Black hole photon spheres or light rings are closely linked to astronomical phenomena, such as the gravitational waves and the shadow in spherically symmetric or axi-symmetric spacetime. In Cunha and Herdeiro [Phys. Rev. Lett. 124, 181101 (2020)], a topological argument was applied for the four dimensional stationary, axi-symmetric, asymptotically flat black hole, and the result indicates that at least, there exists one standard light ring outside the black hole horizon for each rotation sense. Inspired by it, in this paper, we would like to consider a similar issue for a nonrotating, static, spherically symmetric black hole not only with asymptotically flat behavior, but also with AdS and dS behaviors. Following Duan's topological current $ϕ$-mapping theory, the topological current and charge for the photon spheres are introduced. The topological current is nonzero only at the zero point of the vector field determining the location of the photon sphere. So each photon sphere can be assigned a topological charge. Considering the full exterior region, we find the total topological charge always equals -1. This result confirms that there exists at least one standard photon sphere outside of the black hole not only in asymptotically flat spacetime, but also in asymptotically AdS and dS spacetime. Then we apply the study to the dyonic black holes. We observe that even when more photon spheres are included, the total topological charge stays unchanged. Moreover, for a naked singularity, it has a vanishing topological charge, indicating that the black hole and naked singularity are in different topological classes. It is expected that this novel topological argument could provide an insightful idea on the study of the black hole photon spheres or light rings, and further cast new light on the black hole astronomical phenomena.
We study exact solutions of the Einstein-Maxwell equations for the interior gravitational field of static spherically symmetric charged compact spheres. The spheres consist of an anisotropic fluid with a charge distribution that gives rise to a static radial electric field. The density of the fluid has the form $ρ(r) = ρ_o + αr^2$ (here $ρ_o$ and $α$ are constants) and the total charge $q(r)$ within a sphere of radius $r$ has the form $q = βr^3$ (with $β$ a constant). We evaluate the critical values of $M/R$ for these spheres as a function of $Q/R$ and compare these values with those given by …
We study exact solutions of the Einstein-Maxwell equations for the interior gravitational field of static spherically symmetric charged compact spheres. The spheres are composed of a perfect fluid with a charge distribution that creates a static radial electric field. The inertial mass density of the fluid has the form $ρ(r) = ρ_o + αr^2$ ($ρ_o$ and $α$ are constants) and the total charge $q(r)$ within a sphere of radius $r$ has the form $q = βr^3$ ($β$ is a constant). We evaluate the critical values of $M/R$ for these spheres as a function of $Q/R$ and compare these values with those given by…
We consider spherically symmetric and static charged black holes in Einstein-Gauss-Bonnet-Maxwell gravities in general $D\ge 5$ dimensions and study their photon spheres and black hole shadows. We show that they all satisfy the sequence of inequalities recently proposed relating a black hole's horizon, photon sphere, shadow and its mass.
We study the spontaneous symmetry breaking of O(3) scalar field on a fuzzy sphere $S_F^2$. We find that the fluctuations in the background of topological configurations are finite. This is in contrast to the fluctuations around a uniform configuration which diverge, due to Mermin-Wagner-Hohenberg-Coleman theorem, leading to the decay of the condensate. Interesting implications of enhanced topological stability of the configurations are pointed out.
We introduce the quasi-topological electromagnetism which is defined to be the squared norm of the topological 4-form $F\wedge F$. A salient property is that its energy-momentum tensor is of the isotropic perfect fluid with the pressure being precisely the opposite to its energy density. It can thus provide a model for dark energy. We study its application in both black hole physics and cosmology. The quasi-topological term has no effect on the purely electric or magnetic Reissner-Nordström black holes, the dyonic solution is however completely modified. We find that the dyonic black holes can…
We find an exact solution for the stability limit of relativistic charged spheres for the case of constant gravitational mass density and constant charge density. We argue that this provides an absolute stability limit for any relativistic charged sphere in which the gravitational mass density decreases with radius and the charge density increases with radius. We then provide a cruder absolute stability limit that applies to any charged sphere with a spherically symmetric mass and charge distribution. We give numerical results for all cases. In addition, we discuss the example of a neutral sph…
Considering a charged black hole (BH) surrounded by a perfect fluid radiation field (PFRF) in Rastall gravity, we investigate this BH shadow and photon sphere on different spherical accretions backgrounds. The effect of the PFRF parameter/BH charge on the critical impact parameter is studied by investigating the light deflection near this BH. The luminosity of these BH shadows in different spherical accretions is obtained, respectively. It is found that the shadow of this BH with infalling spherical accretion is darker than static spherical accretion, and the photon sphere with infalling spher…
The coupled system of the spherically symmetric Einstein--Maxwell differential equations is solved under two different source conditions: non-zero electric charge and pressure anisotropy. Expressions for the metric functions, and pressures which extend the Tolman VII exact solution are deduced from the new solutions. By applying boundary conditions to these solutions, all integration constants are computed in terms of parameters that are physically meaningful.
Motivated by recent work, nonmonotonic behaviours of photon sphere radius can be used to reflect black hole phase transition for Reissner-Nordstr$\ddot{o}$m-AdS (RN-AdS) black holes, we study the case of five-dimensional charged Gauss-Bonnet-AdS (GB-AdS) black holes in the reduced parameter space. We find that the nonmonotonic behaviours of photon sphere radius still exist. Using the coexistence line calculated from $P-V$ plane, we capture the photon sphere radius of saturated small and large black holes (the boundary of the coexistence phase), then illustrate the reduced coexistence region. T…
Static spherically symmetric anisotropic source has been studied for the Einstein-Maxwell field equations assuming the erstwhile cosmological constant $ Λ$ to be a space-variable scalar, viz., $ Λ= Λ(r) $. Two cases have been examined out of which one reduces to isotropic sphere. The solutions thus obtained are shown to be electromagnetic in origin as a particular case. It is also shown that the generally used pure charge condition, viz., $ ρ+ p_r = 0 $ is not always required for constructing electromagnetic mass models.
A new class of solutions to the coupled, spherically symmetric Einstein-Maxwell equations for a compact material source is constructed. Some of these solutions can be made to satisfy a number of requirements for being physically relevant, including having a causal speed of sound. In the case of vanishing charge these solutions reduce to those found by Bayin and Tolman. Only the latter can be considered as having physically realistic properties.
When the equatorial spin velocity, $v$, of a charged conducting sphere approaches $c$, the Lorentz force causes a remarkable rearrangement of the total charge $q$. Charge of that sign is confined to a narrow equatorial belt at latitudes $b \leqslant \sqrt{3} (1 - v^2/c^2)^{1/2}$ while charge of the opposite sign occupies most of the sphere's surface. The change in field structure is shown to be a growing contribution of the `magic' electromagnetic field of the charged Kerr-Newman black hole with Newton's G set to zero. The total charge within the narrow equatorial belt grows as $(1-v^2/c^2)^{-…
With the question, ``Can relativistic charged spheres form extremal black holes?" in mind, we investigate the properties of such spheres from a classical point of view. The investigation is carried out numerically by integrating the Oppenheimer-Volkov equation for relativistic charged fluid spheres and finding interior Reissner-Nordström solutions for these objects. We consider both constant density and adiabatic equations of state, as well as several possible charge distributions, and examine stability by both a normal mode and an energy analysis. In all cases, the stability limit for these s…
We explore a class of compact charged spheres made of a charged perfect fluid with a polytropic equation of state. The charge density is chosen to be proportional to the energy density. The study is performed by solving the Tolman-Oppenheimer-Volkoff (TOV) equation which describes the hydrostatic equilibrium. We show the dependence of the structure of the spheres for several characteristic values of the polytropic exponent and for different values of the charge densities. We also study other physical properties of the charged spheres, such as the total mass, total charge, radius and sound spee…
We study the static stellar equilibrium configurations ofuncharged and charged spheres composed by a relativistic polytropic fluid, and compare with those of spheres composed by a non-relativistic polytropic fluid, the later case already being studied in a previous work [J. D. Arbañil, P. S. Lemos, V. T. Zanchin, Phys. Rev. D \textbf{88}, 084023 (2013)]. In the relativistic fluid case, a relativistic polytropic equation of state, $p=ωδ^γ$, is assumedd. Here, $δ=ρ-p/(γ-1)$, with $δ$ and $ρ$ being the rest mass density and the energy density, respectively, and $ω$ and $γ$ are respectively the po…
In this article, Einstein-Maxwell space-time has been considered in connection to some of the astrophysical solutions as previously obtained by Tolman (1939) and Bayin (1978). The effect of inclusion of charge into these solutions has been investigated thoroughly and also the nature of fluid pressure and mass density throughout the sphere have been discussed. Mass-radius and mass-charge relations have been derived for various cases of the charged matter distribution. Two cases are obtained where perfect fluid with positive pressures give rise to electromagnetic mass models such that gravitatio…
Multimodal signatures of asymptotic (A)dS Kalb-Ramond black holes: Constraints through the shadow, weak deflection angle, and topological photon spheres
We consider the problem of uniqueness of static and asymptotically flat Einstein-Maxwell spacetimes with a photon sphere $P^3$. We are using a naturally modified definition of a photon sphere for electrically charged spacetimes with the additional property that the one-form $ι_ξF$ is normal to the photon sphere. For simplicity we are restricting ourselves to the case of zero magnetic charge and assume that the lapse function regularly foliates the spacetime outside the photon sphere. With this information we prove that $P^3$ has constant mean curvature and constant scalar curvature. We also de…
In a recent paper by Giuliani and Rothman \cite{GR}, the problem of finding a lower bound on the radius $R$ of a charged sphere with mass M and charge Q<M is addressed. Such a bound is referred to as the critical stability radius. Equivalently, it can be formulated as the problem of finding an upper bound on M for given radius and charge. This problem has resulted in a number of papers in recent years but neither a transparent nor a general inequality similar to the case without charge, i.e., M\leq 4R/9, has been found. In this paper we derive the surprisingly transparent inequality $$\sqrt{M}…
We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions, the topological sphere can be isometrically into Euclidean 3-space and mean curvature of the initial date set does not change sign. The positivity holds if we further assume the image of the topological sphere in Euclidean 3-space has nonnegative mean curvature.
In a static spherically symmetric Einstein-Maxwell spacetime the class of astrophysical solution found out by Ray and Das (2002) and Pant and Sah (1979) are revisited here in connection to the phenomenological relationship between the gravitational and electromagnetic fields. It is qualitatively shown that the charged relativistic stars of Tolman (1939) and Bayin (1978) type are of purely electromagnetic origin. The existence of this type of astrophysical solutions is a probable extension of Lorentz's conjecture that electron-like extended charged particle possesses only `electromagnetic mass'…
