Einstein–Hilbert action

Einstein–Hilbert action
Name	Einstein–Hilbert action
Field	Albert Einstein General relativity Differential geometry
Known for	Action principle for General relativity

Einstein–Hilbert action The Einstein–Hilbert action is the variational functional whose stationary points yield the Einstein field equations of General relativity under the Principle of least action. Introduced in the context of the work of Albert Einstein and David Hilbert during the period of 1915–1916, it provides a compact formulation using the Ricci curvature tensor, the scalar curvature, and the metric tensor on a four-dimensional Lorentzian manifold. The action underpins classical analyses in Cosmology, contributes to path integral approaches to Quantum gravity, and connects to extensions such as Lanczos–Lovelock gravity and f(R) gravity.

Introduction

The Einstein–Hilbert action encapsulates dynamics for the metric tensor on a spacetime manifold by integrating a scalar density formed from the Riemann curvature tensor, the Ricci tensor, and the scalar curvature over a four-dimensional volume element defined by the metric determinant. It plays a central role in derivations involving the Einstein field equations, variational formulations promoted by the Calculus of variations tradition in the works of figures like Joseph-Louis Lagrange, William Rowan Hamilton, and later contributors such as Noether's theorem authors. The formulation interfaces with methods developed in Differential geometry and influences modern pursuits in String theory, Loop quantum gravity, and semiclassical analyses in Hawking radiation studies.

Definition and mathematical formulation

The Einstein–Hilbert integrand is the scalar Lagrangian density L = (c^4/16πG) R √(-g), where R is the scalar curvature constructed from the Riemann curvature tensor via contraction to the Ricci tensor, g is the determinant of the metric tensor, c is the speed of light and G is Newton's gravitational constant. The action S[g] = ∫_M L d^4x is taken over a spacetime manifold M, typically a four-dimensional Lorentzian manifold with orientation and appropriate differentiable structure as in work influenced by Bernhard Riemann and Élie Cartan. One often includes a matter action S_matter coupling fields such as the electromagnetic field, Dirac spinor, Klein–Gordon field, or Yang–Mills theory gauge fields through covariant constructions respecting General covariance as articulated by Albert Einstein and critiqued by contemporaries including Erwin Schrödinger.

Derivation of Einstein field equations

Varying the action with respect to the inverse metric yields the vacuum Einstein field equations with the energy–momentum tensor T_{μν} entering when S_matter is included. The variation invokes identities related to the Bianchi identities and integration by parts on manifolds, drawing on techniques from Tensor calculus and the Levi-Civita connection. The resulting field equations relate the Einstein tensor G_{μν} = R_{μν} - (1/2) R g_{μν} to (8πG/c^4) T_{μν}, a relation central to results such as the Schwarzschild solution, the Friedmann–Lemaître–Robertson–Walker metric used in Big Bang cosmology, and the Kerr metric describing rotating compact objects studied in Black hole thermodynamics.

Variations, boundary terms, and the Gibbons–Hawking–York term

The naive metric variation produces boundary terms requiring careful treatment on manifolds with boundary, leading to the addition of a boundary term now known as the Gibbons–Hawking–York term to ensure a well-posed variational principle for fixed boundary metric. This modification parallels analyses in Dirichlet boundary conditions in the Calculus of variations and appears in path integral formulations by researchers in Stephen Hawking’s school and James Hartle. The boundary term involves the extrinsic curvature K of the boundary and interacts with junction conditions used in models like the Israel junction conditions for thin shells and braneworld scenarios inspired by Randall–Sundrum models.

Extensions and generalizations

Generalizations include higher-derivative actions such as f(R) gravity, Gauss–Bonnet gravity, and Lanczos–Lovelock gravity which add functions of curvature invariants motivated by low-energy limits of String theory and quantum corrections computed in Effective field theory approaches by authors like John Donoghue. Couplings to scalar fields produce scalar–tensor theories such as the Brans–Dicke theory; inclusion of torsion leads to formulations like the Einstein–Cartan theory influenced by Élie Cartan. Discrete and nonperturbative generalizations appear in Regge calculus and Loop quantum gravity where the continuum Einstein–Hilbert action guides spin network and spin foam amplitudes tied to research by Carlo Rovelli and Lee Smolin.

Applications in classical and quantum gravity

Classically, the action underlies derivations of solutions including the Schwarzschild solution, Reissner–Nordström metric, and Kerr–Newman metric, informs studies of gravitational waves confirmed by LIGO and VIRGO collaborations, and structures perturbation theory for cosmological models like ΛCDM. Quantumly, it provides the starting point for perturbative quantization attempts, the background field method used in Renormalization group analyses, and semiclassical path integral methods in Euclidean quantum gravity developed in works by Stephen Hawking and Gary Gibbons. The action is central to black hole entropy calculations stemming from the Bekenstein–Hawking entropy relation and to holographic correspondences exemplified by the AdS/CFT correspondence developed by Juan Maldacena.

Historical context and attribution dispute

The formulation emerged contemporaneously in publications by David Hilbert and Albert Einstein during November 1915; subsequent historical scholarship has examined priority and influence among Hilbert, Einstein, and collaborators including Felix Klein. Archival work and correspondence involving institutions such as the Kaiser Wilhelm Society and analyses by historians like Jürgen Renn and Dieter Hoffmann explore the interplay of ideas across exchanges with figures like Emmy Noether whose theorem linked symmetries to conservation laws. Debates also reference editorial practices of journals such as the Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften and later documentary editions; modern consensus treats the Einstein–Hilbert action as a synthesis arising from overlapping contributions in the milieu of early twentieth-century mathematical physics.

Category:General relativity