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A portion of lecture notes on special and general relativity, focusing on the concepts of horizons and isotropic coordinates. The notes cover definitions, null solutions, the schwarzschild metric, and isotropic coordinates. The document also includes a comparison to electrodynamics and suggested readings for further study.
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1 Horizons 2 1.1 Definitions............................... 2 1.2 Null Solutions............................. 2 1.3 Schwarzschild Metric......................... 2 1.4 Isotropic Coordinates......................... 4
2 Comparison to Electrodynamics 5
3 Next Class 6 3.1 Reading................................ 6
ds^2 = −[1 −
r ]dt^2 +
[1 − 2 Mr ] dr^2 + r^2 dΩ^2 (1)
Making the transformation T 2 − R^2 = [ 2 Mr − 1]e 2 rM^ we obtain:
ds^2 =
r e 2 −Mr (−dT 2 + dR^2 ) + r^2 dΩ^2 (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 Kμ^ = [X, T, 0 , 0]: We can show: KμKμ^ = −X^2 + T 2 =⇒ 0 for X = ±T
Kμ^ orbits cover the killing horizon X^2 − T 2 = 0. −→ The Killing vector is null on the killing horizon. but Kμ∇μKν^ = −κKν^6 = 0 Taking Xμ^ = (T, X, 0 , 0) and U μ^ = (1, 1 , 0 , 0) ∝ Kμ Then:
κ^2 = −
(∇μKν )(∇μKν^ ) (3)
κ^2 = −
(gμρgνσ ∇μKν^ )(∇ρKσ^ ) (4)
κ^2 = −
κ^2 = ± 1 (6) In schwartzchild: We consider the killing vector K = R∇T + T ∇R
From which we can read: dt dV
U e 2 tM
dt dU
U 2 e 2 Mt
Thus we can calculate the killing vector K:
K = −U [ −^2 M U
∂t
∂t
∂t
Kμ^ = (4M, 0 , 0 , 0) (24) From which we calculate:
κ^2 =
∇μKν^ ∇μKν (25)
κ^2 = −
(gμρgνσ ∇μKν^ )(∇ρKσ^ ) (26)
Finally:
κ^2 = −
[g^00 g 00 (∇ 0 K^0 )^2 ] + g^00 g 11 (∇ 0 K^1 )^2 + g^11 g 00 (∇ 1 K^0 )^2 + g^11 g 11 ∇ 1 K (31)^1 )^2
κ^2 = −
r
r^2
(1 − 2 Mr )
( Mr 2 (1 − 2 Mr ))^2 (1 − 2 Mr )^2
κ =
r^4 −→r=2M^1 (33) Defining a ”more useful” form:
k =
∂t
κ^2 =
r^4
κ =
r^2 −→r=2M
The Schwarzschild metric reads:
ds^2 = −
dt^2 + (1 −
)^4 (dR^2 + R^2 dΩ) (37)
Figure 1: Kruskal Mapping
Conformal Mapping: R → (^) R^1 maps the left side of the plot to the right side.
For small MR 1:
ds^2 = −(1 − 2 M R )dt^2 + (1 +^2 M R )(dR^2 + R^2 dΩ^2 ) (38)
Consider a charge density ρ:
Q =
ρdV (39)
Q = 1 4 πε 0
EdΩ (40)
Q = −
∂Σ
d^2 x
γ(2)nμσν F μν^ (41)
But this has no General Relativity equivalent. There exists no volume den- sity for mass (or energy). For r = constant:
σν = (0,
r
nμ = (−
r
We note the following representations for the Energy:
EKomar =
4 πG
∂Σ
d^2 x
γ(2)nμσν ∇μKμ^ (44) (45)
