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The Geodesic Spacetime Equation for Body Motion in Gravitational Fields

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The Geodesic Spacetime Equation for Body Motion in Gravitational Fields The Geodesic Spacetime Equation for Body Motion in Gravitational Fields "The eternal mystery of the world is its comprehensibility" - Albert Einstein ( 1879 - 1955 ) § Einstein's "Law of Equivalence of Acceleration and Gravity" or "The Principle of Equivalence" : Fields of acceleration and fields of gravity are equivalent physical phenomenon. That is, if there's no "upward" acceleration, then there's no "downward" gravity. Preliminary Understanding of the following Mathematical Symbols Naïve realism easily allows ordinary understanding of the mathematical symbols of ; however, for deeper understanding of physical reality it becomes necessary to invent newer and more powerful mathematical tools such as the Christoffel symbol by which a deeper probing of external reality is made entirely possible. Hence, a certain amount of patient indulgence is asked of the serious reader of this mathematical essay since it is strongly suggested that by a thorough reading of these general relativity equations and the tightly woven logic by which they are presented, that with final patience the reader of Relativity Calculator will have a fairly good grasp of the underlying mathematical physics of Einstein's ultimate gravitational equation! A Quick Introductory Meaning of Geodesic Motion Sir Arthur Eddington ( 1882 - 1944 ) on May 29, 1919 tentatively, but empirically, confirmed Einstein's mathematical general relativity conjecture that light passing near the gravity field of the sun was slightly, but convincingly, bent in the spacetime proximate vicinity of the sun, thus showing that on grand cosmic scales there is no such reality for Euclid's straight geometry lines. Likewise, gyroscopic motion will equivalently follow the curvilinear contours of spacetime fabric as influenced by other gravitational fields. In fact, the very best way to "map out" the contours of gravity's spacetime fabric is by setting gyroscopes aloft, so to speak, into the grand cosmos where these gyroscopes will follow the same transit paths by their parallel ( gyroscopic ) transport as their cousin light beams travelling the exact same intervening gravity fields. Naturally, as these gyroscope cousins of light beams trace out spacetime's fabric, their "spin - time" as well as "transit - time" of passage will equivalently also be effected by gravity's field; thus, we arrive at space - time! Hence, the meaning of "geodetic motion" is best understood in terms of the parallel transport of gyroscopes! And the mathematics for such understanding was arrived at in 1854 by Bernhard Riemann in his doctoral "On the Hypothesis That Lie at the Foundations of Geometry", German version; where, the derivation of such geodesic equation for motion in gravity fields is therefore the purpose of this following mathematical exposition: Relativity Science Calculator A Quick Summary for the Geodesic Equation In Riemannian geometry for space as opposed to flat Euclidean metric space, the "shortest curve" or "straightest" possible world timeline with minimum actionable variation in curved spacetime according to the Principle of Least Action between two given fixed points and is determined by the set of solutions to the "timelike" geodesic differential equations, the optimal one of which gives a "stationary" or minimal value to the calculus action integral for the extremal maximum proper transit time for the most efficient use of energy: Relativity Science Calculator Recapitulating A true ( idealized ) geodesic motion for a body object is where a "timelike" world timeline parallel transports, according to the Principle of Least Action, its own tangent vector along the "most consistently stationary" curvilinear spacetime coordinates while maintaining constant magnitude and direction for which Relativity Science Calculator That is, to explicate further, suppose: Relativity Science Calculator NASA's Gravity Probe B Confirms Two Einstein Space-Time Theories See: NASA's Gravity Probe B Confirms Two of Einstein's Space - Time Theories Relativity Science Calculator In other words, the geodesic world timeline is that gyroscopic ( parallel transport ) transit path for any object body or sub - atomic test particle which expends the least amount of energy according to the Principle of Least Action! Hence in Euclidean flat space the shortest distance between fixed points and is exactly equivalent to the shortest world timeline transit time; but in Riemannian curved geometric spacetime with an intervening gravitational mass such as earth's sun within our solar system affecting velocities, it therefore becomes the extremal maximum proper time ( and not minimum straight - line distance! ) which determines the geodesic world timeline connecting fixed points and . Now in order to achieve the derivation for the geodesic spacetime equation describing the minimum world timeline between fixed spacetime points and , we must therefore develop several new "mathematical technologies" as follows: Several New Mathematical Technologies 1). Parallel ( Gyroscopic ) Transport provides the underlying conceptual basis for the Christoffel symbol; additionally, parallel ( gyroscopic ) transport gives an immediate derivation of the geodesic equation which at this time in our mathematical essay will be deferred until later for a deeper understanding of all of these new mathematical constructs; 2). The Christoffel symbol accounts for the rate of change of the local basis ( or its reciprocal ) which varies from infinitesimal world timeline spacetime point to infinitesimal spacetime point in the affine ( affinity ) geometry of curved Riemannian space. In effect, therefore, the Christoffel symbol mathematically represents the intensity of the composite gravitational force field under conditions of parallel ( gyroscopic ) transport; Relativity Science Calculator 3). The non - tensor Christoffel symbol is "arithmetized" by the spacetime gravitational potentials , 2nd - order metric tensor, of the underlying curvilinear coordinates whose tensor proof is proffered later on; 4). The Euler - Lagrange ( or Euler's, Lagrange's ) equation of motion: whose derivation will later on also be given. Parallel ( Gyroscopic ) Transport in an Affinely Connected Parallel Space Demonstrating Gyroscopic Parallel Transport in the International Space Station using "gravity pushing" straws simulating parallel transport - i.e., gravity's parallel transport influence upon a gyroscope § Vector Analysis in terms of Covariant and Contravariant component expansion: Relativity Science Calculator § Vector Analysis in terms of differential calculus: § Vector Analysis in terms of partial differentials: Now the partial differential component expansion for the total differential is given as follows: § Derivation of The Law of Parallel Displacement for Parallel ( Gyroscopic ) Transport: Employing a homogeneous ( constant in magnitude and direction ) vector field , we now consider its parallel displacement between coordinate points to ( using the variational calculus operator - not Kronecker's delta! ) signifying an infinitesimal parallel vector movement along a given - axis for some infinitesimal parallel translation of vector as follows: Therefore, Relativity Science Calculator So, the law of parallel displacement simply becomes Or, equivalently, but only using the components of vector and only using the upper index in for simplicity's sake, we derive and hence also define the following: Relativity Science Calculator However, a simpler and more intuitive parallel transport derivation for a vector field which is homogeneously constant in both magnitude and direction is immediately given as follows: where, nevertheless, the following two terms are missing all of which means that in parallel ( gyroscopic ) transport all vector field component changes ( magnitude, direction ), for whatever "parallel direction" ( or lateral push of the gyroscope ) is invoked, are therefore null or equivalently set to zero which is what we would naïvely expect. Derivation of The Covariant Derivative of the Vector under the Condition of Parallel ( Gyroscopic ) Transport Now that the law of parallel ( gyroscopic ) displacement has been established, namely where the covariant component expansion of vector is parallel transported to along a given - axis which in turn generates a parallel translated vector at coordinate . Therefore, Relativity Science Calculator is some unknown, yet still to be determined "4 x 4 x 4 - component object", system of functions of local ( or its reciprocal ) base vectors. That is, is some arbitrary system of differential functions that characterizes spacetime transport according to the law of parallel displacement and which therefore numerically represents the rate of change of the local ( or its reciprocal ) basis whenever a test particle or object body transits the extremal minimum geodesic path thru intervening gravitational fields for the least action and most efficient expenditure of energy. Furthermore, is linear in both as well as in since the vector sum must also parallel transport according to the same transformation law as for each of the individual vectors. Therefore, these mathematical objects which are dependent on the particular spacetime coordinate system in which they arise, will hence go to null in Euclidean space; that is, in Descartes coordinates. Whereas defines the parallel transport of a covariant vector according to the law of parallel displacement, so also there must analogously be the parallel transport of a contravariant vector under the condition
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