### - Art Gallery -

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)

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

where $$R_{\mu \nu }$$ is the Ricci tensor, R is the Ricci scalar, $$T_{\mu \nu }$$ is the energy–momentum tensor, and $$g_{\mu \nu }$$ 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 $$g_{\mu \nu }$$ 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] $$\eta_{\mu\nu}$$ plus a small perturbation term $$h_{{\mu \nu }}.$$ In other words:

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

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

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

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

Together with the Ricci scalar,

$$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 \( 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 \( h_{{\mu \nu }}$$.

Gauge invariance

The process of decomposing the general spacetime $$g_{\mu \nu }$$ 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 $$h_{{\mu \nu }}$$. 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 $$h_{{\mu \nu }}$$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 $$h_{{\mu \nu }}$$ is represented as being a consequence of the diverse collection of diffeomorphisms on spacetime that leave $$h_{{\mu \nu }}$$sufficiently small. Therefore to continue, it is required that $$h_{{\mu \nu }}$$ 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 $$g_{\mu \nu }$$. With this, the perturbation metric may be defined as the difference between the pullback of $$g_{\mu \nu }$$ and the Minkowski metric:

$$h_{\mu \nu }=(\phi ^{*}g)_{\mu \nu }-\eta _{\mu \nu }.} The diffeomorphisms \( \phi$$ may thus be chosen such that $$|h_{\mu \nu }|\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 ϕ {\displaystyle \phi } \phi , a family of perturbations is given by

{\begin{aligned}h_{\mu \nu }^{(\epsilon )}&=[(\phi \circ \psi _{\epsilon })^{*}g]_{\mu \nu }-\eta _{\mu \nu }\\&=[\psi _{\epsilon }^{*}(\phi ^{*}g)]_{\mu \nu }-\eta _{\mu \nu }\\&=\psi _{\epsilon }^{*}(h+\eta )_{\mu \nu }-\eta _{\mu \nu }\\&=(\psi _{\epsilon }^{*}h)_{\mu \nu }+\epsilon \left[{\frac {(\psi _{\epsilon }^{*}\eta )_{\mu \nu }-\eta _{\mu \nu }}{\epsilon }}\right].\end{aligned}}}

Therefore, in the limit ϵ → 0 {\displaystyle \epsilon \rightarrow 0} \epsilon \rightarrow 0,

$$h_{\mu \nu }^{(\epsilon )}=h_{\mu \nu }+\epsilon {\mathcal {L}}_{\xi }\eta _{\mu \nu }}$$

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 $$h_{{\mu \nu }}:$$

$$h_{\mu \nu }^{(\epsilon )}=h_{\mu \nu }+\epsilon (\partial _{\mu }\xi _{\nu }+\partial _{\nu }\xi _{\mu }),}$$

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 $$h_{{\mu \nu }}$$ distorts measurements of length, it is useful to define the following spatial tensor:

$$s_{ij}=h_{ij}-{\frac {1}{3}}\delta ^{kl}h_{kl}\delta _{ij}}$$

(Note that the indices span only spatial components: $$i,j\in \{1,2,3\}}$$ ). Thus, by using $$s_{ij}$$ , the spatial components of the perturbation can be decomposed as

$$h_{ij}=s_{ij}-\Psi \delta _{ij}}$$

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

$$\nabla ^{2}\xi ^{j}+{\frac {1}{3}}\partial _{j}\partial _{i}\xi ^{i}=-\partial _{i}s^{ij},}$$

then choosing the time component $$\xi ^{0}}$$ to satisfy

$$\nabla ^{2}\xi ^{0}=\partial _{i}h_{0i}+\partial _{0}\partial _{i}\xi ^{i}.}$$

After performing the gauge transformation using the formula in the previous section, the strain becomes spatially transverse:

$$\partial _{i}s_{(\epsilon )}^{ij}=0,}$$

$$\partial _{i}h_{(\epsilon )}^{0i}=0.}$$

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 $$h_{\mu \nu }^{(\epsilon )}}$$ are zero, namely

$$h_{0\nu }^{(\epsilon )}=0.}$$

This can be achieved by requiring the time component of $$\xi ^{\mu }$$ to satisfy

$$\partial _{0}\xi ^{0}=-h_{00}}$$

and requiring the spatial components to satisfy

$$\partial _{0}\xi ^{i}=\partial _{i}\xi ^{0}-h_{0i}.}$$

Harmonic gauge

The harmonic gauge (also referred to as the Lorenz gauge[note 2]) is selected whenever it is necessary to reduce the linearized field equations as much as possible. This can be done if the condition

$$\partial _{\mu }h_{\nu }^{\mu }={\frac {1}{2}}\partial _{\nu }h}$$

is true. To achieve this, ξ$$\xi _{\mu }}$$ is required to satisfy the relation

$$\square \xi _{\mu }=-\partial _{\nu }h_{\mu }^{\nu }+{\frac {1}{2}}\partial _{\mu }h.}$$

Consequently, by using the harmonic gauge, the Einstein tensor $$G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }}$$ reduces to

$$G_{\mu \nu }=-{\frac {1}{2}}\square \left(h_{\mu \nu }^{(\epsilon )}-{\frac {1}{2}}h^{(\epsilon )}\eta _{\mu \nu }\right).}$$

Therefore, by writing it in terms of a "trace-reversed" metric, $${\bar {h}}_{\mu \nu }^{(\epsilon )}=h_{\mu \nu }^{(\epsilon )}-{\frac {1}{2}}h^{(\epsilon )}\eta _{\mu \nu }}$$, the linearized field equations reduce to

$$\square {\bar {h}}_{\mu \nu }^{(\epsilon )}=-16\pi GT_{\mu \nu }.}$$

Which can be solved exactly using the wave solutions that define gravitational radiation.

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

Notes

This 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.

Not to be confused with Lorentz.

References
Sean M. Carroll (2003). Spacetime and Geometry, an Introduction to General Relativity. Pearson. ISBN 978-0805387322.

Relativity
Special relativity
Background

Principle of relativity (Galilean relativity Galilean transformation) Special relativity Doubly special relativity

Fundamental
concepts

Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Relativistic mass

Formulation

Lorentz transformation

Phenomena

Time dilation Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox

Spacetime

Light cone World line Minkowski diagram Biquaternions Minkowski space

General relativity
Background

Introduction Mathematical formulation

Fundamental
concepts

Equivalence principle Riemannian geometry Penrose diagram Geodesics Mach's principle

Formulation

ADM formalism BSSN formalism Einstein field equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation

Phenomena

Gravitational waves: astronomy detectors (LIGO and collaboration Virgo LISA Pathfinder GEO) Hulse–Taylor binary

Other tests: precession of Mercury lensing redshift Shapiro delay frame-dragging / geodetic effect (Lense–Thirring precession) pulsar timing arrays

theories

Solutions

Cosmological: Friedmann–Lemaître–Robertson–Walker (Friedmann equations) Kasner BKL singularity Gödel Milne

Spherical: Schwarzschild (interior Tolman–Oppenheimer–Volkoff equation) Reissner–Nordström Lemaître–Tolman

Axisymmetric: Kerr (Kerr–Newman) Weyl−Lewis−Papapetrou Taub–NUT van Stockum dust discs

Others: pp-wave Ozsváth–Schücking metric

Scientists

Poincaré Lorentz Einstein Hilbert Schwarzschild de Sitter Weyl Eddington Friedmann Lemaître Milne Robertson Chandrasekhar Zwicky Wheeler Choquet-Bruhat Kerr Zel'dovich Novikov Ehlers Geroch Penrose Hawking Taylor Hulse Bondi Misner Yau Thorne Weiss others

Categories
Theory of relativity

Physics Encyclopedia

World

Index