Editing General relativity (section)

== Definition and basic applications ==
{{See also|Mathematics of general relativity|Physical theories modified by general relativity}}

The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.

=== Definition and basic properties ===
General relativity is a [[metric (general relativity)|metric]] theory of gravitation. At its core are [[Einstein's equations]], which describe the relation between the geometry of a four-dimensional [[pseudo-Riemannian manifold]] representing spacetime, and the [[Stress–energy tensor|energy–momentum]] contained in that spacetime.<ref>{{Harvnb|Wald|1984|loc=ch. 4}}, {{Harvnb|Weinberg|1972|loc=ch. 7}} or, in fact, any other textbook on general relativity</ref> Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as [[free-fall]], orbital motion, and [[spacecraft]] [[Trajectory|trajectories]]), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.<ref>At least approximately, cf. {{Harvnb|Poisson|2004a}}</ref> The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist [[John Archibald Wheeler]], spacetime tells matter how to move; matter tells spacetime how to curve.<ref>{{Harvnb|Wheeler|1990|p=xi}}</ref>

While general relativity replaces the [[scalar field|scalar]] gravitational potential of classical physics by a symmetric [[Tensor#As multidimensional arrays|rank]]-two [[tensor]], the latter reduces to the former in certain [[Correspondence principle#Other scientific theories|limiting cases]]. For [[weak-field approximation|weak gravitational fields]] and [[slow-motion approximation|slow speed]] relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.<ref>{{Harvnb|Wald|1984|loc=sec. 4.4}}</ref>

As it is constructed using tensors, general relativity exhibits [[general covariance]]: its laws—and further laws formulated within the general relativistic framework—take on the same form in all [[coordinate system]]s.<ref>{{Harvnb|Wald|1984|loc=sec. 4.1}}</ref> Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is [[Background independence|background independent]]. It thus satisfies a more stringent [[general principle of relativity]], namely that the [[Physical law|laws of physics]] are the same for all observers.<ref>For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see {{Harvnb|Giulini|2007}}</ref> [[Local spacetime structure|Locally]], as expressed in the equivalence principle, spacetime is [[Minkowski space|Minkowskian]], and the laws of physics exhibit [[local Lorentz invariance]].<ref>section 5 in ch. 12 of {{Harvnb|Weinberg|1972}}</ref>

=== Model-building ===
The core concept of general-relativistic model-building is that of a [[solutions of the Einstein field equations|solution of Einstein's equations]]. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-[[Riemannian manifold]] (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.<ref>Introductory chapters of {{Harvnb|Stephani|Kramer|MacCallum|Hoenselaers|2003}}</ref>

Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.<ref>A review showing Einstein's equation in the broader context of other PDEs with physical significance is {{Harvnb|Geroch|1996}}</ref> Nevertheless, a number of [[exact solutions in general relativity|exact solutions]] are known, although only a few have direct physical applications.<ref>For background information and a list of solutions, cf. {{Harvnb|Stephani|Kramer|MacCallum|Hoenselaers|2003}}; a more recent review can be found in {{Harvnb|MacCallum|2006}}</ref> The best-known exact solutions, and also those most interesting from a physics point of view, are the [[Schwarzschild solution]], the [[Reissner–Nordström solution]] and the [[Kerr metric]], each corresponding to a certain type of black hole in an otherwise empty universe,<ref>{{Harvnb|Chandrasekhar|1983|loc=ch. 3,5,6}}</ref> and the [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker]] and [[de Sitter universe]]s, each describing an expanding cosmos.<ref>{{Harvnb|Narlikar|1993|loc=ch. 4, sec. 3.3}}</ref> Exact solutions of great theoretical interest include the [[Gödel metric|Gödel universe]] (which opens up the intriguing possibility of [[time travel]] in curved spacetimes), the [[Taub–NUT space|Taub–NUT solution]] (a model universe that is [[Homogeneity (physics)|homogeneous]], but [[anisotropic]]), and [[anti-de Sitter space]] (which has recently come to prominence in the context of what is called the [[Maldacena conjecture]]).<ref>Brief descriptions of these and further interesting solutions can be found in {{Harvnb|Hawking|Ellis|1973|loc=ch. 5}}</ref>

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by [[numerical integration]] on a computer, or by considering small perturbations of exact solutions. In the field of [[numerical relativity]], powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.<ref>{{Harvnb|Lehner|2002}}</ref> In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as [[naked singularity|naked singularities]]. Approximate solutions may also be found by [[perturbation theory|perturbation theories]] such as [[linearized gravity]]<ref>For instance {{Harvnb|Wald|1984|loc=sec. 4.4}}</ref> and its generalization, the [[post-Newtonian expansion]], both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.<ref>{{Harvnb|Will|1993|loc=sec. 4.1 and 4.2}}</ref> An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.<ref>{{Harvnb|Will|2006|loc=sec. 3.2}}, {{Harvnb|Will|1993|loc=ch. 4}}</ref>