The Einstein-Hilbert Action is a fundamental principle in general relativity that provides the simplest action from which the Einstein field equations can be derived, describing how matter and energy influence spacetime curvature. It is expressed as an integral of the Ricci scalar curvature over a manifold, weighted by the determinant of the metric tensor, and is central to understanding the dynamics of gravitational fields in the framework of modern physics.

Einstein-Hilbert Action

General relativity, formulated by Albert Einstein, is a theory of gravitation that describes gravity as the warping of spacetime by mass and energy, rather than as a force acting at a distance. It fundamentally changed our understanding of the universe, predicting phenomena such as the bending of light around massive objects and the existence of black holes.

general relativity

The Einstein Field Equations are a set of ten interrelated differential equations in Albert Einstein's general theory of relativity that describe how matter and energy in the universe influence the curvature of spacetime. These equations form the core of general relativity, allowing for the prediction of gravitational phenomena such as black holes, gravitational waves, and the expansion of the universe.

Einstein field equations

matter and energy

Spacetime curvature is a fundamental concept in Einstein's General Theory of Relativity, describing how matter and energy influence the geometry of the universe. It explains gravity not as a force, but as a result of objects following the curved paths in spacetime created by mass and energy distributions.

spacetime curvature

integral of the Ricci scalar curvature

A manifold is a topological space that locally resembles Euclidean space, allowing for the application of calculus and other mathematical tools. Manifolds are fundamental in mathematics and physics, providing the framework for understanding complex structures like curves, surfaces, and higher-dimensional spaces.

manifold

determinant of the metric tensor

A gravitational field is a region of space surrounding a mass where another mass experiences a force of attraction. This concept is central to understanding how masses interact in the universe, following the principles of Newton's law of universal gravitation and Einstein's general theory of relativity.

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