Special and General Relativity Lecture Notes:
Day 17 (11/04/08)
Shawn Mitryk
Contents
1 Horizons
1.1 Definitions . . . . . . .
1.2 Null Solutions . . . . .
1.3 Schwarzschild Metric .
1.4 Isotropic Coordinates .
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2
2
2
2
4
2 Comparison to Electrodynamics
5
3 Next Class
3.1 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
1
1
Horizons
1.1
Definitions
• Event Horizon - global
• Apparent Horizon - local (convenient for Numerical Relativity)
• Killing Horizon - Not a ”horizon,” rather a geodesically complete null
surface
1.2
Null Solutions
• Minkowski: X 2 − T 2 = 0
• Schwarzschild: R2 − T 2 = 0
1.3
Schwarzschild Metric
ds2
= −[1 −
1
2M 2
]dt +
dr2 + r2 dΩ2
r
[1 − 2M
]
r
(1)
r
2M we obtain:
Making the transformation T 2 − R2 = [ 2M
r − 1]e
ds2
=
32M 3 −r
e 2M (−dT 2 + dR2 ) + r2 dΩ2
r
(2)
In Mikoswski coordinates there is a killing vector of the form: K = x∂T +T ∂x
This is much like the angular momentum: Pz = x∂y − y∂x
Taking the vector K µ = [X, T, 0, 0]:
We can show: Kµ K µ = −X 2 + T 2 =⇒ 0 for X = ±T
K µ orbits cover the killing horizon X 2 − T 2 = 0.
−→ The Killing vector is null on the killing horizon. but K µ ∇µ K ν = −κK ν 6= 0
Taking X µ = (T, X, 0, 0) and U µ = (1, 1, 0, 0) ∝ K µ
Then:
κ2
=
κ2
=
κ2
=
κ2
=
1
− (∇µ Kν )(∇µ K ν )
2
1
− (g µρ gνσ ∇µ K ν )(∇ρ K σ )
2
1
− (−1 − 1) = 1
2
±1
In schwartzchild:
We consider the killing vector K = R∇T + T ∇R
2
(3)
(4)
(5)
(6)
Noting:
K µ Kµ
32M 3 −r
e 2M (−R2 + T 2 ) = 0@R = ±T
r
=
(7)
Taking the transformation:
V
=
U
=
∂
∂T
=
1
(V − U )
2
1
T − R =⇒ T = (V + U )
2
∂U ∂
∂V ∂
+
∂T ∂U
∂T ∂V
T + R =⇒ R =
(8)
(9)
(10)
Thus we can define the killing vector in the form:
K
K
K
= R∂T + T ∂R
∂
∂
1
∂
∂
1
(V − U )(
+
) + (V + U )(−
+
)
=
2
∂U
∂V
2
∂U
∂V
∂
∂
= −U
+V
∂U
∂V
(11)
(12)
(13)
Then recalling:
V
U
v
=
(14)
e 4M
−e
=
−u
4M
(15)
Thus K becomes:
K
= −U (
∂t ∂
∂r ∂
∂t ∂
∂r ∂
+
)+V(
+
)
∂u ∂t ∂u ∂r
∂v ∂t ∂v ∂r
(16)
We then want to find a function f (r) such that f 0 dr = −U dV − V dU
dr
−V
dr
Thus we find: dv
= −U
f 0 and du = f 0
And finally:
=
−U V
(17)
f (r)
=
v−u
(18)
f (r)
=
f (r)
Taking the ratio:
Differentiating:
V
U
e
4M
r
r
[
− 1]e− 2M
2M
(19)
t
= −e 2M
dV
V
− 2 dU
U
U
3
t
=
e 2M
dt
2M
(20)
From which we can read:
dt
dV
dt
dU
2M
−2M
=
t
V
U e 2M
−2M
2M V
=
t
2
U
U e 2M
=
=
(21)
(22)
Thus we can calculate the killing vector K:
= −U [
K
2M ∂
∂
−2M ∂
]+V[
] = 4M
U ∂t
V ∂t
∂t
K µ = (4M, 0, 0, 0)
(23)
(24)
From which we calculate:
κ2
=
κ2
=
−1
∇µ K ν ∇µ Kν
2
1
− (g µρ gνσ ∇µ K ν )(∇ρ K σ )
2
(25)
(26)
∇0 K 0
=
Γ000 K 0
(27)
∇0 K
1
=
Γ100 K 0
(28)
∇1 K
0
=
(29)
∇1 K
1
=
Γ001 K 0
Γ101 K 0
(30)
Finally:
κ2
=
κ2
=
κ =
1
1 2
)
− [g 00 g00 (∇0 K 0 )2 ] + g 00 g11 (∇0 K 1 )2 + g 11 g00 (∇1 K 0 )2 + g 11 g11 ∇1 K(31)
2
2
( M2 (1 − 2M
1
2M 2 M
4M
r ))
− [−(1 −
) ( 2
)2 + − r
]
(32)
2M
2M
2
r
r (1 − r )
(1 − r )2
16M 4
−→r=2M 1
r4
(33)
Defining a ”more useful” form:
k
=
κ2
=
κ =
1.4
∂
∂t
M2
r4
M
1
−→r=2M
2
r
4M
(34)
(35)
(36)
Isotropic Coordinates
The Schwarzschild metric reads:
ds2
=
−
(1 −
(1 +
M 2
2R )
dt2
M 2
)
2R
+ (1 −
4
M 4
) (dR2 + R2 dΩ)
2R
(37)
Figure 1: Kruskal Mapping
Conformal Mapping:
R → R1 maps the left side of the plot to the right side.
For small
M
R
1:
ds2
2
= −(1 −
2M 2
2M
)dt + (1 +
)(dR2 + R2 dΩ2 )
R
R
(38)
Comparison to Electrodynamics
Consider a charge density ρ:
Z
Q =
ρdV
Z
1
Q =
EdΩ
4πε0
Z
p
Q = −
d2 x γ (2) nµ σν F µν
(39)
(40)
(41)
∂Σ
But this has no General Relativity equivalent. There exists no volume density for mass (or energy).
For r = constant:
r
2M
, 0, 0)
(42)
σν = (0, 1 −
r
r
2M
nµ = (− 1 −
, 0, 0, 0)
(43)
r
We note the following representations for the Energy:
Z
p
1
EKomar =
d2 x γ (2) nµ σν ∇µ K µ
4πG ∂Σ
(44)
(45)
5
Note that K ν is a timelike killing vector →r→∞ 1
Another form:
EADM
=
1
16πG
Z
∂Σ
p
d2 x γ (2) σi (∂j hji − ∂i hjf )
where gµν = nµν + hµν has a solution if h is of the order
enough for integral to converge).
1
rn
(46)
(falls off fast
Finally we also note the angular momentum:
Z
p
1
d2 x γ (2) nµ σν ∇µ Rµ
JKomar =
8πG ∂Σ
(47)
(48)
where Rµ is the rotational killing vector
3
∂
∂φ
Next Class
3.1
Reading
• Ch. 33 in Gravitation (Misner, Thorne, and Wheeler)
• Section 6.7 and Chapter 7 in Spacetime and Geometry (Carroll)
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