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{{Short description|Linear perturbations to solutions of nonlinear Einstein field equations}}{{General relativity sidebar |equations}}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) where R_{munu} is the Ricci tensor, R is the Ricci scalar, T_{munu} is the energy–momentum tensor, and g_{munu} 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 g_{munu} do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the Minkowski metricThis is assuming that the background spacetime is flat. Perturbation theory applied in spacetime that is already curved can work just as well by replacing this term with the metric representing the curved background. eta_{munu} plus a small perturbation term h_{munu}. In other words: In this regime, substituting the general metric g_{munu} for this perturbative approximation results in a simplified expression for the Ricci tensor: where h = eta^{munu}h_{munu} is the trace of the perturbation, partial_mu denotes the partial derivative with respect to the x^mu coordinate of spacetime, and square = eta^{munu}partial_mupartial_nu 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 h_{munu}.

Gauge invariance

The process of decomposing the general spacetime g_{munu} 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 h_{munu}. 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 h_{munu} 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 h_{munu} is represented as being a consequence of the diverse collection of diffeomorphisms on spacetime that leave h_{munu} sufficiently small. Therefore to continue, it is required that h_{munu} be defined in terms of a general set of diffeomorphisms then select the subset of these that preserve the small scale that is required by the weak-field approximation. One may thus define phi to denote an arbitrary diffeomorphism that maps the flat Minkowski spacetime to the more general spacetime represented by the metric g_{munu}. With this, the perturbation metric may be defined as the difference between the pullback of g_{munu} and the Minkowski metric: The diffeomorphisms phi may thus be chosen such that |h_{munu}| ll 1.Given then a vector field xi^mu defined on the flat, background spacetime, an additional family of diffeomorphisms psi_epsilon may be defined as those generated by xi^mu and parameterized by epsilon > 0. These new diffeomorphisms will be used to represent the coordinate transformations for "infinitesimal shifts" as discussed above. Together with phi, a family of perturbations is given by h^{(epsilon)}_{munu} &= [(phicircpsi_epsilon)^*g]_{munu} - eta_{munu} &= [psi^*_epsilon(phi^*g)]_{munu} - eta_{munu} &= psi^*_epsilon(h + eta)_{munu} - eta_{munu} &= (psi^*_epsilon h)_{munu} + epsilonleft[frac{(psi^*_epsiloneta)_{munu} - eta_{munu}}{epsilon}right].end{align}Therefore, in the limit epsilonrightarrow 0, where mathcal{L}_xi is the Lie derivative along the vector field xi_mu.The Lie derivative works out to yield the final gauge transformation of the perturbation metric h_{munu}: which precisely define the set of perturbation metrics that describe the same physical system. In other words, it characterizes the gauge symmetry of the linearized field equations.

Choice of gauge

By exploiting gauge invariance, certain properties of the perturbation metric can be guaranteed by choosing a suitable vector field xi^mu.

Transverse gauge

To study how the perturbation h_{munu} distorts measurements of length, it is useful to define the following spatial tensor: (Note that the indices span only spatial components: i,jin{1,2,3}). Thus, by using s_{ij}, the spatial components of the perturbation can be decomposed as where Psi = frac{1}{3}delta^{kl}h_{kl}.The tensor s_{ij} is, by construction, traceless and is referred to as the strain since it represents the amount by which the perturbation stretches and contracts measurements of space. In the context of studying gravitational radiation, the strain is particularly useful when utilized with the transverse gauge. This gauge is defined by choosing the spatial components of xi^mu to satisfy the relation then choosing the time component xi^0 to satisfy After performing the gauge transformation using the formula in the previous section, the strain becomes spatially transverse: with the additional property:

Synchronous gauge

The synchronous gauge simplifies the perturbation metric by requiring that the metric not distort measurements of time. More precisely, the synchronous gauge is chosen such that the non-spatial components of h^{(epsilon)}_{munu} are zero, namely This can be achieved by requiring the time component of xi^mu to satisfy and requiring the spatial components to satisfy

Harmonic gauge

The harmonic gauge (also referred to as the Lorenz gaugeNot to be confused with Lorentz.) is selected whenever it is necessary to reduce the linearized field equations as much as possible. This can be done if the condition is true. To achieve this, xi_mu is required to satisfy the relation Consequently, by using the harmonic gauge, the Einstein tensor G_{munu} = R_{munu} - frac{1}{2}Rg_{munu} reduces to Therefore, by writing it in terms of a "trace-reversed" metric, bar{h}^{(epsilon)}_{munu} = h^{(epsilon)}_{munu} - frac{1}{2}h^{(epsilon)}eta_{munu}, the linearized field equations reduce to Which can be solved exactly using the wave solutions that define gravitational radiation.

See also

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• Correspondence principle
• Gravitoelectromagnetism
• Lanczos tensor
• Parameterized post-Newtonian formalism
• Post-Newtonian expansion
• Quasinormal mode

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Notes

Further reading

• BOOK, Sean M. Carroll, Spacetime and Geometry, an Introduction to General Relativity, 2003, Pearson, 978-0805387322,

{{Theories of gravitation}}{{Relativity}}