Linearized gravity

In the theory ofgeneral relativity,linearized gravityis the application ofperturbation theoryto themetric tensorthat describes the geometry ofspacetime.As a consequence, linearized gravity is an effective method for modeling the effects of gravity when thegravitational fieldis weak. The usage of linearized gravity is integral to the study ofgravitational wavesand weak-fieldgravitational lensing.

Weak-field approximation

TheEinstein field equation(EFE) describing the geometry ofspacetimeis given as (usingnatural units)

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

where$R_{\mu \nu }$is theRicci tensor,$R$is theRicci scalar,$T_{\mu \nu }$is theenergy–momentum tensor,and$g_{\mu \nu }$is thespacetimemetric tensorthat represents the solutions of the equation.

Although succinct when written out usingEinstein notation,hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric which render the prospect of findingexact solutionsimpractical in most systems. However, when describing particular systems for which thecurvatureof spacetime is small (meaning that terms in the EFE that arequadraticin$g_{\mu \nu }$do not significantly contribute to the equations of motion), one can model the solution of the field equations as being theMinkowski metric$\eta _{\mu \nu }$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 thetraceof 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 thed'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 orderpartial differential equationin 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 ofgauge symmetryis 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 describesis.

To capture this formally, the non-uniqueness of the perturbation$h_{\mu \nu }$is represented as being a consequence of the diverse collection ofdiffeomorphismson 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 thepullbackof$g_{\mu \nu }$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$\phi$,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$\epsilon \rightarrow 0$,

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

where${\mathcal {L}}_{\xi }$is theLie derivativealong the vector field$\xi _{\mu }$.

The Lie derivative works out to yield the finalgauge transformationof 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,tracelessand is referred to as thestrainsince it represents the amount by which the perturbationstretches and contracts measurements of space.In the context of studyinggravitational radiation,the strain is particularly useful when utilized with thetransverse 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:

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

with the additional property:

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

Synchronous gauge

Thesynchronous gaugesimplifies 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

Theharmonic gauge(also referred to as theLorenz gauge) 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,$\xi _{\mu }$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 thewave solutionsthat definegravitational radiation.

See also

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

Further reading

• Sean M. Carroll (2003).Spacetime and Geometry, an Introduction to General Relativity.Pearson.ISBN978-0805387322.