Linearized gravity

Linearized gravity is an approximation scheme in general relativity in which the nonlinear contributions from the spacetime metric are ignored. This allows the study of many problems to be simplified.

The method

In linearized gravity, the metric tensor of spacetime $g$ is treated as a sum of a solution of Einstein's equations (usually the Minkowski space) and a perturbation $h$.

$g\,=\eta +h$

where η is the nondynamical background metric that is 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.

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 linearized EFE is also a solution. The idea of 'ignoring the nonlinear part' is thus encapsulated in this linearization procedure.

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.

This approximation is also known as the weak-field approximation as it is only valid for tiny h's.

Weak-field approximation

In a weak-field 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)

$\delta _{\vec {\xi }}h=\delta _{\vec {\xi }}g-\delta _{\vec {\xi }}\eta ={\mathcal {L}}_{\vec {\xi }}g={\mathcal {L}}_{\vec {\xi }}\eta +{\mathcal {L}}_{\vec {\xi }}h=\left[\xi _{\nu ;\mu }+\xi _{\mu ;\nu }+\xi ^{\alpha }h_{\mu \nu ;\alpha }+\xi _{;\mu }^{\alpha }h_{\alpha \nu }+\xi _{;\nu }^{\alpha }h_{\mu \alpha }\right]dx^{\mu }\otimes dx^{\nu }$

Where ${\mathcal {L}}$ 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 the weak field limit, this gauge transformation simplifies to

$\delta _{\vec {\xi }}h_{\mu \nu }\approx \left({\mathcal {L}}_{\vec {\xi }}\eta \right)_{\mu \nu }=\xi _{\nu ;\mu }+\xi _{\mu ;\nu }$

The weak-field approximation is useful in finding the values of certain constants, for example in the Einstein field equations and in the Schwarzschild metric.

See also

• Parameterized post-Newtonian formalism
• Correspondence principle
• Gravitomagnetism
• Linearised Einstein field equations

References

• Stephani, Hans (1990). General Relativity: An Introduction to the Theory of the Gravitational Field,. Cambridge: Cambridge University Press. ISBN 0-521-37941-5.
• Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem (1965). Introduction to General Relativity. New York: McGraw-Hill. ISBN 0-07-000423-4.{{cite book}}: CS1 maint: multiple names: authors list (link)