Evolucion de Ondas Gravitacionales

8/17/2019 Evolucion de Ondas Gravitacionales

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ua∂ aub = 0 ua

∂ a

ua∂ aub = qmF

bcu

c

q

m

F ab F ab

γ s(t) s ∈ R γ t (t, s) → γ s(t) Σ γ

s(t) T a =(∂/∂t)a

T a∇aT b = 0 ∇a gab X a = (∂/∂s)a

aa = −RcbdaX bT cT d,

Rcbd

Rab = ΣcRacbc = Σc

∂

∂xcΓcab − ∂

∂xa (ΣcΓ

ccb) + Σd,c

ΓdabΓ

cdc − ΓdcbΓcda

.

∇a tb ∇atb = ∂ atb + Γbactc Γbac

Γbac = 12

Σdgbd∂gcd∂xa

+ ∂gad∂xc

+ ∂gac∂xd

∇agbc = 0


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γ s T a

X a

R4

ηab

gab

M

gab

gab

gab ua ρ P F ab

ηab gab ∂ a ηab

∇a gab

ua

ua∇aub = 0.


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ab = ua∇aub

f b = mab

m

m

q

F ab

ua∇aub = q m

F bcuc,

gab F bc = gbdF dc

pa = mua.

E = − paυa, υa

T ab

T ab = ρuaub + P (gab + uaub)

∇aT ab = 0,

ua∇aρ + (ρ + P ) ∇aua = 0

(P + ρ) ua∇aub + (gab + uaub) ∇aP = 0.

υa ∇a T abυb = 0 ∇aυb = 0 ∇(aυb) = 0

υa υaυa = −

1 ∇(aυb) = 0 ∇(aυb) = 0 υa

gab


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∇(aυb) ≈ 0

ηab →gab, ∂ a → ∇a

∇a∇aφ − m2φ = 0.

T ab = ∇aφ∇bφ − 12

gab∇cφ∇cφ + m2φ2

∇aT ab = 0

∇a∇aφ − m2φ − αRφ = 0, α α = 1/6

φ

−−→x · −→∇−→∇φ −→x

−Rcbdaυcxbυd υa xa

Rcbdaυcυd ←→ ∂ b∂ aφ.


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∇2φ = 4πρ, ρ G = c = 1

T ab

T abυaυb ←→ ρ,

υa Rcad

aυcυd =4πT cdυ

cυd Rcd = 4πT cd

∇cT cd = 0

∇c Rcd − 12 gcdR = 0 Rcd 4πT cd ∇dR = 0 R T = T aa

Gab ≡ Rab − 12

Rgab = 8πT ab,

Gab

R = −8πT

Rab = 8π

T ab − 1

2gabT

.

Gab + Λgab = 8πT ab,

Λ

Λ

Λ = 0

∇[aRbc]d e = 0 ⇒∇aRbcda +∇bRcd −∇cRbd = 0 ⇒∇aRc

a +∇bRcb −∇cR = 0


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gab gab

Rµν gµν gµν gµν

gµν

∂ a∂ aAb = −4πjb

T ab ja

ja Aa T ab T ab gab

∇aT ab = 0

M ψ S [ψ] ψ S M ψλ ψ0

dψdλ |λ=0

δψ

dS dλ λ = 0

ψ0 χ ψ

Auxx + 2Buxy + C uyy + D ux + E uy + F = 0 Z =

A B

B C

f : U ⊂ Rn → R C ∞


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ψ (k, l) χ (l, k)

dS

dλ = ˆ

M

χδψ.

S

ψ0 χ

S

χ =

δS

δψ

ψ0

.

S

S [ψ] =

ˆ M

L [ψ],

L ψ

L |x= L ψ(x), ∇ψ(x),..., ∇kψ(x) . S ψ

S δS

δψ

ψ

= 0,

ψ S L L

U M ψλ

gab M

abcd gab a1...ana1...an = (−1)sn! n s = 1

L

L

T

(k, l)

V

T : V ∗(1)× ...× V ∗

(k)× V (1)× ...× V (l) → R

V

V ∗


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eabcd = e[abcd] M M eabcd abcd M eabcd

abcd = f eabcd.

f abcd

a1...anb1...bn = (−1)s n!δ [a1b1δ a2 b2 ...δ an]bn

∇ba1...an = 0. f

√ −g

g

gab

=√ −gεabcddxadxbdxcdxd ∼

√ −gd4x.

eabcd M T a...bc...d

T a...bc...d =√ −g T a...bc...d,

T a...bc...d eabcd S

eabcd L

dS dλ eabcd S

LG =√ −g R

S

gab

=

ˆ LGe,

e = eabcddxadxbdxcdxd = d4x

gab gab δgab = dgab/dλ

gab

dLdλ

= √ −g (δRab) gab + √ −g Rabδgab + Rδ √ −g .

gabδRab = ∇aυa,


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υa =

∇b (δgab)

−gcd

∇a (δgcd) .

tr

dA

dt A−1

=

1

det (A)

d

dt [det (A)] .

δ √ −g = 1

2

√ −g gabδgab = −12

√ −g gabδgab.

dS Gdλ

=

ˆ dLG

dλ =

ˆ ∇aυa

√ −g e +ˆ

Rab + 1

2Rgab

δgab

√ −g e.

∇aυa =

√ −g e

δS Gδgab

=√ −g

Rab − 1

2Rgab

∇a

Rab gab

ϕG

gab, ∇a

=

ˆ √ −g Rabgabe,

gab ∇a

∇cgab = 0

L LG LM

γ ab = dgabdλ

λ=0

0 = Ṙac =

−120gbd0∇a0∇cγ bd −

120gbd0∇b

0∇dγ ac + 0gbd0∇b

0∇(cγ a)d


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L = LG + αM LM , αM LG

S

S M S gab

Gab = Rab + 1

2Rgab = 8πT ab,

T ab

T ab = −αM 8π

1√ −gδS M δgab

.

S M f : M → M S M

gab, ψ

= S M

f ∗λg

ab, f ∗λψ

f ∗λ : V p → V p

0 = dS M

dλ =

ˆ δS M

δgab δgab +

ˆ δS M

δψ δψ,

δgab δgab = ∇ awb wa ψ δS M /δψ |ψ= 0

wa

0 =

ˆ √ −gT ab∇(awb)e =ˆ

T ab∇awb = −ˆ

(∇aT ab) wb,

∇aT ab = 0.

T ab

T ab

S G

∇aGab = 0.

ψ


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t ta Σt t ta∇at = 1 ta Σt

Σ0 t ta

M abcd Σt

(3)abc = dabcnd

nd Σt Ltabcd = 0 Lt(3)abc = 0 Lt t ta Σt Σt Σ0 Σ0 eabcd M

Lteabcd = 0

x1, x2, x3 t ta = (∂/∂t)a e d4x Σt

(3)eabc = edabctd M eabcd Σt

(3)abc eabcd M π Σt

q Σt ψ q V ∗q q

V ∗q

δq q Σt (k, l) π (k, l) Σt π δq R δq → ´

Σtπδq

π

ψ Σt

H [q, π] Σt

H =

ˆ Σt

H,

H q π

q̇ ≡ tq = δH δπ

M

R (k, l)


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π̇ ≡ tπ = −δH δq

ψ

q

ψ Σt

q

L q π ψ Σt

π = ∂ L∂ q̇

.

q̇ q π

H(q, π) = πq̇ − L,

q̇ = q̇ (q, π) L H

J =

ˆ t2t1

H dt =

ˆ t2t1

dt

ˆ Σt

H = −S +ˆ t2t1

dt

ˆ Σt

πq̇.

ψ δψ = 0 t = t1 t = t2

dJ

dλ =

ˆ t2t1

dt

ˆ Σt

δH

δq δq +

δ H

δπ δπ

=

ˆ t2t1

dt

ˆ Σt

[πδ q̇ + q̇δπ ] − dS dλ

=

ˆ t2t1

dt

ˆ Σt

[−π̇δq + q̇δπ ] − dS dλ

,

δS/δψ = 0 H ψ

t

ta

M

ta∇at = 1

t

ta

gab ta Σt t N


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N ≡ −gabtanb = (na∇at)−1 N a

N a ≡ habtb, na Σt hab = gab + nanb Σt N τ t Σt N

a

Σt ta

N N a ta

na = 1

N (ta − N a) ,

N

N a.

gab = hab − nanb = hab − N −2 (ta − N a)

tb − N b

.

hab

N N a = habN b

hab gab

hachcb Σt hab∇bt = 0 hab hab N a = habN b (hab, N, N a) gab

eabcd

Lteabcd = 0 (3)abc =√

h(3)

eabc h hµν hab

(3)eabcd ±1 √

−g = N

√ h.

(hab, N, N a) K ab

hab ḣab = Lthab

K ab = 1

2 nhab =

1

2 [nc∇chab + hac∇bnc + hcb∇anc]


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= 1

2N −1 [N nc∇chab + hac∇b (N nc) + hcb∇a (N nc)]

= 1

2N −1hachbd [ thab + N hcd]

= 1

2N −1

ḣab − DaN b − DbN a

,

Da Σt hab

R

R = 2

Gabnanb − Rabnanb

.

(i) Rabcd = ha

f hbghc

khjdRfgk

j − K acK bd + K bcK ad

(ii) DaK ab − DbK aa = Rcdndhcb

Gabnanb =

1

2

(3)R − K abK ab + K 2

,

K = K aa

Rabnanb = Racb

cnanb

= −na (∇a∇c − ∇c∇a) nc

= (∇ana) (∇cnc) − (∇cna) (∇anc) − ∇a (na∇cnc) + ∇c (na∇ana)

= K 2 − K acK ac − ∇a (na∇cnc) + ∇c (na∇anc) .

LG =√

hN

(3)R + K abK ab − K 2

.

hab

πab = ∂ LG∂ ḣab

=√

h

K ab − Khab . LG N N a


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N N a hab Σt

HG = πab

ḣab − LG

= −h1/2N (3) R + N h−1/2

πabπab − 12

π2

+ 2πabDaN b

= h1/2

N

−(3)R + h−1πabπab − 1

2h−1π2

− 2N a

Da

h−1/2πab

+ 2Da

h−1/2N bπab

,

π = πaa

H G =´ HG(3)e H G N

N a

−(3)R + h−1πabπab − 12

h−1π2 = 0

Da

h−1/2πab

= 0,

N N a

H G

ḣab = δH Gδπab

= 2h−1/2N

πab − 12

habπ

+ 2D(aN b)

π̇ab = −δH Gδhab = −N h1/2(3)Rab − 12 (3)Rhab+ 12 N h−1/2habπcdπcd − 12 π2−2N h−1/2

πacπc

b − 12

ππab

+ h1/2

DaDbN − habDcDcN

+h1/2Dc

h−1/2N cπab

− 2πc(aDcN b),

Rab = 0

N N a

hab ψ Σt hab ψ∗hab


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h̃ab Σt

wa

Σt πab ˆ πab

δhab + D(awb)

=

ˆ πabδhab,

πab

Da

h−1/2πab

= 0.

Σt Σ


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gab Gab = 8πT ab


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Σt

t

p, q

∈ Σt

gab p q

ua

p sa1 , s

a2 ∈ V p ua p gab

p

u

a

p

s

a

1

s

a

2

ua


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Σt ua

Σt ua

Σt

gab hab(t) Σt gab p ∈ Σt Σt hab p ∈ Σt q ∈ Σt Σt

(3)Rabcd = K δ c[aδ

db].

(3)

Rabcd = Khc[ahb]d.

K Σt K

K

K

Σt K K R

4

x2 + y2 + z2 + w2 = R2.

ds2 = dψ2 + sin2ψ

dθ2 + sin2θ dφ2

.

K = 0

ds2 = dx2 + dy2 + dz2.

ds2 = dψ2 + sinh2ψ

dθ2 + sin2θ dφ2

.


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ua

gab

gab = −uaub + hab(t), t hab(t) Σt

τ

ds2 = −dτ 2 + a2(τ )dψ2 + sin2ψ dθ2 + sin2θ dφ2 ,ds2 = −dτ 2 + a2(τ )dx2 + dy2 + dz2 ,

ds2 = −dτ 2 + a2(τ )dψ2 + sinh2ψ dθ2 + sin2θ dφ2 .

T ab

T ab = ρuaub,

ρ

P = ρ/3

T ab

T ab

T ab = ρuaub + P (gab + uaub).

Gab 8πT ab


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Gabuaub = 8πρ

Gabsasb = 8πP,

sa Gabu

aub Gabsasb a(τ )

ds2 = −dτ 2 + a2(τ )dx2 + dy2 + dz2 .

Γτ xx = Γτ yy = Γ

τ zz = aȧ

Γxxτ = Γyyτ = Γ

yτy = Γ

zτz = Γ

zτz = Γ

xxτ = ȧ/a,

ȧ = da/dτ

Rabuaub = −3ä/a

Rabsasb = a−2Rxx =

ä

a + 2

ȧ2

a2.

R = −Rabuaub + 3Gabsasb = 6

ä

a +

ȧ2

a2

Gabuaub = Rabu

aub + 1

2R = 3ȧ2/a2 = 8πρ

Gabsasb = Rabs

asb − 12

R = −2 äa − ȧ

2

a2 = 8πP.

3ä/a = −4π (ρ + 3P ) .

3ȧ2/a2 = 8πρ − 3k/a2

3ä/a = −4π (ρ + 3P ) , k = +1 k = 0 k = −1

ρ > 0 P 0 ä < 0


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τ R R

υ ≡ dRdτ

= R

a

da

dτ = H R,

H (τ ) = ȧ/a

υ

R

ȧ > 0 ä < 0

T = a/ȧ = H −1 a = 0

H −1

a2

τ ä

ρ̇ + 3 (ρ + P ) ȧ/a = 0.

P = 0

ρa3 = constante,

P = ρ/3

ρa4 = constante.

a → 0

k = 0 −1 ȧ

P 0 ρ a−3 ρa2 → 0


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a → ∞ k = 0 ȧ τ → ∞ k = −1 ȧ → 1 τ → ∞

k = +1

a

ac a ac a

ac τ → ∞ ä k = +1 ac

ρ

ȧ2 − C/a + k = 0,

C = 8πρa3/3

ȧ2

− C /a2

+ k = 0,

C = 8πρa4/3

a(τ )

P = 0 P = 13 ρ

k = +1 a = 12 C (1 − cosη) a =√

C

1 −

1 − τ /√ C 21/2

τ = 12 C (1 − sinη) k = 0 a = (9C/4)

1/3 τ 2/3 a = (4C )1/4 τ 1/2

k = −1 a = 12 C (coshη − 1) a =√

C

1 + τ /√

C 2

− 11/2

τ = 12 C (sinhη − η)

a (τ ) τ


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P 1 τ 1

ω1 P 2

τ 2 ω2

ka

ua

ω = −kaua.

P 1

P 2

ka


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ξ a ka Σ1

P 1 ka

Σ2 P 2

ω2ω1

= a (τ 1)

a (τ 2).

z

z = λ2 − λ1

λ1=

ω2ω1

− 1 = a (τ 1)a (τ 2)

− 1.

τ 2 − τ 1 ≈ R R

a (τ 2) ≈ a (τ 1) + (τ 2 − τ 1) ȧ,

z ≈ ȧa

R = H R

a (τ ) τ

P

P P P

ds2 = −dτ 2 + a2 (τ ) dx2 + dy2 + dz2 .


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t =

ˆ dτ

a (τ ),

ds2 = a2 (t)−dt2 + dx2 + dy2 + dz2 .

ds2 = −dt2 + dx2 + dy2 + dz2.

P

t

τ → 0

a (τ ) ατ α τ → 0

t −∞

t = constante

k = 0 a (τ ) ∝ τ 2/3 a (τ ) P > 0 τ → 0

τ → 0 a (τ ) k


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a

a

a a → 0

tE a

tE ∼ a/ȧ = 2τ.


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tI ∝ a3/σ ∝ τ 3/2/σ (T ) ,

σ (T )

σ

tI tE tI > tE

(T → ∞) (ρ → ∞) (tI > tE )

a

10−43

G /c3

≈ 10−33cm.


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S M S t S t = 0 xa (a = 1, 2, 3)

ds2

= −(N 2

− N aN a

)dt2

+ 2N adxa

dt + habdxa

dxb

,

N N a N N a xa S 3

S =

ˆ (LG + Lm)d

3x.dt,

LG = 116π

N (GabcdK abK cd + h1/2(3)R),

K ab = 1

2N

−∂hab

∂t + 2N (a|b)

Evolucion de Ondas Gravitacionales

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