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Concept
Linearized gravity
Summary
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.
The Einstein field equation (EFE) describing the geometry of spacetime is given as (using natural units)
where is the Ricci tensor, is the Ricci scalar, is the energy–momentum tensor, and is the spacetime metric tensor that represent 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 do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the Minkowski metric plus a small perturbation term . In other words:
In this regime, substituting the general metric for this perturbative approximation results in a simplified expression for the Ricci tensor:
where is the trace of the perturbation, denotes the partial derivative with respect to the coordinate of spacetime, and is the d'Alembert operator.
Together with the Ricci scalar,
the left side of the field equation reduces to
and thus the EFE is reduced to a linear, second order partial differential equation in terms of .
The process of decomposing the general spacetime 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 . In order to capture this phenomenon, the application of gauge symmetry is introduced.
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