# Linearized gravity

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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](/source/general_relativity), '''linearized gravity''' is the application of [perturbation theory](/source/perturbation_theory) to the [metric tensor](/source/Metric_tensor_(general_relativity)) that describes the geometry of [spacetime](/source/spacetime). As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the [gravitational field](/source/gravitational_field) is weak. The usage of linearized gravity is integral to the study of [gravitational waves](/source/gravitational_waves) and weak-field [gravitational lensing](/source/gravitational_lensing).

== Weak-field approximation ==
The [Einstein field equation](/source/Einstein_field_equation) (EFE) describing the geometry of [spacetime](/source/spacetime) using the MTW sign convention, including the metric signature {{math|(−+++)}}, is
: <math>R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \kappa T_{\mu\nu}</math>
where <math>R_{\mu\nu}</math> is the [Ricci tensor](/source/Ricci_tensor), <math>R</math> is the [Ricci scalar](/source/Ricci_scalar), <math>T_{\mu\nu}</math> is the [energy–momentum tensor](/source/energy%E2%80%93momentum_tensor), <math>\kappa</math> is the [Einstein gravitational constant](/source/Einstein_gravitational_constant), and <math>g_{\mu\nu}</math> is the [spacetime](/source/spacetime) [metric tensor](/source/metric_tensor) that represents the solutions of the equation.

Although succinct when written out using [Einstein notation](/source/Einstein_notation), hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric tensor that render the prospect of finding [exact solutions](/source/Exact_solutions_in_general_relativity) impractical in most systems. However, when describing systems for which the [curvature](/source/curvature) of spacetime is small (meaning that terms in the EFE that are [quadratic](/source/Quadratic_function) in <math>g_{\mu\nu}</math> do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the [Minkowski metric](/source/Minkowski_metric)<ref group="note">This assumes that the background spacetime is flat. Perturbation theory applied in a spacetime that is already curved can work just as well when this term is replaced with the metric representing the curved background.</ref> <math>\eta_{\mu\nu}</math> plus a small perturbation term <math>h_{\mu\nu}</math>. In other words:
: <math>g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1.</math>

In this regime, substituting the general metric <math>g_{\mu\nu}</math> for this perturbative approximation results in a simplified expression for the Ricci tensor:
: <math>R_{\mu\nu} = \frac{1}{2}(\partial_\sigma\partial_\mu h^\sigma_\nu + \partial_\sigma\partial_\nu h^\sigma_\mu - \partial_\mu\partial_\nu h - \square h_{\mu\nu}),</math>
where <math>h = \eta^{\mu\nu}h_{\mu\nu}</math> is the [trace](/source/Trace_(linear_algebra)) of the perturbation, <math>\partial_\mu</math> denotes the partial derivative with respect to the <math>x^\mu</math> coordinate of spacetime, and <math>\square = \eta^{\mu\nu} \partial_\mu \partial_\nu</math> is the [d'Alembert operator](/source/d'Alembert_operator).

Together with the Ricci scalar,
: <math>R = \eta_{\mu\nu}R^{\mu\nu} = \partial_\mu\partial_\nu h^{\mu\nu} - \square h,</math>
the left side of the field equation reduces to
: <math>R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{1}{2}(\partial_\sigma\partial_\mu h^\sigma_\nu + \partial_\sigma\partial_\nu h^\sigma_\mu - \partial_\mu\partial_\nu h - \square h_{\mu\nu} - \eta_{\mu\nu}\partial_\rho\partial_\lambda h^{\rho\lambda} + \eta_{\mu\nu}\square h).</math>
and thus the EFE is reduced to a linear second order [partial differential equation](/source/partial_differential_equation) in terms of <math>h_{\mu\nu}</math>.

=== Gauge invariance ===
The process of decomposing the general spacetime <math>g_{\mu\nu}</math> into the Minkowski metric plus a perturbation term is not unique. This is because different choices for coordinates may give different forms for <math>h_{\mu\nu}</math>. In order to capture this phenomenon, the concept of [gauge symmetry](/source/Gauge_theory) 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 <math>h_{\mu\nu}</math> 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 <math>h_{\mu\nu}</math> is represented as being a consequence of the diverse collection of [diffeomorphisms](/source/diffeomorphisms) on spacetime that leave <math>h_{\mu\nu}</math> sufficiently small. Therefore, it is required that <math>h_{\mu\nu}</math> 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 <math>\phi</math> to denote an arbitrary diffeomorphism that maps the flat Minkowski spacetime to the more general spacetime represented by the metric <math>g_{\mu\nu}</math>. With this, the perturbation metric may be defined as the difference between the [pullback](/source/Pullback_(differential_geometry)) of <math>g_{\mu\nu}</math> and the Minkowski metric:
: <math>h_{\mu\nu} = (\phi^*g)_{\mu\nu} - \eta_{\mu\nu}.</math>
The diffeomorphisms <math>\phi</math> may thus be chosen such that {{tmath| \vert h_{\mu\nu} \vert \ll 1 }}.

Given then a vector field <math>\xi^\mu</math> defined on the flat background spacetime, an additional family of diffeomorphisms <math>\psi_\epsilon</math> may be defined as those generated by <math>\xi^\mu</math> and parameterized by {{tmath| \epsilon > 0 }}. These new diffeomorphisms will be used to represent the coordinate transformations for "infinitesimal shifts" as discussed above. Together with {{tmath| \phi }}, a family of perturbations is given by
: <math>\begin{align}
h^{(\epsilon)}_{\mu\nu} &= [(\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{align}</math>
Therefore, in the limit {{tmath| \epsilon\rightarrow 0 }},
: <math>h^{(\epsilon)}_{\mu\nu} = h_{\mu\nu} + \epsilon\mathcal{L}_\xi\eta_{\mu\nu}</math>
where <math>\mathcal{L}_\xi</math> is the [Lie derivative](/source/Lie_derivative) along the vector field {{tmath| \xi_\mu }}.

The Lie derivative works out to yield the final ''gauge transformation'' of the perturbation metric {{tmath| h_{\mu\nu} }}:
: <math>h^{(\epsilon)}_{\mu\nu} = h_{\mu\nu} + \epsilon(\partial_\mu\xi_\nu + \partial_\nu\xi_\mu),</math>
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 {{tmath| \xi^\mu }}.

==== Transverse gauge ====
To study how the perturbation <math>h_{\mu\nu}</math> distorts measurements of length, it is useful to define the following spatial tensor:
: <math>s_{ij} = h_{ij} - \frac{1}{3}\delta^{kl}h_{kl}\delta_{ij}</math>
(Note that the indices span only spatial components: {{tmath| i,j\in\{1,2,3\} }}). Thus, by using {{tmath| s_{ij} }}, the spatial components of the perturbation can be decomposed as
: <math>h_{ij} = s_{ij} - \Psi\delta_{ij}</math>
where <math>\Psi = \frac{1}{3}\delta^{kl}h_{kl}</math>.

The tensor <math>s_{ij}</math> is, by construction, [trace](/source/trace_(linear_algebra))less and is referred to as the ''strain'' since it represents the amount by which the perturbation [stretches and contracts measurements of space](/source/Gravitational_wave). In the context of studying [gravitational radiation](/source/gravitational_waves), the strain is particularly useful when utilized with the ''transverse gauge.'' This gauge is defined by choosing the spatial components of <math>\xi^\mu</math> to satisfy the relation
: <math>\nabla^2\xi^j + \frac{1}{3}\partial_j\partial_i\xi^i = -\partial_i s^{ij},</math>
then choosing the time component <math>\xi^0</math> to satisfy
: <math>\nabla^2\xi^0 = \partial_i h_{0i} + \partial_0\partial_i\xi^i.</math>

After performing the gauge transformation using the formula in the previous section, the strain becomes spatially transverse:
: <math>\partial_i s^{ij}_{(\epsilon)} = 0,</math>
with the additional property:
: <math>\partial_i h^{0i}_{(\epsilon)} = 0.</math>

==== 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 <math>h^{(\epsilon)}_{\mu\nu}</math> are zero, namely
: <math>h^{(\epsilon)}_{0\nu} = 0.</math>
This can be achieved by requiring the time component of <math>\xi^\mu</math> to satisfy
: <math>\partial_0\xi^0 = -h_{00}</math>
and requiring the spatial components to satisfy
: <math>\partial_0\xi^i = \partial_i\xi^0 - h_{0i}.</math>

==== Harmonic gauge ====
The ''[harmonic gauge](/source/Harmonic_coordinate_condition)'' (also referred to as the ''[Lorenz gauge](/source/Lorenz_gauge_condition)''<ref group="note">Not to be confused with Lorentz.</ref>) is selected whenever it is necessary to reduce the linearized field equations as much as possible. This can be done if the condition
: <math>\partial_\mu h^\mu_\nu = \frac{1}{2}\partial_\nu h</math>
is true. To achieve this, <math>\xi_\mu</math> is required to satisfy the relation
: <math>\square\xi_\mu = -\partial_\nu h^\nu_\mu + \frac{1}{2}\partial_\mu h.</math>

Consequently, by using the harmonic gauge, the [Einstein tensor](/source/Einstein_tensor) <math>G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}</math> reduces to
: <math>G_{\mu\nu} = -\frac{1}{2}\square\left(h^{(\epsilon)}_{\mu\nu} - \frac{1}{2}h^{(\epsilon)}\eta_{\mu\nu}\right).</math>
Therefore, by writing it in terms of a "trace-reversed" metric, {{tmath|1= \bar{h}^{(\epsilon)}_{\mu\nu} = h^{(\epsilon)}_{\mu\nu} - \frac{1}{2}h^{(\epsilon)}\eta_{\mu\nu} }}, the linearized field equations reduce to
: <math>\square \bar{h}^{(\epsilon)}_{\mu\nu} = -2\kappa T_{\mu\nu}.</math>
This can be solved exactly, to produce the [wave solutions](/source/wave_equation) that define [gravitational radiation](/source/gravitational_wave).

== See also ==
{{cols}}
* [Correspondence principle](/source/Correspondence_principle)
* [Gravitoelectromagnetism](/source/Gravitoelectromagnetism)
* [Lanczos tensor](/source/Lanczos_tensor)
* [Parameterized post-Newtonian formalism](/source/Parameterized_post-Newtonian_formalism)
* [Post-Newtonian expansion](/source/Post-Newtonian_expansion)
* [Quasinormal mode](/source/Quasinormal_mode)
{{colend}}

== Notes ==
{{reflist|group="note"|2}}

== Further reading ==
* {{cite book |author=Carroll |first=Sean M. |title=Spacetime and Geometry: An Introduction to General Relativity |title-link=Spacetime and Geometry |year=2003 |publisher=Addison-Wesley |isbn=978-0-805-38732-2}}

== External links ==
* {{wikiquote-inline}}

{{theories of gravitation}}
{{relativity}}

Category:Mathematics of general relativity
Category:General relativity

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Adapted from the Wikipedia article [Linearized gravity](https://en.wikipedia.org/wiki/Linearized_gravity) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Linearized_gravity?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.