THE SCHWARZSCHILD SPACETIME
Rosemberg Toala Enriquez
The Schwarzchild metric is given by
ds2 = −(1 −
2m 2
2m −1 2
)dt + (1 −
) dr + r2 dθ2 + r2 sin2 (θ)dφ2 ,
r
r
and is a vacuum solution of Einstein equations. The constant m can be regarded as a the total
mass of the system, so it is regarded as modelling the vacuum space surrounding a massive
spherical (non-rotating) body.
Geodesics in Schwarzschild spacetime
Recall that ∂t and ∂φ are Killing vector fields so they determine constants of motion. It
follows that a geodesic γ(τ ) = (t(τ ), r(τ ), θ(τ ), φ(τ )) is completely determined by the equations
θ = θ0 ,
hγ̇, ∂t i = E,
hγ̇, ∂φ i = J,
hγ̇, γ̇i = ,
where E and J are constants and = −1 or 0 if γ is time-like or null, respectively.
We can then assume θ = π/2, hence
2m
−ṫ 1 −
= E
r
r2 φ̇ = J
−1
2m
2m
+ ṙ2 1 −
+ r2 φ̇2 = −ṫ2 1 −
r
r
From these, we can deduce a differential equation for r only:
2
2m
J
2
ṙ + 1 −
− = E2
r
r2
Or equivalenty,
1
r4
2 ṙ
2m
1
E2
+ 1−
−
=
r
r2 J 2
J2
φ̇
Making the substitution u = 1/r, we can regard u function of φ:
2
du
2m
E2 + = 2mu3 − u2 − 2 u +
=: f (u)
dφ
J
J2
1
(1)
We will now consider only time-like bounded geodesics.
Let u1 , u2 , u3 be the roots of f . One of them must be real, say u3 . The following definitions
are motivated by the conic equation in polar coordinates: Note that u1 + u2 = 2l is real, and
1
satisfies uu21 = 1−e
e = uu21 −u
, hence:
+u2
1+e
1
(1 − e)
l
1
=
(1 + e)
l
1
2
=
−
2m
l
u1 =
u2
u3
We have two cases:
1. If e is real (assume e 6= 0), we make the substitution u = 1l (1 + e cos(χ))1 , so equation (1)
becomes:
1/2
6m 2me
dχ
= ± 1−
−
cos(χ)
dφ
l
l
1/2 χ 1/2
6m 2me
1 − k 2 cos2 ( )
−
= ± 1−
l
l
2
where k 2 =
4 me
l
.
1− 6m
+ 2me
l
l
The fact that k is well defined follows by an appropriate labelling
of the roots, that is, in this case we can assume u1 ≤ u2 ≤ u3 , thus 1l (1 + e) ≤
which implies 1 − 6m
+ 2me
≥0
l
l
1
2m
−
2
l
Note also that the minimal and maximal values of u are (1 − e)/l and (1 + e)/l and
correspond to χ = π and χ = 0 respectively. Hence, we obtain φ in terms of χ as a Jacobi
elliptical function:
Z π
dγ
1
p
φ = q
2me χ
1 − k 2 cos2 ( γ2 )
+
1 − 6m
l
l
Z 1 (π−χ)
2
2
dγ
p
= q
1 − k 2 sin2 (γ)
0
+ 2me
1 − 6m
l
l
2. If e is purely imaginary, then ẽ = ie is real and we henceforth drop the tilde. We make the
substitution u = 1l (1 + e tan( 2ξ )), so equation (1) becomes:
1/2
dχ
2me
ξ 6m
2 ξ
= ± 4 cos
tan +
−1
dξ
2
l
2
l
1/2
2me
6m
= ± 2
sin ξ + (
− 1)(cos ξ − 1)
l
l
1
The resulting geodesic is bounded if e < 1, however the analysis holds in general, we just have to be careful
with the domain of definition of χ.
2
6m
−1
l
2me
l
with ξ0 ≤ ξ ≤ π, where tan ξ20 = −
1
2m
−
2
l
, since we are interested in bounded geodesics,
≤ u ≤ ∞.
Period of a circular orbit. Kepler Law.
Consider an equatorial circular geodesic γ = (t(τ ), r, π/2, θ(τ )). The r-component of the
geodesic equation reduces to Γrtt ṫ2 + Γrθθ θ̇2 = 0, explicitly
θ̇2 =
M 2
ṫ
r3
From here we can compute the period as measured by asymptotic time t,
r
r3
2π
= 2π
.
T =
dθ/dt
M
It is easy to integrate the geodesic equation by using the condition hγ̇, γ̇i = −1, that is,
2M 2
− 1−
ṫ + r2 θ̇2 = −1
r
From these two we obtain
3M − 12
t(τ ) = 1 −
τ + t0
r
21
M
θ(τ ) =
τ + θ0
r2 (r − 3M )
Hence the period measured by proper time is
2π
Tp =
= 2π
θ̇
r
3
r2 (r − 3M )
M
Redshift
Consider an observer γ2 and a source γ1 are connected by a null geodesic η. We know that the
emitted and received frequencies at the events P1 =‘γ1 sending a signal’ and P2 =‘γ2 receiving
that signal’ are given by
ωe = −hη̇, γ˙1 iP1
ωr = −hη̇, γ˙2 iP2
respectively.
Let us assume that both γi = (ti (τ ), ri , π/2, φi ) are at rest, that is, tangent to the Killing
field ∂t . Then the formula for redshift is relatively simple since:
ωe
hη̇, γ˙1 iP1
=
ωr
hη̇, γ˙2 iP2
ṫ1 hη̇, ∂t iP1
=
t˙2 hη̇, ∂t iP2
1+z =
But hη̇, ∂t i = E0 is constant. We assume τ to measure proper time, that is, the tangent vectors
to the curves are normalised. Hence ṫi = q 1 2m and
1−
ri
q
1−
1+z = q
1−
4
2m
r2
2m
r1
.
Some numerical values
To get an idea, we will state the values for some cases. We take 1M = 1.5 cG2 km, c = 3 × 105
km/s and 1 kpc = 3 × 1016 km.
1. Consider 2m = 2 × 1011 M = 3 × 1011 km, r1 = 10 kpc= 3 × 1017 km.
So, for an observer at infinity, we have that
1
1+z = q
1−
2m
r1
1 2m 3
+
=1+
2 r1
8
2m
r1
2
+ ...
1
= 1 + 10−6 + ...
2
We can compare with the Doppler effect in Minkowski spacetime for a source moving at a
velocity v = 300 km/s=10−3 c
p
1 + vc
v 1 v2
+
=
1
+
+ ...
1+z = p
c 2 c2
1 − vc
1
= 1 + 10−3 + 10−6 + ...
2
And with the cosmological redshift for a linear expansion. Taking H0 = 69 km/s/Mpc
= 13 × 10−5 c Mpc−1 and a source at a distant D = 4 Mpc we have:
1 + z = 1 + H0 D
= 1 + 5.2 × 10−4
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Now we will work out the case when γ1 = (t1 (τ ), r1 , π/2, φ1 (τ )) is orbiting around the center
of the galaxy. That is, is a time-like geodesic and for simplicity we will assume it circular. We
have then
1+z =
=
q
Recall that E1 = (1 −
2m
)(1
r1
+
J12
),
r2
ωe
hη̇, γ˙1 iP1
=
ωr
hη̇, γ˙2 iP2
ṫ1 hη̇, ∂t iP1 + φ̇1 hη̇, ∂φ iP1
t˙2 hη̇, ∂t iP2
the conservation laws for ṫ and φ̇ gives
q

J12
r
1
+
2
2m 
J0 J1 
r1
q
1+z = 1−
−
r2
E0 r12
1 − 2m
r1
Also, assume that light follows a practically radial path, that is J0 is small, so in the following
computations we will ignore the second term (this assumption must be corrected, since we would
expect the redshift to depend on the sign of J1 (approaching or moving away) but I don’t know
what would be a sensitive value for EJ00 ).
Consider again a source at r1 = 10 kpc=3 × 1017 km, with radial velocity J1 /r1 = 300 km/s
= 10−3 c, then
q
J2
1 + r12
1
1+z = q
2m
1 − r1
= 1 + 10−6 + ...
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Kruskal extension and Penrose diagram
Observe that the coordinates (t, r, θ, φ) are valid only in the chart U = R × (2m, ∞) × S 2 , since
the metric becomes degenerate at r = 2m, however it is possibly to extend the Schwarzschild
R dr
= r + 2m log(r − 2m) and new coordinates
solution by defining r∗ = A(r)
v = t + r∗ ,
w = t − r∗ ,
v 0 = exp(v/4m),
1 0
t0 =
(v + w0 ),
2
w0 = exp(w/4m),
1
x0 = (v 0 − w0 ),
2
In the coordinates (t0 , x0 , θ, φ) the metric takes the form
ds2 = exp(−r/2m)
16m2
(−dt02 + dx02 ) + r2 dθ2 + r2 sin2 (θ)dφ2 ,
r
where r is defined implicitly by
t02 − x02 = −(r − 2m) exp(r/2m).
In this form it can be observed that the image of U corresponds only to I = |x0 | < t0 , whereas the
metric is non-degenerate in the region t02 − x02 < 2m, hence this new metric extends smoothly
the original, and it is called Kruskal extension.
From this last form it is possible to perform a similar process as before to embed the extended
Schwarzschild solution into another manifold, this is done by defining the null coordinates
√
v 00 = arctan(v 0 / 2m),
√
w00 = arctan(w0 / 2m)
And then
t00 = (v 00 + w00 )/2,
x00 = (v 00 − w00 )/2.
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t00
i+
J+
r
0
i
i+
r=0
=
J+
IIB
2M
r
I0
=
2M
IIW
J−
i−
r=0
i0
I
x00
J−
i−
Figure 1: Penrose-Carter diagram for Schwarzschild space in (t00 , x00 ) coordinates. Dashed lines
correspond to {r = constant} and dotted to {t = constant}. Region I is the domain of the
original coordinates. Region I 0 is isometric to region I, so is regarded as “another universe”.
Region IIB is called a Black Hole, since it is impossible for any point in its interior to send
signals to the exterior. Indeed, any future-directed time-like or null curve will inevitable fall into
the singularity r = 0 within a finite proper time. Region IIW is called a White Hole since it has
the opposite behaviour, any event in its interior is eventually seen by an observer in I (or in I 0 ).
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