique sous sa forme matricielle.\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;\">C&#8217;est pratique, car \u00e0 partir de l\u00e0, il est facile de g\u00e9n\u00e9raliser pour des dimensions plus grandes. Par exemple, un point avec des coordonn\u00e9es <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c)<\/span><\/bdi> dans le syst\u00e8me form\u00e9 par les 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>, observ\u00e9 depuis un autre syst\u00e8me form\u00e9 par les 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>, qui se distingue du syst\u00e8me original par une rotation d&#8217;un angle <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> par rapport au plan <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}\\hat{y}<\/span><\/bdi>, serait:<\/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=\"Rotations 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=\"Rotations 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;\">\u00c0 partir de cela, nous obtenons les diff\u00e9rentes matrices de transformation de rotations pour chacun des plans spatiaux.<\/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{Rotation d&#039;un angle }\\theta\\\\ \\text{sur le plan }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{Rotation d&#039;un angle }\\theta\\\\ \\text{sur le plan }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{Rotation d&#039;un angle }\\theta\\\\ \\text{sur le plan }xz \\end{array} \\end{array} <\/span>\n<p><\/bdi><\/p>\n<p style=\"text-align:justify;\">Pour calculer la transformation inverse de ces transformations de rotation, il suffit de remplacer <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">-\\theta<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Rotations Spatiales pour des \u00c9v\u00e9nements avec des Coordonn\u00e9es Espace-Temps<\/h3>\n<p style=\"text-align:justify;\">\nDe m\u00eame que nous avons g\u00e9n\u00e9ralis\u00e9 de deux \u00e0 trois dimensions, nous pouvons \u00e9tendre cela \u00e0 quatre dimensions. Pour rester coh\u00e9rent avec le langage de la relativit\u00e9 sp\u00e9ciale, il est important de comprendre la signification de chaque coordonn\u00e9e. G\u00e9n\u00e9ralement, les coordonn\u00e9es espace-temps sont exprim\u00e9es de la mani\u00e8re suivante:\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;\">\nIci, les exposants ne d\u00e9signent pas des puissances, mais indiquent les caract\u00e9ristiques de chaque coordonn\u00e9e. La coordonn\u00e9e avec un exposant 0 repr\u00e9sente la dimension temporelle, tandis que les coordonn\u00e9es avec des exposants 1, 2 et 3 correspondent aux dimensions spatiales. Cela \u00e9tant dit, les rotations purement spatiales dans l&#8217;espace-temps de Minkowski se d\u00e9crivent par les relations suivantes:<\/p>\n<p><bdi><\/p>\n<p style=\"text-align:center;\"><strong>Rotation par rapport au plan 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>Rotation par rapport au plan 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>Rotation par rapport au plan 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;\">Ces transformations conservent exactement les m\u00eames propri\u00e9t\u00e9s que leurs homologues en trois dimensions.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Rotations hyperboliques de l&#8217;espace-temps<\/h2>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Introduction du param\u00e8tre de vitesse<\/h3>\n<p style=\"text-align:justify;\">La similarit\u00e9 entre les transformations de Lorentz et une rotation spatiale peut \u00eatre obtenue en introduisant ce que nous appelons <strong>param\u00e8tre de vitesse<\/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;\">\u00c9tant donn\u00e9 que <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}\\in]-1,1[<\/span>, on a <span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}\\in\\mathbb{R}<\/span>. De plus, notons qu&#8217;\u00e0 partir de cela on aura que <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>. Cela s&#8217;obtient \u00e0 partir des calculs suivants:<\/p>\n<p style=\"text-align:justify;\">Il est clair que <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_{ss^\\prime_x}= \\text{argtanh}(\\beta_{ss^\\prime_x})<\/span><\/bdi> \u00e9quivaut \u00e0 dire <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x} =\\tanh(\\psi_{ss^\\prime_x})<\/span><\/bdi>; et donc: <\/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;\">Comme le facteur gamma et le cosinus hyperbolique sont toujours sup\u00e9rieurs ou \u00e9gaux \u00e0 1, il est finalement d\u00e9montr\u00e9 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 mani\u00e8re similaire en continuant les calculs pr\u00e9c\u00e9dents, on a:<\/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;\">Et donc <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>Formuler les Rotations Espace-Temps comme des Rotations Hyperboliques<\/h3>\n<p style=\"text-align:justify;\">\n\u00c0 ce stade, nous pouvons maintenant r\u00e9\u00e9crire le facteur associ\u00e9 au boost de vitesse et le facteur gamma en utilisant le param\u00e8tre de vitesse dans les transformations de Lorentz. En consid\u00e9rant deux syst\u00e8mes inertiels <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> en configuration standard, o\u00f9 au second est appliqu\u00e9 un boost sur l&#8217;axe <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}<\/span>, on a:\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;\">Ce syst\u00e8me d&#8217;\u00e9quations admet la repr\u00e9sentation matricielle suivante:<\/p>\n<p style=\"text-align:center;\"><strong>Rotation Hyperbolique de l&#8217;Espace-Temps sur le Plan 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 mani\u00e8re analogue, nous avons des rotations hyperboliques sur chacun des plans espace-temps:<\/p>\n<p style=\"text-align:center;\"><strong>Rotation Hyperbolique de l&#8217;Espace-Temps sur le Plan 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>Rotation Hyperbolique de l&#8217;Espace-Temps sur le Plan 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;\">\nPar leur forme et leurs propri\u00e9t\u00e9s alg\u00e9briques, ces transformations sont tr\u00e8s similaires \u00e0 une rotation spatiale, sauf qu&#8217;au lieu d&#8217;utiliser des fonctions trigonom\u00e9triques, elles utilisent des fonctions hyperboliques. Bien qu&#8217;elles ne soient pas des rotations au sens strict, elles maintiennent une certaine analogie avec les rotations examin\u00e9es au d\u00e9but. Par exemple, de mani\u00e8re similaire \u00e0 ce qui se passe avec les rotations, la transformation inverse s&#8217;obtient en rempla\u00e7ant le param\u00e8tre de vitesse correspondant <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">-\\psi<\/span>. Ces transformations sont parfois appel\u00e9es <strong>rotations hyperboliques<\/strong>, et le param\u00e8tre de vitesse est \u00e9galement connu sous le nom d&#8217;<strong>angle hyperbolique<\/strong>.\n<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclusions<\/h2>\n<p style=\"text-align:justify;\">\n        Jusqu&#8217;\u00e0 pr\u00e9sent, nous avons abord\u00e9 de mani\u00e8re exhaustive le concept des rotations dans l&#8217;espace-temps de Minkowski, ce qui nous permet d&#8217;avoir une compr\u00e9hension plus approfondie des transformations de Lorentz. \u00c0 travers cette \u00e9tude, nous avons atteint les points cl\u00e9s suivants:\n    <\/p>\n<ul>\n<li><strong>R\u00e9interpr\u00e9tation des Transformations de Lorentz<\/strong>: Nous avons appris \u00e0 visualiser et \u00e0 comprendre les transformations de Lorentz non seulement comme des changements de coordonn\u00e9es dus \u00e0 diff\u00e9rents r\u00e9f\u00e9rentiels, mais aussi comme des rotations dans l&#8217;espace-temps.<\/li>\n<li><strong>Compr\u00e9hension des Rotations dans l&#8217;Espace-Temps de Minkowski<\/strong>: Nous avons examin\u00e9 en d\u00e9tail les rotations dans l&#8217;espace \u00e0 quatre dimensions de Minkowski.<\/li>\n<li><strong>Exploration des Rotations Hyperboliques de l&#8217;Espace-Temps<\/strong>: Enfin, nous avons introduit le concept de rotations hyperboliques de l&#8217;espace-temps, en examinant leurs similitudes avec les rotations spatiales habituelles.<\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Rotations Hyperboliques de l&#8217;Espace-Temps R\u00e9sum\u00e9: Dans ce cours, nous examinerons comment les transformations de Lorentz peuvent \u00eatre r\u00e9interpr\u00e9t\u00e9es comme des transformations de rotations de l&#8217;espace-temps. Nous commencerons par examiner les rotations dans l&#8217;espace de Minkowski \u00e0 quatre dimensions, en distinguant les rotations purement spatiales de celles impliquant des axes espace-temps. OBJECTIFS D&#8217;APPRENTISSAGE: \u00c0 la fin [&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":26,"footnotes":""},"categories":[647,703],"tags":[],"class_list":["post-27328","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physique","category-relativite"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Rotations Hyperboliques de l&#039;Espace-Temps - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez comment les transformations de Lorentz peuvent \u00eatre r\u00e9interpr\u00e9t\u00e9es comme des rotations hyperboliques de l&#039;espace-temps.\" \/>\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\/fr\/rotations-hyperboliques-de-lespace-temps\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Rotations Hyperboliques de l&#039;Espace-Temps\" \/>\n<meta property=\"og:description\" content=\"D\u00e9couvrez comment les transformations de Lorentz peuvent \u00eatre r\u00e9interpr\u00e9t\u00e9es comme des rotations hyperboliques de l&#039;espace-temps.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/rotations-hyperboliques-de-lespace-temps\/\" \/>\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=\"2024-01-07T13:00:51+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-15T05:56:56+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=\"Rotations Hyperboliques de l&#039;Espace-Temps\" \/>\n<meta name=\"twitter:description\" content=\"D\u00e9couvrez comment les transformations de Lorentz peuvent \u00eatre r\u00e9interpr\u00e9t\u00e9es comme des rotations hyperboliques de l&#039;espace-temps.\" \/>\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=\"9 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Rotations Hyperboliques de l&#8217;Espace-Temps\",\"datePublished\":\"2024-01-07T13:00:51+00:00\",\"dateModified\":\"2024-08-15T05:56:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/\"},\"wordCount\":2313,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/01\\\/rotacioneshiperbolicas-1.jpg\",\"articleSection\":[\"Physique\",\"Relativit\u00e9\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/rotations-hyperboliques-de-lespace-temps\\\/\",\"name\":\"Rotations Hyperboliques de l'Espace-Temps - 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