{"id":27293,"date":"2023-12-26T13:00:01","date_gmt":"2023-12-26T13:00:01","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27293"},"modified":"2024-06-30T21:17:56","modified_gmt":"2024-06-30T21:17:56","slug":"the-minkowski-spacetime","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-minkowski-spacetime\/","title":{"rendered":"The Minkowski Spacetime"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>The Spacetime of Special Relativity<\/h1>\n<p class=\"eq\"><em><strong>Summary:<\/strong><br \/>\nIn this class, we will review Lorentz Transformations in the context of special relativity, challenging the notion of absolute time and establishing the constancy of the speed of light in all inertial frames. It explores how these transformations connect the space and time coordinates of an event from different inertial frames. This study delves into the symmetry between temporal and spatial coordinates and presents the <strong>Minkowski Spacetime,<\/strong> a fundamental model in special relativity that combines space and time into a four-dimensional structure. It is shown that, unlike pure space and time lengths, spacetime lengths remain constant under Lorentz transformations, which has significant implications for theoretical physics and our understanding of the universe.<\/br><\/em><\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, the student will be able to:<\/p>\n<ol>\n<li><strong>Understand<\/strong> the concept of Minkowski Spacetime and how this model combines space and time into a four-dimensional structure.<\/li>\n<li><strong>Apply<\/strong> Lorentz Transformations to calculate changes in the space and time coordinates of an event from different inertial frames.<\/li>\n<li><strong>Analyze<\/strong> the relationship between time dilation and space contraction, understanding how these effects result from the relationship between an observer&#8217;s speed and the speed of light.<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>Review of Lorentz Transformations<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Minkowski Spacetime<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>What happens to space, time, and spacetime lengths under Lorentz transformations?<\/strong><\/a><br \/>\n<a href=\"#4\">Development for pure time lengths<\/a><br \/>\n<a href=\"#5\">Development for Pure Space Lengths<\/a><br \/>\n<a href=\"#6\">Development for Spacetime Lengths<\/a><br \/>\n<a href=\"#7\"><strong>Conclusions<\/strong><\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6tVlrcyVV8g?si=FUG1kS6GfPgp7Boh\" 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=\"1\"><\/a><\/p>\n<h2>Review of Lorentz Transformations<\/h2>\n<p style=\"text-align:justify;\">In special relativity, the idea of absolute time is discarded. Instead, it is established that the speed of light, <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span>, is constant in all inertial frames. This change, combined with the principle of relativity, leads us to Lorentz Transformations. These transformations connect the coordinates of an event observed from two different inertial frames. This topic is explored in detail in the class on <a href=\"http:\/\/toposuranos.com\/material\/es\/las-transformaciones-de-lorentz-de-la-relatividad-especial\/\" rel=\"noopener\" target=\"_blank\">Lorentz Transformations in Special Relativity<\/a>.<\/p>\n<p style=\"text-align:justify;\">Considering 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, where their axes and origins coincide at <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime =0<\/span>, and a photon emitted at <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime = 0<\/span> from the origin, the space and time coordinates of the photon in each frame must satisfy the equation:<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\nc^2t^2 - x^2 - y^2 - z^2 = c^2{t^\\prime}^2 - {x^\\prime}^2 - {y^\\prime}^2 - {z^\\prime}^2 = 0.\n\n<\/span><\/bdi><\/p>\n<p style=\"text-align:justify;\">From this equation and the principle of relativity, we derive the well-known Lorentz transformations:<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nct^\\prime &amp;= \\gamma_{ss^\\prime_x}(ct - \\beta_{ss^\\prime_x} x), \\\\\n\nx^\\prime &amp;= \\gamma_{ss^\\prime_x}(x - \\beta_{ss^\\prime_x} ct), \\\\\n\ny^\\prime &amp;= y, \\\\\n\nz^\\prime &amp;= z.\n\n\\end{array}\n\n<\/span><\/bdi><\/p>\n<p style=\"text-align:justify;\">Where <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x} =v_{ss^\\prime_x}\/c<\/span> is the <strong>velocity boost<\/strong> acquired by <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> when moving relative to <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> at a speed <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} = 1\/\\sqrt{1-\\beta_{ss^\\prime_x}^2}<\/span> is the associated <strong>Lorentz factor<\/strong>. This Lorentz transformation in the direction <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> simplifies to the Galilean transformation when <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x} \\ll c<\/span>.<\/p>\n<p style=\"text-align:justify;\">Similar to Galilean transformations, there is a symmetry that facilitates calculating the inverse transformation, simply by swapping terms and considering that <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x} = -\\beta_{s^\\prime s_x}<\/span>:<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n ct &amp;= \\gamma_{ss^\\prime_x}(ct^\\prime + \\beta_{ss^\\prime_x} x^\\prime),\\\\\n\n  x &amp;= \\gamma_{ss^\\prime_x}(x^\\prime + \\beta_{ss^\\prime_x} ct^\\prime),\\\\\n\n  y &amp;= y^\\prime, \\\\\n\n  z &amp;= z^\\prime.\n\n\\end{array}<\/span><\/bdi><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Minkowski Spacetime<\/h2>\n<p style=\"text-align:justify;\">\nLorentz transformations reveal that space and time coordinates are intrinsically intertwined. This relationship is particularly evident in the symmetry between <span class=\"katex-eq\" data-katex-display=\"false\">ct<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>. Considering two events, <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>, with coordinates <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(ct_A, x_A, y_A, z_A)<\/span><\/bdi> and <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(ct_B, x_B, y_B, z_B)<\/span><\/bdi>. In the frame <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, we define the quadratic distance as follows:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\Delta s^2 &amp;= c^2(t_B - t_A)^2 - (x_B - x_A)^2 - (y_B - y_A)^2 - (z_B - z_A)^2 \\\\ \\\\\n\n&amp;= c^2\\Delta t^2 - \\Delta x^2 - \\Delta y^2 - \\Delta z^2 \\\\ \\\\\n\n&amp;= c^2\\Delta t^2 - (\\Delta x^2 + \\Delta y^2 + \\Delta z^2)\n\n\\end{array}<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nThe spacetime distance, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta s<\/span>, is written as <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta s = \\sqrt{c^2\\Delta t^2 - (\\Delta x^2 + \\Delta y^2 + \\Delta z^2)}<\/span>. Here, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t<\/span> represents a temporal length and <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r = \\sqrt{\\Delta x^2 + \\Delta y^2 + \\Delta z^2}<\/span> is a spatial length.\n<\/p>\n<p style=\"text-align:justify;\">\nThe <strong>Minkowski Spacetime<\/strong>, characterized by this notion of spacetime distance <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta s<\/span>, is fundamental in special relativity. It was introduced by <a href=\"https:\/\/es.wikipedia.org\/wiki\/Hermann_Minkowski\" rel=\"noopener\" target=\"_blank\">Hermann Minkowski<\/a> and is distinguished from spatial and temporal coordinates by being invariant under Lorentz transformations.\n<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta s = \\Delta s^\\prime<\/span><\/bdi><\/p>\n<p style=\"text-align:justify;\">\nIn this model, space and time combine into a four-dimensional continuum. Unlike Euclidean geometry, Minkowski spacetime geometry is pseudo-Euclidean due to the negative signs in its spatial components. However, for a constant time <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span>, Minkowski&#8217;s spatial geometry remains Euclidean.\n<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>What happens to space, time, and spacetime lengths under Lorentz transformations?<\/h2>\n<p style=\"text-align:justify;\">As previously mentioned, spacetime lengths <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta s<\/span><\/bdi> are invariant under Lorentz transformations, but in addition to this, space and time lengths change under these transformations. What we will do next is the step-by-step demonstration of these facts.<\/p>\n<p><p style=\"text-align:justify;\">First, let&#8217;s recall the events <bdi><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/bdi> and <bdi><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/bdi> considered at the beginning with their respective spacetime coordinates concerning the system <bdi><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/bdi>:<\/p>\n<ul>\n<li> <strong>Event <bdi><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/bdi>:<\/strong> <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(ct_A,x_A, y_A, z_A)<\/span><\/bdi><\/li>\n<li> <strong>Event <bdi><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/bdi>:<\/strong> <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(ct_B,x_B, y_B, z_B)<\/span><\/bdi><\/li>\n<\/ul>\n<p style=\"text-align:justify;\">For these developments, we will use without loss of generality the Lorentz transformations for systems <bdi><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/bdi> and <bdi><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span><\/bdi> in standard configuration where <bdi><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span><\/bdi> moves with velocity <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_{ss^\\prime_x}= v_{ss^\\prime_x} \\hat{x} = \\beta_{ss^\\prime_x}c \\hat{x}<\/span><\/bdi> relative to <bdi><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/bdi> <\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nct^\\prime &amp;= \\gamma_{ss^\\prime_x}(ct - \\beta_{ss^\\prime_x} x), \\\\\n\nx^\\prime &amp;= \\gamma_{ss^\\prime_x}(x - \\beta_{ss^\\prime_x} ct), \\\\\n\ny^\\prime &amp;= y, \\\\\n\nz^\\prime &amp;= z.\n\n\\end{array}\n\n<\/span><\/bdi><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Development for pure time lengths<\/h3>\n<p style=\"text-align:justify;\">\nSuppose the events <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>, observed from the reference frame <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, are separated only by time, like the ticks of a clock. In this case, the elapsed time between a tick-tock is calculated as follows:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\Delta t = c(t_B - t_A)<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nOn the other hand, the temporal separation between the same pair of events observed from <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> will be:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\Delta t^\\prime = c(t^\\prime_B - t^\\prime_A)<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nThese temporal separations are related through Lorentz transformations as follows:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nc\\Delta t^\\prime &amp;= c(t^\\prime_B - t^\\prime_A) \\\\ \\\\\n\n&amp;= ct^\\prime_B - ct^\\prime_A \\\\ \\\\\n\n&amp;= \\gamma_{ss^\\prime_x}(ct_B - \\beta_{ss^\\prime_x} x_B) - \\gamma_{ss^\\prime_x}(ct_A - \\beta_{ss^\\prime_x} x_A) \\\\ \\\\\n\n&amp;= \\gamma_{ss^\\prime_x}c \\Delta t - \\gamma_{ss^\\prime_x} \\beta_{ss^\\prime_x} \\Delta x\n\n\\end{array}\n\n<\/span>\n<\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nNow, since the events <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> are separated only in time for the observer in <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, we have <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = 0<\/span>. Therefore:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\Delta t^\\prime = \\gamma_{ss^\\prime_x} \\Delta t}<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nIt is important to note that:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} = \\dfrac{1}{\\sqrt{1 - \\beta^2_{ss^\\prime_x}}} \\in [1, +\\infty[<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nThis is because <span class=\"katex-eq\" data-katex-display=\"false\">\\beta^2_{ss^\\prime_x} = \\dfrac{v^2_{ss^\\prime_x}}{c^2} \\in [0,1[<\/span>.\n<\/p>\n<p style=\"text-align:justify;\">\nIn simple terms, if an observer in <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> measures a time interval <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t<\/span> like the tick-tock of a clock, an observer in <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> will measure this same interval as <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x} \\Delta t<\/span>, which is greater than or equal to <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t<\/span>. This effect, known as time dilation, indicates how time extends between inertial observers experiencing a velocity boost <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}<\/span>. Therefore, the passage of time is not the same for all inertial observers, showing that time lengths are not invariant under Lorentz transformations.\n<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Development for Pure Space Lengths<\/h3>\n<p style=\"text-align:justify;\">\nSuppose that events <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> are separated only in space, like the ends of a ruler. We assume, without loss of generality, that this ruler is oriented 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>. Then, we have:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_B - x_A<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nSeen from <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, this spatial separation would be:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x^\\prime = x^\\prime_B - x^\\prime_A<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nApplying Lorentz transformations, we can establish the relationship between both observations:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\Delta x^\\prime &amp;= x^\\prime_B - x^\\prime_A \\\\ \\\\\n\n&amp;= \\gamma_{ss^\\prime}(x_B - \\beta_{ss^\\prime_x} ct_B) - \\gamma_{ss^\\prime}(x_A - \\beta_{ss^\\prime_x} ct_A) \\\\ \\\\\n\n&amp;= \\gamma_{ss^\\prime} \\Delta x - \\gamma_{ss^\\prime}\\beta_{ss^\\prime_x} c \\Delta t\n\n\\end{array}\n\n<\/span>\n<\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nSince the events <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> are simultaneous for <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, it follows that <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t = 0<\/span>, and therefore:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\Delta x^\\prime = \\gamma_{ss^\\prime} \\Delta x}<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nFor example, if we place a ruler of length <span class=\"katex-eq\" data-katex-display=\"false\">l_0<\/span> inside a train car (observer <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>), which moves relative to us (observer <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>), and the ruler is aligned with the direction of motion, the observed length will be:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n &amp; l_0 = \\gamma_{ss^\\prime} l \\\\ \\\\\n\n\\equiv &amp; l = \\dfrac{l_0}{\\gamma_{ss^\\prime}} \\leq l_0.\n\n\\end{array}\n\n<\/span>\n<\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nThis means we will perceive the ruler&#8217;s length as shorter than it actually is. This phenomenon is known as <strong>Lorentz contraction<\/strong> and demonstrates that space intervals are not conserved under Lorentz transformations.\n<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Development for Spacetime Lengths<\/h3>\n<p style=\"text-align:justify;\">\nAfter analyzing how pure space and pure time lengths transform, let&#8217;s now examine the behavior of spacetime lengths under Lorentz transformations. Recall that a spacetime length, observed by the observer <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> for two events <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>, is expressed as follows:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\Delta s^\\prime &amp;= \\sqrt{c^2\\Delta t^{\\prime 2} - (\\Delta x^{\\prime 2} + \\Delta y^{\\prime 2} + \\Delta z^{\\prime 2})} \\\\ \\\\\n\n&amp;= \\sqrt{c^2 (t^{\\prime 2}_B - t^{\\prime 2}_A) - \\left[(x^{\\prime 2}_B - x^{2}_A) +  (y^{\\prime 2}_B - y^{\\prime 2}_A) + (z^{2}_B - z^{2}_A) \\right]}\n\n\\end{array}\n\n<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nNext, we will see how these lengths relate after applying Lorentz transformations, in the case that <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> has a velocity boost <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_{ss^\\prime_x}<\/span> concerning <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>.\n<\/p>\n<p style=\"text-align:center;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\color{black}\n\n\\begin{array}{rl}\n\n\\Delta s^{\\prime 2} &amp;= (c^2 t^{\\prime 2}_B - c^2 t^{\\prime 2}_A) - \\left[(x^{\\prime 2}_B - x^{\\prime 2}_A) +  (y^{\\prime 2}_B - y^{2}_A) + (z^{\\prime 2}_B - z^{2}_A) \\right] \\\\ \\\\ \\\\\n\n&amp;= \\left[\\gamma_{ss^\\prime_x}(ct_B - \\beta_{ss^\\prime_x} x_B)\\right]^2 -  \\left[\\gamma_{ss^\\prime_x}(ct_A - \\beta_{ss^\\prime_x} x_A)\\right]^2 + \\cdots \\\\ \\\\\n\n&amp; \\cdots -\\left\\{ \\left( \\left[\\gamma_{ss^\\prime_x}(x_B - \\beta_{ss^\\prime_x} ct_B)\\right]^2 - \\left[\\gamma_{ss^\\prime_x}(x_A - \\beta_{ss^\\prime_x} ct_A)\\right]^2 \\right) + (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \\right\\} \\\\ \\\\ \\\\\n\n&amp;=  \\gamma_{ss^\\prime_x}^2 (ct_B - \\beta_{ss^\\prime_x} x_B)^2 -  \\gamma_{ss^\\prime_x}^2(ct_A - \\beta_{ss^\\prime_x} x_A)^2 + \\cdots \\\\ \\\\\n\n&amp; \\cdots -\\left\\{ \\gamma_{ss^\\prime_x}^2(x_B - \\beta_{ss^\\prime_x} ct_B)^2 - \\gamma_{ss^\\prime_x}^2(x_A - \\beta_{ss^\\prime_x} ct_A)^2  + (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \\right\\} \\\\ \\\\ \\\\\n\n&amp;=    \\color{red}\\gamma_{ss^\\prime_x}^2 c^2 t_B^2 \\color{black} - \\cancel{2  \\gamma_{ss^\\prime_x}^2 \\beta_{ss^\\prime_x} c t_B x_B} + \\color{green}\\gamma_{ss^\\prime_x}^2\\beta_{ss^\\prime_x}^2 x_B^2\\color{black} + \\cdots \\\\ \\\\\n\n&amp; \\cdots   - \\color{blue}\\gamma_{ss^\\prime_x}^2 c^2 t_A^2\\color{black} + 2 \\cancel{\\gamma_{ss^\\prime_x}^2 \\beta_{ss^\\prime_x} c t_A x_A} - \\color{purple}\\gamma_{ss^\\prime_x}^2\\beta_{ss^\\prime_x}^2 x_A^2\\color{black} + \\cdots \\\\ \\\\\n\n&amp; \\cdots  - \\color{green} \\gamma_{ss^\\prime_x}^2x_B^2 \\color{black} + \\cancel{2 \\gamma_{ss^\\prime_x}^2 \\beta_{ss^\\prime_x} ct_B x_B} - \\color{red}\\gamma_{ss^\\prime_x}^2 \\beta_{ss^\\prime_x}^2 c^2t_B^2 \\color{black}+ \\cdots \\\\ \\\\\n\n&amp; \\cdots  + \\color{purple}\\gamma_{ss^\\prime_x}^2x_A^2\\color{black}- \\cancel{2 \\gamma_{ss^\\prime_x}^2 \\beta_{ss^\\prime_x} ct_A x_A} + \\color{blue}\\gamma_{ss^\\prime_x}^2 \\beta_{ss^\\prime_x}^2 c^2t_A^2 \\color{black} + \\cdots \\\\ \\\\\n\n&amp; \\cdots - \\left\\{  (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \\right\\} \\\\ \\\\ \\\\\n\n&amp;= \\color{red}\\gamma_{ss^\\prime_x}^2 (1-  \\beta_{ss^\\prime_x}^2)c^2 t_B^2\\color{black} - \\color{blue}\\gamma_{ss^\\prime_x}^2 (1-  \\beta_{ss^\\prime_x}^2)c^2 t_A^2 \\color{black} + \\cdots \\\\ \\\\\n\n&amp; \\cdots - \\color{green}\\gamma_{ss^\\prime_x}^2(1-\\beta_{ss^\\prime_x}^2)x_B^2\\color{black} + \\color{purple}\\gamma_{ss^\\prime_x}^2(1-\\beta_{ss^\\prime_x}^2)x_A^2  \\color{black} + \\cdots \\\\ \\\\\n\n&amp; \\cdots - \\left\\{  (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \\right\\} \\\\ \\\\ \\\\\n\n\\end{array}\n\n<\/span><\/bdi><\/p>\n<p><p style=\"text-align:justify;\">\nFinally, remembering that <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_{ss^\\prime_x}^2 = 1\/(1-\\beta_{ss^\\prime_x}^2)<\/span>, we obtain the following:\n<\/p>\n<p style=\"text-align:center;\">\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\Delta s^{\\prime 2} &amp;= c^2 t_B^2 - c^2 t_A^2 - x_B^2 + x_A^2 - \\left\\{ (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \\right\\} \\\\ \\\\\n\n&amp;= c^2 (t_B^2 - t_A^2) - \\left\\{ (x_B^2 - x_A^2) + (y^{2}_B - y^{2}_A) + (z^{2}_B - z^{2}_A) \\right\\} \\\\ \\\\\n\n&amp;= c^2 \\Delta t^2 - (\\Delta x^2 + \\Delta y^2 + \\Delta z^2) \\\\ \\\\\n\n&amp;= \\Delta s^2\n\n\\end{array}\n\n<\/span><\/bdi>\n<\/p>\n<p style=\"text-align:justify;\">\nWith this, we have demonstrated that, unlike pure time and space lengths, spacetime lengths remain constant under Lorentz transformations.\n<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<a name=\"7\"><\/a><\/p>\n<h2>Conclusions<\/h2>\n<p style=\"text-align:justify;\">\nThe study of Lorentz Transformations in special relativity reveals fundamental aspects about the nature of space and time. By discarding the notion of absolute time, these transformations show us a universe where the speed of light remains constant in all inertial frames. This leads to a profound interrelationship between spatial and temporal coordinates, as manifested in the symmetry between <span class=\"katex-eq\" data-katex-display=\"false\">ct<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>.\n<\/p>\n<p style=\"text-align:justify;\">\nLorentz Transformations not only change our perception of motion and speed but also introduce concepts like time dilation and space contraction. These effects are direct consequences of the relationship between an observer&#8217;s speed and the speed of light. For example, time dilation demonstrates that time passes at different rates for observers in relative motion, challenging our intuition of a universal time.\n<\/p>\n<p style=\"text-align:justify;\">\nAt the heart of these transformations lies Minkowski Spacetime, a model that merges space and time into a four-dimensional structure. This model is not only crucial for Einstein&#8217;s theory of special relativity but also lays the foundation for a more advanced understanding of physics, including general relativity and modern cosmology.\n<\/p>\n<p style=\"text-align:justify;\">\nIn summary, Lorentz Transformations are not only an essential component in theoretical physics but also offer a window into a deeper understanding of the universe we live in, challenging and enriching our understanding of reality.\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Spacetime of Special Relativity Summary: In this class, we will review Lorentz Transformations in the context of special relativity, challenging the notion of absolute time and establishing the constancy of the speed of light in all inertial frames. It explores how these transformations connect the space and time coordinates of an event from different [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25975,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[635,691],"tags":[],"class_list":["post-27293","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 Minkowski Spacetime - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover the keys to Minkowski Spacetime, Lorentz Transformations, and the true Nature of Space and Time\" \/>\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\/en\/the-minkowski-spacetime\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Minkowski Spacetime\" \/>\n<meta property=\"og:description\" content=\"Discover the keys to Minkowski Spacetime, Lorentz Transformations, and the true Nature of Space and Time\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/the-minkowski-spacetime\/\" \/>\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=\"2023-12-26T13:00:01+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-30T21:17:56+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/ESPACIOTIEMPO-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 Minkowski Spacetime\" \/>\n<meta name=\"twitter:description\" content=\"Discover the keys to Minkowski Spacetime, Lorentz Transformations, and the true Nature of Space and Time\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/ESPACIOTIEMPO.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\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Minkowski Spacetime\",\"datePublished\":\"2023-12-26T13:00:01+00:00\",\"dateModified\":\"2024-06-30T21:17:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/\"},\"wordCount\":2619,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/12\\\/ESPACIOTIEMPO.jpg\",\"articleSection\":[\"Physics\",\"Relativity\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-minkowski-spacetime\\\/\",\"name\":\"The Minkowski Spacetime - 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