{"id":25603,"date":"2022-07-12T00:00:22","date_gmt":"2022-07-12T00:00:22","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25603"},"modified":"2024-05-21T09:42:28","modified_gmt":"2024-05-21T09:42:28","slug":"galileos-transformations-and-their-limitations","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/galileos-transformations-and-their-limitations\/","title":{"rendered":"Galileo&#8217;s Transformations and Their Limitations"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Galileo&#8217;s Transformations and Their Limitations<\/h1>\n<p class=\"eq\"><em><strong>Summary:<\/strong><br \/>\nThe principle of relativity suggests that observations depend on the inertial frame, but in such a way that physical laws remain consistent. An initial and intuitive approach to this principle comes from Galileo&#8217;s Transformations, which model how observations change between inertial reference frames in classical mechanics. In this class, we will study these transformations and their properties, and we will also see how they fail when applied to the phenomenon of wave propagation.<\/br><\/em><\/p>\n<p><strong>LEARNING OBJECTIVES<\/strong><br \/>\nUpon completing this class, students will be able to:\n<\/p>\n<p><\/center><\/p>\n<ol>\n<li><strong>Recognize<\/strong> the fundamental concepts of Galileo&#8217;s Transformations, including their basic formulation and underlying principles.<\/li>\n<li><strong>Analyze<\/strong> the Galilean geometry of space and time and its separation in the framework of classical mechanics.<\/li>\n<li><strong>Evaluate<\/strong> the limitations of Galileo&#8217;s Transformations when applied to phenomena such as wave propagation and their relevance in the advancement towards the theory of special relativity.<\/li>\n<\/ol>\n<p><center><\/p>\n<p class=\"indx\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>Formulation of Galileo&#8217;s Transformations<\/strong><\/a><br \/>\n<a href=\"#2\">The Inverse Transformation<\/a><br \/>\n<a href=\"#3\">Absolute Time and the Sum of Velocities<\/a><br \/>\n<a href=\"#4\">Galilean Geometry of Space and Time<\/a><br \/>\n<a href=\"#5\"><strong>Galileo&#8217;s Relativity and Physical Laws<\/strong><\/a><br \/>\n<a href=\"#6\">Applied to Newtonian Dynamics<\/a><br \/>\n<a href=\"#7\">Applied to Wave Propagation<\/a><br \/>\n<a href=\"#8\">What Effect Do Galileo&#8217;s Transformations Have on Wave Propagation?<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ku-9nbTSaJg?si=1dmuCtjzBPUy14v_\" 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>Formulation of Galileo&#8217;s Transformations<\/h2>\n<p style=\"text-align:justify;\">Newtonian physics rests on the principle of relativity modeled through Galileo&#8217;s Transformations, where time is established as a universal coordinate for all inertial observers; that is: <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime<\/span>. Under this statement, the linear transformation relating observations from two inertial reference frames <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> reviewed in the class on <a href=\"http:\/\/toposuranos.com\/material\/es\/el-principio-de-relatividad-especial\/?fbclid=IwAR2_lpF1hyUlbSzFQ5B5OQpPu1Vhc_20zTGu8D4pxKPsSvzZRvLSzPdwQXU\" rel=\"noopener\" target=\"_blank\">the principle of special relativity<\/a> takes the form of a linear transformation:<\/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,\\\\ x^\\prime &amp;= Dt + Ex,\\\\ y^\\prime &amp;= y, \\\\ z^\\prime &amp;=z,\n\n\\end{array}<\/span> <strong>[1]<\/strong><\/p>\n<p style=\"text-align:justify;\">it takes the following form when the inertial frames <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> are in standard configuration and <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> moves with velocity <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}\\hat{x}<\/span> relative to <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>\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=\"coordinate transformations\" 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=\"coordinate transformations\" 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><\/center><\/p>\n<p><a name=\"eq2\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rlr}\n\n{}t^\\prime &amp;= t  \\\\ x^\\prime &amp;= x - v_{ss^\\prime_x}t \\\\ y^\\prime &amp;= y \\\\ z^\\prime &amp;= z\n\n\\end{array}\n\n<\/span> <strong>[2]<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>The Inverse Transformation<\/h3>\n<p style=\"text-align:justify;\">From a kind of algebraic symmetry, we can write the inverse transformation:<\/p>\n<p><a name=\"eq3\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nt &amp;= t^\\prime \\\\ x &amp;= x^\\prime + v_{ss^\\prime_x}t \\\\ y &amp;= y^\\prime \\\\ z &amp;= z^\\prime \\end{array}\n\n<\/span> <strong>[3]<\/strong><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Absolute Time and the Sum of Velocities<\/h3>\n<p style=\"text-align:justify;\">From the first equation of Galileo&#8217;s transformations (either of the two, <a href=\"#eq2\">[2]<\/a> or <a href=\"#eq3\">[3]<\/a>) it is evident that the temporal coordinate of an event does not depend on the frame from which it is observed, while the second allows obtaining what is commonly understood as \u00abcommon sense\u00bb associated with the sum of velocities. If a particle moves with constant velocity <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}<\/span> 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 its velocity in <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> is determined by<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle v^\\prime_x = \\frac{dx^\\prime}{dt^\\prime} = \\frac{dx^\\prime}{dt} = \\frac{d}{dt}\\left(x - v_{ss^\\prime_x} t \\right) = v_x - v_{ss^\\prime_x}<\/span>\n<p style=\"text-align:justify;\">Deriving in this last expression shows that the acceleration of any particle is the same in <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> and in <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, that is: <span class=\"katex-eq\" data-katex-display=\"false\">dv^\\prime_x\/dt^\\prime = dv_x\/dt<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Galilean Geometry of Space and Time<\/h3>\n<p style=\"text-align:justify;\">If we consider two events <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> that have coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(t_A,x_A,y_A,z_A)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">(t_B,x_B,y_B,z_B),<\/span> respectively. It is easy to see that the quantities <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t = t_B - t_A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2 = \\Delta x^2 + \\Delta y^2 + \\Delta z^2<\/span> are separately invariant under Galileo&#8217;s transformations, leading us to consider space and time as separate entities. On the other hand, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2<\/span> suggests that this is a geometric property of space itself. We recognize <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2<\/span> as the square of the distance between events in Euclidean space. This defines the geometry of space and time in the context of Newtonian mechanics.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Galileo&#8217;s Relativity and Physical Laws<\/h2>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Applied to Newtonian Dynamics<\/h3>\n<p style=\"text-align:justify;\">In the previous section, we saw that, in the context of Newtonian physics, any two different inertial frames will always observe the same accelerations. This, coupled with Newton&#8217;s second law, implies that all inertial frames will always observe the same dynamics. That is:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F_x = m\\frac{dv_x}{dt}= m\\frac{dv^\\prime_x}{dt^\\prime} = F^\\prime_x.<\/span> <\/p>\n<p style=\"text-align:justify;\">This last expression tells us that <strong>physics does not change when performing Galileo&#8217;s transformations,<\/strong> which is equivalent to saying that: physics is the same for all inertial observers.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Applied to Wave Propagation<\/h3>\n<p style=\"text-align:justify;\">While the persistence of physics in the face of changes in inertial observers is expected, first because it is what we observe when moving, and second because it is what has been obtained through previous calculations, it is not always the case. The most notable case of a phenomenon that does not preserve under Galilean transformations is the propagation of waves; generally, the equation that models the propagation of a wave <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> in space and time is in the form<\/p>\n<p><a name=\"eq4\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\nabla^2 \\psi = \\frac{1}{v_0^2}\\frac{\\partial^2 \\psi}{\\partial t^2}<\/span> <strong>[4]<\/strong><\/p>\n<p style=\"text-align:justify;\">where <span class=\"katex-eq\" data-katex-display=\"false\">v_0<\/span> is the wave propagation speed.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h4>What Effect Do Galilean Transformations Have on Wave Propagation?<\/h4>\n<p style=\"text-align:justify;\">For this, there is a short and a long answer. The short answer is that \u00abeven observing the same phenomenon, different inertial observers will see &#8216;different physics'\u00bb. The long answer involves examining how the wave propagation equation changes when the Galilean transformation is applied; to do this, we first take the equation <a href=\"#eq3\">[4]<\/a> and expand it over each of its coordinates obtaining:<\/p>\n<p><a name=\"eq5\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial ^2 \\psi}{\\partial x^2} + \\frac{\\partial ^2 \\psi}{\\partial y^2} + \\frac{\\partial ^2 \\psi}{\\partial z^2} = \\frac{1}{v_0^2} \\frac{\\partial ^2 \\psi}{\\partial t^2}.<\/span> <strong>[5]<\/strong><\/p>\n<p>With this equation at hand, we must now use the equations from <a href=\"#eq3\">[3]<\/a> to re-express the derivatives in the other inertial frame.<\/p>\n<h5>Transformation of the First Derivatives<\/h5>\n<p style=\"text-align:justify;\">Following the expressions from <a href=\"#eq3\">[3]<\/a> and deriving each variable with respect to the prime variables, we obtain:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial x^\\prime}{\\partial x} = \\frac{\\partial y^\\prime}{\\partial y}= \\frac{\\partial z^\\prime}{\\partial z}= \\frac{\\partial t^\\prime}{\\partial t}= 1<\/span>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial x^\\prime}{\\partial t} = - v_{x_0}<\/span>\n<p style=\"text-align:justify;\">While all others are nullified:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial t^\\prime}{\\partial x} = \\frac{\\partial t^\\prime}{\\partial y} = \\frac{\\partial t^\\prime}{\\partial z} = \\frac{\\partial x^\\prime}{\\partial y} = \\frac{\\partial x^\\prime}{\\partial z} = \\frac{\\partial y^\\prime}{\\partial x} = \\frac{\\partial y^\\prime}{\\partial z} = \\frac{\\partial y^\\prime}{\\partial t} = \\frac{\\partial z^\\prime}{\\partial x} = \\frac{\\partial z^\\prime}{\\partial y} = \\frac{\\partial z^\\prime}{\\partial t} = 0\n\n<\/span>\n<p style=\"text-align:justify;\">With this at hand, we can now calculate the derivatives of <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> through the chain rule:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial x} = \\frac{\\partial \\psi}{\\partial x^\\prime} \\underbrace{\\frac{\\partial x^\\prime}{\\partial x}}_{=1} + \\frac{\\partial \\psi}{\\partial y^\\prime} \\underbrace{\\frac{\\partial y^\\prime}{\\partial x}}_{=0} +\n\n\\frac{\\partial \\psi}{\\partial z^\\prime} \\underbrace{\\frac{\\partial z^\\prime}{\\partial x}}_{=0} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\underbrace{\\frac{\\partial t^\\prime}{\\partial x}}_{=0} =  \\frac{\\partial \\psi}{\\partial x^\\prime}.<\/span>\n<p>And analogously for the other two spatial variables:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial y} =  \\frac{\\partial \\psi}{\\partial y^\\prime}.<\/span>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial z} =  \\frac{\\partial \\psi}{\\partial z^\\prime}.<\/span>\n<p style=\"text-align:justify;\">However, the temporal derivative will show some differences:<\/p>\n<p id=\"eq\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\frac{\\partial \\psi}{\\partial t} &amp;= \\displaystyle\\frac{\\partial \\psi}{\\partial x^\\prime}\\underbrace{\\frac{\\partial x^\\prime}{\\partial t}}_{=-v_{x_0}} + \\frac{\\partial \\psi}{\\partial y^\\prime}\\underbrace{\\frac{\\partial y^\\prime}{\\partial t}}_{=0} + \\frac{\\partial \\psi}{\\partial z^\\prime}\\underbrace{\\frac{\\partial z^\\prime}{\\partial t}}_{=0} + \\frac{\\partial \\psi}{\\partial t^\\prime}\\underbrace{\\frac{\\partial t^\\prime}{\\partial t}}_{=1}\\\\ &amp;=\\displaystyle -v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime},\n\n\\end{array}\n\n<\/span>\n<h5>Transformation of the Second Derivatives<\/h5>\n<p style=\"text-align:justify;\">For the spatial part, we can continue without major difficulties, the results are:<\/p>\n<p><a name=\"eq6\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial x^2} = \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2}.<\/span> <strong>[6]<\/strong><\/p>\n<p><a name=\"eq7\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial y^2} = \\frac{\\partial^2 \\psi}{\\partial {y^\\prime}^2}<\/span> <strong>[7]<\/strong><\/p>\n<p><a name=\"eq8\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial z^2} = \\frac{\\partial^2 \\psi}{\\partial {z^\\prime}^2}<\/span> <strong>[8]<\/strong><\/p>\n<p style=\"text-align:justify;\">But the temporal part, as we could already anticipate from the first derivatives, shows great differences:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle\\frac{\\partial^2 \\psi}{\\partial t^2} &amp;=\\displaystyle \\frac{\\partial}{\\partial t}\\left( -v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right)\\\\ &amp; \\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial t} \\left(\\frac{\\partial \\psi}{\\partial x^\\prime} \\right) + \\frac{\\partial }{\\partial t} \\left(\\frac{\\partial \\psi}{\\partial t^\\prime} \\right)\\\\ &amp;\\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial x^\\prime} \\left(\\frac{\\partial \\psi}{\\partial t} \\right) + \\frac{\\partial }{\\partial t^\\prime} \\left(\\frac{\\partial \\psi}{\\partial t} \\right)\\\\ &amp;\\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial x^\\prime} \\left(-v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right) + \\frac{\\partial }{\\partial t^\\prime} \\left(-v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right) \\end{array}<\/span>\n<p><a name=\"eq9\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle\\frac{\\partial^2 \\psi}{\\partial t^2}  = v_{x_0}^2 \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} - 2v_{x_0}\\frac{\\partial^2 \\psi}{\\partial x^\\prime \\partial t^\\prime} + \\frac{\\partial^2 \\psi}{\\partial {t^\\prime}^2}.<\/span> <strong>[9]<\/strong><\/p>\n<h5>Applying Galileo&#8217;s Transformations to Wave Propagation<\/h5>\n<p style=\"text-align:justify;\">Thus, it is possible to perform the Galileo transformation on the wave propagation equation by replacing equations [<a href=\"#eq6\">6<\/a>,<a href=\"#eq7\">7<\/a>,<a href=\"#eq8\">8<\/a>] and [<a href=\"#eq9\">9<\/a>] on [<a href=\"#eq5\">5<\/a>], resulting in:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} + \\frac{\\partial^2 \\psi}{\\partial {y^\\prime}^2} + \\frac{\\partial^2 \\psi}{\\partial {z^\\prime}^2} = \\frac{1}{v_0^2} \\left(\\color{red}{ v_{x_0}^2 \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} - 2v_{x_0}\\frac{\\partial^2 \\psi}{\\partial x^\\prime \\partial t^\\prime}} + \\frac{\\partial^2 \\psi}{\\partial {t^\\prime}^2} \\right).\n\n<\/span> <strong>[10]<\/strong><\/p>\n<p style=\"text-align:justify;\">It is observed that the form of wave propagation does not hold under Galileo transformations due to the appearance of the additional terms marked in red. Although this does not have major consequences for now, in future classes, we will see that this is precisely the point that \u00abbreaks\u00bb, so to speak, with classical physics, giving way to special relativity.<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclusions<\/h2>\n<p style=\"text-align:justify;\">\n        Galileo&#8217;s Transformations, fundamental in classical mechanics, establish a framework for understanding how observations change between different inertial reference frames. Through this study, we have recognized the concept of absolute time and the addition of velocities as pillars of the Galilean geometry of space and time. However, we have discovered significant limitations of these transformations, especially in their application to wave propagation. This analysis underscores the need for a more complex approach to describe the physical universe, leading us towards special relativity and beyond classical intuition. In summary, while Galileo&#8217;s Transformations provide a solid foundation in classical physics, their inadequacy in certain phenomena highlights the constant evolution of our understanding of the universe.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Galileo&#8217;s Transformations and Their Limitations Summary: The principle of relativity suggests that observations depend on the inertial frame, but in such a way that physical laws remain consistent. An initial and intuitive approach to this principle comes from Galileo&#8217;s Transformations, which model how observations change between inertial reference frames in classical mechanics. In this class, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25587,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":42,"footnotes":""},"categories":[635,691],"tags":[],"citadela-post-location":[],"class_list":["post-25603","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 v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Galileo&#039;s Transformations and Their Limitations - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover Galileo&#039;s Transformations in Classical Mechanics and Their Intriguing Limitations in This Detailed Analysis.\" \/>\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\/galileos-transformations-and-their-limitations\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Galileo&#039;s Transformations and Their Limitations\" \/>\n<meta property=\"og:description\" content=\"Discover Galileo&#039;s Transformations in Classical Mechanics and Their Intriguing Limitations in This Detailed Analysis.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/galileos-transformations-and-their-limitations\/\" \/>\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=\"2022-07-12T00:00:22+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-21T09:42:28+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/07\/TRAGALILEOFLL-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Galileo&#039;s Transformations and Their Limitations\" \/>\n<meta name=\"twitter:description\" content=\"Discover Galileo&#039;s Transformations in Classical Mechanics and Their Intriguing Limitations in This Detailed Analysis.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/07\/TRAGALILEOFLL.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=\"8 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Galileo&#8217;s Transformations and Their Limitations\",\"datePublished\":\"2022-07-12T00:00:22+00:00\",\"dateModified\":\"2024-05-21T09:42:28+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/\"},\"wordCount\":1877,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2022\\\/07\\\/TRAGALILEOFLL.jpg\",\"articleSection\":[\"Physics\",\"Relativity\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/galileos-transformations-and-their-limitations\\\/\",\"name\":\"Galileo's Transformations and Their Limitations - 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