physics - Deriving the Schwarzschild solution

The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see general relativity). It is worthwhile deriving this metric in some detail; the following is a reasonably rigorous derivation that is not always seen in the textbooks.

Contents

1 Assumptions and notation

2 Diagonalising the metric

3 Simplifying the components

4 Using the field equations to find A(r) and B(r)

5 Using the Weak-Field Approximation to find K and S

6 Dispensing with the static assumption - Birkhoff's theorem

Assumptions and notation

We use coordinates $\left(r, \theta, \phi, t \right)$ labelled 1 to 4 respectively and begin with the metric in its most general form. The solution is assumed to be spherically symmetric, static and vacuum. More precisely, these assumptions may be stated as follows:

(1) A spherically symmetric spacetime is one in which each point of the spacetime lies on a 2-sphere.

(2) A static spacetime is one for which all metric components are independent of the time coordinate $t$ (so that $\frac {\part g_{ab}}{\part x^4}=0$ ) and hence the geometry of the spacetime is unchanged under a time-reversal $t \rightarrow -t$ .

(3) A vacuum solution is one which satisfies the equation $T a b = 0$ , which, from the Einstein field equations (with zero cosmological constant) imply that $R a b = 0$ .

Diagonalising the metric

The first simplification to be made is to diagonalise the metric. Under the coordinate transformation, $r' = r,θ' = θ,φ' = φ, t' = - t$ , all metric components should remain the same. The metric components $g a 4$ ( $a \ne 4$ ) change under this transformation as:

$g_{a4}'=\frac{\part x^{c}}{\part x^{'a}} \frac{\part x^{d}}{\part x^{'4}} g_{cd}= -g_{a4}$ ( $a \ne 4$ )

But, as we expect $g a 4' = g a 4$ , we then have

g a 4 = 0

( $a \ne 4$ )

Similarly, the coordinate transformations $(r, \theta, \phi, t) \rightarrow (r, \theta, -\phi, t)$ and $(r, \theta, \phi, t) \rightarrow (r, -\theta, \phi, t)$ respectively give:

g a 3 = 0

( $a \ne 3$ )

g a 2 = 0

( $a \ne 2$ )

Putting all these together, we have

g a b = 0

( $a \ne b$ )

and hence the metric must be of the form:

d s 2 = g 11 d r 2 + g 22 d θ 2 + g 33 d φ 2 + g 44 d t 2

where the four metric components are independent of the time coordinate $t$ (by the static assumption).

Simplifying the components

On each hypersurface of constant $t$ , constant $θ$ and constant $φ$ (i.e., on each radial line), we expect the metric component $g 11$ to only depend on $r$ (by spherical symmetry). Hence, we must have:

$g_{11}=A\left(r\right)$

Similarly, on each hypersurface of constant $r$ , constant $θ$ and constant $φ$ (i.e., for each value of $t$ ), we expect the metric component $g 44$ to only dependend on $r$ (by spherical symmetry). Hence:

$g_{44}=B\left(r\right)$

On the hypersurfaces of constant $t$ and constant $r$ , we require the metric to be that of a 2-sphere:

$dl^2=r_{0}^2 (d \theta^2 + sin^2 \theta d \phi^2)$

Choosing one of these hypersurfaces (the one with radius $r 0$ , say), we expect the metric components restricted to this hypersurface (which we denote by $\tilde{g}_{22}$ and $\tilde{g}_{22}$ ) to be independent of $θ$ and $φ$ (again, by spherical symmetry). Hence, comparing the forms of the metric on this hypersurface, we have:

$\tilde{g}_{22}\left(d \theta^2 + \frac{\tilde{g}_{33}}{\tilde{g}_{22}} d \phi^2 \right) = r_{0}^2 (d \theta^2 + sin^2 \theta d \phi^2)$

This immediately gives

$\tilde{g}_{22}=r_{0}^2$ and $\tilde{g}_{33}=r_{0}^2 sin ^2 \theta$

But we require this to be true for each hypersurface; hence,

g 22 = r 2

and

g 33 = r 2 s i n 2 θ

Thus, we have the metric in the form:

$ds^2=A\left(r\right)dr^2+r^2d \theta^2+r^2 sin^2 \theta d \phi^2 + B\left(r\right) dt^2$

with $A$ and $B$ as yet undetermined functions of $r$ . Note that $A$ and $B$ must both be nowhere zero (otherwise the metric would be singular at those points where at least one of them is zero).

Using the field equations to find $A (r)$ and $B (r)$

To determine $A$ and $B$ , we employ the vacuum field equations:

R a b = 0

We end up with the 3 differential equations:

$4 \dot{A} B^2 - 2 r \ddot{B} AB +r \dot{A} \dot{B}B +r \dot{B} ^2 A=0$

$r \dot{A}B +2 A^2 B - 2AB -r \dot{B} A=0$

$- 2 r \ddot{B} AB +r \dot{A} \dot{B}B +r \dot{B} ^2 A -4\dot{B} AB=0$

The first and last of these equations give:

$\dot{A}B +A \dot{B}=0 \Rightarrow A(r)B(r) =K$

where $K$ is a non-zero real constant. Substituting $A (r) B (r) = K$ into the second equation and tidying up, we find:

$r \dot{A} =A(1-A)$

which has general solution

$A(r)=\left(1+\frac{1}{Sr}\right)^{-1}$

for some non-zero real constant $S$ . Hence, we have the metric for a static, spherically symmetric vacuum solution in the form:

$ds^2=\left(1+\frac{1}{S r}\right)^{-1}dr^2+r^2(d \theta^2 +sin^2 \theta d \phi^2)+K \left(1+\frac{1}{S r}\right)dt^2$

Note that the spacetime represented by the above metric is asymptotically flat, i.e. as $r \rightarrow \infty$ , the metric approaches that of the Minkowski metric and the spacetime manifold resembles that of Minkowski space.

Using the Weak-Field Approximation to find $K$ and $S$

To identify the constants $K$ and $S$ , use is made of the weak-field approximation; using this, the $g 44$ components can be compared:

$g_{44}=-c^2\left(1 +\frac{1}{Sr}\right) \approx -c^2+\frac{2Gm}{r} = -c^2 \left(1-\frac{2Gm}{c^2 r} \right)$

where $G$ is the gravitational constant, $m$ is the mass of the gravitational source and $c$ is the speed of light. We thus find that:

$K = - c 2$ and $\frac{1}{S}=-\frac{2Gm}{c^2}$

Hence, we have:

$A(r)=\left(1-\frac{2Gm}{c^2 r}\right)^{-1}$ and $B(r)=-c^2 \left(1-\frac{2Gm}{c^2 r}\right)$

So, we may finally write the Schwarzschild metric in the form:

$ds^2=\left(1-\frac{2Gm}{c^2 r}\right)^{-1}dr^2+r^2(d \theta^2 +sin^2 \theta d \phi^2)-c^2 \left(1-\frac{2Gm}{c^2 r}\right)dt^2$

Dispensing with the static assumption - Birkhoff's theorem

In deriving the Schwarzschild metric, we assumed that the metric was vacuum, spherically symmetric and static. In fact, the static assumption is redundant, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is static; then one obtains the Schwarzschild solution. Birkhoff's theorem has the consequence that any pulsating star which remains spherically symmetric cannot generate gravity waves (as the region exterior to the star must remain static).

Categories: General relativity