Wave Zone Physics: Radiative losses › ~cmw › PHZ7608 › Lecture6.pdfWave Zone Physics:...
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Wave Zone Physics: Radiative losses Balance equations (-g)t ↵β LL := c 4 16⇡G ( @ λ g ↵β @ μ g λμ - @ λ g ↵λ @ μ g βμ + 1 2 g ↵β g λμ @ ⇢ g λ⌫ @ ⌫ g μ⇢ - g ↵λ g μ⌫ @ ⇢ g β⌫ @ λ g μ⇢ - g βλ g μ⌫ @ ⇢ g ↵⌫ @ λ g μ⇢ + g λμ g ⌫⇢ @ ⌫ g ↵λ @ ⇢ g βμ + 1 8 ( 2g ↵λ g βμ - g ↵β g λμ )( 2g ⌫⇢ g σ⌧ - g ⇢σ g ⌫⌧ ) @ λ g ⌫⌧ @ μ g ⇢σ ) hhn j ii =0 , hhn j n k ii = 1 3 δ jk , hhn j n k n n ii =0 , hhn j n k n n n p ii = 1 15 ( δ jk δ np + δ jn δ kp + δ jp δ kn ) Tools: @ j h ↵β = - 1 c N j @ ⌧ h ↵β + O(R -2 ) g ↵β = ⌘ ↵β - h ↵β @ β h ↵β =0= 1 c ( @ ⌧ h ↵0 - N j @ ⌧ h ↵j ) + O(R -2 ) dE dt = -c I 1 (-g )t 0k LL dS k dP j dt = - I 1 (-g )t jk LL dS k dJ jk dt = - I 1 ⇥ x j (-g )t kn LL - x k (-g )t jn LL ⇤ dS n
Transcript of Wave Zone Physics: Radiative losses › ~cmw › PHZ7608 › Lecture6.pdfWave Zone Physics:...
Wave Zone Physics: Radiative losses Balance equations
(�g)t↵�LL :=c4
16⇡G
(@�g
↵�@µg�µ � @�g
↵�@µg�µ +
1
2g↵�g�µ@⇢g
�⌫@⌫gµ⇢
� g↵�gµ⌫@⇢g�⌫@�g
µ⇢ � g��gµ⌫@⇢g↵⌫@�g
µ⇢ + g�µg⌫⇢@⌫g
↵�@⇢g�µ
+1
8
�2g↵�g�µ � g↵�g�µ
��2g⌫⇢g�⌧ � g⇢�g⌫⌧
�@�g
⌫⌧@µg⇢�
)
hhnjii = 0 ,
hhnjnkii = 1
3�jk ,
hhnjnknnii = 0 ,
hhnjnknnnpii = 1
15
��jk�np + �jn�kp + �jp�kn
�
Tools:
@jh↵� = �1
cNj@⌧h
↵� +O(R�2)
g↵� = ⌘↵� � h↵�
@�h↵� = 0 =
1
c
�@⌧h
↵0 �Nj@⌧h↵j�+O(R�2)
dE
dt
= �c
I
1(�g)t0kLL dSk
dP
j
dt
= �I
1(�g)tjkLL dSk
dJ
jk
dt
= �I
1
⇥x
j(�g)tknLL � x
k(�g)tjnLL⇤dSn
Wave Zone Physics: Energy flux dE
dt= c
Z@0⌧
00d3x
= �c
I(�g)t0jLLdSj
= � c3R2
16⇡G
I ⇥(@th+)
2 + (@th⇥)2⇤d⌦
= � G
5c5...Ihpqi ...
Ihpqi +O(c�7)
q Called the quadrupole formula for energy flux q Also known as the “Newtonian” order contribution q Also a flux of angular momentum dJ/dt and of linear momentum dP/dt
For a 2-body system: dE
dt= � 8
15⌘2
c3
G
✓Gm
c2r
◆4 �12v2 � 11r2
�
!/P
P
Energy flux: eccentric orbit
dE
dt
t/P
F =
32
5
⌘2c5
G
✓Gm
c2p
◆5
(1 + e cos�)41 + 2e cos�+
1
12
e2(1 + 11 cos
2 �)
�
dP
dt= �192⇡
5
✓GMc3
2⇡
P
◆5/3
F (e)
F (e) =1 + 73
24e2 + 37
96e4
(1� e2)7/2
Energy flux and binary pulsars Orbit-averaged flux
Period decrease
q “Newtonian” GW flux q 2.5 PN correction to Newtonian
equations of motion q PN corrections can be
calculated, now reaching 4 PN order
PSR 1913+16 Hulse-Taylor binary pulsar
E / a�1 / P�2/3
M ⌘ ⌘3/5m = chirpmass
dE
dt= �32
5⌘2
c5
G
✓Gm
c2a
◆5
F (e)
Energy flux and GW interferometers
From Blanchet, Living Reviews in Relativity 17, 2 (2014)
⌫ = ⌘ = M1M2/(M1 +M2)2
x = �
2/3 = (Gm⌦/c3)2/3 ⇠ (v/c)2
For a circular orbit, to 3.5 PN order:
dE
dt= �
r ≈ 5M
Baker et al. gr-qc/0612024
Energy flux and GW interferometers
Wave Zone Physics: Momentum flux
dP j
dt= � c2R2
32⇡G
ZN j
h�@⌧h+
�2+
�@⌧h⇥
�2id⌦
= �G
c7
✓2
63
(4)
I
hjpqi ...Ihpqi � 16
45✏jpq
...J
hpri ...Ihqri
◆
dP j
dt=
8
105�⌘2
c
G
✓Gm
c2r
◆4 vj✓50v2 � 38r2 + 8
Gm
r
◆
� rnj
✓55v2 � 45r2 + 12
Gm
r
◆�
� =m1 �m2
m1 +m2
⌘ =m1m2
(m1 +m2)2
q Note: recoil is opposite to velocity of lightest particle
q Final v is that when flux turns off (eg merger of 2 BH)
Radiation of momentum and the recoil of massive black holes
General Relativity § Interference between quadrupole and higher moments
Peres (62), Bonnor & Rotenberg (61), Papapetrou (61), Thorne (80) § “Newtonian effect” for binaries
Fitchett (83), Fitchett & Detweiler (84) § 1 PN & 2PN correction terms
Wiseman (92), Blanchet, Qusailah & CMW (05)
Astrophysics § MBH formation by mergers is affected if BH ejected from early galaxies
Schnittman (07) § Ejection from dwarf galaxies or globular clusters § Displacement from center could affect galactic core
Merritt, Milosavljevic, Favata, Hughes & Holz (04) Favata, Hughes & Holz (04)
J0927+2943 - 2600 km/s? Komossa et al 2008
How black holes get their kicks Favata, et al
Getting a kick out of numerical relativity Centrella, et al
A swift kick in the as trophysical compact object
Boot, Foot, et al
Total recoil: the maximum kick…. Gonzalez et al
Recoil velocity as a function of mass ratio
€
η =m1m2
(m1 + m2)2
=X
(1+ X)2
X = 0.38 Vmax = 250 ± 50km/s
Blanchet, Qusailah & CW (2005)
X=1/10 V = 70 ± 15 km/s
Comparing 2PN kick with numerical relativity
Baker et al (GSFC), gr-qc/0603204
Wave Zone Physics: Radiation reaction Loss of energy at order c-5 implies that the dynamics of a system cannot be conservative at 2.5 PN order
There must be a radiation reaction force F that dissipates energy according to
X
A
FA · vA =dE
dt
To find this force, we return to the near-zone and iterate the relaxed Einstein equations 3 times to find the metric to 2.5 PN order
That metric is inserted into the equations of motion r�T↵� = 0
There are Newtonian, 1 PN, 2 PN, 2.5 PN, …. terms (no 1.5 PN!) Happily, to find the leading 2.5 PN contributions, it is not necessary to calculate the 2 PN terms explicitly (though that has been done)
Wave Zone Physics: Radiation reaction Illustration: expand h00 in the near zone: Newtonian plus
corrections up to 2.5 PN order within τ00
No 0.5 PN term: conservation of M
1 PN correction d2X/dt2
Pure function of time – a coordinate effect
2 PN term
2.5 PN term
h00N =
4G
c4
⇢Z
M
⌧00
|x� x
0| d3x0 +
1
2c2@2
@t2
Z
M⌧00|x� x
0| d3x0
� 1
6c3@3
@t3
Z
M⌧00|x� x
0|2 d3x0 +1
24c4@4
@t4
Z
M⌧00|x� x
0|3 d3x0
� 1
120c5@5
@t5
Z
M⌧00|x� x
0|4 d3x0i+O(c�6)
�
Wave Zone Physics: Radiation reaction
Pulling all the contributions together, we find the equations of hydrodynamics to 2.5 PN order
⇢⇤dv
dt= ⇢⇤rU �rp+O(c�2) +O(c�4) + f
Where f is a radiation reaction force density. For body A
FA =
Z
A⇢
⇤fd3x
a[rr] =8
5⌘(GM)2
c5r3
✓3v2 +
17
3
GM
r
◆rn�
✓v2 + 3
GM
r
◆v
�For a 2-body system, this leads to a radiation-reaction contribution
This is harmonic gauge (also called Damour-Deruelle gauge)
Wave Zone Physics: Radiation reaction Alternative gauge: the Burke-Thorne gauge. All RR effects embodied in a modification of the Newtonian potential
U ! U � G
5c5d
5I
hjki
dt
5x
jx
k
a[rr] =8
5⌘(GM)2
c5r3
✓18v2 +
2
3
GM
r� 25r2
◆rn�
✓6v2 � 2
GM
r� 15r2
◆v
�For a two body system
In any gauge, orbital damping precisely matches wave-zone fluxes:
dE
dt= � 8
15⌘2
c3
G
✓Gm
c2r
◆4 �12v2 � 11r2
�,
dJj
dt= �8
5⌘2
c
G
✓Gm
c2r
◆3
hj
✓2v2 � 3r2 + 2
Gm
r
◆,
dP j
dt=
8
105�⌘2
c
G
✓Gm
c2r
◆4 vj✓50v2 � 38r2 + 8
Gm
r
◆
� rnj
✓55v2 � 45r2 + 12
Gm
r
◆�
Wave Zone Physics: Radiation reaction
Check dE/dt:
d
dt
✓v2r
r2
◆=
1
r3
✓v4 � v2
GM
r� 3v2r2 � 2r2
GM
r
◆
dE
dt= µv · a[rr] = 8
5⌘2m
(Gm)2
c5r3
✓3v2 +
17
3
Gm
r
◆r2 �
✓v2 + 3
Gm
r
◆v2�
But
dE⇤
dt= � 8
15⌘2
c3
G
✓Gm
c2r
◆4 �12v2 � 11r2
�
E⇤ = E +8
5⌘Gµm
r
✓Gm
c2r
v2r
c3
◆
Thus
Wave Zone Physics: Radiation reaction
dp
dt= �64
5⌘c
✓GM
c2p
◆3
(1� e2)3/2✓1 +
7
8e2◆,
de
dt= �304
15⌘c
e
p
✓GM
c2p
◆3
(1� e2)3/2✓1 +
121
304e2◆
Radiation reaction causes 2-body orbits to inspiral and circularize
Inserting aRR into the Lagrange planetary equation as a disturbing force and integrating over an orbit
q The Hulse-Taylor binary pulsar will circularize and merge within 300 Myr; the double pulsar within 85 Myr
q This is short compared to the age of galaxies (5 – 10 Gyr) q There must be NS-NS binaries merging today (possibly
even NS-BH and BH-BH binaries) q The inspiral of compact binaries is a leading potential
source of GW for interferometers
p=a(1-e2)
e
e0⇠
✓p
p0
◆19/12
