In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing.

Weak-field approximation

The Einstein field equations (EFE) describing the geometry of spacetime using the MTW sign convention, including the metric signature , is

is the Ricci tensor is the Ricci scalar is the energy–momentum tensor is the Einstein gravitational constant 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 do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the Minkowski metric. In other words:

for this perturbative approximation results in a simplified expression for the Ricci tensor:

is the trace is the d'Alembert operator.

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.

Gauge invariance

. In order to capture this phenomenon, the concept of gauge symmetry is introduced.

is not consistently defined between different coordinate systems, the overall system which it describes is.

is represented as being a consequence of the diverse collection of diffeomorphisms. With this, the perturbation metric may be defined as the difference between the pullback and the Minkowski metric:

may thus be chosen such that .

and parameterized by . These new diffeomorphisms will be used to represent the coordinate transformations for "infinitesimal shifts" as discussed above. Together with , a family of perturbations is given by

Therefore, in the limit ,

is the Lie derivative along the vector field .

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.

Choice of gauge

By exploiting gauge invariance, certain properties of the perturbation metric can be guaranteed by choosing a suitable vector field .

Transverse gauge

distorts measurements of length, it is useful to define the following spatial tensor:

(Note that the indices span only spatial components: ). Thus, by using , the spatial components of the perturbation can be decomposed as

.

is, by construction, tracelessless 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. to satisfy the relation

to satisfy

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

with the additional property:

Synchronous gauge

The synchronous gauge are zero, namely

to satisfy

and requiring the spatial components to satisfy

Harmonic gauge

The harmonic gauge (also referred to as the Lorenz gauge) is selected whenever it is necessary to reduce the linearized field equations as much as possible. This can be done if the condition

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.

See also

• Correspondence principle

• Gravitoelectromagnetism

• Lanczos tensor

• Parameterized post-Newtonian formalism

• Post-Newtonian expansion

• Quasinormal mode

Notes

Further reading

External links