clifford will washington university john archibald wheeler

Clifford WillWashington University

John Archibald WheelerInternational School on Astrophysical Relativity

Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:

A Post-Newtonian ApproachA Post-Newtonian Approach

Interferometers Around The WorldInterferometers Around The WorldLIGO Hanford 4&2 km

LIGO Livingston 4 km

GEO Hannover 600 m

TAMA Tokyo300 m

Virgo Cascina 3 km

LISA: a space interferometer

for 2015

Inspiralling Compact Binaries - Strong-gravity GR tests?Inspiralling Compact Binaries - Strong-gravity GR tests?

Fate of the binary pulsar in 100 MyGW energy loss drives pair toward merger

LIGO-VIRGO Last few minutes (10K cycles)

for NS-NS40 - 700 per year by 2010BH inspirals could be more numerous

LISAMBH pairs(105 - 107 Ms) in galaxiesWaves from the early universe

Last 7 orbits

A chirp waveform

•The advent of gravitational-wave astronomy

•The problem of motion & radiation - a history

•Post-Newtonian gravitational radiation

•Testing gravity and measuring astrophysical parametersusing gravitational waves

•GW recoil - Do black holesget kicked out of galaxies?

•Interface between PN gravity and numerical relativity

Clifford Will, J. A. Wheeler School on Astrophysical Relativity



The problem of motion & radiationThe problem of motion & radiation Geodesic motion 1916 - Einstein - gravitational radiation (wrong by factor 2) 1916 - De Sitter - n-body equations of motion 1918 - Lense & Thirring - motion in field of spinning body 1937 - Levi-Civita - center-of-mass acceleration 1938 - Eddington & Clark - no acceleration 1937 - EIH paper & Robertson application 1960s - Fock & Chandrasekhar - PN approximation 1967 - the Nordtvedt effect 1974 - numerical relativity - BH head-on collision 1974 - discovery of PSR 1913+16 1976 - Ehlers et al - critique of foundations of EOM 1976 - PN corrections to gravitational waves (EWW) 1979 - measurement of damping of binary pulsar orbit 1990s - EOM and gravitational waves to HIGH PN order

Driven by requirements for GW detectors(v/c)12 beyond Newtonian gravity

The post-Newtonian approximationThe post-Newtonian approximation

€

ε ~ (v /c)2 ~ (Gm /rc 2) ~ ( p /ρc 2)

gμν = η μν + εh(1)μν + ε 2h(2)

μν +K

Gμν = 8πTμν (G = c =1)

Tμν = ρ uμ uν + p(uμ uν + gμν )

DIRE: Direct integration of the relaxed Einstein equationsDIRE: Direct integration of the relaxed Einstein equations

€

Gμν = 8πTμν

€

hμν ≡ η μν − −ggμν

∂ν hμν = 0

q hμν = −16π (−g)(T μν + t μν )

hμν = 4τ μν (t− | x − ′ x |, ′ x )

| x − ′ x |C∫ d3 ′ x

∇ν T μν = 0, or ∂ν τ μν = 0

Einstein’s Equations

“Relaxed” Einstein’s Equations

PN equations of motion for compact binariesPN equations of motion for compact binaries

€

a = −m

r3x +1PN +1PNSO +1PNSS + 2PN + 2.5PN

+ 3PN

+ 3.5PN

+ 3.5PNSO

+ 3.5PNSS

B F SB F S

W B W B

W W

W W (in progress)(in progress)

B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schäfer, Jaranowski W = WUGRAV

Gravitational energy flux for compact binariesGravitational energy flux for compact binaries

€

˙ E = ˙ E quad +1PN

+1PNSO +1PNSS

+1.5PN

+ 2PN

+ 2.5PN

+ 3PN

+ 3.5PN

WW

B W B W

B WB W

B B

B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schäfer, Jaranowski W = WUGRAV

B B

B B

Wagoner & CW 76Wagoner & CW 76

€

h(t) ≈ A(t) e iΦ(t )

€

˜ h ( f ) = h(t)e2πiftdt−∞

∞

∫ ≈ ˜ A ( f )e iΨ( f )

Gravitational Waveform and Matched FilteringGravitational Waveform and Matched Filtering

Quasi-Newtonian approximation

Fourier transform

€

h(t) s(t) + n(t)[ ]

Matched filtering - schematic

€

Ψ( f ) = 2πftc − Φc − π /4

+3

128u−5 / 3 1[

+20

9

743

336+

11

4η

⎛

⎝ ⎜

⎞

⎠ ⎟η −2 / 5u2 / 3

−16πu

+10305673

1016064+

5429

1008η +

617

144η 2 ⎛

⎝ ⎜

⎞

⎠ ⎟η −4 / 5u4 / 3

+ O(u5) ]

GW Phasing as a precision probe of gravityGW Phasing as a precision probe of gravity

N

1PN

1.5PN

2PN

Measure chirp mass M

Measure m1 & m2

“Tail” term - test GR

Test GR

M = m1+m2 = m1m2/M2 M = 3/5M

u = Mf ~ v3

€

Ψ( f ) = 2πftc − Φc − π /4

+3

128u−5 / 3 1[

+20

9

743

336+

11

4η

⎛

⎝ ⎜

⎞

⎠ ⎟η −2 / 5u2 / 3

−16πu

+10305673

1016064+

5429

1008η +

617

144η 2 ⎛

⎝ ⎜

⎞

⎠ ⎟η −4 / 5u4 / 3

+ O(u5) ]

GW Phasing: Bounding scalar-tensor gravityGW Phasing: Bounding scalar-tensor gravity

N

1PN

1.5PN

2PN

€

−5

84

ΔS2

ωη 2 / 5u−2 / 3

Self-gravity difference

Coupling constant

M = m1+m2 = m1m2/M2 M = 3/5M

u = Mf ~ v3

Bounding masses and scalar-tensor theory with LISABounding masses and scalar-tensor theory with LISA

NS + 103 Msun BHSpins aligned with LSNR = 10104 binary Monte Carlo____ = one detector------ = two detectors

Solar system bound

Berti, Buonanno & CW (2005)

Speed of Waves and Mass of the GravitonSpeed of Waves and Mass of the Graviton

Why Speed could differ from “1”

massive graviton: vg2 = 1 - (mg/Eg)2

g coupling to background fields: vg = F(,K,H)

gravity waves propagate off the brane

ExamplesGeneral relativity. For <<R, GW follow geodesics of background spacetime, as do photons (vg = 1)

Scalar-tensor gravity. Tensor waves can have vg ≠ 1, if scalar is massiveMassive graviton theories with background metric. Circumvent vDVZ theorem. Visser (1998), Babak & Grishchuk (1999,2003)

Possible Limits

€

1− vg ≈ 5 ×10−17 200Mpc

DΔta − (1+ Z)Δte[ ]

D = distance of source, Z = redshift, Δta (Δte ) = time difference in seconds

Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binaries

t

x

Detector

Source

(CW, 1998)

Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binaries

m1 m2Distance(Mpc)

Bound on g (km)

Ground-Based (LIGO/VIRGO)

1.4 1.4 300 4.6 X 1012

10 10 1500 6.0 X 1012

Space-Based (LISA)

107 107 3000 6.9 X 1016

105 105 3000 2.3 X 1016

Other methods Comments Bound on g (km)

Solar system 1/r2 law

Assumes direct link between static g

and wave g

3 X 1012

Galaxies & clusters Ditto 6 X 1019

CWDB phasing LISA (Cutler et al) 1 X 1014

Radiation of momentum and the recoil of Radiation of momentum and the recoil of massive black holesmassive 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 correction term

Wiseman (92)

Astrophysics

MBH formation by mergers could terminate if BH ejected from early galaxiesEjection from dwarf galaxies or globular clustersDisplacement from center could affect galactic core

Merritt, Milosavljevic, Favata, Hughes & Holz (04)Favata, Hughes & Holz (04)

Radiation of momentum to 2PN orderRadiation of momentum to 2PN orderBlanchet, Qusailah & CW (2005)

Calculate relevant multipole moments to 2PN order quadrupole, octupole, current quadrupole, etc

Calculate momentum flux for quasi-circular orbit [x=(m)2/3≈(v/c)2]recoil = -flux

€

dP

dt= −

464

105

δm

mη 2x11/ 2 1+ −

452

87−

1139

522η

⎛

⎝ ⎜

⎞

⎠ ⎟x +

309

58πx 3 / 2

⎡

⎣ ⎢

+ −71345

22968+

36761

2088η +

147101

68904η 2 ⎛

⎝ ⎜

⎞

⎠ ⎟x 2

⎤

⎦ ⎥λ

Integrate up to ISCO (6m) for adiabatic inspiralMatch quasicircular orbit at ISCO to plunge orbit in SchwarzschildIntegrate with respect to “proper ” to horizon (x -> 0)

Recoil velocity as a function of mass ratioRecoil velocity as a function of mass ratio

€

=m1m2

(m1 + m2)2

=X

(1+ X)2

X = 0.38Vmax = 250 ± 50km/s

V/c ≈ 0.043 X2

Blanchet, Qusailah & CW (2005)

X=1/10V = 70 ± 15 km/s

Radiation of momentum to 2PN orderRadiation of momentum to 2PN orderBlanchet, Qusailah & CW (2005)

Checks and testsVary matching radius between inspiral and plunge (5.3m-6m) --- 7%

Vary matching method --- 10 %

Vary energy damping rate from N to 2PN --- no effect

Vary cutoff: (a) r=2(m+) (b) r=2m --- 1%

Add 2.5PN, 3PN and 3.5PN terms: a2.5PNx5/2 + a3PNx3 + a3.5PNx7/2

and vary coefficients between +10 and -10 --- ±30 %

or an rms error of ±20 %

Maximum recoil velocity: Range of EstimatesMaximum recoil velocity: Range of Estimates

0 100 200 300 400

Favata, Hughes & Holtz (2004)

Campanelli (Lazarus) (2005)

Blanchet, Qusailah & CW (2005)

Damour & Gopakumar (2006)

Baker et al (2006)

Getting a kick from numerical relativityGetting a kick from numerical relativity

Baker et al (GSFC), gr-qc/0603204

The end-game of gravitational radiation reactionThe end-game of gravitational radiation reaction

Evolution leaves quasicircular orbitdescribable by PN approximation

Numerical models start with quasi-equilibrium (QE) stateshelical Killing vector stationary in rotating framearbitrary rotation states (corotation, irrotational)

How well do PN and QE agree?surprisingly well, but some systematic differences exist

Develop a PN diagnostic for numerical relativityelucidate physical content of numerical models“steer” numerical models toward more realistic physics

T. Mora & CMW, PRD 66, 101501 (2002) (gr-qc/0208089) PRD 69, 104021 (2004) (gr-qc/0312082)

€

∂ /∂t + Ω∂ /∂φ

Ingredients of a PN DiagnosticIngredients of a PN Diagnostic

3PN point-mass equations

•Derived by 3 different groups, no undetermined parameters

Finite-size effects

•Rotational kinetic energy (2PN)

•Rotational flattening (5PN)

•Tidal deformations (5PN)

•Spin-orbit (3PN)

•Spin-spin (5PN)€

ERot ≈ mR2ω2 ≈ ENq2(m /r)2

€

EFlat ≈ δIω2 ≈ ω4R5 ≈ ENq5(m /r)5

€

ETide≈(′ δ m)2/R≈ENq

5(m/r)

5

€

ESO ≈ LS /r3 ≈ ENq2(m /r)3

€

ESS ≈ S1S2 /r3 ≈ ENq4 (m /r)5

€

E = EN 1+m

r+

m

r

⎛

⎝ ⎜

⎞

⎠ ⎟2

+m

r

⎛

⎝ ⎜

⎞

⎠ ⎟3 ⎧

⎨ ⎩

⎫ ⎬ ⎭

J = JN 1+m

r+

m

r

⎛

⎝ ⎜

⎞

⎠ ⎟2

+m

r

⎛

⎝ ⎜

⎞

⎠ ⎟3 ⎧

⎨ ⎩

⎫ ⎬ ⎭

““Eccentric” orbits in relativistic systems. IIEccentric” orbits in relativistic systems. II

Relativistic GravityDefine “measurable” eccentricity and semilatus rectum:

€

e ≡Ω p − Ωa

Ω p + Ωa

ζ ≡m

p≡

mΩ p + mΩa

2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4 / 3

=mΩa

(1− e)2

⎛

⎝ ⎜

⎞

⎠ ⎟2 / 3

Plusses: Exact in Newtonian limit Constants of the motion in absence of radiation reaction

Connection to “measurable” quantities ( at infinity) “Easy” to extract from numerical data e ˙ 0 naturally under radiation reactionMinuses Non-local Gauge invariant only through 1PN order

““Eccentric” orbits in relativistic systems. IIIEccentric” orbits in relativistic systems. III

3PN ADMenergy andangular momentumat apocenter

Tidal and rotational effectsTidal and rotational effects

Use Newtonian theory; add to 3PN & Spin resultsstandard textbook machinery (eg Kopal 1959, 1978)multipole expansion -- keep l=2 & 3direct contributions to E and Jindirect contributions via orbit perturbations

Dependence on 4 parameters

€

=I / MR2

q = R / M

k2,k3 : apsidal constants

kl =0 , point mass3

4(l −1), homogeneous

⎧ ⎨ ⎪

⎩ ⎪

Corotating Black Holes - Meudon DataCorotating Black Holes - Meudon Data

Energy of Corotating Neutron Stars - Numerical vs. PNEnergy of Corotating Neutron Stars - Numerical vs. PN

Simulations by Miller, Suen & WUGRAV

Energy of irrotational neutron stars - PN vs Meudon/TokyoEnergy of irrotational neutron stars - PN vs Meudon/Tokyo

Data from Taniguchi& GourgoulhonPRD 68, 124025 (2003)

=2q=8.3


=1.8q=7.1

=2.0q=7.1

=2.25q=7.1

=2.5q=7.1


=2q=8.3

=2q=7.1

=2q=6.25 =2

q=5.6

Concluding remarksConcluding remarks

PN theory now gives results for motion and radiation

through 3.5 PN order

Many results verified by independent groups

Spin and finite-size effects

More “convergent” series?

Measurement of GW chirp signals may give tests of

fundamental theory and astrophysical parameters

PN theory may provide robust estimates of strong-

gravity phenomena

Kick of MBH formed from merger

Initial states of compact binaries near ISCO

It is difficult to think of any occasion in the history of It is difficult to think of any occasion in the history of astrophysics when three stars at once shone more brightly in astrophysics when three stars at once shone more brightly in the sky than our three stars do at this conference today. First is the sky than our three stars do at this conference today. First is X-ray astronomy. It brings rich information about neutron stars. X-ray astronomy. It brings rich information about neutron stars. It begins to speak to us of the first identifiable black hole on the It begins to speak to us of the first identifiable black hole on the books of science. Second is gravitational-wave astronomy. It books of science. Second is gravitational-wave astronomy. It has already established upper limits on the flux of gravitational has already established upper limits on the flux of gravitational waves at selected frequencies…. At the fantastic new levels of waves at selected frequencies…. At the fantastic new levels of sensitivity now being engineered, it promises to pick up signals sensitivity now being engineered, it promises to pick up signals every few weeks from collapse events in nearby galaxies. Third every few weeks from collapse events in nearby galaxies. Third is black-hole physics. It furnishes the most entrancing is black-hole physics. It furnishes the most entrancing applications we have ever seen of Einstein’s geometric account applications we have ever seen of Einstein’s geometric account of gravitation. It offers for our study, both theoretical and of gravitation. It offers for our study, both theoretical and observational, a wealth of fascinating new effects. observational, a wealth of fascinating new effects.

John A. Wheeler, IAU Symposium, Warsaw, 1973

clifford will washington university john archibald wheeler

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