{"id":25513,"date":"2022-07-11T00:00:51","date_gmt":"2022-07-11T00:00:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25513"},"modified":"2024-05-21T09:42:46","modified_gmt":"2024-05-21T09:42:46","slug":"the-principle-of-special-relativity","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/the-principle-of-special-relativity\/","title":{"rendered":"The Principle of Special Relativity"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>The Principle of Relativity<\/h1>\n<p><\/p>\n<p class=\"eq\"><em><strong>Summary:<\/strong> The principle of relativity asserts that observations depend on the inertial frame, but in such a way that physical laws remain consistent. This class will present the concepts of inertial reference frame and the foundations that allow for the transformations between the coordinates observed by different inertial reference frames in the contexts of Newtonian physics and special relativity.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, students will be able to:\n<\/p>\n<ol>\n<li><strong>Describe<\/strong> the fundamental concepts of the principle of relativity and inertial reference frames.<\/li>\n<li><strong>Explain<\/strong> the importance of the inertial reference frame in the context of the principle of relativity and differentiate between Newtonian physics and special relativity.<\/li>\n<li><strong>Apply<\/strong> Lorentz and Galileo transformations to solve simple problems and demonstrate how observations change between different inertial reference frames.<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>The Inertial Reference Frame<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>The Principle of Relativity in Newtonian Physics and Special Relativity<\/strong><\/a><br \/>\n<a href=\"#3\">Simplifying Transformations Between Inertial Reference Frames<\/a><br \/>\n<a href=\"#4\">Lorentz and Galileo Transformations<\/a><br \/>\n<a href=\"#5\"><strong>Conclusions<\/strong><\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/Z1IWvfnSNe8?si=VxQeeMn9VC8NhFi_\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Inertial Reference Frame<\/h2>\n<p>When doing physics, it&#8217;s always possible to choose the reference frame from which events will be measured, and these frames can differ both in orientation and relative motion. Among all possible reference frames, there is a special class that allows us to do physics as we know it: the <strong>inertial reference frames.<\/strong> A reference frame is said to be inertial when it satisfies <strong>Newton&#8217;s first law,<\/strong> which states that in the absence of external agents, particles maintain their state of motion and therefore:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^2}{dt^2} = \\frac{dy^2}{dt^2} = \\frac{dz^2}{dt^2} = 0.<\/span>\n<p>From this, it follows that, in the absence of gravity, if two frames <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> are inertial, then <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> can only differ from <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> in:<\/p>\n<ul>\n<li>A translation,<\/li>\n<li>A rotation,<\/li>\n<li>Relative motion between both frames at a constant speed.<\/li>\n<\/ul>\n<p>The concept of inertial frame is fundamental to the <strong>principle of relativity,<\/strong> which states that the laws of physics have the same form in all inertial frames. This principle applies equally in both Newtonian physics and special relativity.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>The Principle of Relativity in Newtonian Physics and Special Relativity<\/h3>\n<p>Newtonian and special relativistic descriptions differ in how the coordinates of an event, relative to an inertial system, relate to those of another inertial system.<\/p>\n<p>Consider two Cartesian inertial frames <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> in \u00abstandard configuration,\u00bb that is, where <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> moves along the <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> axis of <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> at a constant speed <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_{ss^\\prime} = v_{ss^\\prime_x} \\hat{x}<\/span>, and the respective axes of <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> are aligned and coincide at <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime = 0<\/span>.<\/p>\n<figure>\n<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png\" alt=\"\" width=\"1374\" height=\"741\" class=\"aligncenter size-full wp-image-25502 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png\" alt=\"\" width=\"1374\" height=\"741\" class=\"aligncenter size-full wp-image-25502 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png 1374w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-300x162.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-1024x552.png 1024w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-768x414.png 768w\" sizes=\"(max-width: 1374px) 100vw, 1374px\" \/><\/noscript><br \/>\n<\/figure>\n<p>Then it is found that, if there is a linear transformation relating the coordinates of an event seen from <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, then they are related through the following system of linear equations<\/p>\n<p><a name=\"eq1\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nt^\\prime &amp;= At + Bx,\\\\\n\nx^\\prime &amp;= Dt + Ex,\\\\\n\ny^\\prime &amp;= y, \\\\\n\nz^\\prime &amp;= z,\n\n\\end{array}\\;\\;\\;[\\triangle]<\/span>\n<p>where <span class=\"katex-eq\" data-katex-display=\"false\">A,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">B,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">E<\/span> are constants to be determined.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Simplifying Transformations Between Inertial Reference Frames<\/h3>\n<p>These transformations can be simplified if we make the following observations:<\/p>\n<ul>\n<li>\n<p>Since the transformations must hold for any <span class=\"katex-eq\" data-katex-display=\"false\">x^\\prime<\/span>, it follows that if we place the event at the origin of <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, we will have <span class=\"katex-eq\" data-katex-display=\"false\">x^\\prime =0<\/span>. This implies that the event moves along with <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> and its position relative to <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> will be <span class=\"katex-eq\" data-katex-display=\"false\">x=v_{{ss^\\prime}_x}t.<\/span>\n<p>Substituting <span class=\"katex-eq\" data-katex-display=\"false\">x=v_{{ss^\\prime}_x}t.<\/span> into the second equation of <a href=\"#eq1\"><span class=\"katex-eq\" data-katex-display=\"false\">[\\triangle]<\/span><\/a> results in <span class=\"katex-eq\" data-katex-display=\"false\">D=-Ev_{{ss^\\prime}_x}.<\/span>\n<p>Similarly, the transformations also hold for any <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, so if we place the event at the origin of <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, we will find that, seen from <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, its position is <span class=\"katex-eq\" data-katex-display=\"false\">x^\\prime = -v_{{ss^\\prime}_x}t^\\prime<\/span>.<\/p>\n<p>Substituting this into the first and second equations of <a href=\"#eq1\"><span class=\"katex-eq\" data-katex-display=\"false\">[\\triangle]<\/span><\/a> leads to <span class=\"katex-eq\" data-katex-display=\"false\">t^\\prime=At<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">-v_{{ss^\\prime}_x}t^\\prime =Dt<\/span>. Dividing these two equations concludes that <span class=\"katex-eq\" data-katex-display=\"false\">D=-v_{{ss^\\prime}_x}A<\/span>.<\/p>\n<\/li>\n<li>\n<p>Therefore, the only way to reconcile the previous points is to impose that <span class=\"katex-eq\" data-katex-display=\"false\">A=E,<\/span>, and with this, the transformations are reduced to:<\/p>\n<p><a name=\"eq2\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nt^\\prime &amp;= At + Bx,\\\\\n\nx^\\prime &amp;= A(x - v_{ss^\\prime_x} t), \\\\\n\ny^\\prime &amp;= y, \\\\\n\nz^\\prime &amp;= z.\n\n\\end{array}\\;\\;\\;[2]<\/span>\n<\/li>\n<\/ul>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Lorentz and Galileo Transformations<\/h3>\n<p>In the case of Newtonian physics, we have Galilean relativity, where time elapses in the same way for all inertial frames of reference, and therefore <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime.<\/span> As a consequence, <span class=\"katex-eq\" data-katex-display=\"false\">A=1<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B=0.<\/span> This leads to the well-known <strong>Galilean transformations<\/strong> which allow transforming observations between two inertial frames.<\/p>\n<p><a name=\"eq2\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nt^\\prime &amp;= t\\\\\n\nx^\\prime &amp;= x - v_{ss^\\prime_x} t, \\\\\n\ny^\\prime &amp;= y, \\\\\n\nz^\\prime &amp;= z.\n\n\\end{array}\\;\\;\\;[3]<\/span>\n<\/li>\n<\/ul>\n<p>On the other hand, in the case of relativistic physics, we have Einstein&#8217;s Principle of Relativity, which instead considers the speed of light in vacuum to be the same in all inertial frames. This leads to the well-known Lorentz Transformations of special relativity, which, as we will see in later entries, take the following form:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nct^\\prime &amp;=\\gamma_x \\left( ct - \\beta_x x \\right) \\\\\n\nx^\\prime &amp;= \\gamma_x(x - \\beta_x ct) \\\\\n\ny^\\prime &amp;= y \\\\\n\nz^\\prime &amp;= z\n\n\\end{array}\n\n<\/span>\n<p>where <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_x=v_{ss^\\prime_x}\/c<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma= 1\/\\sqrt{1-\\beta_x^2}.<\/span>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Conclusions<\/h2>\n<p>The Principle of Relativity not only revolutionizes our understanding of the universe but also challenges our most fundamental perceptions of time and space. Through the analysis of inertial frames of reference, we have seen how the laws of physics maintain their constant form, regardless of the observer, in both Newtonian physics and special relativity. The Lorentz and Galileo transformations uniquely illustrate the subtle and profound differences between these two approaches. This principle, which lies at the heart of modern physics, is essential not only for the theoretical understanding of physical phenomena but also for practical applications ranging from GPS technology to space exploration. By unraveling the complexities of the Principle of Relativity, we take one step closer to understanding the intricate fabric of the cosmos and our place within it.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Principle of Relativity Summary: The principle of relativity asserts that observations depend on the inertial frame, but in such a way that physical laws remain consistent. This class will present the concepts of inertial reference frame and the foundations that allow for the transformations between the coordinates observed by different inertial reference frames in [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25506,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[635,691],"tags":[],"class_list":["post-25513","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physics","category-relativity"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Principle of Special Relativity - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover the mysteries of the Principle of Relativity and how it transforms our perception of time and space.\" \/>\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\/en\/the-principle-of-special-relativity\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Principle of Special Relativity\" \/>\n<meta property=\"og:description\" content=\"Discover the mysteries of the Principle of Relativity and how it transforms our perception of time and space.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/the-principle-of-special-relativity\/\" \/>\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=\"2022-07-11T00:00:51+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-21T09:42:46+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/RELATIVIDAD-ESPECIAL-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The Principle of Special Relativity\" \/>\n<meta name=\"twitter:description\" content=\"Discover the mysteries of the Principle of Relativity and how it transforms our perception of time and space.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/RELATIVIDAD-ESPECIAL.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\\\/en\\\/the-principle-of-special-relativity\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-principle-of-special-relativity\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Principle of Special Relativity\",\"datePublished\":\"2022-07-11T00:00:51+00:00\",\"dateModified\":\"2024-05-21T09:42:46+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-principle-of-special-relativity\\\/\"},\"wordCount\":1109,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-principle-of-special-relativity\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/12\\\/RELATIVIDAD-ESPECIAL.jpg\",\"articleSection\":[\"Physics\",\"Relativity\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-principle-of-special-relativity\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-principle-of-special-relativity\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-principle-of-special-relativity\\\/\",\"name\":\"The Principle of Special Relativity - 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