astrofisica relativista - sergio mendoza.pdf
TRANSCRIPT
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Figura I.1:
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§3. Astrofsica relativista
∼ /
Figura I.3:
Figura I.4:
Figura I.5:
Figura I.6:
Figura I.7:
Figura I.8:
Figura I.9:
†
m 1 m 2 m 2 = 6 m 1 = 5
F1/F2 = 1/100 F1 F2 m 1 m 2
m − M = 5
|r1− r2| ,
r .
r2 er,
Figura I.11:
Figura I.12:
Figura I.13:
Estrella fija
θ
1, 2, 3
<
>
δαλ δαµ
δβλ δβµ
<
>
†
<
>
†
del observador
(+, −, −, −)†
y1)2 + (z 2− z 1)2
2( t)2− ( l)2.
s 2 = c 2t 2− l 2
2t 2 > 0.
†
†
(−, +, +, +)
<
>
t 2.
1 =
t2
t1
t
1 −
t r
c 2 t2 − l2
ct
x
Ψ
= x/t
γ = 1 1 −
<
>
c θ
1 − w2/c2 , A1 =
, A2 = A 2 A3 = A 3.
†
<
>
aib i = aibi
3.
§
δip δir
k s
l s
α
fα fα = 1 2 αβγ fβγ
†
≡ Sikl.
.
J = ∂(x 0, x 1, x 2, x 3)
∂(x0, x1, x2, x3) x 0
x 1 x 2
<
>
<
>
pi pi = m c2
,
e
ELECTROMAGNETICO. 87
= e
p
v
c
∂f
(∂Ak/∂xm) xm
B2− E2 = , E B = .
†
kl
S
= − 1
S
S =
∂T ki ∂xk
q∂Λ/∂
q −Λ
ω = e+ p
∂T ki ∂xk
<
>
w = + pV
σ
u i δli− u i u l
2 ρ
2 + ρ,
G
M ≈ ρl3 p ≈ ρc2
l ≈ c√ Gρ
W = m
†
α ≡ κ (n + 1)
ν)
ν)
1
2
∼
p + D −→ He3 + γ
He3 + He3 −→ He4 + 2p,
− G M2
M2 ≈ 1M
e− + p → n + ν.
−
(κ + 1) p1 .
4π (κ + 1) ρ1
m 1r1 + m 2r2 = 0
P K1 ≡ 1 i
K1/K2
∞ c2(r )
<
>
∂(ρ µ)/∂t = ρ∂ µ/∂t+ µ∂ρ/∂t = −ρ ν∂ µ/∂x ν−∂p/∂xµ−
µ∂(ρ ν)/∂x ν
∂ρ µ
Πµν
Πµν = pδµν + ρ µ ν − σ µν = −σµν + ρ µ ν σµν = − pδµν + σ
µν
v = eµ xµ/ t x = r θ −
y = r +
r = σ xy|=0
ax( = 0) = ar ay( = 0) = a a = axex+ ayey = arer+ ae
σ r
∂r .
∂r r.
∂
2 ρ r ar = 0 ar
r
r .
<
>
∞
|r2− r1| (r2− r1),
m 2( p) .
†
<
>
t, z = z ,
x0 = ct, x1 = x, x2 = y, x3 = z
<
>
+, −, −, −
−, +, +, +
<
>
a
Figura VI.1:
Rθ = d
R2 = d2
θ2 = 1
4 l2 +
†
†
xk = (x0, x1, x2, x3)
m,
δkl
∂(x 0, x 1, x 2, x 3) =
∂xi
x k
Epqrs =Eabcdgapgbqgcrgds = 1√ −g
x 2 x 3
∂ (x0, x1, x2, x3) =
√ −g .
x0
gmp
x0
,
<
>
xp ∂2xk
xl,
xm.
xl.
<
>
i m.
Πik;l = ∂Πik
... ...m...
m...
<
>
∂x m
∂x r
∂glp
∂gkl
gik gik gik
gik gik = g
gikgik = δ p p = 4 gik gik = −gik gik
§
xl = 0.
g00 ≈ 1 + 2φ
§
δu i = u i u i
xi xi+ xi u i+δu i = u i+ u i
u i xi
1
Riklm = −Rkilm = −Rikml = Rlmik.
− ∂Γ lil ∂xk
l km.
δgik
ζ = R θ
2 = θ2 +
2θ 2,
κ
,
l2 = x21 + x22 + x23 + (x1 x1 + x2 x2 + x3 x3)2
R2
2 3 ≈ R2
∂xλ = γαλδβη
<
>
χ, θ,
R2
M
gµν
2
η
§52 PARTICULAS Y RAYOS DE LUZ CERCA DE AGUJEROS NEGROS227
α −1 τ 2 + (α c)−2
u 1
R +
3GM
2c2R2
1 +
1
<
>
.
∼
3r
<
>
×
<
>
§ α
§
.
.
. .
=
<
>
l2 = x2
t = t2− t1
<
>
z = ∞ R = 0
a ≡ 0
2H0(0− 1)3/2 .
α ≡ 0
2H0(1 −0)3/2 .
l
†
ρ
ν = ν0(1 + z )
<
>
32πGe0
3c2
0h 2 ,
z ≤ 4× 1040h 2
v = v 0 + δv , ρ = ρ0 + δρ, p = p0 + δp, φ = φ0 + δφ.
Tarea 24
r/ t = 0
ω2 = c2k 2− 4πGρ0
4πGρ0
κ = 4/3
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§
<
>
Figura I.1:
<
>
§3. Astrofsica relativista
∼ /
Figura I.3:
Figura I.4:
Figura I.5:
Figura I.6:
Figura I.7:
Figura I.8:
Figura I.9:
†
m 1 m 2 m 2 = 6 m 1 = 5
F1/F2 = 1/100 F1 F2 m 1 m 2
m − M = 5
|r1− r2| ,
r .
r2 er,
Figura I.11:
Figura I.12:
Figura I.13:
Estrella fija
θ
1, 2, 3
<
>
δαλ δαµ
δβλ δβµ
<
>
†
<
>
†
del observador
(+, −, −, −)†
y1)2 + (z 2− z 1)2
2( t)2− ( l)2.
s 2 = c 2t 2− l 2
2t 2 > 0.
†
†
(−, +, +, +)
<
>
t 2.
1 =
t2
t1
t
1 −
t r
c 2 t2 − l2
ct
x
Ψ
= x/t
γ = 1 1 −
<
>
c θ
1 − w2/c2 , A1 =
, A2 = A 2 A3 = A 3.
†
<
>
aib i = aibi
3.
§
δip δir
k s
l s
α
fα fα = 1 2 αβγ fβγ
†
≡ Sikl.
.
J = ∂(x 0, x 1, x 2, x 3)
∂(x0, x1, x2, x3) x 0
x 1 x 2
<
>
<
>
pi pi = m c2
,
e
ELECTROMAGNETICO. 87
= e
p
v
c
∂f
(∂Ak/∂xm) xm
B2− E2 = , E B = .
†
kl
S
= − 1
S
S =
∂T ki ∂xk
q∂Λ/∂
q −Λ
ω = e+ p
∂T ki ∂xk
<
>
w = + pV
σ
u i δli− u i u l
2 ρ
2 + ρ,
G
M ≈ ρl3 p ≈ ρc2
l ≈ c√ Gρ
W = m
†
α ≡ κ (n + 1)
ν)
ν)
1
2
∼
p + D −→ He3 + γ
He3 + He3 −→ He4 + 2p,
− G M2
M2 ≈ 1M
e− + p → n + ν.
−
(κ + 1) p1 .
4π (κ + 1) ρ1
m 1r1 + m 2r2 = 0
P K1 ≡ 1 i
K1/K2
∞ c2(r )
<
>
∂(ρ µ)/∂t = ρ∂ µ/∂t+ µ∂ρ/∂t = −ρ ν∂ µ/∂x ν−∂p/∂xµ−
µ∂(ρ ν)/∂x ν
∂ρ µ
Πµν
Πµν = pδµν + ρ µ ν − σ µν = −σµν + ρ µ ν σµν = − pδµν + σ
µν
v = eµ xµ/ t x = r θ −
y = r +
r = σ xy|=0
ax( = 0) = ar ay( = 0) = a a = axex+ ayey = arer+ ae
σ r
∂r .
∂r r.
∂
2 ρ r ar = 0 ar
r
r .
<
>
∞
|r2− r1| (r2− r1),
m 2( p) .
†
<
>
t, z = z ,
x0 = ct, x1 = x, x2 = y, x3 = z
<
>
+, −, −, −
−, +, +, +
<
>
a
Figura VI.1:
Rθ = d
R2 = d2
θ2 = 1
4 l2 +
†
†
xk = (x0, x1, x2, x3)
m,
δkl
∂(x 0, x 1, x 2, x 3) =
∂xi
x k
Epqrs =Eabcdgapgbqgcrgds = 1√ −g
x 2 x 3
∂ (x0, x1, x2, x3) =
√ −g .
x0
gmp
x0
,
<
>
xp ∂2xk
xl,
xm.
xl.
<
>
i m.
Πik;l = ∂Πik
... ...m...
m...
<
>
∂x m
∂x r
∂glp
∂gkl
gik gik gik
gik gik = g
gikgik = δ p p = 4 gik gik = −gik gik
§
xl = 0.
g00 ≈ 1 + 2φ
§
δu i = u i u i
xi xi+ xi u i+δu i = u i+ u i
u i xi
1
Riklm = −Rkilm = −Rikml = Rlmik.
− ∂Γ lil ∂xk
l km.
δgik
ζ = R θ
2 = θ2 +
2θ 2,
κ
,
l2 = x21 + x22 + x23 + (x1 x1 + x2 x2 + x3 x3)2
R2
2 3 ≈ R2
∂xλ = γαλδβη
<
>
χ, θ,
R2
M
gµν
2
η
§52 PARTICULAS Y RAYOS DE LUZ CERCA DE AGUJEROS NEGROS227
α −1 τ 2 + (α c)−2
u 1
R +
3GM
2c2R2
1 +
1
<
>
.
∼
3r
<
>
×
<
>
§ α
§
.
.
. .
=
<
>
l2 = x2
t = t2− t1
<
>
z = ∞ R = 0
a ≡ 0
2H0(0− 1)3/2 .
α ≡ 0
2H0(1 −0)3/2 .
l
†
ρ
ν = ν0(1 + z )
<
>
32πGe0
3c2
0h 2 ,
z ≤ 4× 1040h 2
v = v 0 + δv , ρ = ρ0 + δρ, p = p0 + δp, φ = φ0 + δφ.
Tarea 24
r/ t = 0
ω2 = c2k 2− 4πGρ0
4πGρ0
κ = 4/3