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Encyclopedia > Linearized gravity
It has been suggested that Weak-field approximation be merged into this article or section. (Discuss)

Linearized gravity is an approximation scheme in general relativity in which the nonlinear contributions from the spacetime metric are ignored. This simplifies the study of many problems. Image File history File links Please see the file description page for further information. ... The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ...

In linearised gravity, the metric tensor of spacetime g is treated as a sum of a solution of Einstein's equations (usually the Minkowksi metric) and a perturbation h. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ...

where η is the nondynamical background metric that we are perturbing about and h represents the deviation of the true metric (g) from flat spacetime.

The perturbation is treated using the methods of perturbation theory. The adjective "linearized" means that all terms of order higher than one (quadratic in h, cubic in h etc...) in the perturbation are ignored. This article describes perturbation theory as a general mathematical method. ... f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ... Cubic can mean several things: cubic polynomial, a polynomial with a degree of at most three. ...

Applications

The Einstein field equations, being nonlinear in the metric, are difficult to solve exactly and the above perturbation scheme allows one to obtain linearised Einstein field equations. These equations are linear in the metric and the sum of two solutions of the linearised EFE is also a solution. The idea of 'ignoring the nonlinear part' is thus encapsulated in this linearisation procedure. For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energy-momentum tensor (by the field equations). ... This article is in need of attention from an expert on the subject. ...

The method is used to derive the Newtonian limit, including the first corrections, much like for a derivation of the existence of gravitational waves that led, after quantization, to gravitons. This is why the conceptual approach of linearized gravity is the canonical one in particle physics, string theory, and more generally quantum field theory where classical (bosonic) fields are expressed as coherent states of particles. In physics, gravitational radiation is energy that is transmitted through waves in the gravitational field of space-time, according to Albert Einsteins theory of general relativity: The Einstein field equations imply that any accelerated mass radiates energy this way, in the same way as the Maxwell equations that any... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ... In physics, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. ... Particles erupt from the collision point of two relativistic (100GeV) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of particle physics. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. ...

This approximation is also known as the weak-field approximation as it is only valid for tiny h's. The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ...

In this approximation, the gauge symmetry is associated with diffeomorphisms with small "displacements" (diffeomorphisms with huge displacements obviously violate the weak field approximation), which has the exact form (for infinitesimal transformations) In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

where is the Lie derivative and we used the fact that η doesn't transform (by definition). Note that we are raising and lowering the indices with respect to η and not g and taking the covariant derivatives (Levi-Civita connection) with respect to η. This is the standard practice in linearized gravity. The way of thinking in linearized gravity is this: the background metric η IS the metric and h is a field propagating over the spacetime with this metric. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...

In the weak field limit, this gauge transformation simplifies to Results from FactBites:

 Report Dynamical meteorology, DM (1043 words) On the non-linear behaviour of gravity modes in an equatorial á-plane shallow water-model, with respect to linear and non-linear normal mode initialization. The inclusion of non-zero tendencies of the gravity modes in the normal mode procedure. Theory and observations of gravity waves in an inversion layer.
 Einstein's field equation - Wikipedia, the free encyclopedia (1086 words) For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields (i.e. Einstein's equation reduces to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. Mathematicians usually refer to manifolds with a vanishing Ricci tensor as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.
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