PREAMBLE: RIEMANN’S DE

THE ISOMONODROMY METHODFOR BLACK HOLE SCATTERING

Bruno Carneiro da Cunha (UFPE) Physics and Mathematics of Complex Systems - IIP - UFRN - 08/04/2016

Fábio Novaes (UFPE and IIP-UFRN)

Amílcar Queiroz (UnB)

Monica Guica (NORDITA - Uppsala)

José Julián Barragán-Amado (UFPE and Groningen)

Elisabetta Pallante (Groningen)

Filipe Rudrigues (UFPE)

PREAMBLE: RIEMANN’S DE

Frobenius solutions for Hypergeometric:

Connection Formulae:

Connection matrices

Monodromy matrices

Defined up to conjugation

3 parameters read from DE Sufficient to determine 3 connection matrices

Physical Problem: scalar perturbations of BTZ

Each Frobenius solution correspondto a purely incoming/outgoing wave

at critical point.

Monodromy problem solves scattering:

but

Conformally coupled scalar in (A)dS-4

1404.5188

A word about the angular equation

Spheroidal harmonics

Eigenvalue problem

Padé (rational) approximants

0.2 0.4 0.6 0.8 1.0 a/L

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

Cl

aω=0aω=0.3aω=0.6aω=0.9

0.2 0.4 0.6 0.8 1.0 aω

-1.5

-1.0

-0.5

Cl

a/L=0a/L=0.3a/L=0.6a/L=0.9

Eigenvalue problem also solved by monodromy!

Quantization condition!

The radial equation:

Heun’s differential equation

Scalar field in (A)dS5

JBA/EP, to appear

Radial equation also of Heun type: The Brücker Equation

de Sitter

The Brücker Equation

anti-de Sitter

The Heun Equation

de Sitter

Local solutions: plane waves Monodromy

uin

+

(r) =1

T uin

C (r) +RT uout

C (r)

Extra symmetry (time-reversal) allows for normalization

Connection problem (almost) solves scattering problem!

1st order Fuchsian system - SU(2) field of 4 monopoles:

Many potentials for the same equation!

“holomorphic flat connection"

Can embed system into flat holomorphic connection

Isomonodromy deformations!

making use of obvious conserved quantities:

Claim: can find parameters so that:

1508.04046

Schlesinger equations: Painlevé VI transcendent:

Actually Hamiltonian system:

Hamiltonian structure is inherited from:

The Chern-Simons form:

Monodromy parameters form a set of Darboux coordinates in the space of flat connections.

Initial value problem for tau:

“Effective potential”: tau function

Formally can be inverted to give monodromy data in terms of ODE parameters

Deform to

Study behavior in terms of composite monodromy

Jimbo’s strategy:

1142 MlCHIO JlMBO

= (± i sin TTGF cos nalt — cos ndt cos nO^ — cos n60 cosn91)e±nia

+ i sin TC<T cos naQ1 + cos 7i0f cos nQl + cos 710^ cos 7i00.

When o- = o-f0 = 0, the parametrization for M(f), M(0) is obtained by settings = l + s1a and letting <7-»0. Namely we introduce st e C by

, _! / fi*'9* °(1.7) M^ = Ct | o ^_?r.0 t

x sin - (00 + 0r) sin ( - 00 + 0r) + TT sin ?r0t sin -

x sin -5- (00 +0,) sin 4- (-^o+ ^) +7r sin ;r0r sin -5- (00

sn - co-

(C(°))21 =/-i ^ sin -- ( - 00 + 0,) sin (^ - 0,)

= Sl sin i(- 00 + 0() sin-J (0^ + 0,)

Now we can state the main result for PVI.

Theorem 1.1. Under the assumptions (A.l)PVi— (A.3)PVI, we have the

following asymptotic expansion of the t function as f->0.

(1.9) t(0 ~ const. i("2-»§-e?)/4

16(J2(l+(7)2BOUNDARY CONDITION FOR PAINLEVE EQUATIONS 1143

where a=£Q and s is related to s in (1.8) through

(1.10)

(1.9)' t(0 ~ const,

x l - i%-

wf^/7 O=l — SIL — log t, an^/ St is given by

(1.10)' s1 =

= -7- logT(x) denotes the diGamma function.More precisely, for any (^»0, f/iere exists an 8>0 swc^i t/ia^ (1.7)-(1.7)'

holds as t-*Q in the sector (tEC\0<\t\<s, |argr|«p] .

Higher order expansion is determined from the equation (1.3).The t function is uniquely specified by the exponents 9 v and the monodromy

matrices M(v). In order to signify the dependence on them, we employ thenotation r(f; 00> Ot, 9l9 0^; M^\ M«\ M^\ Actually it is invariant underthe joint similarity transformation M^-WPM^P"-1 (v = 0, f, 1) so that the

(Semiclassical) Liouville theory Primary operators

Null state (zero norm)

Ward identity

Accessory parameters are generated by correlators:

is called the “semiclassical Liouville profile"

Where did that come from?

Huge gauge symmetry… How to fix?

Make the gauge potential an “oper":

Reduction of action:

Background with 4 "conical" singularities:

conformal blocks

tau function gives insights for intermediate states in CFT

IntroductionIsomonodromy picture

CFT pictureConformal blocks and special PVI solutions

Painleve VI and isomonodromyJimbo’s formula and beyondGeneral case

Higher order corrections

⌧(t) ⇠ t��0��t⇣1 + B1(✓, �)t+B2(✓, �)t2. . .

⌘+

+C±1 t��±1��0��t⇣1 + B(±1)

1 (✓, �)t + . . .⌘

+ C±2t��±2��0��t

⇣1 + . . .

⌘

with

B1(✓, �) =(�� 0 + �t)(�� 1 + �1)

2��,

B2(✓, �) =(�� 0 + �t)(�� 0 + �t + 1)(�� 1 + �1)(�� 1 + �1 + 1)

4��(2�� + 1)

+

h(1 + 2��) (�0 + �t) + ��(�� + 1)� 3(�0 ��t)2

ih(1 + 2��) (�1 + �1) + ��(�� + 1)� 3(�1 ��1)2

i

2 (2�� + 1) (4�� 1)2,

B(±1)1 (✓, �) = B1(✓, � ± 1).

Observation. PVI tau function is a linear combination of c = 1 conformal blocks:

⌧(t) =X

n2ZCnt

��+n��0��tB (✓, � + n, t)

Graphical representation of B (✓, � + n, t):

Oleg Lisovyy Conformal field theory of Painleve VI

Gamayun, Iorgov, Lisovyy 1207.0787 1302.1832

Expansions for tau exist following the proof of AGT conjecture

was noted and explored [38, 39] and arguments were given for the same happening

for non-extremal black holes [40]. The picture arising from our results are however

more intricate: one can use the isomonodromy flow to relate the scattering of fields

at a particular non-zero value of the accessory parameter t0 – corresponding to the

physical, rotating black hole – to a “confluent” Heun equation where t0 ! 0. In this

limit, the position of the apparent singularity � also coalesces with a regular singular

point (3.42) – which would correspond to an extremal black hole – but in general the

accessory parameters diverge. The interpretation for this symmetry in terms of four-

point functions is not clear, but it seems that, if there is a conformal description of

the non-extremal black hole, the states and primaries involved will not be in the same

SL(2,C) invariant state as the one used to describe the extremal black hole, as the

accessory parameters diverge. We believe that the results presented here will not only

be useful to further the studies in astrophysical applications but also will help clarify

the more technical issues listed above.

Acknowledgements

The authors would like to thank Mark Mineev-Weinstein, Seung-Yeop Lee, Marc Casals,

Monica Guica, Geo↵rey Compere, Amılcar de Queiroz and A. P. Balachandran for use-

ful discussions and comments. Fabio Novaes acknowledges partial support from CNPq

and the Science without Borders initiative process 400635/2012-7. BCdC acknowledges

partial support from PROPESQ-UFPE.

A The Painleve VI ⌧-function and asymptotics

Here we present more information about Painleve VI ⌧ -function and the asymptotic

expansion of �1t when t0 goes to zero. First, let us remind of the general expansion of

the Painleve VI which has been given in [6, 7] in terms of the c = 1 conformal blocks.

The expression is5

⌧(t) =X

n2Z

C(~✓, �0t + n)snt(�+2n)2/4�(✓0�✓t)2/4B(~✓, �0t + n; t), (A.1)

5In the references, the monodromy parameters are defined with an extra factor of 2: {✓i,�ij}there !{✓i/2,�ij/2}here.

– 26 –

where the structure constants C are products of the Barnes functions (defined by the

functional relation G(z + 1) = �(z)G(z), �(z) being the Euler gamma):

C(~✓, �) =

Q✏,✏

0=± G(1 + 12(✓t + ✏✓0 + ✏

0�))G(1 + 1

2(✓1 + ✏✓1 + ✏

0�))

Q✏=± G(1 + ✏�)

, (A.2)

and B have the structure of conformal blocks, given by the combinatorial series:

B(~✓, �; t) = (1� t)✓t✓1/2X

�,µ2�

B�,µ

(~✓, �)t|�|+|µ|, (A.3)

summing over pairs of Young tableaux �, µ with

j

i

�

02 = 3

�2 = 5

h(2, 2) = 5

|�| = 15

Figure 2. A sample Young tableau � = {7, 5, 2, 1} and the relevant quantities for thecombinatorial ⌧ -function expansion.

B�,µ

(~✓, �) =Y

(i,j)2�

((✓t

+ � + 2(i� j))2 � ✓

20)((✓1 + � + 2(i� j))2 � ✓

21)

16h2�

(i, j)(�0j

+ µ

i

� i� j + 1 + �)2⇥

Y

(i,j)2µ

((✓t

� � + 2(i� j))2 � ✓

20)((✓1 � � + 2(i� j))2 � ✓

21)

16h2µ

(i, j)(�i

+ µ

0j

� i� j + 1� �)2, (A.4)

where (i, j) denotes the box in the Young tableau �, �i

the number of boxes in row i,

�

0j

the number of boxes in column j and h

�

(i, j) = �

i

+ �

0j

� i� j + 1 its hook length

(see Fig. 2).

– 27 –



C(~✓, �) =

Q✏,✏

0=± G(1 + 12(✓t + ✏✓0 + ✏

0�))G(1 + 1

2(✓1 + ✏✓1 + ✏

0�))

Q✏=± G(1 + ✏�)

, (A.2)


B(~✓, �; t) = (1� t)✓t✓1/2X

�,µ2�

B�,µ

(~✓, �)t|�|+|µ|, (A.3)


j

i

�

02 = 3

�2 = 5

h(2, 2) = 5

|�| = 15


B�,µ

(~✓, �) =Y

(i,j)2�

((✓t

+ � + 2(i� j))2 � ✓

20)((✓1 + � + 2(i� j))2 � ✓

21)

16h2�

(i, j)(�0j

+ µ

i

� i� j + 1 + �)2⇥

Y

(i,j)2µ

((✓t

� � + 2(i� j))2 � ✓

20)((✓1 � � + 2(i� j))2 � ✓

21)

16h2µ

(i, j)(�i

+ µ

0j

� i� j + 1� �)2, (A.4)



�

0j


�

(i, j) = �

i

+ �

0j


(see Fig. 2).

– 27 –



C(~✓, �) =

Q✏,✏

0=± G(1 + 12(✓t + ✏✓0 + ✏

0�))G(1 + 1

2(✓1 + ✏✓1 + ✏

0�))

Q✏=± G(1 + ✏�)

, (A.2)


B(~✓, �; t) = (1� t)✓t✓1/2X

�,µ2�

B�,µ

(~✓, �)t|�|+|µ|, (A.3)


j

i

�

02 = 3

�2 = 5

h(2, 2) = 5

|�| = 15


B�,µ

(~✓, �) =Y

(i,j)2�

((✓t

+ � + 2(i� j))2 � ✓

20)((✓1 + � + 2(i� j))2 � ✓

21)

16h2�

(i, j)(�0j

+ µ

i

� i� j + 1 + �)2⇥

Y

(i,j)2µ

((✓t

� � + 2(i� j))2 � ✓

20)((✓1 � � + 2(i� j))2 � ✓

21)

16h2µ

(i, j)(�i

+ µ

0j

� i� j + 1� �)2, (A.4)



�

0j


�

(i, j) = �

i

+ �

0j


(see Fig. 2).

– 27 –

Kerr and Painlevé V

Same history, confluent Heun. Isomonodromy also gives solution for scattering in terms of Painlevé V tau:

1506.06588

Confluent Heun (standard form):

Stokes phenomenon near infinity:

1508.01342Invariant monodromy data

NOT Hamilton-Jacobi, tau function!

Also solved by AGT instantons, irregular CFT blocks

⌧ -function:

⌧(t) = t

(✓0�✓t)2/4[⌧(t)]�1

, (6.1)

and assume ✓i, � and ✓i�✓j, ��✓i are not integers. The expansion for the tau-function

is of the form

⌧(t, ~✓) =X

n2Z

C({✓i}, 12

� + n)snt(1

2

�+n)2B({✓i}, 12

� + n; t), (6.2)

where the irregular conformal block B is given as a power series over the set of Young

tableaux Y:

B({✓i}, 12

�; t) = e

�1

2

✓ttX

�,µ2Y

B�,µ({✓i}, 12

�)t|�|+|µ|, (6.3)

with coe�cients

B�,µ =Y

(i,j)2�

(�1

2

✓1 + 1

2

� + i� j)((12

✓t +1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

�(i, j)(�0j + µi � i� j + 1 + �)

⇥

Y

(i,j)2µ

(�1

2

✓1 � 1

2

� + i� j)((12

✓t � 1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

µ(i, j)(�i + µ

0j � i� j + 1 + �)

, (6.4)

where � denotes a Young tableau, �i is the number of boxes in row i, �0j is the number

of boxes in column j and h�(i, j) = �i + �

0j � i� j +1 is the hook length related to the

box (i, j) 2 �. The structure constants C are rational products of Barnes functions

C({✓i}, �) =Y

✏=±

G(1� 1

2

✓1 + ✏

1

2

�)G(1 + 1

2

✓t +1

2

✓

0

+ ✏

1

2

�)G(1 + 1

2

✓t � 1

2

✓

0

+ ✏

1

2

�)

G(1 + ✏�),

(6.5)

where G(z) is defined by the functional equation G(1+z) = �(z)G(z). The parameters

� and s in (6.2) are related to the “constants of integration” of the Painleve V equation.

In our treatment, they are functions of the Stokes parameters s1

and s

2

. The parameter

� is given by (3.10), whereas the expression for s in (6.2) is rather long and involved.

We will outline the procedure in Section 10 of [33] to compute it. Let M

0

, Mt be

the monodromy matrices for the hypergeometric equations. They are of the form

Mi = E

�1

i e

i⇡✓i�3Ei with Ei given by equation (10.15) in [33]. They satisfy MtM0

=

– 14 –

⌧ -function:

⌧(t) = t

(✓0�✓t)2/4[⌧(t)]�1

, (6.1)


is of the form

⌧(t, ~✓) =X

n2Z

C({✓i}, 12

� + n)snt(1

2

�+n)2B({✓i}, 12

� + n; t), (6.2)


tableaux Y:

B({✓i}, 12

�; t) = e

�1

2

✓ttX

�,µ2Y

B�,µ({✓i}, 12

�)t|�|+|µ|, (6.3)

with coe�cients

B�,µ =Y

(i,j)2�

(�1

2

✓1 + 1

2

� + i� j)((12

✓t +1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

�(i, j)(�0j + µi � i� j + 1 + �)

⇥

Y

(i,j)2µ

(�1

2

✓1 � 1

2

� + i� j)((12

✓t � 1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

µ(i, j)(�i + µ

0j � i� j + 1 + �)

, (6.4)





C({✓i}, �) =Y

✏=±

G(1� 1

2

✓1 + ✏

1

2

�)G(1 + 1

2

✓t +1

2

✓

0

+ ✏

1

2

�)G(1 + 1

2

✓t � 1

2

✓

0

+ ✏

1

2

�)

G(1 + ✏�),

(6.5)




and s

2

. The parameter



0

, Mt be


Mi = E

�1

i e


=

– 14 –

⌧ -function:

⌧(t) = t

(✓0�✓t)2/4[⌧(t)]�1

, (6.1)


is of the form

⌧(t, ~✓) =X

n2Z

C({✓i}, 12

� + n)snt(1

2

�+n)2B({✓i}, 12

� + n; t), (6.2)


tableaux Y:

B({✓i}, 12

�; t) = e

�1

2

✓ttX

�,µ2Y

B�,µ({✓i}, 12

�)t|�|+|µ|, (6.3)

with coe�cients

B�,µ =Y

(i,j)2�

(�1

2

✓1 + 1

2

� + i� j)((12

✓t +1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

�(i, j)(�0j + µi � i� j + 1 + �)

⇥

Y

(i,j)2µ

(�1

2

✓1 � 1

2

� + i� j)((12

✓t � 1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

µ(i, j)(�i + µ

0j � i� j + 1 + �)

, (6.4)





C({✓i}, �) =Y

✏=±

G(1� 1

2

✓1 + ✏

1

2

�)G(1 + 1

2

✓t +1

2

✓

0

+ ✏

1

2

�)G(1 + 1

2

✓t � 1

2

✓

0

+ ✏

1

2

�)

G(1 + ✏�),

(6.5)




and s

2

. The parameter



0

, Mt be


Mi = E

�1

i e


=

– 14 –

⌧ -function:

⌧(t) = t

(✓0�✓t)2/4[⌧(t)]�1

, (6.1)


is of the form

⌧(t, ~✓) =X

n2Z

C({✓i}, 12

� + n)snt(1

2

�+n)2B({✓i}, 12

� + n; t), (6.2)


tableaux Y:

B({✓i}, 12

�; t) = e

�1

2

✓ttX

�,µ2Y

B�,µ({✓i}, 12

�)t|�|+|µ|, (6.3)

with coe�cients

B�,µ =Y

(i,j)2�

(�1

2

✓1 + 1

2

� + i� j)((12

✓t +1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

�(i, j)(�0j + µi � i� j + 1 + �)

⇥

Y

(i,j)2µ

(�1

2

✓1 � 1

2

� + i� j)((12

✓t � 1

2

� + i� j)2 � 1

4

✓

2

0

)

h

2

µ(i, j)(�i + µ

0j � i� j + 1 + �)

, (6.4)





C({✓i}, �) =Y

✏=±

G(1� 1

2

✓1 + ✏

1

2

�)G(1 + 1

2

✓t +1

2

✓

0

+ ✏

1

2

�)G(1 + 1

2

✓t � 1

2

✓

0

+ ✏

1

2

�)

G(1 + ✏�),

(6.5)




and s

2

. The parameter



0

, Mt be


Mi = E

�1

i e


=

– 14 –

properties of space-time in the strong-field, high-velocityregime and confirm predictions of general relativity for thenonlinear dynamics of highly disturbed black holes.

II. OBSERVATION

On September 14, 2015 at 09:50:45 UTC, the LIGOHanford, WA, and Livingston, LA, observatories detected

the coincident signal GW150914 shown in Fig. 1. The initialdetection was made by low-latency searches for genericgravitational-wave transients [41] and was reported withinthree minutes of data acquisition [43]. Subsequently,matched-filter analyses that use relativistic models of com-pact binary waveforms [44] recovered GW150914 as themost significant event from each detector for the observa-tions reported here. Occurring within the 10-ms intersite

FIG. 1. The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, rightcolumn panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series are filteredwith a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and band-rejectfilters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra. Top row, left: H1 strain. Top row, right: L1 strain.GW150914 arrived first at L1 and 6.9þ0.5

−0.4 ms later at H1; for a visual comparison, the H1 data are also shown, shifted in time by thisamount and inverted (to account for the detectors’ relative orientations). Second row: Gravitational-wave strain projected onto eachdetector in the 35–350 Hz band. Solid lines show a numerical relativity waveform for a system with parameters consistent with thoserecovered from GW150914 [37,38] confirmed to 99.9% by an independent calculation based on [15]. Shaded areas show 90% credibleregions for two independent waveform reconstructions. One (dark gray) models the signal using binary black hole template waveforms[39]. The other (light gray) does not use an astrophysical model, but instead calculates the strain signal as a linear combination ofsine-Gaussian wavelets [40,41]. These reconstructions have a 94% overlap, as shown in [39]. Third row: Residuals after subtracting thefiltered numerical relativity waveform from the filtered detector time series. Bottom row:A time-frequency representation [42] of thestrain data, showing the signal frequency increasing over time.

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

061102-2

propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of

GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being

produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]

M ¼ ðm1m2Þ3=5

ðm1 þm2Þ1=5¼ c3

G

!5

96π−8=3f−11=3 _f

"3=5

;

where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.

III. DETECTORS

Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO

detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational

waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes

FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).


061102-3

For robustness and validation, we also use other generictransient search algorithms [41]. A different search [73] anda parameter estimation follow-up [74] detected GW150914with consistent significance and signal parameters.

B. Binary coalescence search

This search targets gravitational-wave emission frombinary systems with individual masses from 1 to 99M⊙,total mass less than 100M⊙, and dimensionless spins up to0.99 [44]. To model systems with total mass larger than4M⊙, we use the effective-one-body formalism [75], whichcombines results from the post-Newtonian approach[11,76] with results from black hole perturbation theoryand numerical relativity. The waveform model [77,78]assumes that the spins of the merging objects are alignedwith the orbital angular momentum, but the resultingtemplates can, nonetheless, effectively recover systemswith misaligned spins in the parameter region ofGW150914 [44]. Approximately 250 000 template wave-forms are used to cover this parameter space.The search calculates the matched-filter signal-to-noise

ratio ρðtÞ for each template in each detector and identifiesmaxima of ρðtÞwith respect to the time of arrival of the signal[79–81]. For each maximum we calculate a chi-squaredstatistic χ2r to test whether the data in several differentfrequency bands are consistent with the matching template[82]. Values of χ2r near unity indicate that the signal isconsistent with a coalescence. If χ2r is greater than unity, ρðtÞis reweighted as ρ ¼ ρ=f½1þ ðχ2rÞ3&=2g1=6 [83,84]. The finalstep enforces coincidence between detectors by selectingevent pairs that occur within a 15-ms window and come fromthe same template. The 15-ms window is determined by the10-ms intersite propagation time plus 5 ms for uncertainty inarrival time of weak signals. We rank coincident events basedon the quadrature sum ρc of the ρ from both detectors [45].To produce background data for this search the SNR

maxima of one detector are time shifted and a new set ofcoincident events is computed. Repeating this procedure∼107 times produces a noise background analysis timeequivalent to 608 000 years.To account for the search background noise varying across

the target signal space, candidate and background events aredivided into three search classes based on template length.The right panel of Fig. 4 shows the background for thesearch class of GW150914. The GW150914 detection-statistic value of ρc ¼ 23.6 is larger than any backgroundevent, so only an upper bound can be placed on its falsealarm rate. Across the three search classes this bound is 1 in203 000 years. This translates to a false alarm probability< 2 × 10−7, corresponding to 5.1σ.A second, independent matched-filter analysis that uses a

different method for estimating the significance of itsevents [85,86], also detected GW150914 with identicalsignal parameters and consistent significance.

When an event is confidently identified as a realgravitational-wave signal, as for GW150914, the back-ground used to determine the significance of other events isreestimated without the contribution of this event. This isthe background distribution shown as a purple line in theright panel of Fig. 4. Based on this, the second mostsignificant event has a false alarm rate of 1 per 2.3 years andcorresponding Poissonian false alarm probability of 0.02.Waveform analysis of this event indicates that if it isastrophysical in origin it is also a binary black holemerger [44].

VI. SOURCE DISCUSSION

The matched-filter search is optimized for detectingsignals, but it provides only approximate estimates ofthe source parameters. To refine them we use generalrelativity-based models [77,78,87,88], some of whichinclude spin precession, and for each model perform acoherent Bayesian analysis to derive posterior distributionsof the source parameters [89]. The initial and final masses,final spin, distance, and redshift of the source are shown inTable I. The spin of the primary black hole is constrainedto be < 0.7 (90% credible interval) indicating it is notmaximally spinning, while the spin of the secondary is onlyweakly constrained. These source parameters are discussedin detail in [39]. The parameter uncertainties includestatistical errors and systematic errors from averaging theresults of different waveform models.Using the fits to numerical simulations of binary black

hole mergers in [92,93], we provide estimates of the massand spin of the final black hole, the total energy radiatedin gravitational waves, and the peak gravitational-waveluminosity [39]. The estimated total energy radiated ingravitational waves is 3.0þ0.5

−0.5M⊙c2. The system reached apeak gravitational-wave luminosity of 3.6þ0.5

−0.4 × 1056 erg=s,equivalent to 200þ30

−20M⊙c2=s.Several analyses have been performed to determine

whether or not GW150914 is consistent with a binaryblack hole system in general relativity [94]. A first

TABLE I. Source parameters for GW150914. We reportmedian values with 90% credible intervals that include statisticalerrors, and systematic errors from averaging the results ofdifferent waveform models. Masses are given in the sourceframe; to convert to the detector frame multiply by (1þ z)[90]. The source redshift assumes standard cosmology [91].

Primary black hole mass 36þ5−4M⊙

Secondary black hole mass 29þ4−4M⊙

Final black hole mass 62þ4−4M⊙

Final black hole spin 0.67þ0.05−0.07

Luminosity distance 410þ160−180 Mpc

Source redshift z 0.09þ0.03−0.04


061102-7

Observation of Gravitational Waves from a Binary Black Hole Merger

B. P. Abbott et al.*

(LIGO Scientific Collaboration and Virgo Collaboration)(Received 21 January 2016; published 11 February 2016)

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-WaveObservatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards infrequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0 × 10−21. It matches the waveformpredicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of theresulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and afalse alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greaterthan 5.1σ. The source lies at a luminosity distance of 410þ160

−180 Mpc corresponding to a redshift z ¼ 0.09þ0.03−0.04 .

In the source frame, the initial black hole masses are 36þ5−4M⊙ and 29þ4

−4M⊙, and the final black hole mass is62þ4

−4M⊙, with 3.0þ0.5−0.5M⊙c2 radiated in gravitational waves. All uncertainties define 90% credible intervals.

These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first directdetection of gravitational waves and the first observation of a binary black hole merger.

DOI: 10.1103/PhysRevLett.116.061102

I. INTRODUCTION

In 1916, the year after the final formulation of the fieldequations of general relativity, Albert Einstein predictedthe existence of gravitational waves. He found thatthe linearized weak-field equations had wave solutions:transverse waves of spatial strain that travel at the speed oflight, generated by time variations of the mass quadrupolemoment of the source [1,2]. Einstein understood thatgravitational-wave amplitudes would be remarkablysmall; moreover, until the Chapel Hill conference in1957 there was significant debate about the physicalreality of gravitational waves [3].Also in 1916, Schwarzschild published a solution for the

field equations [4] that was later understood to describe ablack hole [5,6], and in 1963 Kerr generalized the solutionto rotating black holes [7]. Starting in the 1970s theoreticalwork led to the understanding of black hole quasinormalmodes [8–10], and in the 1990s higher-order post-Newtonian calculations [11] preceded extensive analyticalstudies of relativistic two-body dynamics [12,13]. Theseadvances, together with numerical relativity breakthroughsin the past decade [14–16], have enabled modeling ofbinary black hole mergers and accurate predictions oftheir gravitational waveforms. While numerous black holecandidates have now been identified through electromag-netic observations [17–19], black hole mergers have notpreviously been observed.

The discovery of the binary pulsar systemPSR B1913þ16by Hulse and Taylor [20] and subsequent observations ofits energy loss by Taylor and Weisberg [21] demonstratedthe existence of gravitational waves. This discovery,along with emerging astrophysical understanding [22],led to the recognition that direct observations of theamplitude and phase of gravitational waves would enablestudies of additional relativistic systems and provide newtests of general relativity, especially in the dynamicstrong-field regime.Experiments to detect gravitational waves began with

Weber and his resonant mass detectors in the 1960s [23],followed by an international network of cryogenic reso-nant detectors [24]. Interferometric detectors were firstsuggested in the early 1960s [25] and the 1970s [26]. Astudy of the noise and performance of such detectors [27],and further concepts to improve them [28], led toproposals for long-baseline broadband laser interferome-ters with the potential for significantly increased sensi-tivity [29–32]. By the early 2000s, a set of initial detectorswas completed, including TAMA 300 in Japan, GEO 600in Germany, the Laser Interferometer Gravitational-WaveObservatory (LIGO) in the United States, and Virgo inItaly. Combinations of these detectors made joint obser-vations from 2002 through 2011, setting upper limits on avariety of gravitational-wave sources while evolving intoa global network. In 2015, Advanced LIGO became thefirst of a significantly more sensitive network of advanceddetectors to begin observations [33–36].A century after the fundamental predictions of Einstein

and Schwarzschild, we report the first direct detection ofgravitational waves and the first direct observation of abinary black hole system merging to form a single blackhole. Our observations provide unique access to the

*Full author list given at the end of the article.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PRL 116, 061102 (2016)Selected for a Viewpoint in Physics

PHY S I CA L R EV I EW LE T T ER Sweek ending

12 FEBRUARY 2016

0031-9007=16=116(6)=061102(16) 061102-1 Published by the American Physical Society

I

II

IV

III

V IV'V'

VI

VIIVIII

Scattering between different asymptotic regions

Sum over different n: illusion of unitarity!

Important applications in adS/CFT

Killing-Yano, twistor programme and separability

Many cases of interest in 4d: Kerr, Kerr-Newman, Kerr-NUT-(a)dS, etc.

Exact expressions in terms of PVI and PV tau-function

Method works for all spin perturbations

Possible new, interesting expansions for the scattering coefficients

Method works for many exact solutions of Einstein’s Equations

Knowledge of conformal blocks helps uniformization problems

PUZZLES

1. Hamiltonian structure:

Isomonodronic flow = "Yangian"

However, established that tau function is generating fcnl:

which is related to the parameters of the dynamical system K,µ,� by

d

dt

log ⌧(t, {✓i

}) = K(�, µ, t) +✓0✓t

t

+✓1✓t

t� 1� 1(�� t)

t(t� 1)� �(�� 1)µ

t(t� 1)

=�(�� 1)(�� t)

t(t� 1)

µ

2 �✓✓0

�

+✓1

�� 1+

✓

t

�� t

◆µ+

12

�(�� 1)

�+

✓0✓t

t

+✓1✓t

t� 1,

(3.44)

where it is assumed that K(t),�(t), µ(t) satisfy the equations of motion. Inspecting

the parameters, we can arrive at the more direct correspondence [31]:

K(�(t), µ(t); t, ✓0, ✓1, ✓t, ✓1) =d

dt

log ⌧(t; ✓0, ✓1, ✓t� 1, ✓1� 1)� ✓0(✓t � 1)

t

� ✓1(✓t � 1)

t� 1.

(3.45)

The ⌧ -function plays a central role in the theory of integrable systems, being interpreted

in generic grounds as a generating functional, and its existence stems from a zero

curvature condition. Despite the arguments, the ⌧ -function also depends on the trace

of the composite monodromy operators M0Mt

, M1Mt

and M1M

t

. With the initial

conditions set by (3.41), we have

t(t� 1)d

dt

log ⌧(t, ~✓,~�)

��t=t0

= t0✓t✓1 + (t0 � 1)✓0✓t + t0(t0 � 1)K0

d

dt

t(t� 1)

d

dt

log ⌧(t, ~✓,~�)

��t=t0

= (✓0 + ✓1 + 1)✓t =✓

t

2(✓0 + ✓1 � ✓

t

+ ✓1),

(3.46)

where ~

✓ = {✓0, ✓1, ✓t0 , ✓1} and ~� = {�01, �1t} parametrize the invariant monodromy

data – see Appendix A. The function ⇣ = t(t � 1) d

dt

log ⌧(t) obeys the second order

di↵erential equation:

⇣t(t� 1)⇣ 00

⌘2

= �2 det

0

B@2✓20 t⇣

0 � ⇣ ⇣

0 + ✓

20 + ✓

2t

+ ✓

21 � ✓

21

t⇣

0 � ⇣ 2✓2t

(t� 1)⇣ 0 � ⇣

⇣

0 + ✓

20 + ✓

2t

+ ✓

21 � ✓

21 (t� 1)⇣ 0 � ⇣ 2✓21

1

CA ,

(3.47)

sometimes called the “�-form” of Painleve VI equations [7, 31, 32]. Although the

initial value problem is well-posed from the ODE perspective, with those conditions

specifying an unique solution of the �-form of the Painleve VI equation (not to confuse

– 18 –

Why?

2. Classical limit and conformal blocks

So, higher order terms like are also present

Why the full Verma module, if we are doing semiclassical?

Computed from the (semiclassical) OPE

3. Last (but far from least)

Scalar field in (pure) AdS:

Define the "boundary operator"

We have the “bulk field” associated to the boundary profile:

with conformal dimension associated to the mass:So, can rework the OPE as (FGG 71 and FGGP 72):

For any CFT, have:

A LOT of work will get you the semiclassical conformal block Needs Lorentz signature.

For BTZ, whole story(BTZ is secretly AdS)

MG, to appear

So, clear semiclassical conformal block picture from AdS/CFT

Why going to higher dimensions — and adding mass scales — suddenly is equivalent to “going quantum-

mechanical”?

THANK YOU!

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