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The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the principle of least action. With the , the gravitational part of the action is given as〔Richard P. Feynman, Feynman Lectures on Gravitation, Addison-Wesley, 1995, ISBN 0-201-62734-5, p. 136, eq. (10.1.2)〕 : where is the determinant of the metric tensor matrix, is the Ricci scalar, and , where is the gravitational constant and is the speed of light in vacuum. The integral is taken over the whole spacetime if it converges. If it does not converge, 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 first proposed by David Hilbert in 1915. == Discussion == The derivation of equations from an action has several advantages. First of all, it allows for easy unification of general relativity with other classical field theories (such as Maxwell theory), which are also formulated in terms of an action. In the process the derivation from an action identifies a natural candidate for the source term coupling the metric to matter fields. Moreover, the action allows for the easy identification of conserved quantities through Noether's theorem by studying symmetries of the action. In general relativity, the action is usually assumed to be a functional of the metric (and matter fields), and the connection is given by the Levi-Civita connection. The Palatini formulation of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integral spin. The Einstein equations in the presence of matter are given by adding the matter action to the Hilbert–Einstein action. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Einstein–Hilbert action」の詳細全文を読む スポンサード リンク
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