In the theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, linearized gravity is the application of
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
to the
metric tensor that describes the geometry of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. 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
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
and weak-field
gravitational lensing.
Weak-field approximation
The
Einstein field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
(EFE) describing the geometry of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
is given as (using
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
)
:
where
is the
Ricci tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
,
is the
Ricci scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
,
is the
energy–momentum tensor Energy–momentum may refer to:
* Four-momentum
* Stress–energy tensor
* Energy–momentum relation
{{dab ...
, and
is the
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
metric tensor that represent 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 which render the prospect of finding
exact solutions impractical in most systems. However, when describing particular systems for which the
curvature of spacetime is small (meaning that terms in the EFE that are
quadratic in
do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
[This is assuming that the background spacetime is flat. Perturbation theory applied in spacetime that is already curved can work just as well by replacing this term with the metric representing the curved background.] plus a small perturbation term
. In other words:
:
In this regime, substituting the general metric
for this perturbative approximation results in a simplified expression for the Ricci tensor:
:
where
is the
trace of the perturbation,
denotes the partial derivative with respect to the
coordinate of spacetime, and
is the
d'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
.
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
.
Gauge invariance
The process of decomposing the general spacetime
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
. 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
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
is represented as being a consequence of the diverse collection of
diffeomorphisms
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two man ...
on spacetime that leave
sufficiently small. Therefore to continue, it is required that
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
to denote an arbitrary diffeomorphism that maps the flat Minkowski spacetime to the more general spacetime represented by the metric
. With this, the perturbation metric may be defined as the difference between the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
of
and the Minkowski metric:
:
The diffeomorphisms
may thus be chosen such that
.
Given then a vector field
defined on the flat, background spacetime, an additional family of diffeomorphisms
may be defined as those generated by
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
,
:
where
is the
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
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
To study how the perturbation
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
:
where
.
The tensor
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:
:
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
are zero, namely
:
This can be achieved by requiring the time component of
to satisfy
:
and requiring the spatial components to satisfy
:
Harmonic gauge
The ''
harmonic gauge'' (also referred to as the ''Lorenz gauge''
[Not to be confused with Lorentz.]) is selected whenever it is necessary to reduce the linearized field equations as much as possible. This can be done if the condition
:
is true. To achieve this,
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
:
Which can be solved exactly using 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
*
{{Relativity
Mathematical methods in general relativity
General relativity