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${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing.

Weak-field approximation

The Einstein field equation (EFE) describing the geometry of spacetime is given as (using natural units)

${\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }=8\pi GT_{\mu \nu }}$

where ${\displaystyle R_{\mu \nu }}$ is the Ricci tensor, ${\displaystyle R}$ is the Ricci scalar, ${\displaystyle T_{\mu \nu }}$ is the energy–momentum tensor, and ${\displaystyle g_{\mu \nu }}$ is the spacetime metric tensor that represents the solutions of the equation.

Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric which render the prospect of finding exact solutions impractical in most systems. However, when describing particular systems for which the curvature of spacetime is small (meaning that terms in the EFE that are quadratic in ${\displaystyle g_{\mu \nu }}$ do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the Minkowski metric^{[note 1]} ${\displaystyle \eta _{\mu \nu }}$ plus a small perturbation term ${\displaystyle h_{\mu \nu }}$. In other words:

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu },\qquad |h_{\mu \nu }|\ll 1.}$

In this regime, substituting the general metric ${\displaystyle g_{\mu \nu }}$ for this perturbative approximation results in a simplified expression for the Ricci tensor:

${\displaystyle R_{\mu \nu }={\frac {1}{2}}(\partial _{\sigma }\partial _{\mu }h_{\nu }^{\sigma }+\partial _{\sigma }\partial _{\nu }h_{\mu }^{\sigma }-\partial _{\mu }\partial _{\nu }h-\square h_{\mu \nu }),}$

where ${\displaystyle h=\eta ^{\mu \nu }h_{\mu \nu }}$ is the trace of the perturbation, ${\displaystyle \partial _{\mu }}$ denotes the partial derivative with respect to the ${\displaystyle x^{\mu }}$ coordinate of spacetime, and ${\displaystyle \square =\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }}$ is the d'Alembert operator.

Together with the Ricci scalar,

${\displaystyle R=\eta _{\mu \nu }R^{\mu \nu }=\partial _{\mu }\partial _{\nu }h^{\mu \nu }-\square h,}$

the left side of the field equation reduces to

${\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }={\frac {1}{2}}(\partial _{\sigma }\partial _{\mu }h_{\nu }^{\sigma }+\partial _{\sigma }\partial _{\nu }h_{\mu }^{\sigma }-\partial _{\mu }\partial _{\nu }h-\square h_{\mu \nu }-\eta _{\mu \nu }\partial _{\rho }\partial _{\lambda }h^{\rho \lambda }+\eta _{\mu \nu }\square h).}$

and thus the EFE is reduced to a linear, second order partial differential equation in terms of ${\displaystyle h_{\mu \nu }}$.

Gauge invariance

The process of decomposing the general spacetime ${\displaystyle g_{\mu \nu }}$ into the Minkowski metric plus a perturbation term is not unique. This is due to the fact that different choices for coordinates may give different forms for ${\displaystyle h_{\mu \nu }}$. In order to capture this phenomenon, the application of gauge symmetry is introduced.

Gauge symmetries are a mathematical device for describing a system that does not change when the underlying coordinate system is "shifted" by an infinitesimal amount. So although the perturbation metric ${\displaystyle h_{\mu \nu }}$ is not consistently defined between different coordinate systems, the overall system which it describes is.

To capture this formally, the non-uniqueness of the perturbation ${\displaystyle h_{\mu \nu }}$ is represented as being a consequence of the diverse collection of diffeomorphisms on spacetime that leave ${\displaystyle h_{}}$Zdroj:https://en.wikipedia.org?pojem=Weak-field_approximation
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