Editing Einstein–Hilbert action (section)

{{Short description|Concept in general relativity}}
{{General relativity sidebar}}
The '''Einstein–Hilbert action''' in [[general relativity]] is the [[action (physics)|action]] that yields the [[Einstein field equations]] through the [[stationary-action principle]]. With the [[Sign convention#Relativity|{{nowrap|(− + + +)}} metric signature]], the gravitational part of the action is given as<ref>{{cite book |first=Richard P. |last=Feynman |title=Feynman Lectures on Gravitation |url=https://archive.org/details/feynmanlectureso0000feyn_g4q1 |url-access=registration |publisher=Addison-Wesley |year=1995 |isbn=0-201-62734-5 |at=p. 136, eq. (10.1.2) }}</ref>

:<math>S = {1 \over 2\kappa} \int R \sqrt{-g} \, \mathrm{d}^4x,</math>

where <math>g=\det(g_{\mu\nu})</math> is the determinant of the [[metric tensor]] matrix, <math>R</math> is the [[Ricci scalar]], and <math>\kappa = 8\pi Gc^{-4}</math> is the [[Einstein gravitational constant]] (<math>G</math> is the [[gravitational constant]] and <math>c</math> is the [[speed of light]] in vacuum). If it converges, the integral is taken over the whole [[spacetime]]. If it does not converge, <math>S</math> is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the [[Euler–Lagrange equation]] of the Einstein–Hilbert action. The action was proposed<ref>{{Citation
|author-first=David
|author-last=Hilbert
|author-link =David Hilbert
|title = Die Grundlagen der Physik
|trans-title= Foundations of Physics
|journal = Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen – Mathematisch-Physikalische Klasse
|volume =3
|issue=
|pages =395–407
|year =1915
|language =German
|url =
|doi =
|jfm = 
}}</ref> by [[David Hilbert]] in 1915 as part of his application of the [[stationary action principle|variational principle]] to a combination of gravity and electromagnetism.<ref>{{Cite book |last=Mehra |first=Jagdish |chapter=Einstein, Hilbert, and the Theory of Gravitation |editor-last=Mehra |editor-first=Jagdish |title=The physicist's conception of nature |date=1987 |publisher=Reidel |isbn=978-90-277-2536-3 |edition=Reprint |location=Dordrecht}}</ref>{{rp|119}}