Editing General relativity (section)

=== Quantum gravity ===
{{Main|Quantum gravity}}
{{See also|String theory|Canonical general relativity|Loop quantum gravity|Causal dynamical triangulation|Causal sets}}
[[File:Calabi yau.jpg|left|thumb|upright|Projection of a [[Calabi–Yau manifold]], one of the ways of [[compactification (physics)|compactifying]] the extra dimensions posited by string theory]]

The demand for consistency between a quantum description of matter and a geometric description of spacetime,<ref>Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. {{Harvnb|Carlip|2001|loc=sec. 2}}</ref> as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.<ref>{{Harvnb|Schutz|2003|p=407}}</ref> Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.<ref name="Hamber 2009">{{Harvnb|Hamber|2009}}</ref><ref>A timeline and overview can be found in {{Harvnb|Rovelli|2000}}</ref>

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.<ref>{{Harvnb|'t Hooft|Veltman|1974}}</ref> Some have argued that at low energies, this approach proves successful, in that it results in an acceptable [[effective field theory|effective (quantum) field theory]] of gravity.<ref>{{Harvnb|Donoghue|1995}}</ref> At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative [[non-renormalizable|non-renormalizability]]").<ref>In particular, a perturbative technique known as [[renormalization]], an integral part of deriving predictions which take into account higher-energy contributions, cf. {{Harvnb|Weinberg|1996|loc=ch. 17, 18}}, fails in this case; cf. {{Harvnb|Veltman|1975}}, {{Harvnb|Goroff|Sagnotti|1985}}; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see {{Harvnb|Hamber|2009}}</ref>

[[File:Spin network.svg|thumb|upright|Simple [[spin network]] of the type used in loop quantum gravity]]
One attempt to overcome these limitations is [[string theory]], a quantum theory not of [[point particle]]s, but of minute one-dimensional extended objects.<ref>An accessible introduction at the undergraduate level can be found in {{Harvnb|Zwiebach|2004}}; more complete overviews can be found in {{Harvnb|Polchinski|1998a}} and {{Harvnb|Polchinski|1998b}}</ref> The theory promises to be a [[theory of everything|unified description]] of all particles and interactions, including gravity;<ref>At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different [[normal mode|modes]] of oscillation of one and the same type of fundamental string appear as particles with different ([[Electric charge|electric]] and other) [[Charge (physics)|charges]], e.g. {{Harvnb|Ibanez|2000}}. The theory is successful in that one mode will always correspond to a [[graviton]], the [[messenger particle]] of gravity, e.g. {{Harvnb|Green|Schwarz|Witten|1987|loc=sec. 2.3, 5.3}}</ref> the price to pay is unusual features such as six [[Superstring theory#Extra dimensions|extra dimensions]] of space in addition to the usual three.<ref>{{Harvnb|Green|Schwarz|Witten|1987|loc=sec. 4.2}}</ref> In what is called the [[second superstring revolution]], it was conjectured that both string theory and a unification of general relativity and [[supersymmetry]] known as [[supergravity]]<ref>{{Harvnb|Weinberg|2000|loc=ch. 31}}</ref> form part of a hypothesized eleven-dimensional model known as [[M-theory]], which would constitute a uniquely defined and consistent theory of quantum gravity.<ref>{{Harvnb|Townsend|1996}}, {{Harvnb|Duff|1996}}</ref>

Another approach starts with the [[canonical quantization]] procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. [[#Evolution equations|evolution equations]] above), the result is the [[Wheeler–deWitt equation]] (an analogue of the [[Schrödinger equation]]) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.<ref>{{Harvnb|Kuchař|1973|loc=sec. 3}}</ref> However, with the introduction of what are now known as [[Ashtekar variables]],<ref>These variables represent geometric gravity using mathematical analogues of [[electric field|electric]] and [[magnetic field]]s; cf. {{Harvnb|Ashtekar|1986}}, {{Harvnb|Ashtekar|1987}}</ref> this leads to a promising model known as [[loop quantum gravity]]. Space is represented by a web-like structure called a [[spin network]], evolving over time in discrete steps.<ref>For a review, see {{Harvnb|Thiemann|2007}}; more extensive accounts can be found in {{Harvnb|Rovelli|1998}}, {{Harvnb|Ashtekar|Lewandowski|2004}} as well as in the lecture notes {{Harvnb|Thiemann|2003}}</ref>

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,<ref>{{Harvnb|Isham|1994}}, {{Harvnb|Sorkin|1997}}</ref> there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman [[Path integral formulation|Path Integral]] approach and [[Regge calculus]],<ref name="Hamber 2009" /> [[Causal dynamical triangulation|dynamical triangulations]],<ref>{{Harvnb|Loll|1998}}</ref> [[causal sets]],<ref>{{Harvnb|Sorkin|2005}}</ref> twistor models<ref>{{Harvnb|Penrose|2004|loc=ch. 33 and refs therein}}</ref> or the path integral based models of [[quantum cosmology]].<ref>{{Harvnb|Hawking|1987}}</ref>

[[File:LIGO measurement of gravitational waves.svg|thumb|Observation of gravitational waves from binary black hole merger GW150914]]
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.<ref>{{Harvnb|Ashtekar|2007}}, {{Harvnb|Schwarz|2007}}</ref>