Gravity and Strings

188 The Schwarzschild black hole
There are many excellent books and reviews on these subjects. We would like to mention
Frolov and Novikov’s book [737], which is the most complete reference on BH physics,
Townsend’s lectures on BHs [903], the books on quantum-field theory (QFT) on curved
spacetimes [157, 936], and the review articles [431, 938].
7.1 Schwarzschild’s solution
To solve the vacuum Einstein equations
Rµν − 1
2 gµν R = 0, ⇒ Rµν = 0. (7.1)
is necessary to make a simplifying Ansatz for the metric. The Ansatz must, at the same
time, reflect the physical properties that we want the solution to enjoy. In this case we want
to obtain the metric in the spacetime outside a massive spherically symmetric body that is
at rest in a given coordinate system. The latter property is contained in the assumption of
staticity 3 of the metric and the first in the assumption of spherical symmetry. 4 Under these
assumptions, the most general metric can always be cast in the form
ds 2 = W (r)(dct) 2 − W −1 (r)dr 2 − R 2 (r)d 2 (2) , (7.2)
where W (r) and R(r) are two undetermined functions of the coordinate r and d 2 (2) is
the metric on the unit 2-sphere S 2 (see Appendix C). On substituting this Ansatz into the
equations of motion one finds (see for instance [932]) a general solution for W and R,
W = 1 + ω/r, R 2 = r 2 , (7.3)
with one integration constant ω. Wesee that the solution is asymptotically flat; i.e. that, as
the coordinate r, approaches infinity, the metric approaches Minkowski’s. Physically, the
requirement of asymptotic flatness means that we are dealing with an isolated system, with
a source of gravitational field confined in a finite volume. The constant ω has dimensions
of length and we will study its meaning in a moment.
The result is Schwarzschild’s solution [840] in Schwarzschild coordinates {t, r,θ,ϕ}:
ds 2 = W (dct) 2 − W −1 dr 2 − r 2 d 2 (2) , W = 1 + ω/r. (7.4)
Let us now review the properties of this solution.
3 That is, the metric admits a timelike Killing vector with the property of hypersurface-orthogonality: the
space can be foliated by a family of spacelike hypersurfaces that are orthogonal to the orbits of the timelike
Killing vector, and can be labeled by the parameter of these orbits, which takes the same value at any point of
each of these hypersurfaces. If the space does not have this property, the explicit dependence of the metric on
the associated time coordinate can always be avoided, but there will always be non-vanishing off-diagonal
terms in the metric mixing time components with space components, breaking at the same time spherical
symmetry: all stationary, spherically symmetric spacetimes are also static.
4 Invariance under the group SO(3) of spatial rotations in d = 4.