{"id":25653,"date":"2022-07-18T00:00:12","date_gmt":"2022-07-18T00:00:12","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25653"},"modified":"2024-05-21T09:42:21","modified_gmt":"2024-05-21T09:42:21","slug":"the-lorentz-transformations-of-special-relativity","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-lorentz-transformations-of-special-relativity\/","title":{"rendered":"The Lorentz Transformations of Special Relativity"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>The Lorentz Transformations of Special Relativity<\/h1>\n<p class=\"eq\"><em><strong>Summary:<\/strong><br \/>\nThe Lorentz transformations allow for the transformation of observed space and time coordinates between two inertial reference frames. In this article, we will review how the Lorentz transformations are derived as a linear coordinate transformation that emerges from considering the speed of light as constant in all inertial reference frames and their convergence to Galilean transformations for speeds small compared to the speed of light.<\/br><\/em><\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nAt the end of this class, the student will be able to:<\/p>\n<ol>\n<li><strong>Recognize<\/strong> key concepts of special relativity, such as the Lorentz Transformations, \u00abvelocity boost\u00bb and \u00abLorentz factor\u00bb.<\/li>\n<li><strong>Understand<\/strong> how the principle that the speed of light is constant in all inertial frames affects the perception of time and space.<\/li>\n<li><strong>Apply<\/strong> the Lorentz Transformations to concrete situations, such as the relationship between inertial frames and the speed of light in different references.<\/li>\n<li><strong>Integrate<\/strong> previous knowledge of Galilean transformations and special relativity to understand how the Lorentz Transformations generalize and converge them.<\/li>\n<li><strong>Decompose<\/strong> the Lorentz Transformations into their fundamental components, such as the constant speed of light and linearity in coordinate transformations.<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>New Considerations<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Derivation of the Lorentz Transformations<\/strong><\/a><br \/>\n<a href=\"#3\">Recap on (linear) Coordinate Transformations<\/a><br \/>\n<a href=\"#4\">Introducing the Speed of Light as a Universal Constant<\/a><br \/>\n<a href=\"#5\">The Velocity Boost and the Lorentz Factor<\/a><br \/>\n<a href=\"#6\">Synthesis of the Lorentz Transformations<\/a><br \/>\n<a href=\"#7\"><strong>The Lorentz Transformations Converge and Generalize the Galilean Transformations<\/strong><\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KQby8yJGTSA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>New Considerations<\/h2>\n<p style=\"text-align:justify;\">As a consequence of what has been seen in <a href=\"http:\/\/toposuranos.com\/material\/en\/the-speed-of-light-and-electromagnetic-waves\/\" rel=\"noopener\" target=\"_blank\">Propagation of Electromagnetic Waves in Vacuum<\/a>, in special relativity it is postulated as a principle that the speed of light <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> is the same for all inertial frames. But such an assumption does not come without implications:<\/p>\n<ol>\n<li>The Galilean transformations must be abandoned as a valid means to transform observations from one inertial frame into another.<\/li>\n<li>The intuitive idea that time flows in the same way for all inertial reference frames must be left behind.<\/li>\n<\/ol>\n<p style=\"text-align:justify;\">It is through these considerations that the <strong>Lorentz transformations<\/strong> are obtained, serving as a correction and generalization for the Galilean transformations, which also works for electromagnetic theory.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Derivation of the Lorentz Transformations<\/h2>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Recap on (Linear) Coordinate Transformations<\/h3>\n<p style=\"text-align:justify;\">Consider two 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 standard configuration such that the second origin moves with a constant velocity <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_0 = v_{x_0}\\hat{x}<\/span> with respect to the first origin. What will be demonstrated next is that if the coordinates of an event seen from two inertial systems <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> are related by a linear transformation as revised in <a href=\"http:\/\/toposuranos.com\/material\/en\/the-principle-of-special-relativity\/\" rel=\"noopener\" target=\"_blank\">The Principle of Relativity<\/a> (specifically, <a href=\"http:\/\/toposuranos.com\/material\/en\/the-principle-of-special-relativity\/#eq2\" rel=\"noopener\" target=\"_blank\">this expression<\/a>) and it is accepted that the speed of light is the same from all inertial frames, then the coordinate transformation corresponds exactly with the Lorentz transformations that we will obtain later.\n<\/p>\n<p><center><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=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png 1374w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-300x162.png 300w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-1024x552.png 1024w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-768x414.png 768w\" sizes=\"(max-width: 1374px) 100vw, 1374px\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align:justify;\">Initially, the coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(t,x)<\/span> of an event seen from <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, and coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(t^\\prime, x^\\prime)<\/span> of the same event seen from <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> moving with a velocity <span class=\"katex-eq\" data-katex-display=\"false\">v_{v}=v_{x_0}\\hat{x}<\/span> relative to <span class=\"katex-eq\" data-katex-display=\"false\">S,<\/span> are related through a linear transformation such that:<\/p>\n<p><a name=\"eq1\"><\/a><a name=\"eq2\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{llr} t^\\prime &amp;= At + Bx, &amp; [1]\\\\ x^\\prime &amp;= Dt + Ex &amp; [2]  \\end{array} <\/span>\n<p style=\"text-align:justify;\" >where <span class=\"katex-eq\" data-katex-display=\"false\">A, B, C<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> are constants to be determined and the coordinates <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span> have been omitted (without loss of generality) for simplicity.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Introducing the Speed of Light as a Universal Constant<\/h3>\n<p style=\"text-align:justify;\">The constants <span class=\"katex-eq\" data-katex-display=\"false\">A, B, D<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">E<\/span> can be determined from these new considerations by invoking some special cases. First of all, we must consider that the coordinate transformation expressed through <a href=\"#eq1\">[1]<\/a> and <a href=\"#eq2\">[2]<\/a> must always work, and as a consequence, it must work in each particular case, and these particular cases are the ones that will be stated below to delve into their form:<\/p>\n<ul>\n<li>\n<p><strong>Consider the event as moving at the speed of light:<\/strong> If it has coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(t,x)<\/span> seen from <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">(t^\\prime, x^\\prime)<\/span> seen from <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime,<\/span> then they must satisfy the relation:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{x^2}{t^2} = c^2 = \\frac{{x^\\prime}^2}{{t^\\prime}^2}.<\/span>\n<p>From this, it is inferred that<\/p>\n<p><a name=\"eq3\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">c^2 t^2 - x^2 = c^2{t^\\prime}^2 - {x^\\prime}^2 = 0\\;\\;\\; [3]<\/span>\n<\/li>\n<li>\n<p><strong>Consider the event as moving along with the inertial frame <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>:<\/strong><\/p>\n<p>If the event has the same coordinates as the origin of the inertial frame <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime,<\/span> then it will occur that <span class=\"katex-eq\" data-katex-display=\"false\">x=v_0 t<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x^\\prime =0.<\/span> Consequently, from equation <a href=\"#eq2\">[2]<\/a> it will be:<\/p>\n<p><a name=\"eq4\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} &amp; 0 = Dt + Ev_0 t \\\\ \\equiv &amp; D = -Ev_0\\:\\:\\;[4] \\end{array}<\/span>\n<\/li>\n<li>\n<p><strong>Finally, consider the event as remaining along with the origin of the inertial frame <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>:<\/strong><\/p>\n<p style=\"text-align:justify;\">In this case, it will have that <span class=\"katex-eq\" data-katex-display=\"false\">x=0<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x^\\prime = -v_0 t^\\prime,<\/span> so from equation <a href=\"#eq2\">[2]<\/a> it will be:<\/p>\n<p><a name=\"eq5\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} &amp;-v_0t^\\prime = Dt\\\\ \\equiv &amp; t= \\displaystyle -\\frac{v_0}{D} t^\\prime\\;\\;\\;[5] \\end{array}<\/span>\n<p style=\"text-align:justify;\">Then, from <a href=\"#eq1\">[1]<\/a> and <a href=\"#eq5\">[5]<\/a> it is concluded that:<\/p>\n<p><a name=\"eq6\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} &amp; t^\\prime = A \\left(\\displaystyle -\\frac{v_0}{D}\\right) t^\\prime + \\underbrace{Bx}_{x=0} \\\\ \\\\ \\equiv &amp; \\displaystyle \\frac{-Av_0}{D} = 1 \\\\ \\\\ \\equiv &amp; D = -Av_0\\;\\;\\;[6] \\end{array}<\/span>\n<\/li>\n<\/ul>\n<p style=\"text-align:justify;\">Finally, from <a href=\"#eq4\">[4]<\/a> and <a href=\"#eq6\">[6]<\/a>: <span class=\"katex-eq\" data-katex-display=\"false\">A = E,<\/span> so the system of equations given by <a href=\"#eq1\">[1]<\/a> and <a href=\"#eq2\">[2]<\/a> is reduced to<\/p>\n<p><a name=\"eq7\"><\/a> <a name=\"eq8\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll} t^\\prime &amp;= At + Bx  &amp;  [7]\\\\ \\\\ x^\\prime &amp;= A(x - v_{x_0} t) &amp; [8] \\end{array}<\/span>\n<p><a name=\"5\"><\/a><\/p>\n<h3>The Velocity Boost and the Lorentz Factor<\/h3>\n<p style=\"text-align:justify;\">Now, replacing <a href=\"#eq7\">[7]<\/a> and <a href=\"#eq8\">[8]<\/a> into <a href=\"#eq3\">[3]<\/a> we get<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} &amp; c^2 (At +Bx)^2 - A^2 (x - v_{x_0} t)^2 = c^2t^2 - x^2\\\\ \\\\ \\equiv\\; &amp; \\color{blue}{(c^2 A^2) t^2} + \\color{red}{(2c^2 AB)xt} \\color{black} + c^2 B^2 x^2 -  A^2 x^2 + \\color{red} {(2A^2v_{x_0})xt} \\color{black}- \\color{blue}{(A^2 v_{x_0}^2) t^2} \\color{black}= \\color{blue}{(c^2) t^2} \\color{black}- x^2. \\end{array}<\/span>\n<p style=\"text-align:justify;\"><span style=\"color:blue\"><strong>from what has been marked in blue we obtain<\/strong><\/span><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} &amp;c^2 A^2 - A^2 v_{x_0}^2 = c^2 \\\\ \\\\ \\equiv\\;&amp; A^2 (c^2 - v_{x_0}^2) = c^2 \\\\ \\\\ \\equiv\\;&amp; \\displaystyle A^2 = \\frac{c^2}{c^2 - v_{x_0}^2} = \\frac{1}{1 - \\frac{v_{x_0}^2}{c^2}}  \\\\ \\equiv\\;&amp; \\displaystyle A = \\frac{1}{\\sqrt{1 - \\frac{v_{x_0}^2}{c^2}}} \\end{array}<\/span>\n<p><p style=\"text-align:justify;\">This is generally written replacing <span class=\"katex-eq\" data-katex-display=\"false\">A=\\gamma_x<\/span> (Lorentz contraction factor) and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_x = v_{x_0}\/c<\/span> (velocity boost), resulting in the form:<\/p>\n<p><a name=\"eq9\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle A = \\gamma_x = \\frac{1}{\\sqrt{1 - \\beta_x^2}},\\;\\;\\;[9]<\/span>\n<p style=\"text-align:justify;\">And replacing <a href=\"#eq9\">[9]<\/a> into <a href=\"#eq2\">[2]<\/a> we get:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">x^\\prime = \\gamma_x(x - \\beta_x ct)<\/span>\n<p style=\"text-align:justify;\"><span style=\"color:red\"><strong>from what has been marked in red we obtain<\/strong><\/span><\/p>\n<p><a name=\"eq10\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rll} &amp;2c^2 AB + 2A^2v_{x_{x_0}} = 0&amp; \\\\ \\\\ \\equiv\\;&amp; cB^2 + Av_{x_0} = 0 &amp; \\\\ \\\\ \\equiv\\;&amp; B=\\displaystyle -\\frac{1}{c^2}Av_{x_0} = -\\frac{\\gamma_x v_{x_0}}{c^2}&amp; \\\\ \\\\ \\equiv\\;&amp; B=\\displaystyle -\\frac{\\gamma_x \\beta_x}{c} &amp; [10] \\end{array}<\/span>\n<p style=\"text-align:justify;\">thus, replacing <a href=\"#eq9\">[9]<\/a> and <a href=\"#eq10\">[10]<\/a> into <a href=\"#eq7\">[7]<\/a> we get<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} &amp;t^\\prime =\\displaystyle \\gamma_x t -\\frac{\\gamma_x \\beta_x}{c} \\\\ \\\\ \\equiv\\; &amp;t^\\prime =\\displaystyle \\gamma_x \\left( t -\\frac{\\beta_x x}{c}\\right)\\\\ \\\\ \\equiv\\; &amp;ct^\\prime =\\displaystyle \\gamma_x \\left( ct - \\beta_x x \\right) \\end{array}<\/span>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Synthesis of the Lorentz Transformations<\/h3>\n<p style=\"text-align:justify;\">Finally, the linear transformation that models the change of coordinates between the systems <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> is given by the following expressions.<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}ct^\\prime &amp;=\\gamma_x \\left( ct - \\beta_x x \\right) \\\\ x^\\prime &amp;= \\gamma_x(x - \\beta_x ct) \\end{array}<\/span>\n<p style=\"text-align:justify;\">This system of transformations can be expressed in matrix form as follows<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left(\\begin{matrix}ct^\\prime \\\\ x^\\prime \\\\ y^\\prime \\\\ z^\\prime \\end{matrix}\\right) = \\left( \\begin{matrix}\\gamma_x &amp; -\\gamma_x\\beta_x &amp; 0 &amp; 0 \\\\ -\\gamma_x\\beta_x &amp; \\gamma_x &amp; 0 &amp; 0 \\\\  0 &amp; 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 1 \\end{matrix} \\right) \\left(\\begin{matrix} ct \\\\ x \\\\ y \\\\ z \\end{matrix} \\right)\n\n<\/span>\n<p style=\"text-align:justify;\">This is known as the Lorentz Transformations of special relativity<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>The Lorentz Transformations Converge and Generalize the Galilean Transformations<\/h2>\n<p style=\"text-align:justify;\">The convergence of the Lorentz transformations to the Galilean transformations is observed when seeing what happens with the Lorentz transformations when the velocity between inertial reference frames is much less than the speed of light. When this occurs, we have:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> |v_{x_0}| \\ll c \\longrightarrow \\left\\{\\begin{matrix}\\beta_x = \\frac{v_{x_0}}{c} \\approx 0 \\\\ \\\\ \\gamma_x = \\sqrt{1-\\beta_x} \\approx 1 \\\\ \\\\ \\gamma_x \\beta_x c = v_{x_0} \\gamma_x \\approx v_{x_0} \\end{matrix}\\right.  <\/span>\n<p style=\"text-align:justify;\">So:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left(\\begin{matrix}ct^\\prime \\\\ x^\\prime \\\\ y^\\prime \\\\ z^\\prime \\end{matrix}\\right) = \\left( \\begin{matrix}\\gamma_x &amp; -\\gamma_x\\beta_x &amp; 0 &amp; 0 \\\\ -\\gamma_x\\beta_x &amp; \\gamma_x &amp; 0 &amp; 0 \\\\  0 &amp; 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; 1 \\end{matrix} \\right) \\left(\\begin{matrix} ct \\\\ x \\\\ y \\\\ z \\end{matrix} \\right) = \\left(\\begin{matrix} \\gamma_x ct  -\\gamma_x \\beta_x x \\\\ -\\gamma_x \\beta_x c t + \\gamma_x x \\\\ y \\\\ z \\end{matrix} \\right) \\approx \\left(\\begin{matrix} ct \\\\ -v_{x_0}t + x \\\\ y \\\\ z \\end{matrix}\\right)<\/span>\n<p style=\"text-align:justify;\">That is:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} t^\\prime &amp;\\approx t \\\\ x^\\prime &amp;\\approx x - v_{x_0}t \\\\ y^\\prime &amp;\\approx y \\\\ z^\\prime &amp;\\approx z \\end{array}\n\n<\/span>\n<p style=\"text-align:justify;\">which exactly matches the Galilean transformations. Through this, it is corroborated that the Lorentz Transformations generalize the Galilean transformations for velocities close to the speed of light and converge to the Galilean transformations when velocities are much lower than the speed of light.<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclusions<\/h2>\n<p style=\"text-align:justify;\">\n        We have explored in depth the Lorentz Transformations, a fundamental pillar of Einstein&#8217;s Special Relativity theory. Through careful decomposition and analysis, we have seen how these transformations naturally arise from the postulation of the constancy of the speed of light in all inertial frames. We have demonstrated the relevance of the Lorentz Transformations, not only as a generalization and correction of the Galilean transformations, but also as an essential framework for understanding physical phenomena in the realm of relativity and electromagnetic theory.\n    <\/p>\n<p style=\"text-align:justify;\">\n        Understanding these subjects will help students familiarize themselves with key concepts of modern physics, such as the \u00abvelocity boost\u00bb and the \u00abLorentz factor\u00bb, and apply these ideas to concrete situations in the realm of relativity. Additionally, we have seen how, in the limit of velocities much lower than the speed of light, the Lorentz Transformations converge to the Galilean Transformations, thus demonstrating their versatility and universality in the study of the dynamics of moving bodies.\n    <\/p>\n<p style=\"text-align:justify;\">\n        In summary, the Lorentz Transformations not only represent a significant theoretical achievement in physics, but also provide an indispensable tool for understanding and practical application of the principles of special relativity in various scientific and technological contexts.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Lorentz Transformations of Special Relativity Summary: The Lorentz transformations allow for the transformation of observed space and time coordinates between two inertial reference frames. In this article, we will review how the Lorentz transformations are derived as a linear coordinate transformation that emerges from considering the speed of light as constant in all inertial [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25569,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":13,"footnotes":""},"categories":[635,691],"tags":[],"class_list":["post-25653","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 Lorentz Transformations of Special Relativity - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Understand in detail the Lorentz Transformations in Special Relativity, the &#039;Velocity Boost&#039; and the &#039;Lorentz Factor\" \/>\n<meta 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