Editing General relativity (section)

=== Geometry of Newtonian gravity<!--'Einstein's elevator experiment' redirects here--> ===
[[File:Elevator gravity.svg|thumb|According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8&nbsp;m/s<sup>2</sup> (the acceleration due to gravity on the surface of the Earth).]]
At the base of [[classical mechanics]] is the notion that a [[physical body|body]]'s motion can be described as a combination of free (or [[inertia]]l) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second [[Newton's laws of motion|law of motion]], which states that the net [[force]] acting on a body is equal to that body's (inertial) [[mass]] multiplied by its [[acceleration]].<ref>{{Harvnb|Arnold|1989|loc=ch. 1}}</ref> The preferred inertial motions are related to the geometry of space and time: in the standard [[frame of reference|reference frames]] of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are [[geodesic]]s, straight [[world lines]] in [[curved spacetime]].<ref>{{Harvnb|Ehlers|1973|pp=5f}}</ref>

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as [[electromagnetism]] or [[friction]]), can be used to define the geometry of space, as well as a time [[coordinate]]. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of [[Loránd Eötvös|Eötvös]] and its successors (see [[Eötvös experiment]]), there is a universality of free fall (also known as the weak [[equivalence principle]], or the universal equality of inertial and passive-gravitational mass): the trajectory of a [[test body]] in free fall depends only on its position and initial speed, but not on any of its material properties.<ref>{{Harvnb|Will|1993|loc=sec. 2.4}}, {{Harvnb|Will|2006|loc=sec. 2}}</ref> A simplified version of this is embodied in '''Einstein's elevator experiment'''<!--boldface per WP:R#PLA-->, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.<ref>{{Harvnb|Wheeler|1990|loc=ch. 2}}</ref>

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific [[connection (mathematics)|connection]] which depends on the [[gradient]] of the [[gravitational potential]]. Space, in this construction, still has the ordinary [[Euclidean geometry]]. However, space''time'' as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not [[integrable systems|integrable]]. From this, one can deduce that spacetime is curved. The resulting [[Newton–Cartan theory]] is a geometric formulation of Newtonian gravity using only [[Covariance and contravariance of vectors#Informal usage|covariant]] concepts, i.e. a description which is valid in any desired coordinate system.<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.2}}, {{Harvnb|Havas|1964}}, {{Harvnb|Künzle|1972}}. The simple thought experiment in question was first described in {{Harvnb|Heckmann|Schücking|1959}}</ref> In this geometric description, [[tidal effect]]s—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.<ref>{{Harvnb|Ehlers|1973|pp=10f}}</ref>