In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime.
The Einstein field equation (EFE) describing the geometry of spacetime is given as where
is the spacetime metric tensor that represents 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 tensor that render the prospect of finding exact solutions impractical in most systems.
However, when describing systems for which the curvature of spacetime is small (meaning that terms in the EFE that are quadratic in
In other words: In this regime, substituting the general metric
for this perturbative approximation results in a simplified expression for the Ricci tensor: where
Together with the Ricci scalar, the left side of the field equation reduces to and thus the EFE is reduced to a linear second order partial differential equation in terms of
into the Minkowski metric plus a perturbation term is not unique.
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.
is represented as being a consequence of the diverse collection of diffeomorphisms on spacetime that leave
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.
With this, the perturbation metric may be defined as the difference between the pullback of
defined on the flat background spacetime, an additional family of diffeomorphisms
These new diffeomorphisms will be used to represent the coordinate transformations for "infinitesimal shifts" as discussed above.
The Lie derivative works out to yield the final gauge transformation of the perturbation metric
: 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
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
to satisfy the relation then choosing the time component
to satisfy After performing the gauge transformation using the formula in the previous section, the strain becomes spatially transverse: with the additional property: 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
to satisfy and requiring the spatial components to satisfy 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.
is required to satisfy the relation Consequently, by using the harmonic gauge, the Einstein tensor
reduces to Therefore, by writing it in terms of a "trace-reversed" metric,
, the linearized field equations reduce to This can be solved exactly, to produce the wave solutions that define gravitational radiation.