Editing General relativity (section)

=== Relativistic generalization ===
[[File:Light cone.svg|thumb|left|upright|[[Light cone]]]]
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a [[limiting case (philosophy of science)|limiting case]] of (special) relativistic mechanics.<ref>Good introductions are, in order of increasing presupposed knowledge of mathematics, {{Harvnb|Giulini|2005}}, {{Harvnb|Mermin|2005}}, and {{Harvnb|Rindler|1991}}; for accounts of precision experiments, cf. part IV of {{Harvnb|Ehlers|Lämmerzahl|2006}}</ref> In the language of [[symmetry]]: where gravity can be neglected, physics is [[Lorentz invariance|Lorentz invariant]] as in special relativity rather than [[Galilean invariance|Galilei invariant]] as in classical mechanics. (The defining symmetry of special relativity is the [[Poincaré group]], which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the [[speed of light]], and with high-energy phenomena.<ref>An in-depth comparison between the two symmetry groups can be found in {{Harvnb|Giulini|2006}}</ref>

With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each [[event (relativity)|event]] {{math|A}}, there is a set of events that can, in principle, either influence or be influenced by {{math|A}} via signals or interactions that do not need to travel faster than light (such as event {{math|B}} in the image), and a set of events for which such an influence is impossible (such as event {{math|C}} in the image). These sets are [[frame of reference|observer]]-independent.<ref>{{Harvnb|Rindler|1991|loc=sec. 22}}, {{Harvnb|Synge|1972|loc=ch. 1 and 2}}</ref> In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a [[conformal structure]]<ref>{{Harvnb|Ehlers|1973|loc=sec. 2.3}}</ref> or conformal geometry.

Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global [[inertial frame]]s. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight [[time-like]] lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.4}}, {{Harvnb|Schutz|1985|loc=sec. 5.1}}</ref>

A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of [[preferred frame]]s. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. [[#Gravitational time dilation and frequency shift|below]]). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.<ref>{{Harvnb|Ehlers|1973|pp=17ff}}; a derivation can be found in {{Harvnb|Mermin|2005|loc=ch. 12}}. For the experimental evidence, cf. the section [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]], below</ref> The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the [[Equivalence Principle#The Einstein equivalence principle|Einstein equivalence principle]], a crucial guiding principle for generalizing special-relativistic physics to include gravity.<ref>{{Harvnb|Rindler|2001|loc=sec. 1.13}}; for an elementary account, see {{Harvnb|Wheeler|1990|loc=ch. 2}}; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. {{Harvnb|Norton|1985}}</ref>

The same experimental data shows that time as measured by clocks in a gravitational field—[[proper time]], to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the [[Minkowski metric]]. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The [[metric tensor (general relativity)|metric tensor]] that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or [[pseudo-Riemannian]] metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the [[Levi-Civita connection]], and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable [[Local reference frame|locally inertial coordinates]], the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.4}} for the experimental evidence, see once more section [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]]. Choosing a different connection with non-zero [[torsion tensor|torsion]] leads to a modified theory known as [[Einstein–Cartan theory]]</ref>