Linearized gravity

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

${\displaystyle h_{\mu \nu }=(\phi ^{*}g)_{\mu \nu }-\eta _{\mu \nu }.}$

The diffeomorphisms ${\displaystyle \phi }$ may thus be chosen such that ${\displaystyle |h_{\mu \nu }|\ll 1}$.

Given then a vector field ${\displaystyle \xi ^{\mu }}$ defined on the flat, background spacetime, an additional family of diffeomorphisms ${\displaystyle \psi _{\epsilon }}$ may be defined as those generated by ${\displaystyle \xi ^{\mu }}$ and parameterized by ${\displaystyle \epsilon >0}$. These new diffeomorphisms will be used to represent the coordinate transformations for "infinitesimal shifts" as discussed above. Together with ${\displaystyle \phi }$, a family of perturbations is given by

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

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

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

where ${\displaystyle {\mathcal {L}}_{\xi }}$ is the Lie derivative along the vector field ${\displaystyle \xi _{\mu }}$.

The Lie derivative works out to yield the final gauge transformation of the perturbation metric ${\displaystyle h_{\mu \nu }}$:

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

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.

By exploiting gauge invariance, certain properties of the perturbation metric can be guaranteed by choosing a suitable vector field ${\displaystyle \xi ^{\mu }}$.

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

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

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

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

where ${\displaystyle \Psi ={\frac {1}{3}}\delta ^{kl}h_{kl}}$.

The tensor ${\displaystyle s_{ij}}$ 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 ${\displaystyle \xi ^{\mu }}$ to satisfy the relation

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

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

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

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

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

with the additional property:

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

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

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

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

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

and requiring the spatial components to satisfy

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

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

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

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

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

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

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

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

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

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