Sourav Bhattacharya, Amitabha Lahiri
We present a non-perturbative proof of the no hair theorems corresponding to scalar and Proca fields for stationary axisymmetric de Sitter black hole spacetimes. Our method also applies to asymptotically flat and under a reasonable assumption, to asymptotically anti-de Sitter spacetimes.
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We present a proof of the no hair theorems corresponding to free massive non-perturbative Pauli-Fierz spin-2 and perturbative massive spin-1/2 fields for stationary axisymmetric de Sitter black hole spacetimes of dimension four with two commuting Killing vector fields. The applicability of these results for asymptotically flat and anti-de Sitter spacetimes are also discussed.
Numerical Evidence for Non-Axisymmetric Gravitational "Hair" for Extremal Kerr Black Hole Spacetimes with Hyperboloidal Foliations
本文提出极端Kerr黑洞非轴对称引力“毛发”（Ori系数）的表达式，通过数值求解Teukolsky方程，提供其存在及性质的强有力证据。
We give a proof of the non-perturbative no hair theorems for a massive 2-form field with 3-form field strength for general stationary axisymmetric and static (anti)-de Sitter or asymptotically flat black hole spacetimes with some suitable geometrical properties. The generalization of this result for higher form fields is discussed. Next, we discuss the perturbative no hair theorems for massive spin-1/2 fields for general static backgrounds with electric or magnetic charge. Some generalization of this result for stationary axisymmetric spacetimes are also discussed. All calculations are done in…
Aretakis Hair for Extreme Kerr Black Holes with Axisymmetric Scalar Perturbations
本文研究了极端Kerr时空中轴对称标量扰动的演化，测量了由模式耦合激发的模式与视界电荷，发现近场初始数据的Aretakis常数与Ori-Sela前置因子存在线性关系，从而将无毛定理的违反扩展至模式耦合情形。
How Do Axisymmetric Black Holes Grow Monopole and Dipole Hair?
研究了scalar Gauss-Bonnet和dynamical Chern-Simons理论中轴对称黑洞单极和偶极毛发的动力学形成，通过解析和模拟快速旋转中子星坍缩揭示了偶极毛发动力学与事件视界出现的关联。
In this work we check the validity of the no scalar hair theorem in charged axisymmetric stationary black holes for a wide class of scalar tensor theories.
We show that the gravitational phase space for the near-horizon region of a bifurcate, axisymmetric Killing horizon in any dimension admits a 2D conformal symmetry algebra with central charges proportional to the area. This extends the construction of [Haco et. al., JHEP 12, 098 (2018)] to generic Killing horizons appearing in solutions of Einstein's equations, and motivates a holographic description in terms of a 2D conformal field theory. The Cardy entropy in such a field theory agrees with the Bekenstein-Hawking entropy of the horizon, suggesting a microscopic interpretation.
We studied, both analytically and numerically, an Abelian Higgs vortex in the spacetime of stationary axisymmetric black hole being the solution of the low-energy limit of the heterotic string theory, acted as hair on the black hole background. Taking into account the gravitational backreaction of the vortex we show that its influence is far more subtle than causing only a conical defect. As a consequence of its existence, we found that the ergosphere was shifted as well as event horizon angular velocity was affected by its presence. Numerical simulations reveal the strong dependence of the vo…
We present a stationary axisymmetric solution belonging to Carter's family [A] of spaces and representing an anisotropic fluid configuration.
No-hair and almost-no-hair results for static axisymmetric black holes and ultracompact objects in astrophysical environments
本文在静态轴对称情形下推广无毛定理，证明与给定外部引力场相容的黑洞几何仅构成单参数族，并证明在曲率有界条件下，近黑洞的无视界天体的形状偏离必须随趋近黑洞极限而消失。
Two theorems are proved concerning how stationary axisymmetric electrovac spacetimes that are equatorially symmetric or equatorially antisymmetric can be characterized correctly in terms of the Ernst potentials $\E$ and $Φ$ or in terms of axis-data.
We prove that for sub-extremal axisymmetric and stationary black holes with arbitrary surrounding matter the inequality $8π|J|<A$ holds, where $J$ is the angular momentum and $A$ the horizon area of the black hole.
Three natural classes of orthonormal frames, namely Frenet-Serret, Fermi-Walker and parallel transported frames, exist along any timelike world line in spacetime. Their relationships are investigated for timelike circular orbits in stationary axisymmetric spacetimes, and illustrated for black hole spacetimes.
We analyse the expression for the mass of a stationary axisymmetric configuration in general relativity obtained in our previous work [1]. From the generality of our formula and its incompatibility with the corresponding expression in Kerr space-time we argue that a stationary equilibrium distribution of a real matter cannot be a source of the Kerr metric.
It is shown that within conformally flat stationary axisymmetric spacetimes, besides of the static family, there exists a new class of metrics, which is always stationary and axisymmetric. All these spacetimes, the static and the stationary ones, are endowed with an arbitrary function depending on the two non--Killingian coordinates. The explicit form of this function can be determined once the coupled matter, i.e., the energy--momentum tensor is given. One might hope possible extensions of this result to black holes on two--branes in four dimensions.
Einstein's equations for stationary axisymmetric fields are reformulated as the equations for affine geodesics in a two--dimensional space. The affine collineations of this space are investigated and used to relate explicit solutions of Einstein's equations with different physical properties. Particularly, the solutions describing the exterior fields of a dyon and a slowly rotating body are discussed.
It has been shown by one of the authors$^1$ that in isotropic spherical coordinates there is a relation between the mass of a static spherical gravitating body and the pressure distribution inside it. In this paper the result is generalized for the case of stationary axisymmetric configurations.
Clairin's method of obtaining Bäcklund transformations is applied to Einstein's field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid. It is shown that for arbitrary pressure $p$ and mass density $μ$ the method does not give non-trivial Bäcklund transformations, while if $μ+ 3p =0$ it gives the transformation of Ehlers.
In this work we present an approximate solution of the Einstein equations describing a global model for the gravitational field generated by a bounded, self-gravitating stationary and axisymmetric body rotating rigidly with constant angular velocity.
In this second paper devoted to the equatorial symmetry/antisymmetry of stationary axisymmetric electrovac spacetimes we show how two theorems proved in our previous paper (Ernst, Manko and Ruiz 2006 Class. Quantum Grav. 23, 4945) can be utilized to construct exact solutions that are equatorially symmetric or antisymmetric.
Physically admissible choice of the "essential" coordinates identified with components of the metric tensor and co-moving frame of reference reduced to the formulation of the stationary axisymmetric GR problem. Such nontraditional approach allows to obtain new 4-parametric vacuum solution. In special static case this solution becomes the set well-known GR exact solutions for degenerate vacuum gravitational fields, in some special stationary case, after transformations coincides with Kerr solution. In other special stationary case the founded solution turn into modified NUT solution.
Most no-hair theorems involve the assumption that the scalar field is independent of time. Recently in [Phys. Rev. D90 (2014) 041501(R)] the existence of time-dependent scalar hair outside a stationary black hole in general relativity was ruled out. We generalize this work to modified gravities and non-minimally coupled scalar field with an additional assumption that the spacetime is axisymmetric. It is shown that in higher-order gravity such as metric $f(R)$ gravity the time-dependent scalar hair doesn't exist. While in Palatini $f(R)$ gravity and non-minimally coupled case the time-dependent…
In this paper we prove that in a stationary axisymmetric SU(2) Einstein-Yang-Mills theory the most reasonable circularity conditions that can be considered for the Yang-Mills fields imply in fact that the field is of embedded Abelian type, or else that the metric is not asymptotically flat.
Axisymmetric and stationary solutions are constructed to the Einstein--Vlasov and Vlasov--Poisson systems. These solutions are constructed numerically, using finite element methods and a fixed-point iteration in which the total mass is fixed at each step. A variety of axisymmetric stationary solutions are exhibited, including solutions with toroidal, disk-like, spindle-like, and composite spatial density configurations, as are solutions with non-vanishing net angular momentum. In the case of toroidal solutions, we show for the first time, solutions of the Einstein--Vlasov system which contain …
Stationary, axisymmetric, vacuum, solutions of Einstein's equations are obtained as critical points of the total mass among all axisymmetric and $(t,ϕ)$ symmetric initial data with fixed angular momentum. In this variational principle the mass is written as a positive definite integral over a spacelike hypersurface. It is also proved that if absolute minimum exists then it is equal to the absolute minimum of the mass among all maximal, axisymmetric, vacuum, initial data with fixed angular momentum. Arguments are given to support the conjecture that this minimum exists and is the extreme Kerr i…
We reduce Einstein's field equations for the interior of a uniformly rotating, axisymmetric perfect fluid to a system of six second order partial differential equations for the pressure p the energy density $μ$ and four dependent variables.Four of these equations do not depend on p and $μ$ and the other two determine p and $μ$.
In this paper, the problem of finding an axisymmetric stationary spacetime from a specified set of multipole moments, is studied. The condition on the multipole moments, for existence of a solution, is formulated as a convergence condition on a power series formed from the multipole moments. The methods in this paper can also be used to give approximate solutions to any order as well as estimates on each term of the resulting power series.
A formulation for stationary axisymmetric electromagnetic fields in general relativity is derived by casting them into the form of an anisotropic fluid. Several simplifications of the formalism are carried out in order to analyze different features of the fields, such as the derivation of electromagnetic sources for the Maxwell field in the form of thin layers, construction of new solutions, and generation techniques.
A procedure is described for matching a given stationary axisymmetric perfect fluid solution to a not necessarily asymptotically flat vacuum exterior. Using data on the zero pressure surface, the procedure yields the Ernst potential of the matching vacuum metric on the symmetry axis. From this the full metric can be constructed using a variety of well established procedures.
We prove the non existence of massive scalar, vector and tensor hairs outside the surface of a static and stationary compact reflecting star. Our result is the extension of the no hair theorem for black holes to horizonless compact configurations with reflecting boundary condition at the surface. We also generalize the proof for spacetimes with a positive cosmological constant.
We present the fundamentals of the recently proposed geometric description of photon regions in terms of foliation into fundamental photon hypersurfaces, which satisfies the umbilic condition for the subbundle of the tangent bundle defined by the generalized impact parameter.
We apply the energy-momentum tensor which is coordinate independent to calculate the energy content of the axisymmetric solutions. Our results are compared with what have been obtained before within the framework of Einstein general relativity and Møler's tetrad theory of gravitation.
Non-existence of short hairs for static black holes
本文将黑洞无短毛定理推广至静态轴对称有毛黑洞解，证明其“毛发层”必须延伸至最内层光环径向范围之外。
We investigate the existence of the skyrmion field in the background of an asymptotically flat stationary reflecting star. For this purpose, we consider the Einstein Skyrme system for which there is a skyrmion hair in the black hole case. We discuss spherically symmetric skyrmions and employ the hedgehog ansatz for the skyrmion field. We show that, in contrast to the black hole there is no skyrmion hair for a reflecting star.
We present a method for generating exact solutions of Einstein equations in vacuum using harmonic maps, when the spacetime possesses two commutating Killing vectors. This method consists in writing the axisymmetric stationry Einstein equations in vacuum as a harmonic map which belongs to the group SL(2,R), and decomposing it in its harmonic "submaps". This method provides a natural classification of the solutions in classes (Weil's class, Lewis' class etc).
A simple formula, invariant under the duality rotation Φ-> exp(iα) Φ, is obtained for the Poynting vector within the framework of the Ernst formalism, and its application to the known exact solutions for a charged massive magnetic dipole is considered.
In this article we study multipole moments of axisymmetric stationary asymptotically flat spacetimes. We show how the tensorial recursion of Geroch and Hansen can be reduced to a recursion of scalar functions. We also demonstrate how a careful choice of conformal factor collects all moments into one complex valued function on R, where the moments appear as the derivatives at 0. As an application, we calculate the moments of the Kerr solution. We also discuss the freedom in choosing the potential for the moments.
Garcia and Campuzano claim to have found a previously overlooked family of stationary and axisymmetric conformally flat spacetimes, contradicting an old theorem of Collinson. In both these papers it is tacitly assumed that the isometry group is orthogonally transitive. Under the same assumption, we point out here that Collinson's result still holds if one demands the existence of an axis of symmetry on which the axial Killing vector vanishes. On the other hand if the assumption of orthogonal transitivity is dropped, a wider class of metrics is allowed and it is possible to find explicit counte…
We investigate the existence and phenomenology of stable photon orbits (SPOs) in stationary axisymmetric electrovacuum spacetimes in four dimensions. First, we review the classification of equatorial circular photon orbits on Kerr-Newman spacetimes in the charge-spin plane. Second, using a Hamiltonian formulation, we show that Reissner-Nordström di-holes (a family encompassing the Majumdar-Papapetrou and Weyl-Bach special cases) admit SPOs, in a certain parameter regime that we investigate. Third, we explore the transition from order to chaos for typical SPOs bounded within a toroidal region a…
In this paper a new method is derived for constructing electromagnetic surface sources for stationary axisymmetric electrovac spacetimes endowed with non-smooth or even discontinuous Ernst potentials. This can be viewed as a generalization of some classical potential theory results, since lack of continuity of the potential is related to dipole density and lack of smoothness, to monopole density. In particular this approach is useful for constructing the dipole source for the magnetic field. This formalism involves solving a linear elliptic differential equation with boundary conditions at inf…
We present the complete scheme of the application of the one-and two dimensional subspace and subgroups method to five-dimensional gravity with a $G_{3}$ group of motion. We do so in the space time and in the potential space formalisms. From this method one obtains the Kramer, Belinsky-Ruffini, Dobiasch-Maison, Clément, Gross-Perry-Sorkin solutions etc. as special cases.
The tools developed in a preceding article for interpreting spacetime geometry in terms of all possible space-plus-time splitting approaches are applied to circular orbits in some familiar stationary axisymmetric spacetimes. This helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of Abramowicz, de Felice, and others on circular orbits in black hole spacetimes into a more general context.
Recently Kordas (1995, Class. Quantum Grav. 12 2037) and Meinel and Neugebauer (1995, Class. Quantum Grav. 12 2045) studied the conditions for reflection symmetry in stationary axisymmetric space--times in vacuum. They found that a solution to the Einstein field equations is reflectionally symmetric if their Ernst's potential ${\cal E}(ρ=0,z)= e(z)$ on a portion of the positive z-axis extending to infinity satisfies the condition $e_{+}(z){e}_{+}^{*}(-z)=1$. In this note, we formulate an analogous conditions for two complex Ernst potentials in electrovacuum. We also present the special case of…
The structural properties of geodesic currents in an ambient Kerr background is studied from an analytical point of view. The geodesics in the congruence correspond to charged particles that carry energy and angular momentum from the black hole through the Blandford-Znajek mechanism. It is shown that the resulting magnetosphere naturally satisfies the Znajek regularity condition. Particular attention is paid here to the energy extracted by matter currents rather than by electromagnetic Poynting fluxes.
By using the 't Hooft "brick wall" model and the Pauli-Villars regularization scheme we calculate the statistical-mechanical entropy arising from the minimally coupled scalar fields which rotate with the azimuthal angular velocity $Ω_0=Ω_H$ ($Ω_H$ is the angular velocity of the black hole horizon) in the general four-dimensional non-extreme stationary axisymmetric black hole space-time. We also show, for the Kerr-Newman and the Einstein-Maxwell dilaton-axion black holes, that the statistical-mechanical entropy obtained from our derivation and the quantum thermodynamical entropy by the conical …
Stationary, asymptotically flat black holes in scalar-tensor theories of gravity are studied. It is shown that such black holes have no scalar hair and are the same as in General Relativity.
Perturbed stationary axisymmetric isolated bodies, e.g. stars, represented by a matter-filled interior and an asymptotically flat vacuum exterior joined at a surface where the Darmois matching conditions are satisfied, are considered. The initial state is assumed to be static. The perturbations of the matching conditions are derived and used as boundary conditions for the perturbed Ernst equations in the exterior region. The perturbations are calculated to second order. The boundary conditions are overdetermined: necessary and sufficient conditions for their compatibility are derived. The spec…
Nonlinear Stability of Rotating Hairy Black Holes
本文研究了旋转毛黑洞的稳定性，发现当标量场质量次主导时，系统在非线性演化中保持稳定，而标量场质量主导时则在较短时间内出现不稳定性。
We construct stationary black holes in SU(2) Einstein-Yang-Mills theory, which carry angular momentum and electric charge. Possessing non-trivial non-abelian magnetic fields outside their regular event horizon, they represent non-perturbative rotating hairy black holes.
The extended scalar-tensor and vector-tensor theories admit black hole solutions with the nontrivial profiles of the scalar and vector fields, respectively. The disformal transformation maps a solution in a class of the scalar-tensor or vector-tensor theories to that in another class, and hence it can be a useful tool to construct a new nontrivial solution from the known one. First, we investigate how the stationary and axisymmetric solutions in the vector-tensor theories without and with the $U(1)$ gauge symmetry are disformally transformed. We start from a stationary and axisymmetric solutio…
This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along $ρ$ and $z$ directions. The three possibilities of the axial vector depending on the metric function $B$ are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of M$\ddot{o}$ller prescription to…
We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. The SU(2,1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is…
The shadow of a black hole can be one of the strong observational evidences for stationary black holes. If we see shadows at the center of galaxies, we would say whether the observed compact objects are black holes. In this paper, we consider a formula for the contour of a shadow in an asymptotically-flat, stationary, and axisymmetric black hole spacetime. We show that the formula is useful for obtaining the contour of the shadow of several black holes such as the Kerr-Newman black hole and rotating regular black holes. Using the formula, we can obtain new examples of the contour of the shadow…
We investigate the interior of regular axisymmetric and stationary black holes surrounded by matter and find that for non-vanishing angular momentum of the black hole the space time can always be extended regularly up to and including an inner Cauchy horizon. We provide an explicit relation for the regular metric at the inner Cauchy horizon in terms of that at the event horizon. As a consequence, we obtain the universal equality $(8πJ)^2 = A^+ A^-$ where $J$ is the black hole's angular momentum and $A^-$ and $A^+$ denote the horizon areas of inner Cauchy and event horizon, respectively. We als…
We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in Mars and Senovilla, 1994, and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solut…
A class of stationary rigidly rotating perfect fluid coupled with non-linear electromagnetic fields was investigated. An exact solution of the Einstein equations with sources for the Carter B(+) branch was found, for the equation of state $3p + ε= constant$. We use a structural function for the Born-Infeld non-linear electrodynamics which is invariant under duality rotations and a metric possessing a four- parameter group of motions. The solution is of Petrov type D and the eigenvectors of the electromagnetic field are aligned to the Debever-Penrose vectors.
Regular hairy black holes through Minkowski deformation
该研究在事件视界存在性与物质弱能量条件的要求下，利用引力解耦方法得到无曲率奇点的正则有毛黑洞解，明确了对应闵可夫斯基真空形变的物质源及静态、稳态下的最大形变形式。
Photon region (PR) in the strong gravitational field is defined as a compact region where photons can travel endlessly without going to infinity or disappearing at the event horizon. In Schwarzschild metric PR degenerates to the two-dimensional photon sphere $r=3r_g/2$ where closed circular photon orbits are located. The photon sphere as a three-dimensional hypersurface in spacetime is umbilic (its second quadratic form is pure trace). In Kerr metric the equatorial circular orbits have different radii for prograde, $r_p$, and retrograde, $r_r$, motion (where $r$ is Boyer-Lindquist radial varia…
In this paper we consider homothetic Killing vectors in the class of stationary axisymmetric vacuum (SAV) spacetimes, where the components of the vectors are functions of the time and radial coordinates. In this case the component of any homothetic Killing vector along the $z$ direction must be constant. Firstly, it is shown that either the component along the radial direction is constant or we have the proportionality $g_{ϕϕ}\propto g_{ρρ}$, where $g_{ϕϕ}>0$. In both cases, complete analyses are carried out and the general forms of the homothetic Killing vectors are determined. The associated…
In a recent paper Cañate (CQG, {\bf 35}, 025018 (2018)) proved a no hair theorem to static and spherically symmetric or stationary axisymmetric black holes in general $f(R)$ gravity. The theorem applies for isolated asymptotically flat or asymptotically de Sitter black holes and also in the case when vacuum is replaced by a minimally coupled source having a traceless energy momentum tensor. This theorem excludes the case of pure quadratic gravity, $f(R) = R^2$. In this paper we use the scalar tensor representation of general $f(R)$ theory to show that there are no hairy black hole in pure $R^2…
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