The Schwarzschild solution describes spacetime under the influence of a massive, non-rotating, spherically symmetric object. It is considered by some to be one of the simplest and most useful solutions to the Einstein field equations .
Working in a coordinate chart with coordinates
\left(r,\theta,\phi,t\right)
t
\tfrac\partial{\partialt}g\mu=0
t → -t
Tab=0
Rab=0
Rab-\tfrac{R}{2}gab=0
R=0
The first simplification to be made is to diagonalise the metric. Under the coordinate transformation,
(r,\theta,\phi,t) → (r,\theta,\phi,-t)
g\mu
\mu\ne4
g\mu'=
\partialx\alpha | |
\partialx'\mu |
\partialx\beta | |
\partialx'4 |
g\alpha=-g\mu
\mu\ne4
But, as we expect
g'\mu=g\mu
g\mu=0
\mu\ne4
Similarly, the coordinate transformations
(r,\theta,\phi,t) → (r,\theta,-\phi,t)
(r,\theta,\phi,t) → (r,-\theta,\phi,t)
g\mu=0
\mu\ne3
g\mu=0
\mu\ne2
Putting all these together gives:
g\mu=0
\mu\ne\nu
and hence the metric must be of the form:
ds2=g11dr2+g22d\theta2+g33d\phi2+g44dt2
where the four metric components are independent of the time coordinate
t
On each hypersurface of constant
t
\theta
\phi
g11
r
g11
g11=A\left(r\right)
A similar argument applied to
g44
g44=B\left(r\right)
On the hypersurfaces of constant
t
r
2 | |
dl | |
0 |
(d\theta2+\sin2\thetad\phi2)
Choosing one of these hypersurfaces (the one with radius
r0
\tilde{g}22
\tilde{g}33
\theta
\phi
\tilde{g}22\left(d\theta2+
\tilde{g | |
33 |
which immediately yields:
\tilde{g}22
2 | |
=r | |
0 |
\tilde{g}33
2 | |
=r | |
0 |
\sin2\theta
But this is required to hold on each hypersurface; hence,
g22=r2
g33=r2\sin2\theta
An alternative intuitive way to see that
g22
g33
Thus, the metric can be put in the form:
ds2=A\left(r\right)dr2+r2d\theta2+r2\sin2\thetad\phi2+B\left(r\right)dt2
with
A
B
r
A
B
Using the metric above, we find the Christoffel symbols, where the indices are
(1,2,3,4)=(r,\theta,\phi,t)
'
1 | |
\Gamma | |
ik |
=\begin{bmatrix} A'/\left(2A\right)&0&0&0\\ 0&-r/A&0&0\\ 0&0&-r\sin2\theta/A&0\\ 0&0&0&-B'/\left(2A\right)\end{bmatrix}
2 | |
\Gamma | |
ik |
=\begin{bmatrix} 0&1/r&0&0\\ 1/r&0&0&0\\ 0&0&-\sin\theta\cos\theta&0\\ 0&0&0&0\end{bmatrix}
3 | |
\Gamma | |
ik |
=\begin{bmatrix} 0&0&1/r&0\\ 0&0&\cot\theta&0\\ 1/r&\cot\theta&0&0\\ 0&0&0&0\end{bmatrix}
4 | |
\Gamma | |
ik |
=\begin{bmatrix} 0&0&0&B'/\left(2B\right)\\ 0&0&0&0\\ 0&0&0&0\\ B'/\left(2B\right)&0&0&0\end{bmatrix}
To determine
A
B
R\alpha\beta=0
Hence:
\rho | |
{\Gamma | |
\beta\alpha,\rho |
Rtt=-
1 | |
4 |
B' | \left( | |
A |
A' | - | |
A |
B' | + | |
B |
4 | |
r |
\right)-
1 | \left( | |
2 |
B' | |
A |
\right)',
Rrr=-
1 | \left( | |
2 |
B' | |
B |
\right)'-
1 | \left( | |
4 |
B' | |
B |
\right)2+
1 | |
4 |
A' | \left( | |
A |
B' | + | |
B |
4 | |
r |
\right),
R\theta\theta=1-\left(
r | |
A |
\right)'-
r | \left( | |
2A |
A' | + | |
A |
B' | |
B |
\right),
R\phi\phi=
2(\theta)R | |
\sin | |
\theta\theta |
,
where the prime means the r derivative of the functions.
Only three of the field equations are nontrivial (the fourth equation is just
\sin2\theta
4A'B2-2rB''AB+rA'B'B+rB'2A=0
-2rB''AB+rA'B'B+rB'2A-4B'AB=0
rA'B+2A2B-2AB-rB'A=0
Subtracting the first and second equations produces:
A'B+AB'=0 ⇒ A(r)B(r)=K
where
K
A(r)B(r)=K
rA'=A(1-A)
which has general solution:
A(r)=\left(1+ | 1 |
Sr |
\right)-1
for some non-zero real constant
S
| ||||
ds |
\right)-1dr2+r2(d\theta2+\sin2\thetad\phi2)+K\left(1+
1 | |
Sr |
\right)dt2
Note that the spacetime represented by the above metric is asymptotically flat, i.e. as
r → infty
The geodesics of the metric (obtained where
ds
0=\delta\int | ds |
dt |
dt=\delta\int(KE+PEg)dt
(where
KE
PEg
K
S
g44=K\left(1+
1 | |
Sr |
\right) ≈
| ||||
-c |
=-c2\left(1-
2Gm | |
c2r |
\right)
where
G
m
c
K=-c2
1 | =- | |
S |
2Gm | |
c2 |
Hence:
A(r)=\left(1- | 2Gm |
c2r |
\right)-1
B(r)=-c2\left(1-
2Gm | |
c2r |
\right)
So, the Schwarzschild metric may finally be written in the form:
| ||||
ds |
\right)-1dr2+r2(d\theta2+\sin2\thetad\phi2)-c2\left(1-
2Gm | |
c2r |
\right)dt2
Note that:
2Gm | |
c2 |
=rs
is the definition of the Schwarzschild radius for an object of mass
m
| ||||
ds |
\right)-1dr2+r2(d\theta2+\sin2\thetad\phi2)-c
| ||||
\right)dt2
which shows that the metric becomes singular approaching the event horizon (that is,
r → rs
r=0
The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass.[1] Start with the metric with coefficients that are unknown coefficients of
r
-c2=\left({ds\overd\tau}\right)2=A(r)\left({dr\overd\tau}\right)2+r2\left({d\phi\overd\tau}\right)2+B(r)\left({dt\overd\tau}\right)2.
Now apply the Euler–Lagrange equation to the arc length integral
{
\tau2 | |
J=\int | |
\tau1 |
\sqrt{-\left(ds/d\tau\right)2}d\tau.}
ds/d\tau
(ds/d\tau)2,
J
\begin{array}{lcl}A'(r)
r |
2+2r
\phi |
2+B'(r)
t |
2&=&2A'(r)
r |
2+2A(r)\ddot{r}\ 0&=&2r
r |
\phi |
+r2\ddot{\phi}\ 0&=&B'(r)
r |
t |
+B(r)\ddot{t}\end{array}
where dot denotes differentiation with respect to
\tau.
In a circular orbit
r |
=\ddot{r}=0,
2r\phi |
2+B'(r)
t |
2=0\LeftrightarrowB'(r)=-2r
\phi |
| |||
2=-2r(d\phi/dt)2.
Kepler's third law of motion is
T2 | |
r3 |
=
4\pi2 | |
G(M+m) |
.
In a circular orbit, the period
T
2\pi/(d\phi/dt),
\left({d\phi\overdt}\right)2=GM/r3
since the point mass
m
M.
B'(r)=-2GM/r2
B(r)=2GM/r+C,
C
C
M=0,
B(r)=-c2.
C=-c2
B(r)=2GM/r-c2=c2(2GM/c2r-1)=
2(r | |
c | |
s/r |
-1).
When the point mass is temporarily stationary,
r |
=0
\phi |
=0.
t |
2=-c2/B(r)
A(r)=B'(r)
t |
2/(2\ddot{r}).
\ddot{r}
-MG/r2.
A(r)=\left(
-2MG | |
r2 |
\right)\left(
-c2 | |
2MG/r-c2 |
\right)\left(-
r2 | |
2MG |
\right)=
1 | |
1-2MG/(rc2) |
=
1 | |
1-rs/r |
.
The original formulation of the metric uses anisotropic coordinates in which the velocity of light is not the same in the radial and transverse directions. Arthur Eddington gave alternative forms in isotropic coordinates.[2] For isotropic spherical coordinates
r1
\theta
\phi
\theta
\phi
r\geq
2Gm | |
c2 |
r=r1\left(1+
Gm | |
2c2r1 |
\right)2
dr=dr1\left(1-
(Gm)2 | ||||||||||||
|
\right)
\left(1- | 2Gm |
c2r |
\right)=\left(1-
Gm | |
2c2r1 |
\right)2/\left(1+
Gm | |
2c2r1 |
\right)2
Then for isotropic rectangular coordinates
x
y
z
x=r1\sin(\theta)\cos(\phi) ,
y=r1\sin(\theta)\sin(\phi) ,
z=r1\cos(\theta)
The metric then becomes, in isotropic rectangular coordinates:
ds2=\left(1+
Gm | |
2c2r1 |
\right)4(dx2+dy2+dz2)-c2dt2\left(1-
Gm | |
2c2r1 |
\right)2/\left(1+
Gm | |
2c2r1 |
\right)2
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows. Birkhoff's theorem has the consequence that any pulsating star that remains spherically symmetric does not generate gravitational waves, as the region exterior to the star remains static.