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Dynamics of star clusters containing stellar mass black holes:
1. Introduction to Gravitational Waves
July 25, 2017 Bonn
Hyung Mok Lee Seoul National University
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Outline• What are the gravitational waves?
• Generation of gravitational waves
• Detection methods
• Astrophysical Sources
• Detected GW Sources2
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Principle of Equivalence
• Principle of Equivalence: inertial mass = gravitational mass
3
• The trajectory of a particle in gravity does not depend on the mass: geometrical nature
• In a freely falling frame, one cannot feel the gravity.
• However, the presence of the gravity will cause tidal force. Again, this is similar to the difference between flat and curved space.
F = mia = mgg
g = �GM
r3r
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Curved Spacetime
• General geometry of space-time can be characterized by metric
4
• In the absence of gravity, flat spacetime
• The presence of gravity causes deviation from flat spacetime: curved spacetime
ds
2 = gµ⌫dxµdx
⌫
gµ⌫ = ⌘↵� =
0
BB@
�1 0 0 00 1 0 00 0 1 00 0 0 1
1
CCA
ds
2 = �dt
2 + dx
2 + dy
2 + dz
2
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Tidal gravitational forces• Let the positions of the two nearby particles be x and x+𝝌, then the equations of motions are
5
• If 𝝌 is small, one can expand 𝚽
d
2(xi + �
i)
dt
2= ��
ij @�(x+ �)
@x
j
• Then the relative acceletation between two particles becomes
@�(x+ �)
@x
j⇡ @�(x)
@x
j+
@
@x
k
✓@�(x)
@x
j
◆�
k + ...
Tidal acceleration tensor
d
2x
i
dt
2= ��
ij @�(x)
@x
j,
d
2�
i
dt
2= ��
ij
✓@
2�
@x
j@x
k
◆
x
�
k
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
In GR, similar expression can be derived
• "Geodesic deviation" equation
6
• In the weak field limit, v≪c, |𝝫|≪c2. The only remaining components are 𝜇=ν =0. Therefore,
i.e.,
Riemann curvature tensorD
2�
�
D⌧
2= R
�⌫µ⇢
dx
⌫
d⌧
dx
⇢
d⌧
�
µ
d2�i
dt2= Ri
0j0�j
R
i0j0 = � @
2�
@x
i@x
j
• Ricci tensor can be defined by contraction of Riemann tensor:Rµ⌫ = R�
µ�⌫
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Einstein's Field Equation• In Newtonian dynamics, the motion of a particle is governed by the
gravitational potential which can be computed by the Poisson's equation:
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i.e., summation over the quantities that describe the geodesic deviation (=tidal acceleration tensor).
• The Ricci tensor is similarly defined with Φ, i.e., summation of all tidal components.
• In vacuum,
r2� = �
ij
✓@
2�
@x
i@x
j
◆= 4⇡G⇢
Rµ⌫ = 0, similar to r2� = 0
• In the presence of energy (mass, etc.), Einstein's field equation becomes
Rµ⌫ � 1
2Rgµ⌫ = �8⇡G
c4Tµ⌫
Curvature scalar
Energy-momentum tensor
R = Rµµ
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Field equation in weak field limit
• In the weak field limit,
8
• Einstein's field equation becomes
• Perform coordinate transformation,
gµ⌫ = ⌘µ⌫ + hµ⌫ , |hµ⌫ | ⌧ 1
where
flat part small perturbation
⇤hµ⌫ =@
2
@x
�@x
µh
�⌫ � @
2
@x
�@x
⌫h
�µ +
@
2
@x
µ@x
⌫h
�� = �16⇡GSµ⌫
Sµ⌫ ⌘ Tµ⌫ � 1
2⌘µ⌫T
�� and ⇤ = � @2
@t2+r2
x
µ ! x
0µ = x
µ + ⇠
µ
• Then, h
0µ⌫ = hµ⌫ � @⇠µ
@x
⌫� @⇠⌫
@x
µ
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Choice of guage (i.e., coordinates)
• Define a trace-reversed perturbation
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and impose the Lorentz guage condition
then equation for becomes
hµ⌫ = hµ⌫ � 1
2h⌘µ⌫
@hµ⌫
@x⌫= 0
hµ⌫
⇤hµ⌫ = �16⇡GTµ⌫
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Further Properties• In vacuum (i.e., T𝜇𝜈=0),
10
⇤hµ⌫ = 0
• The solution can be written in the form
¯
hµ⌫ = Re{Aµ⌫ exp(ik�x�)}
• One can choose a coordinate system by rotating so that
Aµµ = 0 (traceless), A0i = 0 purely spatial
• The wave is transverse in Lorentz gauge. Therefore the metric perturbation becomes very simple in transverse-traceless (TT) gauge. Also in TT coordinates.hµ⌫ = hµ⌫
i.e, equation for plane waves
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
Effects of GWs in TT gauge• In TT guage, GWs traveling along z-direction can be written with only
two components, h+ and hx: they are called ‘plus’ and ‘cross’ polarizations
11
hTTµ⌫ = h+(t� z)
0
BB@
0 0 0 00 1 0 00 0 �1 00 0 0 0
1
CCA+ h⇥(t� z)
0
BB@
0 0 0 00 0 1 00 1 0 00 0 0 0
1
CCA
• In these coordinates, the line element becomes
ds
2 = �dt
2 + (1 + h+)dx2 + (1� h+)dy
2 + dz
2 + 2h⇥dxdy
• The lengths in x- and y- directions for h+ then oscillate in the following manner
L
x
=
Zx2
x1
p1 + h+dx ⇡ (1 +
1
2h+)Lx0;
L
y
=
Zy2
y1
p1� h+dy ⇡ (1� 1
2h+)Ly0
h+ hx
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Generation of gravitational waves• The wave equation
gives formal solution of
• One can show that
hjk(t,x) =2rd2Ijk(t�r)
dt2
• Using the identities@T
tt
@t
+@T
kt
@x
k= 0, and
@
2T
tt
@t
2=
@
2T
kl
@x
k@x
l
⇤hµ⌫ = �16⇡GTµ⌫
hµ⌫(t,x) = 4
ZTµ⌫(t� |x� x
0|,x0)
|x� x
0| d
3x ⇡ 4
r
ZTµ⌫(t� |x� x
0|,x0)d3x
whereIjk =
Z⇢x
jx
kd
3x
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Order of magnitude estimates of GW amplitude
• Note: the moment of inertia tensor is not exactly the same as the quadrupole moment tensor, but in TT guage, it does not matter.
• Quadrupole kinetic energy
¨Ijk ⇠ (mass)⇥(size)2
(transit time)2 ⇠ quadrupole Kinetic E. = ✏Mc2
• In most cases, 𝝐 is small, but it could become ~0.1
hjk ⇠ 10�22�
✏0.1
� ⇣MM�
⌘⇣100Mpc
r
⌘
Qjk =
Z⇢
✓x
jx
k � 1
3r
2�jk
◆d
3x
Ijk =
Z⇢x
jx
kd
3x
Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
How small is h~10-21
which was the detected amplitude of the GW150914?
14
Simulation created by T. Pyle, Caltech/MIT/LIGO Lab
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Measurement of GWs: 1. Tidal forces• Equation of geodesic deviation becomes GW tidal acceleration:
d
2�x
j
dt
2= �Rj0k0x
k =1
2hjkx
k
• Riemann tensor Rj0k0 is a gravity gradient tensor in Newtonian limit
• Gravity gradiometer can be used as a gravitational wave detector
Rj0k0 = � @
2�
@x
j@x
k= �!
2GWhjk
• Tidal force can cause resonant motion of a metallic bar = bar detector
Force field
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Measurement of GWs: 2. Laser Interferometer
• Consider a simple Michelson interferometer with lx ≈ ly ≈l.
• The phase difference of returning lights reflected by x and y ends
• Toward the laser:
• Toward the photo-detector:
• For small Δ𝜙,
�� = �x
� �y
⇡ 2!0 [lx � ly
+ lh(t)]
Elaser / ei�x + ei�y
EPD / ei�x � ei�y = ei�y (ei�� � 1)
IPD
/ |ei�� � 1|2 ⇡ |��|2 ⇡ 4!20(lx � l
y
)2 + 8!20(lx � l
y
)lh(t)
Time varying part
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Astrophysical sources• Compact Binary Coalescence (neutron stars
and black holes) • Strong signal, but rare • Computable waveforms
• Continuous (~ single neutron stars) • Weak, but could be abundant • Deviation from axis-symmetry is not known
• Burst (supernovae or gamma-ray bursts) • GW amplitudes are now well known, not
frequent • Stochastic
• Superposition of many random sources • Could be useful to understand distant
populations of GW sources or early universe
Black hole binaries
Ijk =
Z⇢x
jx
kd
3x
Confirmation of GW with Binary Pulsar
• Orbit of binary neutron stars shrinks slowly
• Hulse & Taylor received Nobel Prize in 1993 for the discovery of a binary pulsar Weisberg & Taylor 2005
P=7.75 hrs a=1.6 x 106 km M1=1.4 Msun M2=1.35 Msun Time to merge=3x108 yrs
PSR 1913+16
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
About 300 million years after, the following event is expected (Movie credit: Gwanho Park [SNU])
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
The final moment of merger of black holes (Movie credit: Han-Gil Choe [SNU])
20
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Comparison of waveforms between NS and BH mergers: Note differences in frequencies and shape .
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Waveform tells many thingsNeutron star binary
Black hole binary
high f
low f
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
GW from a merging black hole
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LIGO-G1600341
Principles of GW DetectorImage credit: LIGO/T. Pyle
LIGO-G1600341
Gravitational Wave Detector 1990
K. Thorne, R. Weiss, R. Drever
• 2002-2010 Initial LIGO • 2015.9 ~ Advanced LIGO (10times better sensitivity)
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Simple Estimates of Sensitivity of Interferometers
• If the length resolution is 𝛌laser, detectable strain is
• However, due to quantum nature of the photons, the length resolution coud be as small as N -1/2
photons𝛌laser. Thus sentivity could reach
h ⌘ �l
l=
�laser
l=
10�6m
103m= 10�9
• Optical path length can be significantly increased by adopting optical cavity, but should be smaller than GW wavelength (~1000 km for 300 Hz)
h ⇠ �l
leff⇠ �laser
�GW⇠ 10�6m
106m= 10�12
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h ⇠ N�1/2photons
�laser
�GW
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Shot Noise• Collect photons for a time of the order of the period of GW
wave
Nphotons
=Plaser
hc/�laser
⌧ ⇠ Plaser
hc/�laser
1
fGW
⌧ ⇠ 1/fGW
• For 1W laser with λlaser=1 µm, fGW=300Hz, Nphotons=1016
• By adopting high power laser (20W for O1) and power recycling, we can reach ‘astrophysical sensitivity’ of ~10-22.
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h ⇠ �l
leff
⇠N�1/2
photons
�laser
�GW
⇠ 10�8 ⇥ 10�6m
106m= 10�20
LIGO-G1601201-v3
Other Noises• Radiation pressure noise
• Make mirrors heavier • Suspension thermal noise/ mirror coating
brownian noise • Increase beam size, monolithic
suspension structure • Seismic noise
• Multi-stage suspension, underground • Newtonian Noise
• So far difficult to avoid. • Seismic and wind measurement and
careful modeling
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
GW Events from O1/O2 • GW150914
(FAR<6x10-7 yr-1)
• LVT151012 (Candidate, FAR~0.37 yr-1)
• GW151226 (FAR<6x10-7 yr-1)
• GW170104 (<5x10-5 yr-1
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
What made these detections interesting?
• LIGO detected gravitational waves from merger of black hole binaries, instead of neutron star binaries
• Black hole mass range was quite large: GW150914 is composed of 36 and 29 times of the mass of the Sun
• Most of the known black holes are much less massive (~10 times of the sun)
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0
1
2
3
4
5
6
7
5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
Num
ber
Mass(Msun)
BHMassDistribu3on
LVT151012
GWsources
X-rayBinaries
Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
What these results tell us? (Abbott et al., 2016, ApJL, 828, L22; PHYSICAL REVIEW X 6, 041015 (2016)
PRL 118, 221101 (2017))
• Proof of the existence of the black holes
• Existence of stellar mass black holes in binaries
• Individual masses in wide range (7-35 Msun)
• Formation of ~60 Msun BH
• BH appears to be much more frequent than previously thought
• Current estimation of the rate 12-210 yr-1 Gpc-1
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Hyung Mok LeeIMPRS Blackboard Lecture,July 25, 2017
Summary of the GW sources• GW150914:
• First Unambiguous detection of stellar mass black holes and a BH binary
• Accurate measurement of black hole masses (within ~10%) • Higher mass of stellar mass BH than previously thought: low
metallicity environment? • GW151226:
• Lower masses than GW150914, similar to the X-ray binary BH mass
• Lower mass progenitor or high metallicity environment? • GW170104
• High mass (~50 Msun) • Spin may not be aligned
• Origin • Isolated or dynamical?
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Hyung Mok LeeIMPRS Blackboard Lecture, July 25, 2017
References• S. Weinberg, "Gravitation and Cosmology: Principles and applications of the general theory of
relativity", 1972, John Wiley & Sons • J. B. Hartle, "Gravity, an introduction to Einstein's General Relativity". 2003, Addison Wesley • Lecture notes by K. Thorne:
• https://www.lorentz.leidenuniv.nl/lorentzchair/thorne/Thorne1.pdf • http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/ (together with R. Blandford,
Chapters 25 & 27) • Recent LIGO papers
• Abbott et al. 2016, "Abbott et al. 2016, "Observation of Gravitational Waves from a Binary Black Hole Merger", 2016, PRL, 116, 061102
• Abbott et al., 2016, "ASTROPHYSICAL IMPLICATIONS OF THE BINARY BLACK HOLE MERGER GW150914" , ApJL, 828, L22
• Abbott et al. 2016, "GW150914: Implications for the Stochastic Gravitational-Wave Background from Binary Black Holes", PRL, 115, 131102
• Abbott et al. 2016, "GW150914: The Advanced LIGO Detectors in the Era of First Discoveries", 2016, PRL, 116, 131103
• Abbott et al., 2017, “GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence”, PRL, 118, 221101
at Re
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