parenrightbigg
0
0
0
0
1
−
2
GM
rc
2
0
0
0
0
1
r
2
0
0
0
0
1
r
2
sin
2
θ
Since the coefficients are independent of coordinate time
t
, stationary ob
servers will measure physical properties of spacetime to be independent of
time (note: a “static” or “stationary” observer” in a Schwarzschild gravita
tional field is one who is at a constant (
r,θ,φ
)) .
At this point, one may also notice that the metric diverges at
r
=
R
s
,
where
R
s
=
2
GM
c
2
The quantity
R
s
is called the “Schwarzschild radius” or “gravitational radius”
or “Event Horizon” of an object of mass
M
. We can calculate its value for
different types of objects found in the universe. A brief list is given below:
M/M
⊙
R
s
R
star
1
2
.
86 km
7
×
10
5
km
Sun
1
2
.
86 km
7
×
10
3
km
WD
1
2
.
86 km
10 km
NS
2
5
.
72 km
Typical stellar mass black hole
10
8
2
.
86
×
10
8
km
Black hole in nuclei of some galaxies
Ordinary nuclear burning stars (e.g. the Sun), white dwarfs, and neutron
stars, have radii that are larger than the corresponding Schwarzschild radius.
Note, however, that the radius of a one solar mass neutron star is only slightly
above its Schwarzschild radius. That is, this star is already so compressed
that it will only take a small reduction in radius to make it collapse into a
black hole.
Birkhoff’s Theorem (1923)
Birkhoff’s theorem states that “any spherically symmetric solution of the
vacuum field equations must be static and asymptotically flat”.
It follows
that the exterior solution must be given by the Schwarzschild metric, which,
at large distances, will reduce to the Newtonian limit.
130
SPACETIME NEAR A BLACK HOLE: SCHWARZSCHILD METRIC
A corollary states that “the metric inside a spherical cavity inside a spher
ical mass distribution” is the Minkowski metric.
Spatial sections
Spatial sections:
t
section
Take a
t
=
constant
slice, so that
dt
= 0. The proper distance
dl
will satisfy
dl
2
=
ds
2
=
dr
2
1
−
2
GM
c
2
r
+
r
2
d
Ω
2
If
θ
and
φ
are constant, then
dl
2
=
ds
2
=
dr
2
1
−
2
GM
c
2
r
The
proper distance
between spheres of radial coortdinates
r
1
and
r
2
is
D
=
integraldisplay
r
2
r
1
dl
=
integraldisplay
r
2
r
1
dr
radicalbigg
1
−
2
GM
c
2
r
negationslash
=
r
2
−
r
1
thus
r
is
not
the radial distance!
D
r
r
2
1
r

r
2
1
/
D =
Approximately we can write
D
=
integraldisplay
r
2
r
1
bracketleftBigg
1 +
1
2
2
GM
c
2
r
+
o
parenleftbigg
2
GM
c
2
r
parenrightbigg
2
bracketrightBigg
dr
≈
(
r
2
−
r
1
) +
GM
c
2
ln
parenleftbigg
r
2
r
1
parenrightbigg
THE TIME COORDINATE
T
131
At large distances (e.g.
r
1
= 100
R
s
) the radial coordinate distance
d
=
r
2
−
r
1
takes approximately the same value as the proper distance (
D
) (see figure
below). However, near the Schwarzschild radius, the proper distance
D
and
the coordinate distance
d
=
r
2
−
r
1
take quite different values, as shown in
the next figure.