Chapter 9
Spherically Symmetric Gravitational
Fields
Abstract This chapter begins with a derivation of the Schwarzschild solution, the
single most important result of general relativity theory. It describes the gravitational
field of a spherically symmetric body such as the sun. As an application of the
Schwarzschild solution the orbits of planets and the deflection of star light can be
obtained, and the comparison of these with observation gives strong evidence that
the theory is correct.
9.1 The Schwarzschild Solution
We now turn to a study of the full nonlinear theory and obtain its best-known exact
solution, that of Schwarzschild (1916). Because of its importance in physics and in
history we will do this in considerable detail. The Einstein field equations for free
space in (8.27) are a set of 10 partial differential equations. We repeat them explicitly
here
R μν =
β
βν,μ −
β
μν,β +
β
τ μ
τ
βν −
β
τβ
τ
μν = 0.
(9.1)
The first two terms contain second derivatives of the metric tensor, and there are many
terms containing the metric and its first derivatives scattered about. We therefore have
a set of equations that look a bit formidable. We cannot merely stare at them and
write a solution, but instead must ponder the physical context and set the problem up
cleverly to find solutions. The solution of Schwarzschild for the field of a spherically
symmetric body is a beautiful example of this. It was obtained only about a year
after Einstein first presented his vacuum field equations in 1915 (Einstein 1915,
1923; Schwarzschild 1916). It is certainly the most important solution in general
relativity since it represents the exterior field of the sun and other stars (Misner
1973; Adler 1975).
It is natural to use spherical coordinates for the problem. In the absence of gravity
the appropriate metric is
ds
2
= c
2 dt
2
−
dr
2
+ r
2 dθ
2
+ r
2 sin
2
θ dϕ
2
.
(9.2)
© 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_9
125
Spherically Symmetric Gravitational
Fields
Abstract This chapter begins with a derivation of the Schwarzschild solution, the
single most important result of general relativity theory. It describes the gravitational
field of a spherically symmetric body such as the sun. As an application of the
Schwarzschild solution the orbits of planets and the deflection of star light can be
obtained, and the comparison of these with observation gives strong evidence that
the theory is correct.
9.1 The Schwarzschild Solution
We now turn to a study of the full nonlinear theory and obtain its best-known exact
solution, that of Schwarzschild (1916). Because of its importance in physics and in
history we will do this in considerable detail. The Einstein field equations for free
space in (8.27) are a set of 10 partial differential equations. We repeat them explicitly
here
R μν =
β
βν,μ −
β
μν,β +
β
τ μ
τ
βν −
β
τβ
τ
μν = 0.
(9.1)
The first two terms contain second derivatives of the metric tensor, and there are many
terms containing the metric and its first derivatives scattered about. We therefore have
a set of equations that look a bit formidable. We cannot merely stare at them and
write a solution, but instead must ponder the physical context and set the problem up
cleverly to find solutions. The solution of Schwarzschild for the field of a spherically
symmetric body is a beautiful example of this. It was obtained only about a year
after Einstein first presented his vacuum field equations in 1915 (Einstein 1915,
1923; Schwarzschild 1916). It is certainly the most important solution in general
relativity since it represents the exterior field of the sun and other stars (Misner
1973; Adler 1975).
It is natural to use spherical coordinates for the problem. In the absence of gravity
the appropriate metric is
ds
2
= c
2 dt
2
−
dr
2
+ r
2 dθ
2
+ r
2 sin
2
θ dϕ
2
.
(9.2)
© 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_9
125