{"id":116,"date":"2010-05-17T03:11:07","date_gmt":"2010-05-17T02:11:07","guid":{"rendered":"http:\/\/www.quantum-bits.org\/?p=114"},"modified":"2018-06-08T15:01:50","modified_gmt":"2018-06-08T14:01:50","slug":"basic-principles-of-general-relativity","status":"publish","type":"post","link":"https:\/\/www.quantum-bits.org\/?p=116","title":{"rendered":"Basic principles of General Relativity"},"content":{"rendered":"<p>As promised a couple of posts ago, here are a few words on <a href=\"http:\/\/en.wikipedia.org\/wiki\/General_relativity\" target=\"_new\">General Relativity<\/a>. I&#8217;ll try to concentrate on the principles by not focusing (too much) on <a href=\"http:\/\/en.wikipedia.org\/wiki\/Mathematics_of_general_relativity\" target=\"_new\">technicalities<\/a>.<\/p>\n<p>Keeping this post concise is quite a challenge, because of the incredible beauty lying in the physics and mathematics of General Relativity. A thing of beauty is a joy forever&#8230;<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-866\" src=\"https:\/\/www.quantum-bits.org\/wp-content\/uploads\/2010\/05\/general-relativity.jpg\" alt=\"\" width=\"850\" height=\"491\" srcset=\"https:\/\/www.quantum-bits.org\/wp-content\/uploads\/2010\/05\/general-relativity.jpg 1702w, https:\/\/www.quantum-bits.org\/wp-content\/uploads\/2010\/05\/general-relativity-300x173.jpg 300w, https:\/\/www.quantum-bits.org\/wp-content\/uploads\/2010\/05\/general-relativity-1024x591.jpg 1024w\" sizes=\"(max-width: 850px) 100vw, 850px\" \/><\/p>\n<p><strong>Symmetry and relativity<\/strong><\/p>\n<p><a href=\"http:\/\/en.wikipedia.org\/wiki\/Symmetry_in_physics\" target=\"_new\">Symmetries<\/a> and laws of physics are deeply connected. There exists an extremely important result called <a href=\"http:\/\/en.wikipedia.org\/wiki\/Noether%27s_theorem\" target=\"_new\">Noether&#8217;s theorem<\/a>, which states that such symmetries also imply the conservation of physical quantities.<\/p>\n<p>For example, <a href=\"http:\/\/en.wikipedia.org\/wiki\/Energy\" target=\"_new\">energy<\/a> is conserved if the laws describing a system at a given time are still the same at another time: time-translation invariance of the laws of physics implies the conservation of energy. The same link exists between conservation of linear <a href=\"http:\/\/en.wikipedia.org\/wiki\/Momentum\" target=\"_new\">momentum<\/a> and space-translation invariance, or between conservation of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Angular_momentum\" target=\"_new\">angular momentum<\/a> and rotations.<\/p>\n<p>&nbsp;<\/p>\n<div align=\"center\"><img decoding=\"async\" src=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/noether.png\" alt=\"\"><\/div>\n<p>&nbsp;<\/p>\n<p>Symmetry principles are not just statements about how scientists should write laws. They make testable predictions about how nature behaves. Conversely, they constitute a formidable tool to formulate physical theories from experimental data: what truly and only characterize a physical system is what doesn&#8217;t change when one changes his point of view of that system. Give to a theoretical physicist a quantity that is conserved, he&#8217;ll find a symmetry.<\/p>\n<p>That&#8217;s extremely powerful, because symmetries and <a href=\"http:\/\/en.wikipedia.org\/wiki\/Variational_principle\" target=\"_new\">variational principles<\/a> are all that is needed for a theoretician to build accurate descriptions of how Mother Nature works.<\/p>\n<p>Relativity is a particular form of symmetry: a principle of relativity is the requirement that the laws of physics, have the same form independently of frames of reference.<\/p>\n<p>Historically, many forms of these principles were introduced since: <a href=\"http:\/\/en.wikipedia.org\/wiki\/Galileo_Galilei\" target=\"_new\">Galileo<\/a>&#8216;s relativity, <a href=\"http:\/\/en.wikipedia.org\/wiki\/Albert_Einstein\" target=\"_new\">Einstein<\/a>&#8216;s <a href=\"http:\/\/en.wikipedia.org\/wiki\/Special_relativity\" target=\"_new\">Special<\/a> and General relativity, &#8230;<\/p>\n<p><strong>Galilean relativity<\/strong><\/p>\n<p>Let&#8217;s go back to 1632 and Galileo&#8217;s &#8220;<a href=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/galileos_dialogue.png\" rel=\"facebox\">Dialogue Concerning the Two Chief World Systems<\/a>&#8220;. In his seminal work, <a href=\"http:\/\/en.wikipedia.org\/wiki\/Galileo_Galilei\" target=\"_new\">Galileo<\/a> introduced a relativity principle which, in today&#8217;s formulation, could be expressed as follows:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">There exists an <a href=\"http:\/\/en.wikipedia.org\/wiki\/Absolute_space\" target=\"_new\">absolute space<\/a>. A Galilean inertial frame is a reference frame in relative uniform motion to absolute space<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">All inertial frames share a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Absolute_space\" target=\"_new\">universal absolute time<\/a><\/span><\/li>\n<\/ul>\n<p>In the absence of external interactions, free bodies are following straight lines at constant velocity (or are at rest). Any influence that causes a free body to deviate from this state is called a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Force\" target=\"_new\">force<\/a>. Galilean relativity states that the laws of physics take the same form all inertial frame. In simple mathematical terms, the laws of physics are invariant under transformations of this form (<a href=\"http:\/\/en.wikipedia.org\/wiki\/Galilean_transformation\" target=\"_new\">Galilean transformations<\/a>):<\/p>\n<div style=\"position: relative; top: -12px;\" align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cleft%5C%7B+%5Cbegin%7Barray%7D%7Bllll%7D+x%27+%26%3D%26+x+-vt+%5C%5C+y%27+%26%3D%26+y+%5C%5C+z%27+%26%3D%26+z+%5C%5C+t%27+%26%3D%26+t+%5Cend%7Barray%7D+%5Cright.+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\left\\{ \\begin{array}{llll} x&#039; &amp;=&amp; x -vt \\\\ y&#039; &amp;=&amp; y \\\\ z&#039; &amp;=&amp; z \\\\ t&#039; &amp;=&amp; t \\end{array} \\right. ' title='\\left\\{ \\begin{array}{llll} x&#039; &amp;=&amp; x -vt \\\\ y&#039; &amp;=&amp; y \\\\ z&#039; &amp;=&amp; z \\\\ t&#039; &amp;=&amp; t \\end{array} \\right. ' class='latex' \/><\/div>\n<p><strong>Newton&#8217;s law of universal gravitation<\/strong><\/p>\n<p>Let&#8217;s jump to 1687. <a href=\"http:\/\/en.wikipedia.org\/wiki\/Isaac_Newton\" target=\"_new\">Newton<\/a>&#8216;s masterpiece, &#8220;<a href=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/principa.png\" rel=\"facebox\">Philosophi\u00e6 Naturalis Principia Mathematica<\/a>&#8220;, has set the grounds of what is now called <a href=\"http:\/\/en.wikipedia.org\/wiki\/Newtonian_mechanics\" target=\"_new\">classical mechanics<\/a>.<\/p>\n<p>Based on Galileo&#8217;s reflections, Newtons defined three laws:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><strong>Fist law (inertia)<\/strong>: it is a restatement of Galileo&#8217;s relativity. An object at rest tends to stay at rest, or if it is in motion tends to stay in motion with the same speed and in the same direction unless acted upon by a sum of physical forces<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><span style=\"color: #666666;\"><strong>Second law (motion)<\/strong>: a body will accelerate with acceleration proportional to the force and inversely proportional to the mass:<br \/>\n<\/span><\/span><\/p>\n<div style=\"position: relative; top: -5px;\" align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=F_i+%3D+m+%5Cfrac%7Bd%5E2x_i%7D%7Bdt%5E2%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F_i = m \\frac{d^2x_i}{dt^2} ' title='F_i = m \\frac{d^2x_i}{dt^2} ' class='latex' \/><\/div>\n<\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><strong>Third law (reciprocal actions)<\/strong>: every action has a reaction equal in magnitude and opposite in direction.<\/span><\/li>\n<\/ul>\n<p>Newton&#8217;s <a href=\"http:\/\/en.wikipedia.org\/wiki\/Newton%27s_law_of_universal_gravitation\" target=\"_new\">law of universal gravitation<\/a> completes these principles. Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses:<\/p>\n<div align=\"center\"><img decoding=\"async\" src=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/newton-laws.png\" alt=\"\"><\/div>\n<p>The following picture illustrates Newton&#8217;s theory:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><strong>Case A<\/strong>: in absence of external influence, a free body follows a straight line at constant speed<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><strong>Case B<\/strong>: under the influence of gravity, the body follows a trajectory explained by the combination of Newton&#8217;s second law of motion and Newton&#8217;s law of universal gravitation.<\/span><\/li>\n<\/ul>\n<div align=\"center\"><img decoding=\"async\" src=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/newton-gravitation.png\" alt=\"\"><\/div>\n<p>Based on this work, Newton was able to prove that <a href=\"http:\/\/en.wikipedia.org\/wiki\/Kepler%27s_law\" target=\"_new\">Kepler&#8217;s laws of planetary motion<\/a>, which match <a href=\"http:\/\/en.wikipedia.org\/wiki\/Tycho_Brahe\" target=\"_'new'\">Tycho Brahe<\/a>&#8216;s observations, were consequences of his own laws of motion and universal gravitation.<\/p>\n<p><strong>What is wrong with this ?<\/strong><\/p>\n<p>Newton&#8217;s theory is accurate enough for many practical purposes, and is still widely use. Nevertheless, in light of today&#8217;s knowledge, major concerns (experimental and theoretical) with this theory lead scientists to find a totally new <a href=\"http:\/\/en.wikipedia.org\/wiki\/Paradigm\" target=\"_new\">paradigm<\/a>.<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><strong>From an experimental point of view<\/strong>: Newton&#8217;s theory cannot cope with today&#8217;s high precision observations. For example, it doesn&#8217;t fully explain the precession of the perihelion of the planet Mercury, or the deflection of light rays by gravity<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\"><strong>From a theoretical point of view<\/strong>: Newton&#8217;s concept of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Action_at_a_distance_%28physics%29\" target=\"_new\">action at a distance<\/a> is quite unsatisfactory. Furthermore, it require that the gravitational force be transmitted instantaneously, which is known now to be impossible.<\/span><\/li>\n<\/ul>\n<p>To go any further, one has to revise both Galilean relativity (absolute time and space) and Newtonian gravitation (action at a distance ruled by an <a href=\"http:\/\/en.wikipedia.org\/wiki\/Inverse-square_law\" target=\"_new\">inverse square law<\/a>)<\/p>\n<p><strong>Einstein&#8217;s Special Relativity<\/strong><\/p>\n<p>By the end of the 19th century, several <a href=\"http:\/\/en.wikipedia.org\/wiki\/Interferometer\" target=\"_new\">interferometer<\/a>-based experiments (<a href=\"http:\/\/en.wikipedia.org\/wiki\/Michelson-Morley_experiment\" target=\"_new\">Michelson &amp; Morley<\/a>, <a href=\"http:\/\/en.wikipedia.org\/wiki\/Fizeau_experiment\" target=\"_new\">Fizeau<\/a>, &#8230;) were carried out.<\/p>\n<p>The results of these experiments were quite surprising, and, at the time, there were no possible explanation within the corpus of available theories. These experiments are now referred to as &#8220;the kicking-off point for the theoretical aspects of the Second Scientific Revolution&#8221;.<\/p>\n<div align=\"center\"><img decoding=\"async\" src=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/michelson.png\" alt=\"\"><\/div>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">There is no such thing as the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Luminiferous_aether\" target=\"_new\">luminiferous aether<\/a><\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">The speed of light is the same for all inertial observers regardless of the state of motion of the source.<\/span><\/li>\n<\/ul>\n<p>This very last observation is of particular importance, since it is incompatible with the Galilean-transformation invariance requirement. Great minds, such as <a href=\"http:\/\/en.wikipedia.org\/wiki\/Henri_Poincar%C3%A9\" target=\"_new\">Poincar\u00e9<\/a>, <a href=\"http:\/\/en.wikipedia.org\/wiki\/Hendrik_Lorentz\" target=\"_new\">Lorentz<\/a> or Einstein tackled this problem.<\/p>\n<p>Einstein postulated in 1905 the following:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">The speed of light is a fundamental constant (invariant)<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">The physical laws (including the constancy of the speed of light) are independent from the choice of inertial system<\/span><\/li>\n<\/ul>\n<p>To build a new framework for physics, compatible with these propositions, Einstein realized than one has to drop Galilean relativity, and proposed a new form a relativity (now called Special Relativity). Within this theory, all physical laws must be invariant under a new group of symmetry, called <a href=\"http:\/\/en.wikipedia.org\/wiki\/Lorentz_transformation\" target=\"_new\">Lorentz transformations<\/a>:<\/p>\n<div style=\"position: relative; top: -6px;\" align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cleft%5C%7B+%5Cbegin%7Barray%7D%7Bllll%7D+x%27+%26%3D%26+%5Cgamma+%28x+-vt%29+%5C%5C+y%27+%26%3D%26+y+%5C%5C+z%27+%26%3D%26+z+%5C%5C+t%27+%26%3D%26+%5Cgamma+%28t+-+v+x+%2F+c%5E2%29+%5Cend%7Barray%7D+%5Cright.+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\left\\{ \\begin{array}{llll} x&#039; &amp;=&amp; \\gamma (x -vt) \\\\ y&#039; &amp;=&amp; y \\\\ z&#039; &amp;=&amp; z \\\\ t&#039; &amp;=&amp; \\gamma (t - v x \/ c^2) \\end{array} \\right. ' title='\\left\\{ \\begin{array}{llll} x&#039; &amp;=&amp; \\gamma (x -vt) \\\\ y&#039; &amp;=&amp; y \\\\ z&#039; &amp;=&amp; z \\\\ t&#039; &amp;=&amp; \\gamma (t - v x \/ c^2) \\end{array} \\right. ' class='latex' \/><\/div>\n<p>where <span style=\"position: relative; top: +4px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cgamma+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%28v%2Fc%29%5E2%7D%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\gamma = \\frac{1}{\\sqrt{1-(v\/c)^2}} ' title='\\gamma = \\frac{1}{\\sqrt{1-(v\/c)^2}} ' class='latex' \/><\/span> and <span style=\"position: relative; top: 0px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=v+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='v ' title='v ' class='latex' \/><\/span> is the relative velocity between two observers.<\/p>\n<p>These transformations are quite different from Galilean&#8217;s. Time and space coordinates are intertwined, resulting in a radically new concept: <a href=\"http:\/\/en.wikipedia.org\/wiki\/Spacetime\" target=\"new\">spacetime<\/a>. Time and space are not absolute concepts anymore. Speed of light is. I&#8217;ll try to write a post sometimes on all the consequences of this (relativity of simultaneity, length contraction, time dilatation, &#8230;)<\/p>\n<p>Spacetime is a mathematical model that combines space and time into a single continuum. It is a 4-dimensional space (named <a href=\"http:\/\/en.wikipedia.org\/wiki\/Minkowski_space\" target=\"_new\">Minkowski space<\/a>), where each point (called &#8220;event&#8221;) is a set of 4 coordinates:<\/p>\n<div style=\"position: relative; top: -5px;\" align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%28x%5E%5Cmu%29+%3D+%28x%5E0%2Cx%5E1%2Cx%5E2%2Cx%5E3%29+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='(x^\\mu) = (x^0,x^1,x^2,x^3) ' title='(x^\\mu) = (x^0,x^1,x^2,x^3) ' class='latex' \/> &nbsp;where&nbsp; <img src='https:\/\/s0.wp.com\/latex.php?latex=x_0+%5Cequiv+ct%2C%5C%3B+x_1+%5Cequiv+x%2C%5C%3B+x_2+%5Cequiv+y%2C%5C%3B+x_3+%5Cequiv+z+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x_0 \\equiv ct,\\; x_1 \\equiv x,\\; x_2 \\equiv y,\\; x_3 \\equiv z ' title='x_0 \\equiv ct,\\; x_1 \\equiv x,\\; x_2 \\equiv y,\\; x_3 \\equiv z ' class='latex' \/><\/div>\n<p>At this point, for the sake of simplicity and homogeneity, we will introduce &#8220;<a target=\"_new\">God&#8217;s units<\/a>&#8221; where <span style=\"position: relative; top: -4px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=c+%5Cequiv+1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='c \\equiv 1' title='c \\equiv 1' class='latex' \/><\/span>.<\/p>\n<p>In these units, any interval between two events is given by:<\/p>\n<div align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=ds%5E2+%3D+%5Ceta_%7B%5Cmu%5Cnu%7D+dx%5E%5Cmu+dx%5E%5Cnu+%3D+dt%5E2+-+%28dx%5E2+%2B+dy%5E2+%2B+dz%5E2%29+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='ds^2 = \\eta_{\\mu\\nu} dx^\\mu dx^\\nu = dt^2 - (dx^2 + dy^2 + dz^2) ' title='ds^2 = \\eta_{\\mu\\nu} dx^\\mu dx^\\nu = dt^2 - (dx^2 + dy^2 + dz^2) ' class='latex' \/><\/div>\n<p>where:<\/p>\n<div style=\"position: relative; top: -5px;\" align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ceta_%7B%5Cmu%5Cnu%7D+%3D+%5Cleft%5B+%5Cbegin%7Barray%7D%7Bcccc%7D+1+%26+0+%26+0+%26+0+%5C%5C+0+%26+-1+%26+0+%26+0+%5C%5C+0+%26+0+%26+-1+%26+0+%5C%5C+0+%26+0+%26+0+%26+-1%5C%5C+%5Cend%7Barray%7D+%5Cright%5D+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\eta_{\\mu\\nu} = \\left[ \\begin{array}{cccc} 1 &amp; 0 &amp; 0 &amp; 0 \\\\ 0 &amp; -1 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; -1 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; -1\\\\ \\end{array} \\right] ' title='\\eta_{\\mu\\nu} = \\left[ \\begin{array}{cccc} 1 &amp; 0 &amp; 0 &amp; 0 \\\\ 0 &amp; -1 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; -1 &amp; 0 \\\\ 0 &amp; 0 &amp; 0 &amp; -1\\\\ \\end{array} \\right] ' class='latex' \/><\/div>\n<p>Within this formalism, a light cone is the surface describing the temporal evolution of a flash of light in Minkowski spacetime. This can be visualized in 3-space if the two horizontal axes are chosen to be spatial dimensions, while the vertical axis is time:<\/p>\n<div align=\"center\"><img decoding=\"async\" src=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/lightcone.png\" alt=\"\"><\/div>\n<p>This construction brings to spacetime a very particular structure called <a href=\"http:\/\/en.wikipedia.org\/wiki\/Causal_structure\" target=\"_new\">causal structure<\/a>. This is a major &#8220;contributor&#8221; to the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Arrow_of_time\" target=\"_new\">arrow of time<\/a>. Relatively to a given event E, the light cone classifies events into distinct categories:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Events on the future light cone<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Events on the past line cone<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Events inside the future light cone are those affected by material particle emitter at E<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Events inside the past light cone are those that can emit a material particle and affect the present<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">All other events, in the absolute elsewhere, cannot affect or be affected by E<\/span><\/li>\n<\/ul>\n<p>A spacetime interval <span style=\"position: relative; top: -2px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=ds%5E2+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='ds^2 ' title='ds^2 ' class='latex' \/><\/span> is said to be:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Time-like if <span style=\"position: relative; top: -4px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=ds%5E2+%3E+0+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='ds^2 &gt; 0 ' title='ds^2 &gt; 0 ' class='latex' \/><\/span><\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Light-like (or null) if <span style=\"position: relative; top: -2px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=ds%5E2+%3D+0+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='ds^2 = 0 ' title='ds^2 = 0 ' class='latex' \/><\/span><\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Space-like if <span style=\"position: relative; top: -2px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=ds%5E2+%3C+0+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='ds^2 &lt; 0 ' title='ds^2 &lt; 0 ' class='latex' \/><\/span><\/span><\/li>\n<\/ul>\n<p>Similarly, a smooth regular curve will be classified depending on their tangent vectors. Such a curve is:<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Chronological or time-like if the tangent vector is timelike at all point in the curve<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Spacelike if the tangent vector is spacelike at all points in the curve<\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Causal (or non-spacelike) if the tangent vector is timelike or null at all points in the curve<\/span><\/li>\n<\/ul>\n<p>More specifically, a curve is<\/p>\n<ul>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Future-directed if, for every point in the curve, the tangent vector is future-directed <\/span><\/li>\n<li style=\"list-style: square inside; color: #aaaaaa;\"><span style=\"color: #666666;\">Past-directed if, for every point in the curve, the tangent vector is past-directed<\/span><\/li>\n<\/ul>\n<p><strong>Einstein&#8217;s General Relativity<\/strong><\/p>\n<p>It is now time to introduce gravity. Einstein quickly realized than Newtonian gravitation is not compatible with his Special Relativity. On one hand, Newton&#8217;s actions at a distance are instantaneous, on the other, a simple consequence of Lorentzian invariance is that no information can travel faster than light.<\/p>\n<p>In 1915, Einstein introduced its General Relativity theory. In this theory, the effects of gravitation are ascribed to spacetime curvature instead of a force. How he came to this conclusion starting from the equivalence principle (identification of free fall with inertial motion) is in itself a fabulous story.<\/p>\n<p>Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime. These straight paths are called <a href=\"http:\/\/en.wikipedia.org\/wiki\/Geodesic_%28general_relativity%29\" target=\"_new\">geodesics<\/a>. In abscence of gravity, physics is described by Einstein&#8217;s Special Relativity. In Minkowski spacetimes, geodesics are plain straight lines.<\/p>\n<p>A Minkowskian spacetime is actually a flat <a href=\"http:\/\/en.wikipedia.org\/wiki\/Pseudo-Euclidean_space\" target=\"_new\">pseudo-euclidean space<\/a>. Technically speaking, this is the simplest form of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Pseudo-Riemannian_manifold\" target=\"_new\">pseudo-riemannian manifold<\/a>.<\/p>\n<p>Einstein postulated that spacetime is a (pseudo-riemannian) <a href=\"http:\/\/en.wikipedia.org\/wiki\/Manifold\" target=\"_new\">manifold<\/a>. Simply speaking, these are curved spaces which can always be locally approximated by flat pseudo-euclidean spaces (Minkowski spacetimes).<\/p>\n<p>To deal with these geometry, one has to introduce new mathematical objects: <a href=\"http:\/\/en.wikipedia.org\/wiki\/Tensor\" target=\"_new\">tensors<\/a>. Simply speaking, tensors are geometric entities that extend the notion of scalars, vectors and matrices. These are the natural objects to describe quantities in a relativistic way: tensors themselves are independent of a particular choice of coordinate system.<\/p>\n<p>To express his field equations, Einstein introduced a particular tensor, denoted <span style=\"position: relative; top: -2px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T%5E%7B%5Cmu%5Cnu%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='T^{\\mu\\nu} ' title='T^{\\mu\\nu} ' class='latex' \/><\/span> and called <a href=\"http:\/\/en.wikipedia.org\/wiki\/Stress-energy_tensor\" target=\"_new\">stress-energy tensor<\/a>. It is a quantity that describes the density and flux of energy and momentum in spacetime.<\/p>\n<p>Then, he connected stress-energy tensors to local properties of spacetime (curvatures):<\/p>\n<div style=\"position: relative; top: -8px;\" align=\"center\"><span style=\"position: relative; top: 3px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=G_%7B%5Cmu%5Cnu%7D+%2B+%5CLambda+g_%7B%5Cmu%5Cnu%7D+%3D+%5Cfrac%7B8%5Cpi+G%7D%7Bc%5E4%7D+T_%7B%5Cmu%5Cnu%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu} ' title='G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu} ' class='latex' \/><\/span>&nbsp;&nbsp;where&nbsp;&nbsp;<span style=\"position: relative; top: 3px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=G_%7B%5Cmu%5Cnu%7D+%3D+R_%7B%5Cmu%5Cnu%7D+-+%5Cfrac%7B1%7D%7B2%7D+Rg_%7B%5Cmu%5Cnu%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G_{\\mu\\nu} = R_{\\mu\\nu} - \\frac{1}{2} Rg_{\\mu\\nu}' title='G_{\\mu\\nu} = R_{\\mu\\nu} - \\frac{1}{2} Rg_{\\mu\\nu}' class='latex' \/><\/span><\/div>\n<p>Theses field equations mean than energy \/ mass and momentum proportionally curves spacetime. In absence of gravity (A), spacetime is a flat Minkowskian manifold and free bodies follows straight line. In presence of gravity (B), spacetime is curved and free bodies follow geodesics.<\/p>\n<p>This is a big paradigm shift. Instead of thinking of gravity as a force pulling objects (instantaneous action at a distance), hence causing them to travel along curved paths in a 3-dimensional flat space, one has to think of objects traveling along the straightest paths (geodesics) in a 4-dimensional spacetime curved by the presence of energy and momentum. For example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-dimensional spacetime geometry around the star onto 3-dimensional space.<\/p>\n<p>I tried to visually sum up these results in the following picture:<\/p>\n<div align=\"center\"><img decoding=\"async\" src=\"http:\/\/quantum-bits.org\/wp-content\/uploads\/2010\/05\/einstein-gravitation.png\" alt=\"\"><\/div>\n<p>&nbsp;<\/p>\n<p>I guess this will be all for tonight as it is now quite late (3:30am). I&#8217;ll need a shot of caffeine tomorrow morning. But it is already enough to kick Tony Stark&#8217;s ass (see previous posts). More on a the consequences (and limitations) of this formalism in another post &#8230;<\/p>\n<p><span style=\"text-decoration: underline;\">Credits<\/span>: <a href=\"http:\/\/www.jesusda.com\/\" target=\"_new\">jEsuSdA 8 )<\/a> for the rocket icon.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>As promised a couple of posts ago, here are a few words on General Relativity. I&#8217;ll try to concentrate on the principles by not focusing (too much) on technicalities. Keeping &#8230;<\/p>\n","protected":false},"author":1,"featured_media":866,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0},"categories":[6],"tags":[],"_links":{"self":[{"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=\/wp\/v2\/posts\/116"}],"collection":[{"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=116"}],"version-history":[{"count":0,"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=\/wp\/v2\/posts\/116\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=\/wp\/v2\/media\/866"}],"wp:attachment":[{"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=116"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=116"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.quantum-bits.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}