Chapter 11
Linearized General Relativity
and Gravitational Waves
Abstract Gravitational waves are the analog of radio waves in electromagnetic
theory. They were first predicted soon after the advent of general relativity theory, and
after about a century of theoretical research and decades of experimental work they
have been finally detected. In this chapter we develop the theory of the production, the
propagation, and the detection of gravitational waves. Gravitational waves provide
an entirely new observational window on the universe; the mergers of black holes
and neutron stars are the sources of the waves so far observed.
11.1 The Field Equations of the Linearized Theory
In this chapter we will discuss approximate solutions to the Einstein equations, with
an emphasis on gravitational waves. Einstein recognized when he first formulated
his equations that exact solutions would be difficult to obtain since the equations are
nonlinear. To get approximate solutions to the field equations we will linearize them
by assuming, as we did in Chaps. 7 and 8, that the metric is the Lorentz metric plus
a small dimensionless perturbation that describes weak gravity. That is
g μν = η μν + h μν , all h μν 1.
(11.1)
Then the inverse metric is given to lowest order by
g
μν
= η
μν
− h
μν
, h
μν
≡ η
μα h αβ η
βν
.
(11.2)
We will usually call the perturbation h μν the metric field. In this section we will
often not bother to repeat the phrases “approximately equal” or “to lowest order in
the perturbation” but assume that (almost) all of the equations are approximate and
only correct to lowest order. Note moreover that indices may usually be raised and
lowered with the Lorentz metric, appropriate to this approximation, as in (11.2).
Since the Lorentz metric has constant elements many manipulations are thereby
greatly simplified. Many of the algebraic manipulations are the same as in special
relativity.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_11
159
Linearized General Relativity
and Gravitational Waves
Abstract Gravitational waves are the analog of radio waves in electromagnetic
theory. They were first predicted soon after the advent of general relativity theory, and
after about a century of theoretical research and decades of experimental work they
have been finally detected. In this chapter we develop the theory of the production, the
propagation, and the detection of gravitational waves. Gravitational waves provide
an entirely new observational window on the universe; the mergers of black holes
and neutron stars are the sources of the waves so far observed.
11.1 The Field Equations of the Linearized Theory
In this chapter we will discuss approximate solutions to the Einstein equations, with
an emphasis on gravitational waves. Einstein recognized when he first formulated
his equations that exact solutions would be difficult to obtain since the equations are
nonlinear. To get approximate solutions to the field equations we will linearize them
by assuming, as we did in Chaps. 7 and 8, that the metric is the Lorentz metric plus
a small dimensionless perturbation that describes weak gravity. That is
g μν = η μν + h μν , all h μν 1.
(11.1)
Then the inverse metric is given to lowest order by
g
μν
= η
μν
− h
μν
, h
μν
≡ η
μα h αβ η
βν
.
(11.2)
We will usually call the perturbation h μν the metric field. In this section we will
often not bother to repeat the phrases “approximately equal” or “to lowest order in
the perturbation” but assume that (almost) all of the equations are approximate and
only correct to lowest order. Note moreover that indices may usually be raised and
lowered with the Lorentz metric, appropriate to this approximation, as in (11.2).
Since the Lorentz metric has constant elements many manipulations are thereby
greatly simplified. Many of the algebraic manipulations are the same as in special
relativity.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_11
159