Editing Equations of motion (section)

==General relativity==

===Geodesic equation of motion===

[[File:Geodesic deviation on a sphere.svg|200px|thumb|Geodesics on a [[sphere]] are arcs of [[great circle]]s (yellow curve). On a [[Plane (mathematics)|2D]]&ndash;[[manifold]] (such as the sphere shown), the direction of the accelerating geodesic is uniquely fixed if the separation vector {{math|'''ξ'''}} is [[orthogonal]] to the "fiducial geodesic" (green curve). As the separation vector {{math|'''ξ'''<sub>0</sub>}} changes to {{math|'''ξ'''}} after a distance {{math|''s''}}, the geodesics are not parallel (geodesic deviation).<ref name="Gravitation"/>]]

{{Main|Geodesics in general relativity|Geodesic equation}}

The above equations are valid in flat spacetime. In [[curved space|curved]] [[spacetime]], things become mathematically more complicated since there is no straight line; this is generalized and replaced by a ''[[geodesic]]'' of the curved spacetime (the shortest length of curve between two points). For curved [[manifold]]s with a [[metric tensor]] {{math|''g''}}, the metric provides the notion of arc length (see [[line element]] for details).  The [[differential (infinitesimal)|differential]] arc length is given by:<ref name="ParkerEncycl">{{cite book | title = McGraw Hill Encyclopaedia of Physics | edition = second | author = C.B. Parker | year = 1994 | publisher = McGraw-Hill | isbn = 0-07-051400-3 | url = https://archive.org/details/mcgrawhillencycl1993park}}</ref>{{rp|p=[https://archive.org/details/mcgrawhillencycl1993park/page/1199 1199]}}

<math display="block" qid=Q1228250>ds = \sqrt{g_{\alpha\beta} d x^\alpha dx^\beta}</math>

and the geodesic equation is a second-order differential equation in the coordinates.  The general solution is a family of geodesics:<ref name="ParkerEncycl"/>{{rp|p=[https://archive.org/details/mcgrawhillencycl1993park/page/1200 1200]}}

<math display="block">\frac{d^2 x^\mu}{ds^2} = - \Gamma^\mu{}_{\alpha\beta}\frac{d x^\alpha}{ds}\frac{d x^\beta}{ds}</math>

where {{math|''Γ&thinsp;<sup>μ</sup><sub>αβ</sub>''}} is a [[Christoffel symbols#Christoffel symbols of the second kind (symmetric definition)|Christoffel symbol of the second kind]], which contains the metric (with respect to the coordinate system).

Given the [[mass-energy equivalence|mass-energy]] distribution provided by the [[stress–energy tensor]] {{math|''T&thinsp;<sup>αβ</sup>''}}, the [[Einstein field equations]] are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of spacetime is equivalent to a gravitational field (see [[equivalence principle]]). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because [[Fictitious force#Gravity as a fictitious force|gravity is a fictitious force]]. The ''relative acceleration'' of one geodesic to another in curved spacetime is given by the ''[[geodesic deviation equation]]'':

<math display="block">\frac{D^2\xi^\alpha}{ds^2} = -R^\alpha{}_{\beta\gamma\delta}\frac{dx^\alpha}{ds}\xi^\gamma\frac{dx^\delta}{ds} </math>

where {{math|'''ξ'''<sup>''α''</sup> {{=}} ''x''<sub>2</sub><sup>''α''</sup> − ''x''<sub>1</sub><sup>''α''</sup>}} is the separation vector between two geodesics, {{math|{{sfrac|''D''|''ds''}}}} (''not'' just {{math|{{sfrac|''d''|''ds''}}}}) is the [[covariant derivative]], and {{math|''R<sup>α</sup><sub>βγδ</sub>''}} is the [[Riemann curvature tensor]], containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.<ref name="Gravitation">{{cite book | title=Gravitation | author1=J.A. Wheeler | author2=C. Misner | author3=K.S. Thorne | publisher = W.H. Freeman & Co. | year=1973 | isbn=0-7167-0344-0}}</ref>{{rp|pp=34–35}}

For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to [[Newton's law of gravity]].

===Spinning objects===

In general relativity, rotational motion is described by the [[relativistic angular momentum]] tensor, including the [[spin tensor]], which enter the equations of motion under [[covariant derivative]]s with respect to [[proper time]]. The [[Mathisson–Papapetrou–Dixon equations]] describe the motion of spinning objects moving in a [[gravitational field]].