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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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
Can keep going
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
[Blanchet 2006, Liv Rev Rel 9, 4, Eq. (168)]
… and going.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Scott A. Hughes, MIT
Simple chirp of two non-spinning black holes.
GWs with spin vs GWs withoutInfluence of spin strongly imprints the waveform.
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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.
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017
Reduced high-f noise improves ringdown signal: Better mass and spin of final remnant; measure mixture of modes present at end of coalescence.
Scott A. Hughes, MIT
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 fringdown � (0.06� 0.15)c3
GM
Recent Developments in General Relativity, Jerusalem, 22 May 2017
Classical GR has no intrinsic scale: Frequencies which characterize GWs from black hole systems
are determined by the mass scale.
Scott A. Hughes, MIT
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 fringdown � (0.06� 0.15)c3
GM
Recent Developments in General Relativity, Jerusalem, 22 May 2017
Classical GR has no intrinsic scale: Frequencies which characterize GWs from black hole systems
are determined by the mass scale.
Scott A. Hughes, MIT
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
Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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).
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
Instantaneous EMRI amplitude will typically be factor ~10 — 100 smaller than noise!
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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)
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Scott A. Hughes, MIT
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:
Recent Developments in General Relativity, Jerusalem, 22 May 2017
Measure “shape” of Kerr
Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017
From Vigeland & Hughes, PRD 81, 082002 (2010)
Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
Horizon coupling
Flux to infinity has simple behavior:�
dE
dt
��> 0 Always takes energy
away from the binary.
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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.
Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
Horizon coupling
… but adds energy to orbit if hole spins faster than orbit.
If Ωorb < ΩH
�dE
dt
�H
< 0
Recent Developments in General Relativity, Jerusalem, 22 May 2017
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.
Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017
From Hughes, PRD 64, 064004 (2001)
Scott A. Hughes, MIT
Scott A. Hughes, MIT
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.
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
Planet’s fluid is viscous. The bulging response will lag applied tide in time.
Recent 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”:
Scott A. Hughes, MIT
If planet spins slower than orbit’s frequency, the
bulge lags orbit’s position.
Recent 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”:
Scott A. Hughes, MIT
Bulge exerts a torque that takes angular momentum from orbit … slowing it down (and speeding up planet’s spin).
Recent 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”:
Scott A. Hughes, MIT
If planet spins faster than orbit’s frequency, bulge leads orbit’s position.
Recent 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”:
Scott A. Hughes, MIT
Bulge exerts a torque that adds angular momentum to orbit … speeding it up (and slowing down planet’s spin).
Recent 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”:
Scott A. Hughes, MIT
Net effect: Orbit loses energy if Ωorb > ΩP
Orbit gains energy if Ωorb < ΩP
Exactly like the horizon absorption effect.
Recent 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”:
Scott A. Hughes, MIT
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)
Recent Developments in General Relativity, Jerusalem, 22 May 2017Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017
Tidal distortion of horizonsFlux language misleads … but can be recast in a
dual description as tidal deformation of BH.
Scott A. Hughes, MIT
Horizon couplingUsing this language, recast the down-horizon GW
flux describes “tidal bulge” on horizon.
Recent Developments in General Relativity, Jerusalem, 22 May 2017
O’Sullivan & Hughes, PRD 90, 124039 (2014); PRD 94, 044057 (2016).
Scott A. Hughes, MIT
Another null test: Does the horizon coupling agree with the Kerr horizon viscosity?
Scott A. Hughes, MIT
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
Recent Developments in General Relativity, Jerusalem, 22 May 2017