(c)2017 van Putten

Gravitational waves

2

r >> RS

r ~ RS ,RISCO

a ~ cH0

a >> cH0

Edwin Hubble (1889-1953)

repulsive gravitational

interactions (dark energy)

(c)2017 van Putten

Scales in gravitation

3

r >> RS r ~ RS

Einstein’s embedding of Newtonian gravity

(c)2017 van Putten

εMercury /sun ≅ 3×10−8

ε =Rgr

: εPSR1913+16 ≅ 5 ×10−6

εS2/SgrA* ≅ 3×10−4

4

Einstein’s extension of Newton’s theory

H = Ek +U UN = 12m u − u0( )2

UE =UN −Mu3

v = Mεr:

v ''+ v = 1+ 3εv2u0 =Mj2,ε = Mu0

—> strong gravityNewtonian gravity <—

(c)2017 van Putten

(c)2012 van Putten 5

a

Rotating tidal field Φij in binaries

(c)2017 van Putten

Symmetry over π: spin-2

(c)2012 van Putten 6

Gab = 8πTabEinstein’s tensor is an elliptic-hyperbolic operator:

Elliptic part: embedding of Newtonian gravity in curved spacetime

Hyperbolic part (serendipitous!): propagation of spin-2 gravitational waves

Φij + GWs = binary motion

Waves in hyperbolic spacetime

(SO(3,1) four-covariant formulation, van Putten & Eardley 1996)

Gab = 8πTab

!ω aαβ − Racω cαβ − ω c ,∇aω c⎡⎣ ⎤⎦αβ = 16πτ aµν

GR: mixed elliptic (gab) -hyperbolic system(⍵a⍺β)

Gravitational radiation in SO(3,1) are partially analogous to U(1) (EM), except GW are spin-2 and EM waves are spin-1

(c)2016 van Putten

(c)2017 van Putten

http://carina.astro.cf.ac.uk/groups/relativity/research/part4.html

GW polarizations

(c)2017 van Putten

( ) erg/s532 10/3Ω= µGWL

3212

3/121

3/52

3/51

)(

ammmmmm

+=Ω

+=µ

Quadrupole GW luminosity

L0 =c5

G= 3.6 ×1059 erg s−1

Binary of mass m1,m2 at orbital separation a has a chirp mass and angular velocity:

Unit of luminosity (!)

(c)2017 van Putten

erg/s)108.3( of 0.15%

erg/s106.3

33

595

0

×=

×==

sunGW

GW

LL

GcL

Agreement with linearized GW theory ~ 0.1% (Nobel Prize Physics 1993)

LGW observed:

(11x quadrupole LGW by e=0.617)

LGW ~10−29L0

~cm/day

Gentle whispers: PSR 1913+105

11

Φ = 1+ M1

2 r − r1+ M 2

2 r − r2

H-H interactions

Horizon area perturbations in conformal factor at turning points

Black hole binaries

12

SAH ,i =14AH ,i f ξ1,ξ2( ), AH ,i = 16πMi

2, ξi =Mi

a, fi ≅ 1+ ξ j i ≠ j( )

TAH ,i =∂SAH ,i∂Mi

⎛⎝⎜

⎞⎠⎟

−1

−dU = TAHdSAH( )1 + TAHdSAH( )2

U = − 12M1 log f1 ξ2( )− 1

2M 2 log f2 ξ1( )

Gibbs virtual perturbations at the same total energy-at-infinity

Thermodynamic temperature

BH-BH attraction by entropic forces

Bekenstein-Hawking entropy

AAH ,2 = 16πM 2

2 f ξ( ), f ξ( ) = 1+ M1

a+! Newton’s law

van Putten 2012 PRD 85 064046

13

van Putten 2012 PRD 85 064046Head-on mergers

EGW ≤ M280

≅ 0.36%Head-on collision

Classical “almost null result” from numerical relativity To make gravitational waves, let’s turn to BH binaries…

14

Credit: Chris Henze (NASA Ames), from a simulation by J. Centrella, B. Kelly, J. Van Meter, and J. Baker (NASA GSFC)

“We will measure the spins of hundreds of black holes …

From these spins we will learn about the collapse of stellar cores that have exhausted their nuclear fuel, about the chaotic flows of superheated gas that allow black holes to grow… We expect these observations to show incontrovertible evidence of event horizons, the space-time curtains that forever shield a black hole’s interior from any external view.”

astro-ph/1401.3741v1

(c)2017 van Putten

“Somewhere, something incredible is waiting to be known” – Carl Sagan

15

We will measure the spins of hundreds of black holes using X-ray telescopes and gravitational wave observatories, probing both the remnants of long-dead stars in the Milky Way and the supermassive monsters that lie at the hearts of other galaxies. From these spins we will learn about the collapse of stellar cores that have exhausted their nuclear fuel, about the chaotic flows of superheated gas that allow black holes to grow and quasars to shine, and about the fates of black holes after the gravitational mergers of their parent galaxies. We expect these observations to show incontrovertible evidence of event horizons, the space-time curtains that forever shield a black hole’s interior from any external view. But we could find that Einstein’s theory of relativity breaks down at the border of causality, and that in gravitational extremes nature creates an object that is even stranger than a black hole.

We will listen to the cosmic symphony of gravity waves using ears spanning a hundred million kilometers and designed with subatomic sensitivity. Within this symphony we will pick out the crescendos of neutron stars that crash together in galaxies billions of light-years away, mapping the rise and fall of massive stars and measuring the accelerating expansion of the universe with unprecedented accuracy. In the background hum of this symphony, we may detect the quantum noise of the in stationary epoch or the reverberations of colliding domain walls that mark the cracks of primordial symmetries of nature.

“Somewhere, something incredible is waiting to be known” – Carl Sagan

(c)2012 van Putten

LIGO-Virgo and KAGRA GW detectors (all 4 sites, working!)

(c)2014 van Putten 16(c)2015 van Putten

LIGO (Hanford, Louisiana) Virgo (Italian-French)

KAGRA (Japan)

(c)2016 van Putten

http://www.ligo.caltech.edu, http://www.ego-gw.it,http://gwcenter.icrr.u-tokyo.ac.jp/en/

(c)2017 van Putten

L-δL

L+δL

laser

photodiode

Optical Measurement resolution: δL/λ<1e-10

GW amplitude resolution: δL/L<1e-22 (L=4km,λ=400nm)

Freely suspended mirrors

Laser Interferometry

(c)2017 van Putten

Instrument noise

19(c)2017 van Putten

20(c)2017 van Putten

Not so gentle: short GWB150914

(c)2017 van Putten

LGW ~10−3L0

LGW ≅ 200M⊙s−1 ≅ 1023L⊙ ≅ 10

−3L0

Hulse-Taylor type chirp

positive (ascending)

chirp

(c)2017 van Putten

soft merger

(regularised by horizon surfaces with strongly

damped QNR)

23

Progenitor black hole binary and final black hole

LIGO-Virgo arXiv:1602.03842

M1 ≅ 36M⊙

aM

⎛⎝⎜

⎞⎠⎟ 1

≅ 0.31

M 2 ≅ 29M⊙

aM

⎛⎝⎜

⎞⎠⎟ 2

≅ 0.46

M f ≅ 62M⊙

aM

⎛⎝⎜

⎞⎠⎟ f

≅ 0.67

Merger of high-mass slow-spin black holes— mass of tens of solar masses

— dimensionless spin about 1/3

(c)2017 van Putten

(c)2017 van Putten

Lie derivative

Linearized Ricci tensor

Lienerd-Wiechert potential for GWs

Derivation of GW radiation

(c)2017 van Putten

Independent of Christoffel symbols

( ) ( ) ( ) ( ) fufufufufuuw

bb

cc

bb

cc

bb

cc

bb

cc

bb

bb

∂∂−∂∂=∂∂−∂∂=∂

=

ξξξξξ

ξ

],[],[

bc

cbc

cb uuu ξξξ ∂−∂=],[

Vector fields from vector fields

fufu bbb ∂→:

Vector field

Lie derivative (I)

(c)2017 van Putten

( )

ca

aac

c

cbccbb

b

bbb

bbb

b

b

b

b

bbb

bbb

b

b

bbb

bbb

uu

xuxuxx

uuuL

xxx

xx

A

xxx

A

xx

ξξ

εξε

ε

ξεξ

ε

ξεξ

ε

εξ

ε

εξ

∂−∂=

⎥⎦

⎤⎢⎣

⎡−+

∂

∂=

−=

∂+=∂

∂+=⎥

⎦

⎤⎢⎣

⎡

∂

∂=

∂

∂=

∂−=∂

∂−=

∂

∂=

−→

−

→

−

)()(

lim

11

11

''

1

*

0

''1

'''

'''

'

e.g. Stephani, H.

Derivative with associated coordinate transformation along a vector field:

Associated coordinate transformation

Ordinary Lagrangian derivative

Lie derivative (II)

(c)2017 van Putten

ce

ecce

ecc

c

ccc

ee

cc

ee

cccc

ccc

cc

eec

ee

cceec

cc

uvvuvLu

vLuuvuvvLuuLvvuL

vuuvvuvuL

ξξ

ξξ

ξξξ

ξ

ξξξξ

ξ

∂+∂=

+∂−∂=+≡

∂=∂+∂≡

cacac

ca vvvL ξξξ ∂+∂=

Independent of Christoffel symbols

Lie derivative (III)

(c)2017 van Putten

cabc

cbacabc

c

cabc

cbacabc

cab

ggg

ggggL

ξξξ

ξξξξ

∇+∇+∇=

∂+∂+∂=

abbaabgL ξξξ ∇+∇=

0=∇+∇ abba ξξ

bξWe say is a Killing vector whenever

Symmetries

(c)2017 van Putten

If are tangent vectors to two independent coordinates:

( ) ( )0

],[

≡−=∂−=

∂∂−∂∂=∂

sttssaa

ta

cc

aa

cc

aa

bb

fffvfufuvfvufvu

),( bb vu

0],[ =bvu

Lie derivative and coordinates

(c)2017 van Putten

HW exercise: show this.

cc pξ

bξbpIf is a Killing vector and the four-momentum of

a particle moving along a geodesic, then is

conserved along its world-line

Conservation laws

(c)2017 van Putten

( )

[ ] ( )

hhhR

OhggggR

gggg

OR

hhhg

caecadde

acee

ac

caeaccaeaecde

bd

adc

ccde

deddc

dbca

dacb

dabc

abab

ababab

∂∂−∂∂+∂∂−=

+∂∂−⎟⎠

⎞⎜⎝

⎛ −+∂=

∂==Γ

+Γ∂−Γ∂=

=+=

21

21

21

21

21

,

)(

2,,,

,

2

η

ε

ε

ηη

Linearized Riemann and Ricci tensor

(c)2017 van Putten

hhhh

hhhh

accaaee

acee

a

accaaeae

ceae

∂∂−∂∂−∂∂+∂∂

∂∂−∂∂−∂∂+∂∂

41

41

21

21

41

41

21

21

hhh

hhR

abab

eae

cacee

ac

η21

21

)(

−=

∂∂+∂∂−=

:21

)( hh caecadde ∂∂−∂∂η

Linearized Ricci tensor

ab

(c)2017 van Putten

hhhhhR ababeae

cacee

ac η21,

21

)( −=∂∂+∂∂−=

Gravitational radiation is in transverse waves. With

Choose

In transverse traceless gauge

abbaabgL ξξξ ∇+∇=

.21

0,under invariant ,

h

hhhh

aa

aa

baaaaba

b

abbaababbaabab

−=∂+∂

=Δ+→−∂=Δ

∂+∂+=∇+∇+→

ξµ

µµξξξ

ξξξξ

0,21

=∂∂−= RhR acee

ac

linearized Ricci tensor - simplified

(c)2017 van Putten

0,21

=∂∂−= RhR acee

ac

linearized Ricci tensor - d’Alembertian

This is the wave operator acting on linearized perturbations.

!u = −utt + uxx + uyy + uzz

In Minkowski spacetime:

(Geometrical units, c = 1)

(c)2017 van Putten

Plane waves

k

Δϕ = kΔx = π

(c)2017 van Putten

Dispersion relation

!u = −utt + uxx + uyy + uzz

k = (k,0,0)u = u ϕ( ), ϕ =ωt − k i x =ωt − kx

!u = −ω 2 + k2( )u = 0 :ω = k

ω

k

Gravitational waves in vacuum are without dispersion.

(c)2017 van Putten

Can show:

Einstein-Hilbert action

S = R −gd 4x∫

δS = Gabδ gab −gd 4x∫ (e.g. Wald, 1984)

δS = 0 :

Gab = 0

(c)2017 van Putten

∫∫∫ −∂−=−==−= xdghhxdghRSxdgRS ababab

ab4244

321

161

161

ππδ

π

Small amplitude gravitational waves about Minkowski spacetime are a solution of the Einstein equations.

By the linearized Ricci tensor, the Einstein-Hilbert action integrand reduces to

Linearized Einstein-Hilbert action

(c)2017 van Putten

[ ] ( )∫ −∂= xdghhS abcab42

21

161π

[ ] ( )∫ −∂=↔ xdgS c42

21

ϕϕ

Quadratic Lagrangian in standard form

Integration by parts:

In form, identical to the action of scalar wave.

(c)2017 van Putten

××

++

+×

×+ +=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−= ababab eheh

hhhh

h

0

0

Metric fluctuation hab in Coulomb gauge :00 =ah

[ ] ( ) ( )[ ] :161 422∫ −∂+∂= ×+ xdghhhS ccab π

( ) ( )22000

161

×+ ∂+∂=== hhttt ttzzz

π

Time-average/orbital-average

Stress-energy tensor

Expand over these two polarization modes (treating each coefficient as a scalar field):

(c)2017 van Putten

Implications

GWs have a stress-energy tensor in canonical form for both polarizations with vanishing dispersion in vacuum.

This has several consequences:

i) GWs like EM waves are massless and propagate side-by-side with no dispersion at the same velocity of light c;

ii) As GWs carry energy to infinity. the total mass-energy of the source diminishes, observed first in PSR1913+106.

iv) GWs are parameterized by dimensionless strain amplitudes, detected as spin-2 waves from GWB150914 by LIGO.

(c)2017 van Putten

At low energies…A DE described by a cosmological constant (may be variable!) introduces dispersion in GWs and EM waves (by coupling of Riemann-Cartan connections and vector potentials to Rab):

ω = k2 + Λ

!u −m2u = 0 : ω = k2 +m2

Recall the Klein-Gordon equation for a massive scalar field:

Dark energy introduces dispersion in GWs and EM waves with wave numbers on the order of

k ~ Λ

(c)2017 van Putten

Dispersion at long wave lengths

Very long wavelength GWs and EM waves are dispersive.

k ~ Λ :

k = 2πλ, Λ ~ H 2 = RH

−2

λ ~ RH

(c)2017 van Putten

0

02

2

2

2

==→==×=

====→=

∑∑

∑∑∑

Jdtd

dtd

Jjpx

Pdtd

pdtd

mxdtd

mdtdd

xmd

AAA

AA

AA

AA

µµ

No dipole or magnetic monopole radiation

(c)2017 van Putten

baabababab jAThTG πππ 4821:8 22 −=∂↔=∂−=

Radiation from a binary

III ijijTij δ

31

−=

Traceless part (tidal perturbation):

(c)2013 van Putten

kik

jkjk

ijk

ikik

kjij TxTxxTxTxdT ∂+∂−=∂+∂= ∫∫∫ 21

213

( ) ijjiijjiij Idtd

xdxxTxdTxTxxdT 2

23002

0300

03

21

21

21

≡∂=+∂= ∫∫∫

|'|,'),'(4),( 3 xxrxdrtxTr

trhV

TTij

TTij −=−= ∫

Lienard-Wiechert potential

Expand and integrate by parts:

Focus on traceless transverse components of strain:

(c)2017 van Putten

TTij

TTij I

dtd

rh 2

22=

’t Hooft 2002Strain amplitude of radiation field

(c)2017 van Putten

TTjk

TTjk

TTjk

TTjk

z

IdtdI

dtd

dtdEd

ht

ht

T

3

3

3

32

0

81

counting) double avoid to1/2 (extra 32

1

π

π

=Ω

∂

∂

∂

∂=

In each direction, 2 polarization modes. These represent 2/5 times the contributions of all 5 (=6-1) components in traceless part of T

ijI

’t Hooft 2002

Tjk

Tjk

Tjk

TjkGW I

dtdI

dtdI

dtdI

dtdL 3

3

3

3

3

3

3

3

51

814

52

=××=→π

π

GW luminosity

(c)2017 van Putten

( ) ( )

( )

( ) ( ) ( ) 246222

263

3

3

3

3

3

3

3

2

22222

211

21

21213

2123

3

3

3

322cos22cos221

2

:constant a toreduces inertia ofmoment of trace

const.const.const. const.2cos21

const.2cos21

cos

,,,,51

µϕϕµ

ϕµ

ϕµϕ

µϕ

aaIdtd

Idtd

Idtd

Idtd

IIIaI

aamamI

mmmm

aaaamm

tIdtd

Idtd

L

Tjk

Tjk

ijTij

yxxyzzyy

xx

Tjk

TjkGW

Ω=+⎟⎠

⎞⎜⎝

⎛Ω=

=

→

===+−=

+=+=

+=+=

+=ΩΩ==

GW emission in binary motion

(c)2017 van Putten

( ) ( )

( )

( ) ( ) ( ) 246222

263

3

3

3

3

3

3

3

2

22222

211

21

21213

2123

3

3

3

322cos22cos221

2

:constant a toreduces inertia ofmoment of trace

const.const.const. const.2cos21

const.2cos21

cos

,,,,51

µϕϕµ

ϕµ

ϕµϕ

µϕ

aaIdtd

Idtd

Idtd

Idtd

IIIaI

aamamI

mmmm

aaaamm

tIdtd

Idtd

L

Tjk

Tjk

ijTij

yxxyzzyy

xx

Tjk

TjkGW

Ω=+⎟⎠

⎞⎜⎝

⎛Ω=

=

→

===+−=

+=+=

+=+=

+=ΩΩ==

GW emission in binary motion

(c)2017 van Putten

( )5

2321246

3

3

3

3

532

532

51

ammaI

dtdI

dtdL T

jkTjkGW

µµ

+=Ω==

1-595

s erg106.3 ×=Gc

( )25

3/10

532~

532

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛≅Ω=Mm

aMMLGW

δ

( )2/32/1

5/121

5/32

5/31 ,~ −≅Ω+

= aMmmmmM

Gas from disk inhomogeneities

In units of

(c)2017 van Putten

van Putten 2002 ApJ 575 L71

At a critical slenderness: m=1,2,... instability

Ma

mδb

mδ

m=2

Formation of mass-inhomogeneities

High density matter about newly formed black holes

Candidate events for

Low frequency

Inspiral of NS/NS binaries, stellar mass BHs

High frequency

Associated with formation of black holes and neutron stars

Very high frequency

Quasi-normal mode ringing of stellar mass black holes

(c)2012 van Putten 54(c)2015 van Putten

Summary

(c)2017 van Putten

Gravitation in cosmology and astrophysics features the limits of weak and strong gravity

Weak gravity [elliptic part of EE]: dynamics at or below the de Sitter scale of acceleration adS=cH (about 1 Angstrom s-2), where c is the velocity of light and H is the Hubble parameter.

Strong gravity [elliptic-hyperbolic part of EE]: gravitational wave sources, such as GWB 150914 that opens an exciting new field of GW-astronomy, e.g., to probe central engines of GRBs and CC-SNe for newborn rotating black holes interacting with high density matter.