Dynamics of star clusters containing stellar mass black ...Dynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves July 25, 2017 Bonn

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


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


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


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


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�⌫


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:

7

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


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


Choice of guage (i.e., coordinates)

• Define a trace-reversed perturbation

9

and impose the Lorentz guage condition

then equation for becomes

hµ⌫ = hµ⌫ � 1

2h⌘µ⌫

@hµ⌫

@x⌫= 0

hµ⌫

⇤hµ⌫ = �16⇡GTµ⌫


Further Properties• In vacuum (i.e., T𝜇𝜈=0),

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⇤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


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


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


How small is h~10-21

which was the detected amplitude of the GW150914?

14

Simulation created by T. Pyle, Caltech/MIT/LIGO Lab


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


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


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


About 300 million years after, the following event is expected (Movie credit: Gwanho Park [SNU])

19


The final moment of merger of black holes (Movie credit: Han-Gil Choe [SNU])

20


Comparison of waveforms between NS and BH mergers: Note differences in frequencies and shape .

21

Waveform tells many thingsNeutron star binary

Black hole binary

high f

low f


GW from a merging black hole

22

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)


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

25

h ⇠ N�1/2photons

�laser

�GW


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

27


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

28


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)

29

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


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

30


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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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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https://www.lorentz.leidenuniv.nl/lorentzchair/thorne/Thorne1.pdf

http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/

Documents

Dynamics of star clusters containing stellar mass black ...Dynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves July 25, 2017 Bonn