scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A Matrix Method for Quasinormal Modes: Schwarzschild Black Holes in Asymptotically Flat and (Anti-) de Sitter Spacetimes

TL;DR: In this article, the quasinormal modes of Schwarzschild and Schwarzschild (Anti-) de Sitter black holes were studied by a matrix method, which involves discretizing the master field equation and expressing it in form of a homogeneous system of linear algebraic equations.
Abstract: In this work, we study the quasinormal modes of Schwarzschild and Schwarzschild (Anti-) de Sitter black holes by a matrix method. The proposed method involves discretizing the master field equation and expressing it in form of a homogeneous system of linear algebraic equations. The resulting homogeneous matrix equation furnishes a non-standard eigenvalue problem, which can then be solved numerically to obtain the quasinormal frequencies. A key feature of the present approach is that the discretization of the wave function and its derivatives are made to be independent of any specific metric through coordinate transformation. In many cases, it can be carried out beforehand which in turn improves the efficiency and facilitates the numerical implementation. We also analyze the precision and efficiency of the present method as well as compare the results to those obtained by different approaches.

Summary (2 min read)

1. Introduction

  • The second stage consists of a long period dominated by the quasinormal oscillations, where the amplitude of the oscillation decays exponentially in time.
  • Mathematically, the QNMs are governed by the linearized equations of general relativity constraining perturbations around a black hole solution.
  • Due to the difficulty in finding exact solutions to most problems of interest, various approximate methods have been proposed [8].
  • In their approach, the master field equation is presented in terms of linear equations describing 3 N discretized points where the wave function is expanded up to Nth order for each of these points.

2. Matrix method and the eigenvalue problem for quasinormal modes

  • Recently, the authors proposed a non-grid-based interpolation scheme which can be used to solve the eigenvalue problem [1].
  • Based on the information about N scattered data point, Taylor series are carried out for the unknown eigenfunction up to Nth order for each discretized point.
  • It was shown [1] that the boundary conditions can be implemented by properly replacing some of the above equations.
  • In the case of asymptotically flat Schwarzschild spacetime below, the authors choose to replace the first and the last line in the matrix equation and implement the boundary condition by replacing equation (3.11) with equation (3.13).
  • Now the authors apply the above method to investigate the master field equation of for QNM.

4. Quasinormal modes in Schwarzschild de Sitter black hole spacetime

  • Now the authors are in the position to utilize the same numerical procedure to discretize the wave function in the interval ⩽ ⩽y0 1, and solve for the quasinormal frequencies.
  • The obtained the quasinormal frequencies are presented in table 2 compared to those obtained by the WKB method.
  • It is found that the results from the present method are consistent with those from the WKB method.

5. Quasinormal modes in in Schwarzschild anti-de Sitter black hole spacetime

  • 10 Again, the eigenequation can be obtained following the same procedures as before.
  • The calculated quasinormal frequencies are shown in table 3 and compared with the results obtained by the HH method.
  • It is inferred from the results that the present method is in accordance with the HH method.

6. Discussions and outlooks

  • The authors proposed a new interpolation scheme to discretize the master field equation  for the scalar quasinormal modes.
  • It possesses flexibility and therefore the potential to explore some black hole metrics where the applications of other traditional methods become less straightforward.
  • As an example, the quasinormal modes of a rotational black hole is characterized by, besides the quasinormal frequency ω, a second eigenvalue λ whose physical content is associated with the angular quantum number L. Its numerical solution, therefore, involves finding the two eigenvalues, ω and λ, simultaneously.
  • A preliminary attempt [15] shows that the approach proposed in this work, on the other hand, can be applied straightforwardly in a more intuitive fashion.
  • This procedure is identical for most calculations once the grid points are fixed, therefore the efficiency of the method can be significantly improved for such similar problems.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

Classical and Quantum Gravity
PAPER
A matrix method for quasinormal modes:
Schwarzschild black holes in asymptotically flat
and (anti-) de Sitter spacetimes
To cite this article: Kai Lin and Wei-Liang Qian 2017 Class. Quantum Grav. 34 095004
View the article online for updates and enhancements.
Related content
Black hole quasinormal modes using the
asymptotic iteration method
H T Cho, A S Cornell, Jason Doukas et al.
-
Fermion perturbations in string theory
black holes
Owen Pavel Fernández Piedra and
Jeferson de Oliveira
-
Quasinormal Modes of a Noncommutative-
Geometry-Inspired Schwarzschild Black
Hole
Jun Liang
-
Recent citations
Charged scalar fields around Einstein-
power-Maxwell black holes
Grigoris Panotopoulos
-
The matrix method for black hole
quasinormal modes
Kai Lin and Wei-Liang Qian
-
Quasinormal modes of the BTZ black hole
under scalar perturbations with a non-
minimal coupling: exact spectrum
Grigoris Panotopoulos
-
This content was downloaded from IP address 186.217.236.64 on 04/07/2019 at 20:39

1
Classical and Quantum Gravity
A matrix method for quasinormal
modes: Schwarzschild black holes
inasymptotically at and (anti-)
de Sitter spacetimes
KaiLin
1
,
2
and Wei-LiangQian
2
,
3
1
Universidade Federal de Itajubá, Instituto de Física e Química, Itajubá, MG, Brazil
2
Escola de Engenharia de Lorena, Universidade de São Paulo, Lorena, SP, Brazil
3
Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista,
Guaratinguetá, SP, Brazil
E-mail: lk314159@hotmail.com and wlqian@usp.br
Received 11 January 2017, revised 1 March 2017
Accepted for publication 13 March 2017
Published 31 March 2017
Abstract
In this work, we study the quasinormal modes of Schwarzschild and
Schwarzschild (Anti-) de Sitter black holes by a matrix method. The proposed
method involves discretizing the master eld equationand expressing it in the
form of a homogeneous system of linear algebraic equations. The resulting
homogeneous matrix equationfurnishes a non-standard eigenvalue problem,
which can then be solved numerically to obtain the quasinormal frequencies.
A key feature of the present approach is that the discretization of the wave
function and its derivatives is made to be independent of any specic metric
through coordinate transformation. In many cases, it can be carried out
beforehand, which in turn improves the efciency and facilitates the numerical
implementation. We also analyze the precision and efciency of the present
method as well as compare the results to those obtained by different approaches.
Keywords: quasinormal modes, black hole, Schwarzschild spacetime,
deSitter spacetime
(Some guresmay appear in colour only in the online journal)
1. Introduction
A black hole, long considered to be a physical as well as mathematical curiosity, is derived
in general relativity as a generic prediction. Through gravitational collapse, a stellar-mass
black hole can be formed at the end of the life cycle of a very massive star, when its gravity
K Lin and W-L Qian
A matrix method for quasinormal modes: Schwarzschild black holes in asymptotically flat and (anti-) de Sitter spacetimes
Printed in the UK
095004
CQGRDG
© 2017 IOP Publishing Ltd
34
Class. Quantum Grav.
CQG
1361-6382
10.1088/1361-6382/aa6643
Paper
9
1
13
Classical and Quantum Gravity
IOP
2017
1361-6382/17/095004+13$33.00 © 2017 IOP Publishing Ltd Printed in the UK
Class. Quantum Grav. 34 (2017) 095004 (13pp) https://doi.org/10.1088/1361-6382/aa6643

2
overcomes the neutron degeneracy pressure. A crucial feature of a black hole is the existence
of the event horizon, a boundary in spacetime beyond which events cannot affect an outside
observer. Despite its invisible interior, however, the properties of a black hole can be inferred
through its interaction with other matter. By quantum eld theory in curved spacetime, it is
shown that the event horizons emit Hawking radiation, with the same spectrum of black body
radiation at a temperature determined by its mass, charge and angular momentum [2]. The
latter completes the formulation of black hole thermodynamics [3], which describes the prop-
erties of a black hole in analogy to those of thermodynamics by relating mass to energy, area
to entropy, and surface gravity to temperature. Quasinormal modes (QNMs) arise as the tem-
poral oscillations owing to perturbations in black hole spacetime [5]. Owing to the energy loss
through ux conservation, these modes are not normal. Consequently, when writing the oscil-
lation in an exponential form,
()ω texpi
, the frequency of the modes is a complex number.
The real part,
ω
R
, represents the actual temporal oscillation; and the imaginary part,
ω
I
, indi-
cates the decay rate. Therefore, these modes are commonly referred to as quasinormal. The
stability of the black hole spacetime guarantees that all small perturbation modes are damped.
Usually, QNMs can be conditionally divided into three stages. The rst stage involves a short
period of the initial outburst of radiation, which is sensitively dependent on the initial condi-
tions. The second stage consists of a long period dominated by the quasinormal oscillations,
where the amplitude of the oscillation decays exponentially in time. This stage is character-
ized by only a few parameters of the black holes, such as their mass, angular momentum, and
charge. The last stage takes place when the QNMs are suppressed by power-law or exponen-
tial late-time tails. The properties of QNMs have been investigated in the context of the AdS/
CFT correspondence [6, 7]. As a matter of fact, practically every stellar object oscillates, and
oscillations produced by very compact stellar objects and their detection are of vital impor-
tance in physics and astrophysics. In 2015, the rst observation of gravitational waves from
a binary black hole merger was reported [4]. The observation provides direct evidence of the
last remaining unproven prediction of general relativity and reconrms its prediction of space-
time distortion on the cosmic scale.
Mathematically, the QNMs are governed by the linearized equationsof general relativity
constraining perturbations around a black hole solution. The resulting master eld equationis
a linear second order partial differential equation. Due to the difculty in nding exact solu-
tions to most problems of interest, various approximate methods have been proposed [8]. If
the inverse potential, which can be viewed as a potential well, furnishes a well-dened bound
state problem, the QNMs can be evaluated by solving the associated Schrödinger equation. In
particular, when a smooth potential well can be approximated by the Pöschl-Teller potential,
QNM frequencies can be obtained through the known bound states [9]. For general potential
function, approaches such as continued fraction method [10], Horowitz and Hubeny (HH)
method [6], asymptotic iteration method [11] can be utilized. A common feature of the above
methods is that the corresponding master eld equationis obtained by representing the wave
function with power series. Higher precision is therefore achieved by considering higher order
expansions. A semi-analytic technique to obtain the low-lying QNMs is based on a match-
ing of the asymptotic WKB solutions at spatial innity and on the event horizon [12]. The
WKB formula has been extended to the sixth order [13]. Further generalization to a higher
order, however, is not straightforward. Finite difference method is developed to numerically
integrate the master eld equation[14], and the temporal evolution of the perturbation can be
obtained.
In this work, by discretizing the linear partial different equation[1], we transfer the mas-
ter eld equationas well as its boundary conditions into a homogeneous matrix equation. In
our approach, the master eld equationis presented in terms of linear equationsdescribing
K Lin and W-L Qian
Class. Quantum Grav. 34 (2017) 095004

3
N discretized points where the wave function is expanded up to Nth order for each of these
points. This leads to a non-standard eigenvalue problem and can be solved numerically for
the quasinormal frequencies. The present paper is organized as follows. In the next section,
we briey review how to reformulate the master eld equation in terms of a matrix equa-
tionof non-standard eigenvalue problem. In sections3 to 5, we investigate the quasinormal
modes of Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti- de Sitter black hole
spacetime respectively. The precision and efciency of the present approach are studied by
comparing to the results obtained by other methods. Discussions and speculations are given
in the last section.
2. Matrix method and the eigenvalue problem for quasinormal modes
Recently, we proposed a non-grid-based interpolation scheme which can be used to solve the
eigenvalue problem [1]. A key step of the method is to formally discretize the unknown eigen-
function in order to transform a differential equationas well as the boundary conditions into a
homogeneous matrix equation. Based on the information about N scattered data point, Taylor
series are carried out for the unknown eigenfunction up to Nth order for each discretized point.
Then the resulting homogeneous system of linear algebraic equationsis solved for the eigen-
value. Here, we briey describe the discretization procedure. For a univariate function f(x),
one applies the Taylor expansion of a function to N 1 discrete points in a small vicinity of
another point. Without loss of generality, let us expand the function about x
2
to
xxxx,,,,
N13 4
,
and therefore obtains N 1 linear relations between function values and their derivatives as
follows
F∆=MD,
(2.1)
where
(()()()()()()()())
F
∆= −− −−
fx fx fx fx fx fx fx fx
,,,,,,
jN
T
1232
22
(2.2)
=
−−
−−
−−
−−






M
xx
xx xx
i
xx
N
xx
xx xx
i
xx
N
xx
xx xx
i
xx
N
xx
xx xx
i
xx
N
2! 1!
2! 1!
2! 1!
2! 1!
j
jj
i
j
N
N
NN
i
N
N
12
12
2
12 12
1
32
32
2
32 32
1
2
2
2
1
2
2
2
1
() ()
() ()
() ()
()()
(2.3)
(()()()())
() ()

=
Dfxfxfxfx,,,,
,.
kN
T
22
22
(2.4)
Now, the above equationimplies that all the derivatives at x = x
2
can be expressed in terms of
the function values by using the Cramers rule. In particular, we have
() ()/()
() ()/()
=
=
fx MM
fx
MM
detdet ,
detdet
,
21
22
(2.5)
K Lin and W-L Qian
Class. Quantum Grav. 34 (2017) 095004

4
where M
i
is the matrix formed by replacing the ith column of M by the column vector
F
.
Now, by permuting the N points,
xx x,, ,
N12
, we are able to rewrite all the derivatives at the
above N points as linear combinations of the function values at those points. Substituting the
derivatives into the eigenequation, one obtains N equationswith
() ()fx fx,,
N1
as its vari-
ables. It was shown [1] that the boundary conditions can be implemented by properly replac-
ing some of the above equations. Usually, the equations which are closer to the boundary
of the problem are chosen to be replaced, since those equationsare the least precise ones.
For instance, in the case of asymptotically at Schwarzschild spacetime below, we choose to
replace the rst and the last line in the matrix equationand implement the boundary condition
by replacing equation(3.11) with equation(3.13).
Now we apply the above method to investigate the master eld equationof for QNM. For
simplicity, here we only investigate the scalar perturbation in black hole spacetime. According
to the action of the massless scalar eld with minimal coupling in curved four dimensional
spacetime:
()
L
∫∫
=−=−∂Φ∂Φ
µ
µ
Sx
gxgdd
,
44
(2.6)
the equationof motion for the massless scalar eld reads
∇∇Φ=
µν
µν
g 0.
(2.7)
Consider the following static spherical metric
()
()
()θθ
ϕ=− ++ +sFrt
r
Fr
r
dd
d
dsin
d,
22
2
22
22
(2.8)
and rewriting the scalar eld by using the separation of variables ()
()
θ
Φ=
φ
ωϕ
−+
Y e
r
r
tmii
, we
obtain the following well-known Schrödinger-type equation
[()]
φ
ωφ
+− =
r
Vr
d
d
0
2
2
2
(2.9)
where
()
() ()
() ()
=+
+
V
rFr
Fr
r
LL
r
1
2
is the effective potential, and
()
=
r
r
Fr
d
is tortoise coordi-
nate. As discussed below, the boundary conditions in asymptotically at, de Sitter and anti-de
Sitter spacetimes are different. For the interpolation in equation(2.5) to be valid, appropriate
coordinate transformation shall be introduced, which will be discussed in detail in the follow-
ing sections.
3. Quasinormal modes in Schwarzschild black hole spacetime
In Schwarzschild spacetime, one has
()=−
Fr
M
r
1
2
,
(3.1)
and r
h
= 2M corresponds to the event horizon of the black hole. The potential vanishes on
the horizon F(r
h
) = 0 and at innity
r
, therefore, the wave function has the asymptotic
solution
()
()
φ
ω±
r e
i
r
Fr
d
, where ± correspond to the wave travelling in positive and negative
direction respectively. Since the wave function must be an ingoing wave on the horizon and an
outgoing wave at innity, the boundary conditions of equation(2.9) read
K Lin and W-L Qian
Class. Quantum Grav. 34 (2017) 095004

Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the authors show that for the rotating black holes, a moderate source term in the master equation in the Laplace s-domain does not modify the quasinormal modes.
Abstract: In the study of perturbations around black hole configurations, whether an external source can influence the perturbation behavior is an interesting topic to investigate. When the source acts as an initial pulse, it is intuitively acceptable that the existing quasinormal frequencies will remain unchanged. However, the confirmation of such an intuition is not trivial for the rotating black hole, since the eigenvalues in the radial and angular parts of the master equations are coupled. We show that for the rotating black holes, a moderate source term in the master equation in the Laplace s-domain does not modify the quasinormal modes. Furthermore, we generalize our discussions to the case where the external source serves as a driving force. Different from an initial pulse, an external source may further drive the system to experience new perturbation modes. To be specific, novel dissipative singularities might be brought into existence and enrich the pole structure. This is a physically relevant scenario, due to its possible implication in modified gravity. Our arguments are based on exploring the pole structure of the solution in the Laplace s-domain with the presence of the external source. The analytical analyses are verified numerically by solving the inhomogeneous differential equation and extracting the dominant complex frequencies by employing the Prony method.

9 citations

Journal ArticleDOI
TL;DR: In this article , the authors studied the effect of two-photon autocorrelation on the frequency-domain correlation between the quasinormal modes and the characteristic parameters of null geodesics and found that the dominant contributions to the light echoes are from those in the eikonal limit.
Abstract: In this work, we study the black hole light echoes in terms of the two-photon autocorrelation and explore their connection with the quasinormal modes. It is shown that the above time-domain phenomenon can be analyzed by utilizing the well-known frequency-domain relations between the quasinormal modes and characteristic parameters of null geodesics. We found that the time-domain correlator, obtained by the inverse Fourier transform, naturally acquires the echo feature, which can be attributed to a collective effect of the asymptotic poles through a weighted summation of the squared modulus of the relevant Green's functions. Specifically, the contour integral leads to a summation taking over both the overtone index and angular momentum. Moreover, the dominant contributions to the light echoes are from those in the eikonal limit, consistent with the existing findings using the geometric-optics arguments. For the Schwarzschild black holes, we demonstrate the results numerically by considering a transient spherical light source. Also, for the Kerr spacetimes, we point out a potential difference between the resulting light echoes using the geometric-optics approach and those obtained by the black hole perturbation theory. Possible astrophysical implications of the present study are addressed.

7 citations

Journal ArticleDOI
TL;DR: In this paper , the authors analyzed the slow rotation approximation of the rotating black hole in the Einstein-bumblebee gravity, and obtained the master equations for scalar, vector and axial gravitational perturbation, respectively.
Abstract: We have studied the quasinormal modes (QNMs) of a slowly rotating black hole with Lorentz-violating parameter in Einstein-bumblebee gravity. We analyse the slow rotation approximation of the rotating black hole in the Einstein-bumblebee gravity, and obtain the master equations for scalar perturbation, vector perturbation and axial gravitational perturbation, respectively. Using the matrix method and the continuous fraction method, we numerically calculate the QNM frequencies. In particular, for scalar field, it shows that the QNMs up to the second order of rotation parameter have higher accuracy. The numerical results show that, for both scalar and vector fields, the Lorentz-violating parameter has a significant effect on the imaginary part of the QNM frequencies, while having a relatively smaller impact on the real part of the QNM frequencies. But for axial gravitational perturbation, the effect of increasing the Lorentz-violating parameter $\ell$ is similar to that of increasing the rotation parameter $\tilde{a}$.

6 citations

Journal ArticleDOI
TL;DR: In this article, the authors perturb the non-rotating BTZ black hole with a non-minimally coupled massless scalar field, and compute the quasinormal spectrum exactly.
Abstract: We perturb the non-rotating BTZ black hole with a non-minimally coupled massless scalar field, and we compute the quasinormal spectrum exactly. We solve the radial equation in terms of hypergeometric functions, and we obtain an analytical expression for the quasinormal frequencies. In addition, we compare our analytical results with the 6th order semi-analytical WKB method, and we find an excellent agreement. The impact of the nonminimal coupling as well as of the cosmological constant on the quasinormal spectrum is briefly discussed.

6 citations

Journal ArticleDOI
TL;DR: In this article, the photon sphere, shadow radius and quasinormal modes of a four-dimensional charged Einstein-Gauss-Bonnet black hole were derived by the WKB approximation approach and shadow radius, respectively.
Abstract: In this paper, we investigate the photon sphere, shadow radius and quasinormal modes of a four-dimensional charged Einstein-Gauss-Bonnet black hole. The perturbation of a massless scalar field in the black hole's background is adopted. The quasinormal modes are gotten by the $6th$ order WKB approximation approach and shadow radius, respectively. When the value of the Gauss-Bonnet coupling constant increase, the values of the real parts of the quasinormal modes increase and those of the imaginary parts decrease. The coincidence degrees of quasinormal modes derived by the two approaches increases with the increase of the values of the Gauss-Bonnet coupling constant and multiple number. It shows the correspondence between the shadow and test field in the four-dimensional Einstein-Gauss-Bonnet-Maxwell gravity. The radii of the photon sphere and shadow increase with the decrease of the Gauss-Bonnet coupling constant.

5 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, it was shown that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of the super Yang-Mills theory in four dimensions.
Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of ${\cal N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the ${\cal N}=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

14,084 citations

Journal ArticleDOI
TL;DR: In this paper, a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory is introduced to obtain certain Green's functions in 3+1-dimensional N = 4 supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory.

11,887 citations

Journal ArticleDOI
TL;DR: In this article, it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature, which leads to a slow decrease in the mass of the black hole and to its eventual disappearance.
Abstract: In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature\(\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K\) where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.

10,923 citations

Journal ArticleDOI
B. P. Abbott1, Richard J. Abbott1, T. D. Abbott2, Matthew Abernathy1  +1008 moreInstitutions (96)
TL;DR: This is the first direct detection of gravitational waves and the first observation of a binary black hole merger, and these observations demonstrate the existence of binary stellar-mass black hole systems.
Abstract: On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of $1.0 \times 10^{-21}$. It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greater than 5.1 {\sigma}. The source lies at a luminosity distance of $410^{+160}_{-180}$ Mpc corresponding to a redshift $z = 0.09^{+0.03}_{-0.04}$. In the source frame, the initial black hole masses are $36^{+5}_{-4} M_\odot$ and $29^{+4}_{-4} M_\odot$, and the final black hole mass is $62^{+4}_{-4} M_\odot$, with $3.0^{+0.5}_{-0.5} M_\odot c^2$ 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 direct detection of gravitational waves and the first observation of a binary black hole merger.

9,596 citations

Posted Content
TL;DR: In this article, a correspondence between conformal field theory observables and those of supergravity was proposed, where correlation functions in conformal fields are given by the dependence of the supergravity action on the asymptotic behavior at infinity.
Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

8,751 citations