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In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation. As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light. Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves. Mathematical form The Einstein field equations (EFE) may be written in the form: G μ ν + Λ g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }} where G μ ν {\displaystyle G_{\mu \nu }} is the Einstein tensor, g μ ν {\displaystyle g_{\mu \nu }} is the metric tensor, T μ ν {\displaystyle T_{\mu \nu }} is the stress–energy tensor, Λ {\displaystyle \Lambda } is the cosmological constant and κ {\displaystyle \kappa } is the Einstein gravitational constant. The Einstein tensor is defined as G μ ν = R μ ν − 1 2 R g μ ν , {\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },} where Rμν is the Ricci curvature tensor, and R is the scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives. The Einstein gravitational constant is defined as κ = 8 π G c 4 ≈ 2.07665 ( 5 ) × 10 − 43 N − 1 , {\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665(5)\times 10^{-43}\,{\textrm {N}}^{-1},} or m/J , {\displaystyle {\textrm {m/J}},} where G is the Newtonian constant of gravitation and c is the speed of light in vacuum. The EFE can thus also be written as R μ ν − 1 2 R g μ ν + Λ g μ ν = κ T μ ν . {\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.} In standard units, each term on the left has units of 1/length2. The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime. These equations, together with the geodesic equation, which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity. The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom, which correspond to the freedom to choose a coordinate system. Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when Tμν is everywhere zero) define Einstein manifolds. The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor g μ ν {\displaystyle g_{\mu \nu }} , since both the Ricci tensor and scalar curvature depend on th.... Discover the Soner Efe popular books. Find the top 100 most popular Soner Efe books.