learning about astrophysical black holes with gravitational...

Learning about Astrophysical Black Holes with

Gravitational Waves

Recent Developments in General Relativity, Jerusalem, 22 May 2017

Image: Steve Drasco, California Polytechnic State University and MIT

How gravitational waves teach us about black holes and probe strong-field gravity

Scott A. Hughes, MIT

Subramanyan Chandrasekhar The Nora and Edward Ryerson Lecture,

University of Chicago, 22 April 1975.

“In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein’s equations

of general relativity provides the absolutely exact representation

of untold numbers of black holes that populate the universe.”

Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT

“It is well known that the Kerr solution … provides the unique solution for stationary

black holes … in the universe.

But a confirmation of the metric of the Kerr spacetime (or some aspect of it) cannot even be contemplated in the

foreseeable future.”

Subramanyan Chandrasekhar The Karl Schwarzschild Lecture, Astronomischen Gesellschaft, Hamburg, 18 September 1986


Understanding BHs and GWsBoth black holes and gravitational waves are solutions of the vacuum Einstein equations:

G�� = 0To study black holes orbiting

one another and the GWs they generate, “just” need to write down initial data, and solve this equation …

Essentially solved now … after several decades of focused effort.


Result: Gravitational waves carry imprint of orbit dynamics. Waves’ phase comes from kinematics of black holes as they orbit about one another.

Simple limit for intuition: Treat binary’s kinematics with Newtonian gravity, add lowest contribution to waves.

Eorb = �GMµ

2r� =

�GM

r3

dE

dt=

G

5c5

d3Ijk

dt3d3Ijk

dt3=

32G

5c5�6µ2r4

Energy radiated away causes r to slowly decrease, so orbit frequency slowly increases.

Understanding BHs and GWs


Result: Gravitational waves carry imprint of orbit dynamics. Waves’ phase comes from kinematics of black holes as they orbit about one another.

Result:

Frequency sweeps up at a rate controlled by the chirp mass … measure the rate at which the frequency chirps, you measure this mass.

�(t) =

�5

256

�c3

GM

�5/3 1(tc � t)

�3/8

Defined the chirp mass: M � µ3/5M2/5

Understanding BHs and GWs


Beyond the leading bit

Additional terms introduce dependence on other mass terms … can measure more combinations

than just chirp mass from inspiral.

Preceding analysis uses only Newtonian gravity:

Can regard this as the leading piece of full relativistic gravity


Can keep going


[Blanchet 2006, Liv Rev Rel 9, 4, Eq. (168)]

… and going.


GravitomagnetismMagnetic-like contribution to the spacetime

drives magnetic-like precession of binary members’ spins.

Orbital motion contribution.

Contribution from other body’s spin

Leads to new forces, modifying the orbital acceleration felt by each body.


Magnetic-like contribution to the spacetime drives magnetic-like precession of

binary members’ spins.

Angular momentum is globally conserved: J = L + S1 + S2 = constant

Orbital plane precesses to compensate for precession of the individual spins.

Gravitomagnetism

Scott A. Hughes, MIT Recent Developments in General Relativity, Jerusalem, 22 May 2017

Gravitomagnetism

Scott A. Hughes, MIT Recent Developments in General Relativity, Jerusalem, 22 May 2017


Simple chirp of two non-spinning black holes.

GWs with spin vs GWs withoutInfluence of spin strongly imprints the waveform.


Modulated chirp of two rapidly spinning black holes.

RingdownFinal waves: Last bit of radiation to leak out of the system as it settles down to the Kerr state.

Frequency and damping of these modes depend on and thus encode mass and spin of remnant BH.

hring = Ae�t/�ring(Mfin,afin) sin [2�fring(Mfin, afin) + �]

Example waveform: A few final cycles

of inspiral followed by ringdown.


Frequency bands

10s to 100s of Msun: 100s to 10s of Hz.

Right in the sensitive band of LIGO and

other ground-based GW detectors.

Classical GR has no intrinsic scale: Frequencies which characterize GWs from black hole systems

are determined by the mass scale.

finspiral � (0.02� 0.05)c3

GM fringdown � (0.06� 0.15)c3

GM


GW150914Last ~8 cycles, corresponding to moments when binary’s members merged into one:

Observed signal (loud enough to

stand above noise) consistent with template that assumes GR’s black holes.


m1 = (36 ± 4)M☉ m2 = (29 ± 4)M☉ mfin = (62 ± 4)M☉

afin = 0.67 [+0.05/-0.07] Δm = (3 ± 0.5)M☉Scott A. Hughes, MIT

GW151226About 55 cycles detected, corresponding to last several dozen orbits when the binary’s

members were still well separated:

Signal needed correlation with a theoretical template in order to be detected in the noise.


m1 = (14.2 [+8/-4])M☉ m2 = (7.5 ± 2.3)M☉ mfin = (20.8 [+6/-2])M☉

afin = 0.74 ± 0.06 Δm = (1 [+0.1/-0.2])M☉Scott A. Hughes, MIT

Signal versus noiseImproved detectors will enhance our ability to learn about BH properties from the coalescence waves:

Colpi & Sesana, arXiv:1610.05309


Reduced low-f noise improves inspiral signal: More signal in band, enables better masses and spins; better knowledge of position on sky, distance to binary.


Signal versus noiseImproved detectors will enhance our ability to learn about BH properties from the coalescence waves:

Colpi & Sesana, arXiv:1610.05309


Reduced high-f noise improves ringdown signal: Better mass and spin of final remnant; measure mixture of modes present at end of coalescence.


Frequency bands

A few 104 to a few 107

Msun: 10-5 — 1 Hz. Waves in the sensitive band of LISA … can be

“heard” to high redshift.

finspiral � (0.02� 0.05)c3


GM





Frequency bands

~108 through ~1010 Msun: Nanohertz frequencies. Targets for pulsar timing

arrays … can probe massive black hole

mergers to low redshift. Movie courtesy Penn State Gravitational Wave Astronomy Group, http://gwastro.org

finspiral � (0.02� 0.05)c3


GM





http://gwastro.org

Go to space to escape low-frequency noise: sensitive in band ~3×10-5 Hz < f < 1 Hz

A target-rich frequency band.

LISA: tentative 2034 launch


Several million km interferometer antenna in space. ESA mission … details in flux. Working for NASA involvement.

TeV Particle Astrophysics, 13 Sept 2016

LISA metrologyThanks to its much longer arms,

effect of a GW is relatively large: h =�L

L

Ground: h ≲ 10-21, L ~ kilometers: ΔL ≲ 10-3 fm

Space: h ≲ 10-20, L ~ 106 kilometers: ΔL ≲ 10 pmAbout an order of magnitude from fringe shift of original Michelson interferometer.

Measured at DC


using his eyeball.Scott A. Hughes, MIT

LISA noiseFar more challenging: Ensuring the noise budget can

be met for each element of a free-flying constellation of spacecraft.

LISA Pathfinder: Testbed for technologies to demonstrate that free fall, control, and metrology can be done with

the precision needed for LISA.

Launched: 3 Dec 2015 Arrived at L1: 22 Jan 2016

Began science operations: 8 Mar 2016


LISA noiseFar more challenging: Ensuring the noise budget can

be met for each element of a free-flying constellation of spacecraft.

Significantly exceeded

mission spec.LISA is

within reach.

Figure 1 of Armano et al, Phys. Rev. Lett. 116, 231101 (2016).


Science goalsScience reach in this band known for some time.

Track growth and evolution of massive black holes from z ~ 15.

Precisely measure black hole properties, test nature of gravity near them.

Explore dynamical stellar populations around black holes in galaxy centers.

Survey population of stellar-mass compact remnants in Milky Way and into low-z universe.

Constrain or probe exotic physics in the early universe.

Antenna sensitivity to a variety of low-frequency sources.

Taken from Gravitational Observatory Advisory Team (GOAT) report.


Source goalsScience goals met by measuring a range of sources

that oscillate on periods of minutes to hours.Massive black hole binaries: Form as consequence of the hierarchical galaxy growth, in band for months to years.

Extreme mass ratio binaries: Capture of stellar-mass compact body by massive black hole; also in band for months to years.

Compact binaries: Stellar mass binaries in our galaxy (low masses) to z ~ 0.1 (high mass).

Processes in the early universe


Massive black hole scienceGalaxies were built hierarchically: Big galaxies

assembled through repeated mergers of subunits.Evidence from quasars tells us that black holes have

existed since the earliest cosmic times.

Combining these facts indicates that massive black hole mergers should be relatively common. As long as

there are mergers with total masses

(a few) 104 ≤ (1 + z) M/Msun ≤ (a few) 107

they will be in band of a space-based low-f detector, and detectable out to z ~ 15.


Extreme mass ratio inspiralsCapture of stellar mass compact objects onto

relativistic orbits of black holes in galaxy cores.Galactic nucleus

Galaxy Massive Black Hole

Stellar cluster

EMRI setting, courtesy Marc D. FreitagScott A. Hughes, MIT

Similar to galactic center S-stars

Animation courtesy Genzel group, Max-Planck-Institut für Extraterrestrische Physik

Analogous to orbits we see in center of our

galaxy, but much closer to large black hole …

also smaller body must be compact (NS star or small BH) else it will

tidally disrupt.


Relativity viewSpecial limit of two-body problem: One body far more massive than other. Binary dominated by large black hole … GWs encode its properties.

Large mass ratio consequences:

1. Model perturbatively: Can understand system using simpler equations than full Einstein.

2. Evolve slowly: Duration scales as (Mbig/Msmall), small body slowly spirals through strong-field of big BH.

Expect ~105 cycles in band … need very precise models to

accurately match phase.Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT

Need precise wave modelsInstantaneous EMRI amplitude will typically

be factor ~10 — 100 smaller than noise!

Data analysis rule of thumb: Coherently matching wave for N cycles boosts SNR by N1/2.

Need to develop models capable of tracking

system for ~105 orbits deep in Kerr strong field.


Need precise wave models

Data analysis rule of thumb: Coherently matching wave for N cycles boosts SNR by N1/2.

Need to develop models capable of tracking

system for ~105 orbits deep in Kerr strong field.


Instantaneous EMRI amplitude will typically be factor ~10 — 100 smaller than noise!

Need precise wave models

Measure mass, spin, mass ratio: δM/M, δa, δη ~ 10-4 — 10-2

Measure orbit’s geometry: δe0 ~ 10-3 — 10-2

δ(spin direction) ~ a few deg2 δ(orbit plane) ~ 10 deg2

Measure distance to binary: δD/D ~ 0.03 — 0.1

Barack & Cutler PRD 69, 082005 (2004)


Instantaneous EMRI amplitude will typically be factor ~10 — 100 smaller than noise!

Measure “shape” of KerrKerr metric only depends on two parameters … but has a “shape” that can be characterized by

an infinite number of multipole moments.Analogy: Newtonian potential of a gravitating body.

Blm coefficients determine potential’s “shape,” can be mapped by orbits. Connect to an interior description… they tell us how mass is distributed.


Kerr metric only depends on two parameters … but has a “shape” that can be characterized by

an infinite number of multipole moments.Analogy: Newtonian potential of a gravitating body.

Blm coefficients determine potential’s “shape,” can be mapped by orbits. Connect to an interior description… they tell us how mass is distributed.

GRACE gravity modelRecent Developments in General Relativity, Jerusalem, 22 May 2017

Measure “shape” of Kerr


In GR, two families of multipole moments are needed to describe spacetimes:

Ml: “Mass multipole.” For a fluid body, describes angular distribution of mass.

Sl: “Current multipole.” For a fluid body, describes angular distribution of mass flow.

Ml + iSl = M(ia)l

For black hole spacetimes, there is a very simple relation between these moments:


Measure “shape” of Kerr


Orbit spectroscopyKerr black hole orbits are characterized by 3 frequencies, determined by hole’s mass and

spin, and by the orbit geometry:Asymptotes to Kepler’s

law at large r

rmin

fΩr: freq. of radial motion Ωθ: freq. of polar motion Ωφ: freq. of axial motion

Orbits of bodies in strong-field of Kerr spacetime periodic atthese frequencies, and generate gravitational waves with these frequencies.


If multipole moments differ from those of Kerr, then the orbital frequencies will differ.

One example of how orbit frequencies are shifted from the Kerr values, if the black

hole has the “wrong” l = 2 moment.

Orbit spectroscopy


From Vigeland & Hughes, PRD 81, 082002 (2010)


If multipole moments differ from those of Kerr, then the orbital frequencies will differ.

Precision measurement of an inspiral will track

phase through a sequence of orbital frequencies … null hypothesis is that these frequencies will

only depend on the black hole’s mass and spin.

Orbit spectroscopy


Horizon coupling

Flux to infinity has simple behavior:�

dE

dt

��> 0 Always takes energy

away from the binary.


When we calculate waves from a binary, we have two pieces: Flux that goes to infinity, and flux that

goes down the black hole’s event horizon.


Horizon couplingWhen we calculate waves from a binary, we have

two pieces: Flux that goes to infinity, and flux that goes down the black hole’s event horizon.

Horizon flux is a bit weird: Its sign depends on relative frequency of orbit and BH spin.

Horizon takes energy away if orbit is faster

than hole’s spin …

�dE

dt

�H

> 0 If Ωorb > ΩH


Horizon coupling

… but adds energy to orbit if hole spins faster than orbit.

If Ωorb < ΩH

�dE

dt

�H

< 0


When we calculate waves from a binary, we have two pieces: Flux that goes to infinity, and flux that

goes down the black hole’s event horizon.

Horizon flux is a bit weird: Its sign depends on relative frequency of orbit and BH spin.


Example: 1 Msun body spiraling into 106 Msun

black hole; spin 5% max.

Turn off horizon flux, inspiral slightly slowed:

Takes about an extra day over 18 month inspiral.

In accord with intuition: Flux from horizon takes energy from orbit faster.

Horizon coupling


From Hughes, PRD 64, 064004 (2001)



Same system, but with spin 99.8% maximum:With horizon flux Without

Horizon coupling

Horizon flux slows inspiral by up to 4 weeksRecent Developments in General Relativity, Jerusalem, 22 May 2017

Why doesn’t horizon always absorb energy from the energy?

Consider an apparently totally different effect in Newtonian gravity: Tidal orbit coupling.

Consider a “moon” orbiting a fluid “planet”:

Gravity of moon raises tide on planet … planet

bulges in response.


Planet’s fluid is viscous. The bulging response will lag applied tide in time.






If planet spins slower than orbit’s frequency, the

bulge lags orbit’s position.






Bulge exerts a torque that takes angular momentum from orbit … slowing it down (and speeding up planet’s spin).






If planet spins faster than orbit’s frequency, bulge leads orbit’s position.






Bulge exerts a torque that adds angular momentum to orbit … speeding it up (and slowing down planet’s spin).






Net effect: Orbit loses energy if Ωorb > ΩP

Orbit gains energy if Ωorb < ΩP

Exactly like the horizon absorption effect.






Tidal distortion of horizonsFlux language misleads … but can be recast in a

dual description as tidal deformation of BH.Foundations developed by Hawking and Hartle: Key point is similarity between entropy generation in fluids and entropy generation in event horizon.

Fluids: Flow lines sheared by tide. Sheared fluid generates heat, which

generates entropy. Rate set by viscosity.

TfluiddS

dt= 2� �ij�

ij

η = shear viscosity

S = entropy generated in fluid

Tfluid = fluid temperature

Image credit: Wikipedia article “Streamlines, streaklines, and pathlines"

σij = shear of fluid (trace-free part of gradient of fluid velocity)


Foundations developed by Hawking and Hartle: Key point is similarity between entropy generation in fluids and entropy generation in event horizon.

Black hole: Horizon generators sheared by tide. Black hole mechanics: This generates area which (via

Bekenstein and Hawking) is entropy. Constant of

proportionality is viscosity.

THdS

dt= 2�H �µ��µ�

σµν = shear of horizon generators (trace-free part of gradient of

generators’ 4-momentum)ηH = Horizon shear viscosity =

(1/16π)(c3/G)

TH = Bekenstein-Hawking temperature


Tidal distortion of horizonsFlux language misleads … but can be recast in a

dual description as tidal deformation of BH.


Horizon couplingUsing this language, recast the down-horizon GW

flux describes “tidal bulge” on horizon.


O’Sullivan & Hughes, PRD 90, 124039 (2014); PRD 94, 044057 (2016).


Another null test: Does the horizon coupling agree with the Kerr horizon viscosity?


LIGO’s observations have validated the promise of GW astronomy, unveiling new BH populations

and probing gravity as never before.

Strong-field gravity is now becoming a data-driven subject.

The end of the beginning


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