<\/p>\n\n\n\n
General Relativity explains gravity as Space-Time\ncurvature and orbits of planets as geodesics of curved Space-Time. However, this\nconcept is extremely hard to understand and geodesics hard to compute. If we\ncan find an analytical orbit equation for planets like Newtonian orbit equation,\nrelativistic gravity will become intuitive and straightforward so that most\npeople can understand.<\/p>\n\n\n\n
From gravitational force and acceleration,\nI have derived the analytical orbit\nequation for relativistic gravity which is equation (1). Below I will explain the derivation of this equation. Albert Einstein had correctly\npredicted the orbital precession of planet Mercury which had definitively validated\nGeneral Relativity. Equation (2) is the angle of orbital precession that this orbit\nequation gives, which is identical to the one Albert Einstein had given [1][2]<\/sup>.<\/p>\n\n\n\n If this orbit equation gave the same result\nthan Space-Time geodesics, then everyone can compute the orbit of any object in\ngravitational field which obeys General Relativity using personal computer\nrather than big or super computer. Also, everyone can see how gravity leads to Space-Time\ncurvature without the need of knowing Einstein tensor. <\/p>\n\n\n\n The derivation of the orbit equation is rather tedious and lengthy. So, for clarity of the reasoning and explanation, I have collected all the mathematical equations in the last section \u201cDerivation of equations\u201d, in which full details are provided to help readers for checking the validity of my mathematics. <\/p>\n\n\n\n Take an attracting body of mass M around\nwhich orbits a small body of mass m, see Figure\n1. We work with a polar coordinate system of which the body\nM sits at the origin. The position of the body m with respect to M is specified\nby the radial position vector r<\/em><\/strong>, of which the magnitude is r <\/em>and the polarangle is q<\/em>.<\/p>\n\n\n\n Let the frame of reference \u201cframe_m\u201d be an\ninertial fame that instantaneously moves with m. Frame_m is the proper frame of\nm where the velocity of m is 0. So, Newton\u2019s laws apply in this frame. Let a<\/em><\/strong>m<\/sub> be the acceleration vector of m in frame_m\nand the inertial force of m is m\u00b7<\/em><\/strong>a<\/em><\/strong>m<\/sub>, see equation (3). The gravitational force on m is given by equation (4). Equating (4) with (3), we get equation (5), the proper acceleration of m caused by gravitational\nforce in frame_m.<\/p>\n\n\n\n Let \u201cframe_l<\/em>\u201d be the local frame of reference in which M is stationary. In frame_l<\/em> m is under the effect of gravity of M,\nthe velocityvector of m is v<\/em><\/strong>l<\/sub><\/em>\nand the acceleration of m is a <\/sub><\/em><\/strong>l<\/sub><\/em>. As frame_m moves with m, it moves at the velocity v<\/em><\/strong>l<\/sub><\/em> in frame_l.<\/p>\n\n\n\n The acceleration of m in frame_m and frame_l<\/em> are respectivelya<\/em><\/strong>m<\/sub><\/em> and a<\/em><\/strong>l<\/sub><\/em>. To transform a<\/em><\/strong>l<\/sub><\/em> into a<\/em><\/strong>m<\/sub><\/em> we use the transformation of acceleration between relatively moving\nframes which is the equation (18) in \u00abRelativistic\nkinematics<\/a> and gravity<\/a>\u00bb[3]<\/sup>[4]<\/sup>, in which we replace a<\/em><\/strong>1<\/sub> with a<\/em><\/strong>l<\/sub><\/em>, a<\/em><\/strong>2<\/sub> with a<\/em><\/strong>m<\/sub><\/em> and u<\/em><\/strong> with v<\/em><\/strong>l<\/sub><\/em>.\nThen the transformation between a<\/em><\/strong>m<\/sub><\/em> and a <\/sub><\/em><\/strong>l<\/sub><\/em> is equation (6). <\/p>\n\n\n\n Equating (5) with (6) we get equation (7), both sides of which are then dotted by the vector v<\/em><\/strong>l<\/sub><\/em>\u00b7dtl<\/sub><\/em>, see equation (8). On the left hand side of (8), we find dr<\/em>\nthe variation of the radial distance r<\/em>,\nsee (9). On the right hand side, we find the variation vector\nof velocity dv<\/strong>l<\/sub><\/em>, see (10), which, dotted by the velocity vector v<\/em><\/strong>l<\/sub><\/em>, gives dv<\/em><\/strong>l<\/sub>2<\/sup><\/em>\/2 in (11).<\/p>\n\n\n\n Plugging (9) and (11) into (8), we get (12), both sides of which are differential expressions,\nsee (13) and (15). Then, plugging (13) and (15) into (12) gives (16) which is a differential equation. (16) is integrated to give (17), with K<\/em>\nbeing the integration constant. Then, we rearrange (17) to express v<\/em><\/strong>l<\/sub>2<\/sup><\/em>\/c2<\/sup>\nin (18), which relates the local orbital velocity v<\/em><\/strong>l<\/sub><\/em> to the gravitational field of\nM.<\/p>\n\n\n\n The value of eK<\/sup><\/em> is determined at a known point 0 at which the velocity\nis v<\/em><\/strong>0<\/sub><\/em> and the radial distance is r0<\/sub><\/em>, see (19). <\/p>\n\n\n\n \u2026<\/p>\n\n\n\n General\nequation for Space-Time geodesics and orbit equation in relativistic gravity<\/strong> https:\/\/www.academia.edu\/44540764\/Analytical_orbit_equation_for_relativistic_gravity_without_using_Space_Time_geodesics<\/a><\/p>\n\n\n\n https:\/\/pengkuanonphysics.blogspot.com\/2020\/11\/analytical-orbit-equation-for.html<\/a><\/p>\n\n\n\n <\/p>\n\n\n","protected":false},"excerpt":{"rendered":" Orbit equation and orbital precession General Relativity explains gravity as Space-Time curvature and orbits of planets as geodesics of curved Space-Time. However, this concept is extremely hard to understand and geodesics hard to compute. If we can find an analytical … Continue reading
a) Velocity in local frame<\/li><\/ul>\n\n\n\n