View source for Linearized gravity

{{Expert-subject|Physics|date=February 2009}}
{{General relativity|cTopic=Equations}}
'''Linearized gravity''' is an approximation scheme in [[general relativity]] in which the [[nonlinear]] contributions from the [[spacetime]] [[metric tensor (general relativity)|metric]] are ignored, simplifying the study of many problems while still producing useful approximate results. 

==The method==
In linearized gravity the [[metric tensor]], <math>g</math>, of spacetime is treated as a sum of an exact solution of [[Einstein's equations]] (often in Minkowski spacetime) and a [[perturbation theory|perturbation]] <math>h</math>.

:<math>g \, =\eta+h</math>

where η is the nondynamical background metric that is being perturbed about, and <math>h</math> represents the deviation of the true metric (g) from [[Minkowski spacetime|flat spacetime]].

The perturbation is treated using the methods of [[perturbation theory]], "linearized" by ignoring all terms of order higher than one ([[quadratic]] in <math>h</math>, [[Cubic function|cubic]] in <math>h</math> etc...) in the perturbation.

==Applications==
The [[Einstein field equation]]s, being nonlinear in the metric, are difficult to [[Exact solutions|solve exactly]] and the above perturbation scheme allows linearised Einstein field equations to be obtained. 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 wave]]s that led, after [[Quantization (physics)|quantization]], to [[graviton]]s. 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 state]]s of particles.

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

===Weak-field approximation===
In a weak-field approximation, the gauge symmetry is associated with [[diffeomorphism]]s with small "displacements" (diffeomorphisms with large displacements obviously violate the weak field approximation), which has the exact form (for infinitesimal transformations)

:<math>\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^\alpha_{;\mu} h_{\alpha\nu} +  \xi^\alpha_{;\nu} h_{\mu\alpha}\right]dx^\mu \otimes dx^\nu</math>

Where <math>\mathcal{L}</math> is the [[Lie derivative]] and we used the fact that η does not transform (by definition). Note that we are raising and lowering the indices with respect to η and not g and taking the [[covariant derivative]]s ([[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

:<math>\delta_{\vec{\xi}}h_{\mu\nu}\approx \left(\mathcal{L}_{\vec{\xi}}\eta\right)_{\mu\nu}=\xi_{\nu;\mu} + \xi_{\mu;\nu}</math>

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]].

== Linearised Einstein field equations==
The '''linearised Einstein field equations''' (linearised EFE) are an approximation to [[Einstein's field equations]] that is valid for a weak [[gravitational field]] and is used to simplify many problems in [[general relativity]] and to discuss the phenomena of [[gravitational radiation]]. The approximation  can also be used to derive [[Newtonian gravity]] as the weak-field approximation of [[general relativity|Einsteinian gravity]].

The equations are obtained by assuming the [[metric tensor (general relativity)|spacetime metric]] is only slightly different from some baseline metric (usually a [[Minkowski metric]]). Then the difference in the metrics can be considered as a field on the baseline metric, whose behaviour is approximated by a set of linear equations.

=== Derivation for the Minkowski metric ===
Starting with the metric for a [[spacetime]] in the form

:<math>g_{ab} = \eta_{ab} + h_{ab}</math>

where <math>\, \eta_{ab}</math> is the Minkowski metric and <math>\, h_{ab}</math> &mdash; sometimes written as <math>\epsilon \, \gamma_{ab}</math> &mdash; is the deviation of <math>\, g_{ab}</math> from it. <math>h</math> must be negligible compared to <math>\eta</math>: <math>\left| h_{\mu \nu} \right| \ll 1</math> (and similarly for all derivatives of <math>h</math>). Then one ignores all products of <math>h</math> (or its derivatives) with <math>h</math> or its derivatives (equivalent to ignoring all terms of higher order than 1 in <math>\epsilon</math>). It is further assumed in this approximation scheme that all indices of h and its derivatives are raised and lowered with <math>\eta</math>.

The metric h is clearly symmetric, since g and η are. The consistency condition <math>g_{ab}g^{bc}=\delta_a{}^c</math> shows that  

:<math>g^{ab} \, = \eta^{ab} - h^{ab}</math>

The [[Christoffel symbols]] can be calculated as

:<math>2 \Gamma ^a_{bc} = (h^a{}_{b,c}+h^a{}_{c,b}-h_{bc,}{}^a)</math>

where <math>h_{bc,}{}^a \ \stackrel{\mathrm{def}}{=}\  \eta^{ar} h_{bc,r}</math>, and this is used to calculate the [[Riemann tensor]]:

:<math>2R^a{}_{bcd} = 2(\Gamma^a_{bd,c}-\Gamma^a_{bc,d}) 
= \eta^{ae} (h_{eb,dc}+h_{ed,bc}-h_{bd,ec} - h_{eb,cd}-h_{ec,bd}+h_{bc,ed}) =</math>

:<math> = \eta^{ae} (h_{ed,bc}-h_{bd,ec}-h_{ec,bd}+h_{bc,ed})
= h^a_{d,bc} - h_{bd,}{}^ a{}_c + h_{bc,}{}^a{}_d - h^a{}_{c,bd}</math>

Using <math>R_{bd}= \delta  ^c{}_a R^a{}_{bcd}</math> gives

:<math>2R_{bd}= h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \eta ^{rs}</math>

Then the linearized Einstein equations are

:<math> 8\pi T_{bd} \, = R_{bd} - R_{ac} \eta^{ac} \eta_{bd} / 2 </math>

or 

:<math> 8\pi T_{bd} = (h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, r}{}^r - h^r_{s,r}{}^s \eta_{bd})/2  + ( h_{,a}{}^a \eta_{bd} + h_{ac, r}{}^r \eta^{ac} \eta_{bd}) /4 </math>

Or, equivalently:

:<math> 8\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = R_{bd} </math>

:<math> 16\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \eta ^{rs}</math>

== Applications ==

The linearised EFE are used primarily in the theory of [[gravitational radiation]], where the gravitational field far from the source is approximated by these equations. 

==See also==
*[[Correspondence principle]]
*[[Gravitomagnetism]]
*[[Parameterized post-Newtonian formalism]]
*[[Quasinormal mode]]

==References==
* {{cite book | author=Stephani, Hans | title=General Relativity: An Introduction to the Theory of the Gravitational Field, | location=Cambridge | publisher=Cambridge University Press | year=1990 | isbn=0-521-37941-5}}
* {{cite book | author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem | title=Introduction to General Relativity | location=New York | publisher=McGraw-Hill | year=1965 | isbn=0-07-000423-4}}

[[Category:Mathematical methods in general relativity]]
[[Category:General relativity]]

[[es:Aproximación para campos gravitatorios débiles]]
[[fr:Approximation des champs faibles]]
[[it:Approssimazione per i campi gravitazionali deboli]]
[[zh:線性化重力]]
[[uk:Слабке гравітаційне поле]]