We perform an observational test of no-hair theorem using quasi-periodic oscillations within the relativistic precession model. Two well motivated metrics we apply are Kerr-Q and Hartle-Thorne metrics in which the quadrupole is the parameter that possibly encodes deviations from the Kerr black hole. The expressions for the quasi-periodic frequencies are derived before comparing the models with the observation. We encounter a degeneracy in constraining spin and quadrupole parameters that makes it difficult to measure their values. In particular, we here propose a novel test of no-hair theorem by adapting the Hartle-Thorne metric. It turns out that a Kerr black hole is a good description of the central object in GRO J1655$-$40 given the present observational precisions.
One avenue for testing the no-hair theorem is obtained through timing a pulsar orbiting close to a black hole and fitting for quadrupolar effects on the time-of-arrival of pulses. If deviations from the Kerr quadrupole are measured, then the no-hair theorem is invalidated. To this end, we derive an expression for the light travel time delay for a pulsar orbiting in a black-hole spacetime described by the Butterworth-Ipser metric, which has an arbitrary spin and quadrupole moment. We consider terms up to the quadrupole order in the black-hole metric and derive the time-delay expression in a clo…
According to the no-hair theorem, an astrophysical black hole is uniquely described by only two quantities, the mass and the spin. In this series of papers, we investigate a framework for testing the no-hair theorem with observations of black holes in the electromagnetic spectrum. We formulate our approach in terms of a parametric spacetime which contains a quadrupole moment that is independent of both mass and spin. If the no-hair theorem is correct, then any deviation of the black-hole quadrupole moment from its Kerr value has to be zero. We analyze in detail the properties of this quasi-Ker…
According to the no-hair theorem, astrophysical black holes are uniquely described by their mass and spin. In this paper, we review a new framework for testing the no-hair hypothesis with observations in the electromagnetic spectrum. The approach is formulated in terms of a Kerr-like spacetime containing a quadrupole moment that is independent of both mass and spin. If the no-hair theorem is correct, then any deviation from the Kerr metric quadrupole has to be zero. We show how upcoming VLBI imaging observations of Sgr A* as well as spectroscopic observations of iron lines from accreting black…
We discuss the consequences of violating the no-hair theorem on magnetic fields surrounding a black hole. This is achieved by parametrically deforming the Kerr spacetime and studying the effects of such deformations on asymptotically uniform magnetic fields around the black hole. We compute the deformed electromagnetic field for slow spins and small deformation parameter, and show that the correction is of order the deformation parameter and mimics the angular structure of a quadrupole.
According to the no-hair theorem, all astrophysical black holes are fully described by their masses and spins. This theorem can be tested observationally by measuring (at least) three different multipole moments of the spacetimes of black holes. In this paper, we analyze images of black holes within a framework that allows us to calculate observables in the electromagnetic spectrum as a function of the mass, spin, and, independently, the quadrupole moment of a black hole. We show that a deviation of the quadrupole moment from the expected Kerr value leads to images of black holes that are eith…
Constituted with a massive black hole and a stellar mass compact object, Extreme Mass Ratio Inspiral (EMRI) events hold unique opportunity for the study of massive black holes, such as by measuring and checking the relations among the mass, spin and quadrupole moment of a massive black hole, putting the no-hair theorem to test. TianQin is a planned space-based gravitational wave observatory and EMRI is one of its main types of sources. It is important to estimate the capacity of TianQin on testing the no-hair theorem with EMRIs. In this work, we use the analytic kludge waveform with quadrupole…
The detection of gravitational waves from the inspiral of a neutron star or stellar-mass black hole into an intermediate-mass black hole (IMBH) promises an entirely new look at strong-field gravitational physics. Gravitational waves from these intermediate-mass-ratio inspirals (IMRIs), systems with mass ratios from ~10:1 to ~100:1, may be detectable at rates of up to a few tens per year by Advanced LIGO/Virgo and will encode a signature of the central body's spacetime. Direct observation of the spacetime will allow us to use the "no-hair" theorem of general relativity to determine if the IMBH …
According to the no-hair theorem, astrophysical black holes are uniquely described by their masses and spins. An observational test of the no-hair theorem can be performed by measuring at least three different multipole moments of the spacetime of a black hole and verifying whether their values are consistent with the unique combinations of the Kerr solution. In this paper, we study quasi-periodic variability observed in the emission from black holes across the electromagnetic spectrum as a test of the no-hair theorem. We derive expressions for the Keplerian and epicyclic frequencies in a quas…
Observations of the precessing orbits of stars very near the massive black hole in the galactic center could provide measurements of the spin and quadrupole moment of the hole and thereby test the no-hair theorem of general relativity. Since the galactic center is likely to be populated by a distribution of stars and small black holes, their gravitational interactions will perturb the orbit of any given star. We estimate the effects of such perturbations using analytic orbital perturbation theory, and show that for a range of possible stellar distributions, and for an observed star sufficientl…
We describe a new numerical algorithm for ray tracing in the external spacetimes of spinning compact objects characterized by arbitrary quadrupole moments. Such spacetimes describe non-Kerr vacuum solutions that can be used to test the no-hair theorem in conjunction with observations of accreting black holes. They are also appropriate for neutron stars with spin frequencies in the 300-600 Hz range, which are typical of the bursting sources in low-mass X-ray binaries. We use our algorithm to show that allowing for the quadrupole moment of the spacetime to take arbitrary values leads to observab…
The "no-hair" theorem can, in principle, be tested at the center of the Milky Way by measuring the spin and the quadrupole moment of Sgr A$^*$ with the orbital precession of S-stars, measured over their full periods. Contrary to the original method, we show why it is possible to test the no-hair theorem using observations from only a single star, by measuring precession angles over a half-orbit. There are observational and theoretical reasons to expect S-stars to spin rapidly, and we have quantified the effect of stellar spin, via spin-curvature coupling (the leading-order manifestation of the…
The no-hair theorem characterizes the fundamental nature of black holes in general relativity. This theorem can be tested observationally by measuring the mass and spin of a black hole as well as its quadrupole moment, which may deviate from the expected Kerr value. Sgr A*, the supermassive black hole at the center of the Milky Way, is a prime candidate for such tests thanks to its large angular size, high brightness, and rich population of nearby stars. In this review I discuss a new theoretical framework for a test of the no-hair theorem that is ideal for imaging observations of Sgr A* with …
The black hole in the center of the Milky Way, Sgr A*, has the largest mass-to-distance ratio among all known black holes in the Universe. This property makes Sgr A* the optimal target for testing the gravitational no-hair theorem. In the near future, major developments in instrumentation will provide the tools for high-precision studies of its spacetime via observations of relativistic effects in stellar orbits, in the timing of pulsars, and in horizon-scale images of its accretion flow. We explore here the prospect of measuring the properties of the black-hole spacetime using all these three…
We present a family of solutions of Einstein's gravity minimally coupled to a complex, massive scalar field, describing asymptotically flat, spinning black holes with scalar hair and a regular horizon. These hairy black holes (HBHs) are supported by rotation and have no static limit. Besides mass M and angular momentum J, they carry a conserved, continuous Noether charge Q measuring the scalar hair. HBHs branch off from the Kerr metric at the threshold of the superradiant instability and reduce to spinning boson stars in the limit of vanishing horizon area. They overlap with Kerr black holes f…
Capturing the image of the shadow cast by the event horizon of an illuminated black hole is, at the most basic level, an experiment of extreme light deflection in a strongly curved spacetime. As such, the properties of an imaged shadow can be used to probe the general relativistic Kerr nature of astrophysical black holes. As an example of this prospect, it is commonly asserted that a shadow can test the validity of the theory's famous `no hair theorem' for the black hole's mass and spin multipole moments. In this paper, we assess this statement by calculating the shadow's equatorial radius in …
If a class of stars orbits the central black hole in our galaxy in short period (~ 0.1 year), high eccentricity (~ 0.9) orbits, they will experience precessions of their orbital planes induced by both relativistic frame-dragging and the quadrupolar gravity of the hole, at levels that could be as large as 10 microarcseconds per year, if the black hole is rotating faster than 1/2 of its maximum rotation rate. Astrometric observations of the orbits of at least two such stars can in principle lead to a determination of the angular momentum vector J of the black hole and its quadrupole moment Q_2. …
Gravitational-wave (GW) observations of binary black holes offer the best probes of the relativistic, strong-field regime of gravity. Gravitational radiation, in the leading order is quadrupolar. However, non-quadrupole (higher-order) modes make appreciable contribution to the radiation from binary black holes with large mass ratios and misaligned spins. The multipolar structure of the radiation is fully determined by the intrinsic parameters (masses and spin angular momenta of the companion black holes) of a binary in quasi-circular orbit. Following our previous work \cite{Dhanpal:2018ufk}, w…
We propose a novel method to test the binary black hole (BBH) nature of compact binaries detectable by gravitational wave (GW) interferometers and hence constrain the parameter space of other exotic compact objects. The spirit of the test lies in the "no-hair" conjecture for black holes where all properties of a black hole are characterised by the mass and the spin of the black hole. The method relies on observationally measuring the quadrupole moments of the compact binary constituents induced due to their spins. If the compact object is a Kerr black hole (BH), its quadrupole moment is expres…
The advent of the Event Horizon Telescope (EHT), a millimeter-wave very-long baseline interferometric array, has enabled spatially-resolved studies of the sub-horizon-scale structure for a handful of supermassive black holes. Among these, the supermassive black hole at the center of the Milky Way, Sagittarius A* (Sgr A*), presents the largest angular cross section. Thus far, these studies have focused upon measurements of the black hole spin and the validation of low-luminosity accretion models. However, a critical input into the analysis of EHT data is the structure of the black hole spacetim…
In this work we analyze some judiciously chosen solutions of Kerr Black Holes with Scalar Hair (KBHsSH) of special interest for Gravitational Wave (GW) events originated from Extreme Mass Ratio Inspirals (EMRIs). Because of the off-center distribution of energy density, these spacetimes are warped in such a way that not all metric functions behave monotonically on the equatorial plane as in the exterior region of Kerr black holes (KBHs). This has great impact on the orbital parameters, which in turn affects the imprints on signals descendant from EMRIs in a adiabatic evolution. By investigatin…
Spin-induced Quadrupole Moment (SIQM) Test for Eccentric Compact Binaries
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Centre for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras（印度马德拉斯理工学院物理学院弦、引力和宇宙学中心）
Measurability of Quadrupole Deviations from Kerr in Binary black hole Mergers
Shanghai Astronomical Observatory（上海天文台）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院）；International Centre for Theoretical Physics Asia-Pacific（亚太理论物理中心）
We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on ar…
In this work we revisit Wald's cosmic no-hair theorem in the context of accelerating Bianchi cosmologies for a generic cosmic fluid with non-vanishing anisotropic stress tensor and when the fluid energy momentum tensor is of the form of a cosmological constant term plus a piece which does not respect strong or dominant energy conditions. Such a fluid is the one appearing in inflationary models. We show that for such a system anisotropy may grow, in contrast to the cosmic no-hair conjecture. In particular, for a generic inflationary model we show that there is an upper bound on the growth of an…
Theoretical investigations have provided proof-of-principle calculations suggesting measurements of stellar or pulsar orbits near the Galactic Center could strongly constrain the properties of the Galactic Center black hole, local matter, and even the theory of gravity itself. In this work, we develop both a Markov chain Monte Carlo and an analytic model (Fisher matrix) to understand what properties are well-constrained and why. We conclude that existing astrometric measurements cannot constrain the spin of the Galactic Center black hole. Extrapolating to the precision and cadence of future ex…
Scalar fields around compact objects are of interest for scalar-tensor theories of gravity and dark matter models consisting of a massive scalar, e.g. axions. We study the behaviour of a scalar field around a Kerr black hole with non trivial asymptotic boundary conditions - both non zero density and non zero angular momentum. Starting from an initial radially homogeneous configuration, a scalar cloud is accreted, which asymptotes to known stationary configurations over time. We study the cloud growth for different parameters including black hole spin, scalar field mass, and the scalar field de…
We provide strong numerical evidence for a new no-scalar-hair theorem for black holes in general relativity, which rules out spherical scalar hair of static four dimensional black holes if the scalar field theory, when coupled to gravity, satisfies the Positive Energy Theorem. This sheds light on the no-scalar-hair conjecture for Calabi-Yau compactifications of string theory, where the effective potential typically has negative regions but where supersymmetry ensures the total energy is always positive. In theories where the scalar tends to a negative local maximum of the potential at infinity…
Any recipe to grow black hole hair has to circumvent no-hair theorems by violating some of their assumptions. Recently discovered hairy black hole solutions exist due to the fact that their scalar fields don't inherit the symmetries of the spacetime metric. We present here a general analysis of the constraints which limit the possible forms of such a hair, for both the real and the complex scalar fields. These results can be taken as a novel piece of the black hole uniqueness theorems or simply as a symmetry noninheriting Ansätze guide. In addition we introduce new classification of the gravit…
We show that the Riemannian Kerr solutions are the only Riemannian, Ricci-flat and asymptotically flat ${\rm C}^{2}$-metrics $g_{μν}$ on a 4-dimensional complete manifold ${\cal M}$ of topology ${\rm R}^{2} \times {\rm S}^{2}$ which have (at least) a 1-parameter group of periodic isometries with only isolated fixed points ("nuts") and with orbits of bounded length at infinity.
In the context of complex scalar field coupled to Einstein gravity theory, we present a novel family of solutions of Kerr black holes with excited-state scalar hair inspired by the work of Herdeiro and Radu in [Phys.\ Rev.\ Lett.\ {\bf 112}, 221101 (2014)], which can be regarded as numerical solutions of rotating compact objects with excited scalar hair, including boson stars and black holes. In contrast to Kerr black holes with ground state scalar hair, we find that the first-excited Kerr black holes with scalar hair have two types of nodes, including radial $n_r=1$ and angular $n_θ=1$ nodes.…
The evidence for supermassive Kerr black holes in galactic centers is strong and growing, but only the detection of gravitational waves will convincingly rule out other possibilities to explain the observations. The Kerr spacetime is completely specified by the first two multipole moments: mass and angular momentum. This is usually referred to as the ``no-hair theorem'', but it is really a ``two-hair'' theorem. If general relativity is the correct theory of gravity, the most plausible alternative to a supermassive Kerr black hole is a rotating boson star. Numerical calculations indicate that t…
This paper reconsider the problem of a Proca field in the exterior of a static black hole. The original Bekenstein's demonstration on the vanishing of this field, based on an integral identity, is improved by using more natural arguments at the event horizon. In particular, the use of the so-called standard integration measure in the horizon is fully justified. Accordingly, the horizon contribution to the Bekenstein integral identity is more involved and its vanishing can be only established using the related Einstein equations. With the new reasoning the ``no-hair'' theorem for the Proca fiel…
We show that transonic one dimensional flows which are analogous to black holes obey no-hair theorems both at the level of linear perturbations and in non-linear regimes. Considering solutions of the Gross-Pitaevskii (or Korteweg-de Vries) equation, we show that stationary flows which are asymptotically uniform on both sides of the horizon are stable and act as attractors. Using Whitham's modulation theory, we analytically characterize the emitted waves when starting from uniform perturbations. Numerical simulations confirm the validity of this approximation and extend the results to more gene…
