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Photon Spheres and Sonic Horizons in Black Holes from Supergravity and Other Theories
Abstract
We study closed photon orbits in spherically-symmetric static solutions of supergravity theories, a Horndeski theory, and a theory of quintessence. These orbits lie in what we shall call a photon sphere (anti-photon sphere) if the orbit is unstable (stable). We show that in all the asymptotically flat solutions we examine that admit a regular event horizon, and whose energy-momentum tensor satisfies the strong energy condition, there is one and only one photon sphere outside the event horizon. We give an example of a Horndeski theory black hole (whose energy-momentum tensor violates the strong energy condition) whose metric admits both a photon sphere and an anti-photon sphere. The uniqueness and non-existence also holds for asymptotically anti-de Sitter solutions in gauged supergravity. The latter also exhibit the projective symmetry that was first discovered for the Schwarzschild-de Sitter metrics: the unparameterised null geodesics are the same as when the cosmological or gauge coupling constant vanishes. We also study the closely related problem of accretion flows by perfect fluids in these metrics. For a radiation fluid, Bondi's sonic horizon coincides with the photon sphere. For a general polytropic equation of state this is not the case. Finally we exhibit counterexamples to a conjecture of Hod's.
... Here we explore circular null geodesics with a constant radius R which is the radius of a photon sphere. The parameters of photon spheres play a key role for numerous applications of black hole solutions: spectra of quasinormal modes (QNM) in the eikonal approximation, circular orbits for massive particles, black hole shadows etc., see [3,[42][43][44][45] and references therein. ...
... Remark 3. We here do not prove the instability of circular null geodesics. We note that in the special case h 1 = h 2 = 1/2 (or λ 2 1 = λ 2 2 = 3/2), the existence, uniqueness and instability the of photon sphere outside the event horizon was proved in [44]. In this case, the metric (2.2) turns out to be a special case of the metric explored in [44]. ...
... We note that in the special case h 1 = h 2 = 1/2 (or λ 2 1 = λ 2 2 = 3/2), the existence, uniqueness and instability the of photon sphere outside the event horizon was proved in [44]. In this case, the metric (2.2) turns out to be a special case of the metric explored in [44]. ...
Dilatonic black hole dyon-like solutions with two (colored) charges and (electric and magnetic ones) in the gravitational 4d model with two scalar fields, two 2-forms are considered. 2-dimensional dilatonic coupling vectors , , obey .
The circular null geodesics are explored. The master equation for radius
R of photon sphere is derived. The conjecture on existence and uniquenes of the solution of master equation obeying , where is horizon radius, is suggested. This conjecture
is varified for certain special cases, e.g. for symmetric charge configuration: . In the charge symmetric case the relation for spectrum of quasinormal modes for a
test massless scalar field in the eikonal approximation and
an example of circular orbits for a massive particle are presented.
... This is in fact the general feature of our analytic f (F 2 ) theory. The duality transformation (40) implies that G 2 as a function of F 2 is infinitely differentiable for our requirements of both analyticity and weak-field limit (2), and hence vice versa, F 2 as a function of G 2 is also infinitely differentiable. This ensures that our regular magnetic monopole metrics are also the welldefined electrically-charged solutions of the dual theory. ...
... It would thus be problematic if the electric field ψ becomes large at the regular core, since f (G 2 ) is not analytic at large G 2 . However, the duality transformation (40) in this case become ...
... which include those obtained in [39,40]. The relevant energy conditions were established in [41]. ...
We construct regular black holes and horizonless spacetimes that are geodesically complete and satisfy the dominant energy condition from Einstein- f ( F 2 ) gravities with several classes of analytic f ( F 2 ) functions that can be viewed as perturbations to Maxwell’s theory in weak field limit. We establish that regular black holes with special static metric ( g tt g rr = - 1 ) violate the strong energy condition and such a regular black hole with Minkowski core violates the null energy condition. We develop a formalism to perform electromagnetic duality transformations in f ( F 2 ) . We obtain two new explicit examples where the duality is a symmetry. We study the properties of the corresponding dyonic black holes. We study the geodesic motions of a particular class of solutions that we call horizonless or black hole repulsons.
... On the other hand, the authors of [24] considered asymptotically flat and spherically symmetric black hole solutions in various theories and found that there is only one photon sphere outside the event horizon if the dominant energy condition (DEC) and strong energy condition (SEC) are satisfied. This observation then led to the conjecture, which states that a violation of either SEC or DEC is a necessary condition for the existence of double photon spheres outside the event horizon. ...
... This observation then led to the conjecture, which states that a violation of either SEC or DEC is a necessary condition for the existence of double photon spheres outside the event horizon. Additionally, it was proved in [24] that SEC requires a monotonically increasing jg tt j in Einstein's gravity. For dyonic black holes with double photon spheres, jg tt j was shown to have a wiggle outside the event horizon, and hence SEC is violated [17]. ...
... For dyonic black holes with double photon spheres, jg tt j was shown to have a wiggle outside the event horizon, and hence SEC is violated [17]. This discovery is consistent with the conjecture proposed in [24]. Although the existence of a double photon sphere in black hole spacetime is new, it is well known that wormholes can have two photon spheres, one on each side of the wormholes. ...
Recently, asymptotically flat black holes with multiple photon spheres have been discovered and found to produce distinctive observational signatures. In this paper, we focus on whether these black hole solutions are physically viable, e.g., satisfying energy conditions of interest. Intriguingly, black hole and naked singularity solutions with two photon spheres and one antiphoton sphere are shown to exist in physically reasonable models, which satisfy the null, weak, dominant and strong energy conditions. Our findings reveal that black holes with multiple photon spheres may not be frequent, but they are not exotic.
... Therefore the existence of a light-ring (or anti-light-ring [162]) in such a geometry, i.e., a locus r such that ∂ r (f /r 2 ) = 0, is equivalently a locus where ...
... Following the recent construction of geometries, which are solutions to the Einstein's equations with matter fields violating energy conditions, and having multiple light rings [163], we have also investigated the existence of such solutions within our phase space supported by an anisotropic dark matter distribution. We found that, indeed, there exist matter configurations that obey the dominant energy condition, while they admit multiple light rings, or, more precisely, light rings and anti-light rings in the sense of [162]. Such matter configurations span a small region of parameter space, which we consider unphysical due to the very high matter densities involved, but we cannot rule them out just from known energy conditions. ...
We derive the model of the Schwarzschild black hole immersed into a dark matter halo with a relativistic Hernquist profile, the Schwarzschild-Hernquist black hole, and obtain its tidal Love numbers and quasi-normal modes. We thoroughly compare our odd and even parity perturbation equations with the literature and point out that two distinct choices of matter perturbations lead to distinct spectra. We establish that the quasi-normal modes admit qualitatively distinct scaling laws in terms of dark matter densities for non-relativistic and relativistic halos. We develop a stable numerical scheme for computing tidal Love numbers based on asymptotic series expansions. We further comment upon the existence of matter configurations obeying the dominant energy condition that lead to multiple light rings.
... It is found that R s /r ph ≈ 1 in the case of d > 6, while R s /r ph > 1 for d ≤ 6. Remarkably, by using the weak energy condition and the trace of energy-momentum tensor condition, Refs. [36,37] shows a bound between R s and r ph . Recently, Refs. ...
... By using Eqs. (37) and (54), one can show that the apparent radius of shadow equal to the critical impact parameter of critical curve, ...
The behaviors of null geodesics in the spherical symmetric black holes in Einstein–Maxwell-dilaton (EMD) theory with coupling function f ( Φ ) = e - 2 α Φ are meticulously analyzed. We investigate the effects of coupling constant α on the effective potential of photon trajectories within three ranges, namely 0 < α < 1 , α = 1 and α > 1 . We find that the thicknesses of lensing and photon rings are smaller at larger α and fixed electric charge in the unit of mass q , whereas they are larger at fixed α and larger q . This behavior can be described by using the angular Lyapunov exponent γ in the vicinity of the critical curve. Remarkably, the behaviors of photon trajectories are found to be more interesting when α > 1 . Namely, the radius of the black hole shadow R s becomes to be smaller than the photon sphere radius r ph when α > 1 and q > q ∗ . Moreover, R s goes to zero as q saturates the extremal limit, beyond which the photon orbit becomes absent. Furthermore, we construct the optical appearance of black holes surrounded by optically and geometrically thin accretion disk with three cases of Gralla–Lupsasca–Marrone (GLM) emission profile. Our results indicate that the observed flux originating from the lensing and photon rings exhibits suppression as α increases, while it undergoes amplification with the increasing parameter q .
... Recently, in Ref. [40], the authors investigated closed photon orbits in spherically symmetric static solutions of supergravity theories, a Horndeski theory, and a theory of quintessence. These orbits lie in what is called a photon sphere (or anti-photon sphere) if the orbit is unstable (stable). ...
... Refs. [38][39][40]45] also explore geodesics, which may differ in the field of ordinary black holes. Some works explore the motion of test particles and photons around static [46][47][48][49][50] and rotating [51,52] dilatonic black holes in different theories of gravity. ...
A non-extreme dilatonic charged (by two “color electric” charges) black hole solution is examined within a four-dimensional gravity model that incorporates two scalar (dilaton) fields and two Abelian vector fields. The scalar and vector fields interact through exponential terms containing two dilatonic coupling vectors. The solution is characterized by a dimensionless parameter a ( 0 < a < 2 ) , which is a specific function of dilatonic coupling vectors. The paper presents solutions for timelike and null circular geodesics that may play a crucial role in different astrophysical scenarios, including quasinormal modes of various test fields in the eikonal approximation. For a = 1 / 2 , 1 , 3 / 2 , 2 , the radii of the innermost stable circular orbit are presented and analyzed.
... Thus, the WEC is also satisfied on and outside the horizon of the black holes. It was conjectured [20], [21] that "a violation of either the dominant or the strong energy condition is a necessary condition for the existence of an anti-photon sphere outside a regular black hole horizon". Thus, according to our analysis of the energy conditions, the static dyonic black holes in the EMDA theory do not consist anti-photon sphere or stable photon orbits. ...
... 20) where the integration constants are identified as the mass M and the sum of the square of the charges (P 2 + Q 2 ). The black hole resembles the Reissner-Nordström black holes asymptotically. ...
Black holes with dyonic charges in Einstein-Maxwell-dilaton-axion supergravity theory are revisited in the context of black hole shadows. We consider static as well as rotating (namely the dyonic Kerr-Sen) black holes. The matter stress-energy tensor components, sourced by the Maxwell, axion and dilaton fields satisfy the standard energy conditions. The analytical expressions for the horizon and the shadow radius of the static spacetimes demonstrate their dependence on (P, Q the magnetic and electric charges, respectively) and the mass parameter M. The shadow radius lies in the range and there is no stable photon orbit outside the horizon. Further, shadows cast by the rotating dyonic Kerr-Sen black holes are also studied and compared graphically with their Kerr-Newman and Kerr-Sen counterparts. Deviation of the shadow boundary is prominent with the variation of the magnetic charge, for the relatively slowly rotating dyonic Kerr-Sen spacetimes. We test any possible presence of a magnetic monopole charge in the backdrop of recent EHT observations for the supermassive black holes M87 and Sgr A. Deviation from circularity of the shadow boundary () and deviation of the average shadow radius from the Schwarzschild shadow radius (quantified as the fractional deviation parameter ) are the two observables used here. Observational bound on (available only for M87) is satisfied for all theoretically allowed regions of parameter space and thus cannot constrain the parameters. The observational bound on available for Sgr A translates into an upper limit on any possible magnetic monopole charge linked to Sgr A and is given as . Such a constraint on P is however expected to be far more stringent for other astrophysical tests.
... In the present paper a similar study will be made concerning the TCOs of static, spherically symmetric, asymptotically flat 1 + 3 dimensional spacetimes, whose metric functions are at least C 2 smooth, using for this purpose the Jacobi metric. We will also extend the analysis to the number of LRs and their stability, similar to what was done in [21] for BHs. In addition, we also recover the results of recent theorems for stationary spacetimes in the spherically symmetric case [22,23]. ...
... The presence of the latter has been argued to imply the spacetime is unstable [28,29]; in fact this was recently shown in specific examples [30]. Moreover, it has been conjectured that the appearance of more than one LR in BH spacetimes requires the violation of the strong energy condition [21]. ...
The motion of particles on spherical 1 + 3 dimensional spacetimes can, under some assumptions, be described by the curves on a 2-dimensional manifold, the optical and Jacobi manifolds for null and timelike curves, respectively. In this paper we resort to auxiliary 2-dimensional metrics to study circular geodesics of generic static, spherically symmetric, and asymptotically flat 1+3 dimensional spacetimes, whose functions are at least C ² smooth. This is done by studying the Gaussian curvature of the bidimensional equivalent manifold as well as the geodesic curvature of circular paths on these. This study considers both null and timelike circular geodesics. The study of null geodesics through the optical manifold retrieves the known result of the number of light rings (LRs) on the spacetime outside a black hole and on spacetimes with horizonless compact objects. With an equivalent procedure we can formulate a similar theorem on the number of marginally stable timelike circular orbits (TCOs) of a given spacetime satisfying the previously mentioned assumptions.
... In the present paper a similar study will be made concerning the TCOs of static, spherically symmetric, asymptotically flat 1 + 3 dimensional spacetimes, whose metric functions are at least C 2 smooth, using for this purpose the Jacobi metric. We will also extend the analysis to the number of LRs and their stability, similar to what was done in [21] for BHs. In addition, we also recover the results of recent theorems for stationary spacetimes in the spherically symmetric case [22,23]. ...
... The presence of the latter has been argued to imply the spacetime is unstable [28,29]; in fact this was recently shown in specific examples [30]. Moreover, it has been conjectured that the appearance of more than one LR in BH spacetimes requires the violation of the strong energy condition [21]. ...
The motion of particles on spherical dimensional spacetimes can, under some assumptions, be described by the curves on a 2-dimensional manifold, the optical and Jacobi manifolds for null and timelike curves, respectively. In this paper we resort to auxiliary 2-dimensional metrics to study circular geodesics of generic static, spherically symmetric, and asymptotically flat dimensional spacetimes, whose functions are at least smooth. This is done by studying the Gaussian curvature of the bidimensional equivalent manifold as well as the geodesic curvature of circular paths on these. This study considers both null and timelike circular geodesics. The study of null geodesics through the optical manifold retrieves the known result of the number of light rings (LRs) on the spacetime outside a black hole and on spacetimes with horizonless compact objects. With an equivalent procedure we can formulate a similar theorem on the number of marginally stable timelike circular orbits (TCOs) of a given spacetime satisfying the previously mentioned assumption
... By choosing an appropriate sign for ω we get the asymptotic relations (as l → +∞) on real and imaginary parts of complex ω in the eikonal approximation We note that the parameters of the unstable circular null geodesics around stationary spherically symmetric and asymptotically flat black holes are in correspondence with the eikonal part of quasinormal modes of these black holes. See [20][21][22] and references therein. Due to Ref. [23] this correspondence is valid if certain restrictions on perturbations are imposed. ...
... We note that recently, some examples of the violation of the Hod conjecture have been discussed for certain black hole solutions in supergravity and other theories [21]. ...
We consider a family of four-dimensional black hole solutions from Dehnen et al. (Grav Cosmol 9:153 arXiv:gr-qc/0211049 , 2003) governed by natural number q = 1 , 2 , 3 , ⋯ , which appear in the model with anisotropic fluid and the equations of state: p r = - ρ ( 2 q - 1 ) - 1 , p t = - p r , where p r and p t are pressures in radial and transverse directions, respectively, and ρ > 0 is the density. These equations of state obey weak, strong and dominant energy conditions. For q = 1 the metric of the solution coincides with that of the Reissner–Nordström one. The global structure of solutions is outlined, giving rise to Carter–Penrose diagram of Reissner–Nordström or Schwarzschild types for odd q = 2 k + 1 or even q = 2 k , respectively. Certain physical parameters corresponding to BH solutions (gravitational mass, PPN parameters, Hawking temperature and entropy) are calculated. We obtain and analyse the quasinormal modes for a test massless scalar field in the eikonal approximation. For limiting case q = + ∞ , they coincide with the well-known results for the Schwarzschild solution. We show that the Hod conjecture which connect the Hawking temperature and the damping rate is obeyed for all q ≥ 2 and all (allowed) values of parameters.
... A number of studies have calculated the different relationships between these observable phenomena [16][17][18][19][20][21]. These results have confirmed that the photon spheres play an extremely important role in astronomical observations [22,23]. The existence of the photon orbit marks a possible York-Hawking-Page-type phase transition. ...
... (20) into eq. (18), we obtain V eff r ps , m = 3L 2 2m − r ps r 5 ps (22) when r ps /m = 2, q/m = −2/ 3 √ 5, and V eff = 0. Thus, when r ps /m > 2 and V eff < 0, the photon sphere is unstable; otherwise the photon sphere is stable. When q/m < −2/ 3 √ 5, the photon sphere does not exist. ...
This paper was devoted to studying the structure of the photon spheres and time-like circular orbits in the magnetic Gauss–Bonnet black hole space–time. Herein, the relationship between the photon spheres, the time-like circular orbits, and the black hole horizons was analyzed. We found that the photon sphere curve of the black hole ends at the horizon curve of the extreme black hole. The outer photon sphere is unstable and the inner photon sphere is stable. However, given that there is physically no inner photon sphere of the black hole inside the event horizon, the black hole has only one unstable photon sphere. Specifically, compact massive objects have at most two photon spheres, i.e., stable and unstable. Moreover, the curve of extremal stable time-like circular orbits ends at the coincidence of the inner and outer photon spheres, whereas the inner time-like circular orbits cannot be inside the photon spheres. Therefore, the extremal stable time-like circular orbits become multivalued, which is related to the existence of the photon sphere. When the photon sphere exists, the extremal stable time-like circular orbits behave as the innermost stable circular orbits. When the photon sphere does not exist, the extremal stable time-like circular orbits have both an innermost stable circular orbit and an outermost stable circular orbit.
... By choosing an appropriate sign for ω we get the asymptotic relations (as l → +∞) on real and imaginary parts of complex ω in the eikonal approximation We note that the parameters of the unstable circular null geodesics around stationary spherically symmetric and asymptotically flat black holes are in correspondence with the eikonal part of quasinormal modes of these black holes. See [20,21] and references therein. Due to Ref. [22] this correspondence is valid if certain restrictions on perturbations are imposed. ...
... We note that recently, some examples of the violation of the Hod conjecture have been discussed for certain black hole solutions in supergravity and other theories [21]. ...
We consider a family of 4-dimensional black hole solutions from Ref. \cite{DIM} governed by natural number , which appear in the model with anisotropic fluid and the equations of state: , , where and are pressures in radial and transverse directions, respectively, and is the density. These equations of state obey weak, strong and dominant energy conditions. For the metric of the solution coincides with that of the Reissner-Nordstr\"om one. The global structure of solutions is outlined, giving rise to Carter-Penrose diagram of Reissner-Nordstr\"om or Schwarzschild types for odd or even , respectively. Certain physical parameters corresponding to BH solutions (gravitational mass, fluid mass, PPN parameters, Hawking temperature and entropy) are calculated. We obtain and analyse the quasinormal modes for a test massless scalar field in the eikonal approximation. For limiting case , they coincide with the well-known results for the Schwarzschild solution. We show that the Hod conjecture which connect the Hawking temperature and the damping rate is obeyed for all and all (allowed) values of parameters.
... They can reveal non-trivial aspects related to the spacetime singularities, event horizons, causal structures and asymptotic behaviors associated with black holes and ultracompact objects. For instance, a number of recent researches have pointed out that the number and distribution of photon spheres (or light rings) in black hole spacetimes, horizon-less spacetimes produced by ultracompact objects, and naked singularity spacetimes could exhibit entirely different features [37][38][39][40][41][42][43]. ...
Photon sphere has attracted significant attention since the capture of black hole shadow images by Event Horizon Telescope. Recently, a number of studies have highlighted that the number of photon spheres and their distributions near black holes are strongly constrained by black hole properties. Specifically, for black holes with event horizons and proper asymptotic behaviors, the number of stable and unstable photon spheres satisfies the relation n stable - n unstable = - 1 . In this study, we provide a new proof on this relation using a geometric analysis, which is carried out using intrinsic curvatures in the optical geometry of black hole spacetimes. Firstly, we demonstrate the existence of photon spheres near black holes assuming most general asymptotic behaviors (asymptotically flat black holes, asymptotically de-Sitter and anti-de-Sitter black holes). Subsequently, we prove that the stable and unstable photon spheres near black holes must be one-to-one alternatively separated from each other, such that each unstable photon sphere is sandwiched between two stable photon spheres (and each stable photon sphere is sandwiched between two unstable photon spheres). Our analysis in this study is applicable to any spherically symmetric black hole spacetimes.
... We will then show that a large class of spherically symmetric static extremal black hole spacetimes possess a stable null photon sphere on their horizons. (In the literature, a stable photon sphere is also known as "anti-photon sphere" [42]). ...
It has been shown in the literature that the event horizon of an extremal asymptotically flat Reissner-Nordstrom black hole is also a stable photon sphere. We further clarify this statement and give a general proof that this holds for a large class of static spherically symmetric black hole spacetimes with an extremal horizon. In contrast, in the Doran frame, an extremal asymptotically flat Kerr black hole has an unstable photon orbit on the equatorial plane of its horizon. In addition, we show that an extremal asymptotically flat Kerr-Newman black hole exhibits two equatorial photon orbits if a < M/2, one of which is on the extremal horizon in the Doran frame and is stable, whereas the second one outside the horizon is unstable. For a > M/2, there is only one equatorial photon orbit, located on the extremal horizon, and it is unstable. There can be no photon orbit on the horizon of a non-extremal Kerr-Newman black hole.
... In Refs. [35,36], the authors pointed out that the presence of photon orbits signals a possible York-Hawking-Page type phase transition. Using the stable photon orbit, the effect of a negative cosmological constant on the instabilities of black hole spacetimes was investigated in Ref. [37]. ...
We study the relationship between the null geodesics and thermodynamic phase transition for the charged AdS black hole. In the reduced parameter space, we find that there exist non-monotonic behaviors of the photon sphere radius and the minimum impact parameter for the pressure below its critical value. The study also shows that the changes of the photon sphere radius and the minimum impact parameter can serve as order parameters for the small-large black hole phase transition. In particular, these changes have an universal exponent of near the critical point for any dimension d of spacetime. These results imply that there may exist universal critical behavior of gravity near the thermodynamic critical point of the black hole system.
... Indeed, the effective metric for the graphene carriers becomes conformally equivalent to the one of a black hole, while their Whightman correlation function is invariant under this conformal transformation and, thus, display the same thermal behavior as in presence of the black hole. This approach based on conformal transformation has some difficulties pointed in [134] by Cvetic and Gibbons who argued that there is a fundamental geometric obstacle to obtaining a model that extends all the way to the black hole horizon with a finite graphene sheet (for a more advance discussion on the properties of optical metrics and of their relation with cosmological and holographic solutions can be found in [135]). Then, Iorio and Lambiase replayed by showing that a way out to the problem above exists, and different conformal maps can be considered, which allow to reach the horizon on a finite lattice at the price of a non-thermal correction in the Wightman response function [136]. ...
In this paper I construct the naive lattice Dirac Hamiltonian describing the propagation of fermions in a generic 2D optical metric for different lattice and flux-lattice geometries. First, I apply a top-down constructive approach that we first proposed in [Boada {\it et al.,New J. Phys.} {\bf 13} 035002 (2011)] to the honeycomb and to the brickwall lattices. I carefully discuss how gauge transformations that generalize momentum (and Dirac cone) shifts in the Brillouin zone in the Minkowski homogeneous case can be used in order to change the phases of the hopping. In particular, I show that lattice Dirac Hamiltonian for Rindler spacetime in the honeycomb and brickwall lattices can be realized by considering real and isotropic (but properly position dependent) tunneling terms. For completeness, I also discuss a suitable formulation of Rindler Dirac Hamiltonian in semi-synthetic brickwall and -flux square lattices (where one of the dimension is implemented by using internal spin states of atoms as we originally proposed in [Boada {\it et al.,Phys. Rev. Lett. } {\bf 108} 133001 (2012)] and [Celi {\it et al.,Phys. Rev. Lett. } {\bf 112} 043001 (2012)]).
... In [53] it was proposed that such surfaces, which are totally geodesic submanifolds of a suitably defined optical metric, be termed photon surfaces or anti-photon surfaces, depending upon whether the null geodesics were unstable or stable respectively. Because of the misgivings we have voiced ...
The equations of null geodesics in the STU family of rotating black hole solutions of supergravity theory, which may be considered as deformations of the vacuum Kerr metric, are completely integrable. We propose that they be used as a foil to test, for example, with what precision the gravitational field external to the black hole at the centre of our galaxy is given by the Kerr metric. By contrast with some metrics proposed in the literature, the STU metrics satisfy by construction the dominant and strong energy conditions. Our considerations may be extended to include the effects of a cosmological term. We show that these metrics permit a straightforward calculation of the properties of black hole shadows.
... Photon spheres that coincide with an extremal horizon of static, spherically-symmetric black hole space-times were studied in [44]. Photon and antiphoton spheres in static, spherically symmetric-solutions of supergravity theories, a Horndeski theory, and a theory of quintessence were studied in [45]. Gravitational lensing by naked singularities and the formation of photon spheres around them was studied in [46][47][48]. ...
There are circular planar null geodesics at r=3M around a Schwarzschild black hole of mass M. These geodesics form a photon sphere. Null geodesics of the Schwarzschild space-time which do not form the photon sphere are either escape to null infinity or get captured by the black hole. Thus, from the dynamical point of view, the photon sphere represents a smooth basin boundary that separates the basins of escape and capture of the dynamical system governing the null geodesics. Here we consider a Schwarzschild black hole distorted by an external, static, and axisymmetric quadrupolar gravitational field. We study null geodesics around such a black hole and show that the photon sphere transforms into a fractal basin boundary that indicates chaotic behavior of the null geodesics. We calculate the box-counting fractal dimension of the basin boundary and the related uncertainty exponent, which depend on the value of the quadrupole moment.
... For 3M ≥ a, the wormhole possesses a single photon sphere at r ph = 0; for 2M < a < 3M , the wormhole possesses two photon spheres at r ± ph = ± √ 9M 2 − a 2 . In contrast, a minimum point of the effective potential corresponds to stable circular orbits, known as an anti-photon sphere in spherically symmetric spacetimes [88]. These anti-photon spheres have been identified in various spacetimes, including horizonless ultra-compact objects [44] and extreme static black holes [89]. ...
In this paper, we study the observational signatures of traversable Simpson-Visser wormholes illuminated by luminous celestial spheres and orbiting hot spots. We demonstrate that when light sources and observers are on the same side of the wormholes, the images of the wormholes mimic those of black holes. However, when the light sources are positioned on the opposite side from observers, photons traversing the wormhole throat generate distinct observational signatures. Specifically, unlike black hole images, the wormhole images are confined within the critical curve, resulting in smaller centroid variations. Furthermore, the light curve of hot spots can exhibit additional peaks.
... In other words, a local maximum of the effective potential signifies the presence of a photon sphere. In addition, a local potential minimum gives rise to stable circular null geodesics, forming what is known as an "antiphoton sphere" [75]. Such anti-photon spheres have been observed in various spacetimes, including horizonless ultracompact objects [76] and extreme static black holes [77]. ...
This paper investigates the observational signatures of hot spots orbiting scalarized Reissner–Nordström black holes, which have been reported to possess multiple photon spheres. In contrast to the single-photon sphere case, hot spots orbiting black holes with two photon spheres produce additional image tracks in time integrated images capturing a complete orbit of hot spots. Notably, these newly observed patterns manifest as a distinct second-highest peak in temporal magnitudes when observed at low inclination angles. These findings offer promising observational probes for distinguishing black holes with multiple photon spheres from their single-photon sphere counterparts.
... Firstly, people proposed that the photon orbits signals a possible York-Hawking-Page phase transition in refs. [64,65]. Then the more clearly relation between photon orbits and phase transitions of Reissner-Nordstrom-AdS black hole and rotating Kerr-AdS black hole were shown in refs. ...
A bstract
In this work, we investigate the relation between the photon sphere radius and the first-order phase transition for the charged Einstein-power-Yang-Mills AdS black hole. Through the analysis, we find with a certain condition there exist the non-monotonic behaviors between the photon sphere radius, the impact parameter, the non-linear Yang-Mills charge parameter, temperature, and pressure. And both the changes of photon sphere radius and impact parameter before and after phase transition can be regarded as the order parameter, their critical exponents near the critical point are equal to the same value 1/2, just like the ordinary thermal systems. These indicate that there maybe exists a universal relation of gravity nearby the critical point for a black hole thermodynamical system. Furthermore, the effect of impact parameter on the deflect angle is also investigated.
... Firstly, people proposed that the photon orbits signals a possible York-Hawking-Page phase transition in Refs. [65,66]. Then the more clearly relation between photon orbits and phase transitions of Reissner-Nordstrom-AdS black hole and rotating Kerr-AdS black hole were shown in Ref. [67]. ...
In this work, we investigate the relationship between the photon sphere radius and the first-order phase transition for the charged EPYM AdS black hole. Through the analysis, we find with a certain condition there exist the non-monotonic behaviors between the photon sphere radius, the impact parameter, the non-linear YM charge parameter, temperature, and pressure. And both the changes of photon sphere radius and impact parameter before and after phase transition can be regarded as the order parameter, their critical exponents near the critical point are equal to the same value 1/2, just like the ordinary thermal systems. These indicate that there maybe exists a universal relationship of gravity nearby the critical point for a black hole thermodynamical system. Furthermore, the effect of impact parameter on the deflect angle is also investigated.
Chaotic systems near black holes satisfy a universal bound, λ ≤ κ H linking the Lyapunov coefficient λ associated with unstable orbits to surface gravity κ H of the event horizon. A natural question is whether this bound is satisfied by unstable circular null geodesics in the vicinity of black holes. However, there are known cases where this bound is violated. It is intriguing to ask whether there exists an alternative universal bound that is valid in such situations. We show that for any spherically symmetric, static black hole that satisfies Einstein’s equations and the dominant energy condition, there exist other universal bounds relating the Lyapunov coefficient to a generalized notion of surface gravity at the photon sphere. As applications, we show how these bounds also constrain the imaginary part of quasinormal modes in the eikonal regime and how the Lyapunov coefficient relates to the shadow size and the entropy of the horizon.
In this paper, we study the observational signatures of traversable Simpson-Visser wormholes illuminated by luminous celestial spheres and orbiting hot spots. We demonstrate that when light sources and observers are on the same side of the wormholes, the images of the wormholes mimic those of black holes. However, when the light sources are positioned on the opposite side from observers, photons traversing the wormhole throat generate distinct observational signatures. Specifically, unlike black hole images, the wormhole images are confined within the critical curve, resulting in smaller centroid variations. Furthermore, the light curve of hot spots can exhibit additional peaks.
Recently, the ring images or the shadow images of the centers of the galaxy M87 and the Milky way have been reported by Event Horizon Telescope Collaboration. It is believed that the ring images imply that the central objects form unstable light circular orbits. Some of wormholes with Z 2 symmetry against a throat are wrongly excluded from the candidates at the centers of M87 and the Milky way due to the overlooking the unstable light circular orbits on the throat. A general asymptotically-flat, static, and spherical symmetrical wormhole without a thin shell has at least one unstable circular light orbit at the throat or elsewhere. If the wormhole has Z 2 symmetry against the throat, it has the unstable circular light orbits on the throat or it has stable circular light orbits on the throat and unstable ones near the throat. We need to analyze the throat carefully to make sure we do not unfairly rule out the Z 2 -symmetrical wormholes. In this study, we categorize the numbers of the circular light orbits of the Z 2 -symmetrical wormhole and their stability from the derivatives of an effective potential at the throat and we investigate the circular light orbits around a Simpson–Visser black-bounce spacetime, a Damour–Solodukhin wormhole spacetime, a Reissner–Nordström black-hole-like wormhole spacetime or a charged Damour–Solodukhin wormhole spacetime as examples. We give complete treatments including degenerated circular light orbits made from more than one stable and unstable circular light orbits on and off the throat.
By utilizing the AdS/CFT correspondence and wave optics techniques, we conducted an extensive study of the imaging properties of holographic Einstein rings in the context of Anti-de Sitter (AdS) black holes (BHs) in Horndeski theory. Our results indicate that the optical characteristics of these holographic Einstein rings are significantly influenced by the observer's position, the physical parameters of the BH, the nature of the wave source, and the configuration of the optical system. Specifically, when the observer is positioned at the north pole of the AdS boundary, the holographic image prominently displays a ring structure aligning with the BH's photon sphere. We thoroughly analyzed how various physical parameters---including the observation position, event horizon radius, temperature, and the parameter in Horndeski theory---affect the holographic Einstein rings. These parameters play a crucial role in determining the rings' radius and brightness, with variations potentially causing the ring structures to deform or even transform into bright spots. Furthermore, our comparative analysis between wave optics and geometric optics reveals a strong agreement in predicting the positions and brightness of both the photon ring and the Einstein ring. This research offers new insights into the spacetime geometry of BHs in Horndeski theory and proposes a promising framework for exploring the gravitational duals of strongly coupled systems.
The acceleration observed in cosmological expansion is often attributed to negative pressure, potentially arising from quintessence. We explore the relationship between the photon orbit radius and the phase transition of spherical AdS black holes in f (R, T) gravity influenced by quintessence dark energy, specifically Kiselev-AdS black holes in f (R, T) gravity. We are treating the negative cosmological constant Λ as the system's pressure to examine the impact of the state parameter ω and the f (R, T) gravity parameter γ. Interestingly, Kiselev-AdS black holes within the f (R, T) gravity framework exhibit a van der Waals-like phase transition. In contrast, these black holes display a Hawking-Page-like phase transition in general relativity. We demonstrate that below the critical point, the black hole undergoes a first-order vdW-like phase transition, with rps and ups serving as order parameters exhibiting a critical exponent of 1/2, similar to ordinary thermal systems. It suggests that rps and ups can serve as order parameters in characterizing black hole phase transitions, hinting at a potentially universal gravitational relationship near critical points within black hole thermodynamic systems. Investigating the correlation between photon sphere radius and thermodynamic phase transitions provides a valuable means of distinguishing between different gravity theory models, ultimately shedding light on the nature of dark energy. Finally, as γ tends towards zero, our results precisely align with those of Kiselev-AdS black holes.
It has been reported that the photon ring structure in black hole images produces strong and universal interferometric signatures on long interferometric baselines, holding promise for measuring black hole parameters and testing general relativity. This paper investigates the interferometric signatures of black holes with one or two photon spheres, specifically within the framework of Einstein-Maxwell-scalar models. Notably, for black holes possessing two photon spheres, interference between light rays orbiting the inner and outer photon spheres manifests as beat signals in the visibility amplitude, deviating from the universal signatures observed in the single-photon sphere case.
We revisit static, spherically symmetric solutions to anti–de Sitter (AdS)-Einstein gravity with a conformally coupled scalar field (and no self-interaction potential) in four dimensions. We first observe that a convenient choice of coordinates leads to a significant simplification of the field equations, which enables one to identify various roots of the indicial equations and thus distinct branches of solutions. Next, we construct an explicit 2-parameter hairy black hole solution in terms of an infinite power series around the event horizon. The black hole is nonextremal with a regular scalar field on and outside the event horizon, and it reduces to the Schwarzschild-AdS metric in the limit of vanishing hair. Its properties are illustrated for various values of the parameters and compared with previous numerical results by other authors. In addition, the analysis reveals the presence of a photon sphere and how the scalar field affects its size and the angular radius of the corresponding shadow. The thermodynamics of the solution is also briefly discussed.
Kiselev metric in the static and rotating form is widely used to test different aspects of the dark energy ( DE ) effects. We consider a DE Kiselev spacetime, predicting the reduction to the Kerr black hole ( BH ) solution under suitable conditions on the DE parameters and in this frame we study the effects of the dark energy on BHs and disks accretion. Elaborating a close comparison with the limiting vacuum Kerr spacetime, we focus on thick accretion disks around the central BH in the Kiselev solution, both co-rotating and counter-rotating with respect the central BH . We examine different aspects of BH accretion energetics by focusing on quantities related to the accretion rates and cusp luminosity, when considered the DE presence, related to the pure Kerr central BH . Our findings show that in these conditions heavy divergences with respect to the vacuum case are expected for the DE metrics. A known effect of the Kiselev metric is to lead to a false estimation the BH spin, we confirm this characteristic from the fluids dynamics analysis. Remarkably our results show that DE is affecting differently the accretion physics, and particularly the accretion rate, according to the fluid rotation orientation with respect to the central spinning attractor, leading in some cases to an under-estimation of the BH spin mass ratio. These contrasting aspects emerging in dependence on the fluids rotational orientation can be a distinguishing general DE feature which could lead to a revised observational paradigm where DE existence is considered.
In this paper, we study the observational appearance of the asymmetric thin-shell wormhole (ATW) in Horndeski theory by employing the ray-tracing method. We first calculate the effective potential and null geodesic of the ATW, and then we obtain the deflection angle of the photon in the ATW spacetime. Based on the impact parameter of the photon, the trajectory of the photon can be classified into three cases. Two typical emission models of the thin accretion disk are considered to analyze the observational appearance of the ATW. By comparing the observational appearances of the ATW and a black hole with the same mass parameter, we find additional features in the observational appearance of the ATW, such as the “lensing band” and “photon ring group.”
We cut a general, static, spherically symmetric spacetime and paste its copy to make a wormhole with a thin shell of any barotropic fluid in general relativity. We show that the stability of the thin-shell wormhole is characterized by a set of circular photon orbits called an (anti)photon sphere in the original spacetime if a momentum flux passing through a throat is prohibited. Our result will be useful to classify the stability of the thin shell on the throat against linearized spherically symmetric perturbations.
A hairy Schwarzschild black hole describes the deformation of a Schwarzschild black hole due to including additional sources. It is found that depending on the hairy parameters, the photons’ configurations around this black hole can be classified into two cases, corresponding to the hairy Schwarzschild black hole with a single photon sphere and double photon spheres, respectively. We focus on the shadows and images of the hairy Schwarzschild black hole under two types of static thin illuminations conditions: disk accretion and spherical accretion, respectively. Under both illuminations, the two hairy parameters (α and lo) have competitive effects on the shadow and optical appearance image of the hairy Schwarzschild black hole with a single photon sphere. This means that even though the parameters have significant influences on the rings and shadows, its images with certain groups of α and lo could be indistinguishable to that of Schwarzschild black hole, namely, the degeneracies of images raise between the hairy Schwarzschild black hole and Schwarzschild black hole. Moreover, the optical appearance image of the hairy Schwarzschild black hole with double photon spheres will exhibit new additional rings and accretion features, which are not present in the images of a (hairy) Schwarzschild black hole with a single photon sphere. Our theoretical studies on the rings and shadows provide a potential tool to differentiate the hairy Schwarzschild black hole with double photon spheres from the Schwarzschild black hole, but they are not always helpful for the cases with a single photon sphere due to the degeneracies.
The Kiselev model describes a black hole surrounded by a fluid with equations of state p r / ρ = - 1 and p t / ρ = ( 3 w + 1 ) / 2 respectively in radial and tangential directions. It has been extensively studied in the parameter region - 1 < w < - 1 / 3 . If one rids off the black hole and turns to the region - 1 / 3 < w < 0 , i.e. p t > 0 , then a new horizon of black hole type will emerge. This case has been mentioned in Kiselev’s pioneer work but seldom investigated in the literature. Referring to it as reduced Kiselev black hole, we revisit this case with attention to its causal structure, thermodynamics, shadow cast and weak-field limit. An alternative interpretation and extensions of the black hole are also discussed.
Black holes with dyonic charges in Einstein-Maxwell-dilaton-axion supergravity theory are revisited in the context of black hole shadows. We consider static as well as rotating (namely the dyonic Kerr-Sen) black holes. The matter stress-energy tensor components, sourced by the Maxwell, axion and dilaton fields, satisfy the standard energy conditions. The analytical expressions for the horizon and the shadow radius of the static spacetimes demonstrate their dependence on P2+Q2 (P, Q the magnetic and electric charges, respectively) and the mass parameter M. The shadow radius lies in the range 2M<Rshadow<33M and there is no stable photon orbit outside the horizon. Further, shadows cast by the rotating dyonic Kerr-Sen black holes are also studied and compared graphically with their Kerr-Newman and Kerr-Sen counterparts. Deviation of the shadow boundary is prominent with the variation of the magnetic charge, for the relatively slowly rotating dyonic Kerr-Sen spacetimes. We test any possible presence of a magnetic monopole charge in the backdrop of recent EHT observations for the supermassive black holes M87* and SgrA*. Deviation from circularity of the shadow boundary (ΔC) and deviation of the average shadow radius from the Schwarzschild shadow radius (quantified as the fractional deviation parameter δ) are the two observables used here. The observational bound on ΔC (available only for M87*) is satisfied for all theoretically allowed regions of parameter space and thus cannot constrain the parameters. The observational bound on δ available for SgrA* translates into an upper limit on any possible magnetic monopole charge linked to SgrA* and is given as P≲0.873M. Constraints on P obtained from other astrophysical effects are however expected to be far more stringent though rigorous analyses along these lines is lacking in the literature. In addition, future refined imaging (shadow) observations will surely help in improving the bound on P arrived at here.
In this paper, the Gibbons and Werner technique is used to calculate the weak deflection angle for the black hole solution under the effects of Einstein-Gauss-Bonnet gravity. The Gauss-Bonnet theorem in the limits of weak field is used to evaluate the Gaussian optical curvature in order to obtain the results. The visual effects of the deflection angle on the impact parameter is also looked at and the smallest radius in the non-plasma/plasma medium. Moreover, in order to check the consistency of the results concerning the weak deflection angle, the Keeton and Petters approach is applied to study the deflection angle, which is the expansion of series with a single mass variable, which can be directly addressed by using the post-post Newtonian framework. Furthermore, the deflection angle and shadow under the influence of the plasma is examined by using the motion of particle in a non-magnetized plasma and pressure-free plasma medium as described by the new ray-tracing algorithm. It is shown that plasma as well as Einstein-Gauss-Bonnet gravity corrections are affected by shadows.
We summarize our results on the presence and location of horizons in charged black hole solutions of Einstein–Maxwell-dilaton theory with nontrivial dilaton potentials, asymptotically flat or (anti-)de Sitter, as function of the black hole parameters mass, charge and dilaton coupling strength. We observe that there is a value of latter which separates two regions, one where the black hole is Reissner–Nordström-like from a region where it is Schwarzschild-like. We find that for de Sitter and small nonvanishing of the dilaton coupling parameter, the extremal case is not reached by the solution. We also discuss the attractive or repulsive nature of the leading long distance interaction between two such black holes, or a test particle and one black hole, from a worldline effective field theory point of view.
We consider the influence of the higher-order correction to the gravitational action inspired by the Goroff-Sagnotti term upon the Kiselev black hole with ω˜=−2/3 and contrast the thus obtained results with the analogous results obtained for the Schwarzschild–de Sitter black hole. Expressing the perturbed solution in terms of the exact radius of the event horizon and the radius of the cosmological horizon of the unperturbed black hole we calculate corrections to the line element, to the cosmological horizon, and to the surface gravity. It is shown that although in both cases the lukewarm configuration does not exist classically (the equality of the surface gravities is possible only for the merged horizons), the sixth-order term removes the degeneracy of the classical solution and simultaneously shifts the degenerate horizon to a new place in the space of parameters. The lukewarm configuration is characterized by the value of the parameters that classically characterize the extremal solution. It is shown that the Karlhede scalar still may serve as the detector of the event and the cosmological horizon. Finally, we study the complex frequencies of the low-lying fundamental modes of the quasinormal oscillations and argue that they are the best candidates (at least theoretically) to distinguish between different black hole configurations.
The spherical photon orbits around a black hole with constant radii are particularly important in astrophysical observations of the black hole. In this paper, the equatorial and nonequatorial spherical photon orbits around Kerr-Newman black holes are studied. The radii of these orbits satisfy a sextic polynomial equation with three parameters: the rotation parameter u, charge parameter w, and effective inclination angle v. For orbits in the polar plane (v=1), we find that there are two positive solutions to the orbit equation, one is inside and the other is outside the event horizon. Particularly, we obtain the analytical expressions for the radii of these two orbits that are functions of u and w. For orbits in the equatorial plane (v=0) around an extremal Kerr-Newman black hole (u+w=1), we find three positive solutions to the equation and provide analytical formulas for the radii of the three orbits. We also find that a critical value of rotation parameter u=14 exists, above which only one retrograde orbit exists outside the event horizon and below which two orbits (one prograde and one retrograde) exist outside the event horizon. For orbits in the equatorial plane around a nonextremal Kerr-Newman black hole, there are four or two positive solutions to the corresponding sextic equation, which we show numerically. The number of solutions depends on the choice of parameters in different regions of the (u,w) plane. A critical curve in the (u,w) plane that separates these regions is also obtained. On this critical curve, the explicit formulas of three solutions are found. There always exist two equatorial photon orbits outside the event horizon in the nonextremal Kerr-Newman case. For orbits between the above two special planes, there are four or two solutions to the general sextic equation. When the black hole is extremal, we find that there is a critical inclination angle vcr, below which only one orbit exists and above which two orbits exist outside the event horizon for a rapidly rotating black hole u>14. At this critical inclination angle, an exact formula for the radii is also derived. Finally, a critical surface in parameter space of (u, w, v) is shown that separates regions with four or two solutions in the nonextremal black hole case.
Photon spheres play a pivotal role in the imaging of luminous objects near black holes. In this paper, we examine observational appearances of a star freely falling in hairy black holes, which can possess one or two photon spheres outside the event horizon. When there exists a single photon sphere, the total luminosity measured by distant observers decreases exponentially with time at late times. Due to successive arrivals of photons orbiting around the photon sphere different times, a specific observer would see a series of light flashes with decreasing intensity, which share a similar frequency content. Whereas in the case with two photon spheres, photons temporarily trapped between the photon spheres can cause a peak of the total luminosity, which is followed by a slow exponential decay, at late times. In addition, these photons lead to one more series of light flashes seen by the specific observer.
In this work, the gravitational deflection angle of photon in the weak field limit (or the weak deflection angle) and shadow cast by the electrically charged and spherically symmetric static Kiselev black hole (BH) in the string cloud background are investigated. The influence of the BH charge Q, the quintessential parameter and the string cloud parameter a is studied on the weak deflection angle using the Gauss-Bonnet theorem, on the radius of photon spheres and on the size of the BH shadow in the spacetime geometry of the charged-Kiselev BH in string clouds. Moreover, we study the effects of plasma (uniform and non-uniform) on the weak deflection angle and on the shadow cast by the charged-Kiselev BH surrounded by the clouds of strings. In the presence of uniform/nonuniform plasma medium, increase in the cloud of string parameter a, increases the deflection angle . On the other hand decrease in the BH charge Q, decreases the deflection angle. Further we observe that an increase of the BH charge Q causes a decrease in the size of the shadow of the BH. We notice that with increase in the values of the parameters and a, the size of the BH shadow also increases and therefore the intensity of the gravitational field around the charged-Kiselev BH in string clouds increases. Thus the gravitational field of the charged-Kiselev BH in string cloud background would be stronger than the field produced by the pure Reissner-Nordstrom BH. Moreover, we use the data released by the Event Horizon Telescope (EHT) collaboration, for the supermassive BHs M87* and Sgr A*, to obtain constraints on the values of the parameters and a.
Based on the ideas used by Kiselev, we study three black holes surrounded by quintessence and the effects of quintessence on the classical and semiclassical scattering cross-sections. In contrast, the absorption section is studied with the sinc approximation in the eikonal limit. For Schwarzschild, Reissner–Nordström and Bardeen black holes surrounded by quintessence, the critical values of charges, and the normalization factor are obtained. We also described the horizons and the extremal condition of the black holes surrounded by quintessence, by setting the quintessence state parameter in the two particular cases ω=-2/3 and ω=-1/2.
A problem which has now come to a head in general relativity, primarily as a result of the many recent detections of gravitational waves from the coalescence of compact binaries, is that there are very few viable theories against which to “test” it. To perform realistic tests of theories of gravity, we need to be able to look beyond general relativity and evaluate the consistency of a parametrized, physically acceptable, family of black hole metric alternatives with observational data from, especially, gravitational wave detections using, for example, an agnostic Bayesian approach. In this paper we further examine properties of one class of such metrics, which in fact arise as solutions of ungauged supergravity. In particular, we examine the massless, neutral, minimally coupled scalar wave equation in a general stationary, axisymmetric background metric such as that of a charged rotating black hole, when the scalar field is either time independent or in the low-frequency, near-zone limit, with a view to calculating the Love numbers of tidal perturbations, and of obtaining harmonic coordinates for the background metric. For a four-parameter family of charged asymptotically flat rotating black hole solutions of ungauged supergravity theory known as STU black holes, which includes Kaluza-Klein black holes and the Kerr-Sen black hole as special cases, we find that all time-independent solutions, and hence the harmonic coordinates of the metrics, are identical to those of the Kerr solution. In the low-frequency limit we find the scalar fields exhibit the same SL(2,R) symmetry as holds in the case of the Kerr solution. We point out extensions of our results to a wider class of metrics, which includes solutions of Einstein-Maxwell-dilaton theory.
We investigate retrolensing by two photon spheres in a novel black-bounce spacetime suggested by Lobo et al. which can correspond to a Schwarzschild black hole, a regular black hole, and a traversable wormhole including an Ellis-Bronnikov wormhole. In a case, the wormhole has a throat which acts as a photon sphere and it has another photon sphere outside of the throat. With the sun as a light source, an observer, and the wormhole are lined up in this order, sunlight reflected slightly outside of the throat and barely outside and inside of the outer photon sphere can reach the observer. We show that the light rays reflected by the outer photon sphere are dominant in retrolensing light curves in the case.
We demonstrate that the location of a stable photon sphere (PS) in a compact object is not always an edge such as the inner boundary of a black hole shadow, whereas the location of an unstable PS is known to be the shadow edge; notably, in the Schwarzschild black hole. If a static spherically symmetric (SSS) spacetime has the stable outermost PS, the spacetime cannot be asymptotically flat. A nondivergent deflection is caused for a photon traveling around a stable PS, though a logarithmic divergent behavior is known to appear in most of SSS compact objects with an unstable photon sphere. The reason for the nondivergence is that the closest approach of a photon is prohibited in the immediate vicinity of the stable PS when the photon is emitted from a source (or reaches a receiver) distant from a lens object. The finite gap size depends on the receiver and source distances from the lens as well as the lens parameters. The mild deflection angle of light can be approximated by an arcsine function. A class of SSS solutions in Weyl gravity exemplify the nondivergent deflection near the stable outer PS.
It has been shown in the literature that the event horizon of an asymptotically flat extremal Reissner-Nordstr\"om black hole is also a stable photon sphere. We further clarify this statement and give a general proof that this holds for a large class of static spherically symmetric black hole spacetimes with an extremal horizon. In contrast, an asymptotically flat extremal Kerr black hole has an unstable photon orbit on the equatorial plane of its horizon. In addition, we show that an asymptotically flat extremal Kerr-Newman black hole exhibits two equatorial photon orbits if , one of which is on the extremal horizon and is stable, whereas the second one outside the horizon is unstable. For , there is only one equatorial photon orbit, located on the extremal horizon, and it is unstable. There can be no photon orbit on the horizon of a non-extremal Kerr-Newman black hole.
We present explicit solutions of the time-symmetric initial value
constraints, expressed in terms of freely specfiable harmonic functions for
examples of supergravity theories, which emerge as effective theories of
compactified string theory. These results are a prequisite for the study of the
time-evolution of topologically non-trivial initial data for supergravity
theories, thus generalising the "Geometrodynamics" program of Einstein-Maxwell
theory to that of supergravity theories. Specifically, we focus on examples of
multiple electric Maxwell and scalar fields, and analyse the initial data
problem for the general Einstein-Maxwell-Dilaton theory both with one and two
Maxwell fields, and the STU model. The solutions are given in terms of up to
eight arbitrary harmonic functions in the STU model. As a by-product, in order
compare our results with known static solutions, the metric in isotropic
coordinates and all the sources of the non-extremal black holes are expressed
entirely in terms of harmonic functions. We also comment on generalizations to
time-nonsymmetric initial data and their relation to cosmological solutions of
gauged so-called fake supergravities with positive cosmological constant.
Ultracompact objects are self-gravitating systems with a light ring. It was
recently suggested that fluctuations in the background of these objects are
extremely long-lived and might turn unstable at the nonlinear level, if the
object is not endowed with a horizon. If correct, this result has important
consequences: objects with a light ring are black holes. In other words, the
nonlinear instability of ultracompact stars would provide a strong argument in
favor of the "black hole hypothesis," once electromagnetic or
gravitational-wave observations confirm the existence of light rings. Here we
explore in some depth the mode structure of ultracompact stars, in particular
constant-density stars and gravastars. We show that the existence of very
long-lived modes -- localized near a second, stable null geodesic -- is a
generic feature of gravitational perturbations of such configurations. Already
at the linear level, such modes become unstable if the object rotates
sufficiently fast to develop an ergoregion. Finally, we conjecture that the
long-lived modes become unstable under fragmentation via a
Dyson-Chandrasekhar-Fermi mechanism at the nonlinear level. Depending on the
structure of the star, it is also possible that nonlinearities lead to the
formation of small black holes close to the stable light ring. Our results
suggest that the mere observation of a light ring is a strong evidence for the
existence of black holes.
We prove that, in a class of spherically symmetric spacetimes exhibiting
stable trapping of null geodesics, linear waves cannot (uniformly) decay faster
than logarithmically. When these linear waves are treated as a model for
nonlinear perturbations, this slow decay is highly suggestive of nonlinear
instability. We also prove that, in a large class of asymptotically flat,
spherically symmetric spacetimes, logarithmic decay actually holds as a uniform
upper bound. In the presence of stable trapping, this result is therefore the
best one can obtain. In addition, we provide an application of these results to
ultracompact neutron stars, suggesting that all stars with r < 3M might be
unstable.
One of the most remarkable predictions of the general theory of
relativity is the existence of black-hole "photonspheres", compact null
hypersurfaces on which massless particles can orbit the central black
hole. We prove that every spherically-symmetric asymptotically flat
black-hole spacetime is characterized by a photonsphere whose radius is
bounded from above by rγ⩽3M, where M is the total
ADM mass of the black-hole spacetime. It is shown that hairy black-hole
configurations conform to this upper bound. In particular, the null
circular geodesic of the (bald) Schwarzschild black-hole spacetime
saturates the bound.
In this paper we construct asymptotically locally AdS and flat black holes in
the presence of scalar field whose kinetic term is constructed out from a
linear combination of the metric and the Einstein tensor. The field equations
as well as the energy-momentum tensor are second order in the metric and the
field, therefore the theory belongs to the ones defined by Horndeski. We show
that in the presence of a cosmological term in the action, it is possible to
have a real scalar field in the region outside of the event horizon. The
solutions are characterized by a single integration constant, the scalar field
vanishes at the horizon and it contributes to the effective cosmological
constant at infinity. We extend these results to the topological case. The
solution is disconnected from the maximally symmetric AdS background, however,
within this family there exits a gravitational soliton which is everywhere
regular. This soliton is therefore used as a background to define a finite
Euclidean action and to obtain the thermodynamics of the black holes. We extend
the solution to arbitrary dimensions grater than four and show that the
presence of a cosmological term in the action allows to consider the case in
which the standard kinetic term for the scalar its not present. In such
scenario, the solution reduces to an asymptotically locally flat black hole
In this work we study the null geodesics in the background of Reissner-Nordström Anti de Sitter black hole. We compute the exact trajectories in terms of Weierstrass elliptic functions, obtaining a detailed description of the orbits in terms of the charge, mass and the cosmological constant. The trajectories of the photon are classified using the impact parameter.
Particle emission by small black holes in the ultrahigh-temperature
limit is investigated. Black-hole emission and accretion processes are
analyzed by assuming that high-temperature matter is an adiabatic
perfect fluid in local thermal equilibrium at a given temperature, with
the energy density and pressure determined as functions of the conserved
entropy density. The rate of accretion onto a classical black hole
embedded in a background medium of thermal-equilibrium matter with an
asymptotic temperature at large distances equal to the black-hole
temperature is calculated along with the form of the emission flow from
the black hole into an asymptotically empty background, assuming that
the total rate of energy output has the same value as the rate of energy
inflow in the accretion process. An attempt is made to estimate when and
where the matter will be sufficiently opaque for the fluid treatment to
be valid. The total luminosity is estimated for the cases of standard,
hard, and Hagedorn-type (1965) equations of state. Consideration is
given to the observable consequences that would ensue if primordial
fluctuations led to the formation of a cosmological number-density
distribution of black holes in an initial mass range given by a specific
formula.
In this paper we focus on analytical calculations involving null geodesics in
some spherically symmetric spacetimes. We use Weierstrass elliptic functions to
fully describe null geodesics in Schwarzschild spacetime and to derive
analytical formulae connecting the values of radial distance at different
points along the geodesic. We then study the properties of light triangles in
Schwarzschild spacetime and give the expansion of the deflection angle to the
second order in both and M/b where M is the mass of the black hole,
the distance of closest approach of the light ray and b the impact
parameter. We also use the Weierstrass function formalism to analyze other more
exotic cases such as Reissner-Nordstr\om null geodesics and Schwarzschild null
geodesics in 4 and 6 spatial dimensions. Finally we apply Weierstrass functions
to describe the null geodesics in the Ellis wormhole spacetime and give an
analytic expansion of the deflection angle in M/b.
Trajectories of light rays in a static spacetime are described by
unparametrised geodesics of the Riemannian optical metric associated with the
Lorentzian spacetime metric. We investigate the uniqueness of this structure
and demonstrate that two different observers, moving relative to one another,
who both see the universe as static may determine the geometry of the light
rays differently. More specifically, we classify Lorentzian metrics admitting
more than one hyper--surface orthogonal time--like Killing vector and analyze
the projective equivalence of the resulting optical metrics. These metrics are
shown to be projectively equivalent up to diffeomorphism if the static Killing
vectors generate a group , but not projectively equivalent in
general. We also consider the cosmological C--metrics in Einstein--Maxwell
theory and demonstrate that optical metrics corresponding to different values
of the cosmological constant are projectively equivalent.
The Einstein equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. Like black holes in asymptotically flat space, these solutions have thermodynamic properties including a characteristic temperature and an intrinsic entropy equal to one quarter of the area of the event horizon in Planck units. There are however some important differences from the asymptotically flat case. A black hole in anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the anti-de Sitter space. For larger black holes the red-shifted temperature measured at infinity is greater. This means that such black holes have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. It also implies that the canonical ensemble exists for asymptotically anti-de Sitter space, unlike the case for asymptotically flat space. One can also consider the microcanonical ensemble. One can avoid the problem that arises in asymptotically flat space of having to put the system in a box with unphysical perfectly reflecting walls because the gravitational potential of anti-de Sitter space acts as a box of finite volume.
Poletti and Wiltshire have shown that, with the exception of a pure cosmological constant, the solution of a dilaton black hole in the background of de Sitter or anti-de Sitter universe, does not exist in the presence of one Liouville-type dilaton potential. Here with the combination of three Liouville-type dilaton potentials, we obtain the dilaton black hole solutions in the background of de Sitter or anti-de Sitter universe. Comment: 13 pages,to appear in Phys. Rev. D
The Dirichlet boundary-value problem and isoperimetric inequalities for positive definite regular solutions of the vacuum Einstein equations are studied in arbitrary dimensions for the class of metrics with boundaries admitting a U(1) action. We show that in the case of non-trivial bundles Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique. In the case of trivial bundles, there are two Schwarzschild infillings in arbitrary dimensions. The condition of whether a particular type of filling in is possible can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for arbitrary dimension are delineated. For the case of Schwarzschild, analytic formulae for the two infilling black hole masses in arbitrary dimension have been obtained. This should facilitate the study of black hole dynamics/thermodynamics in higher dimensions. We found that all infilling solutions are convex. Thus convexity of the boundary does not guarantee uniqueness of the infilling. Isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are then investigated. In particular, the analogues of Minkowski's celebrated inequality in flat space are found and discussed providing insight into the geometric nature of these Ricci-flat spaces. Comment: 40 pages, 3 figures
The photon sphere concept in Schwarzschild space-time is generalized to a definition of a photon surface in an arbitrary space-time. A photon sphere is then defined as an SO(3)xR-invariant photon surface in a static spherically symmetric space-time. It is proved, subject to an energy condition, that a black hole in any such space-time must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must surround a black hole, a naked singularity or more than a certain amount of matter. A second order evolution equation is obtained for the area of an SO(3)-invariant photon surface in a general non-static spherically symmetric space-time. Many examples are provided. Comment: 24 pages, 5 figures. Uses the following packages: amsmath, amsthm, amssymb, amsfonts, mathrsfs, enumerate, graphicx
We present new static spherically-symmetric exact solutions of Einstein equations with the quintessential matter surrounding a black hole charged or not as well as for the case without the black hole. A condition of additivity and linearity in the energy-momentum tensor is introduced, which allows one to get correct limits to the known solutions for the electromagnetic static field implying the relativistic relation between the energy density and pressure, as well as for the extraordinary case of cosmological constant, i.e. de Sitter space. We classify the horizons, which evidently reveal themselves in the static coordinates, and derive the Gibbons-Hawking temperatures. An example of quintessence with the state parameter w=-2/3 is discussed in detail.
We make use of the fact that the optical geometry near a static non-degenerate Killing horizon is asymptotically hyperbolic to investigate universal features of black hole physics. We show how the Gauss-Bonnet theorem allows certain lensing scenarios to be ruled in or out. We find rates for the loss of scalar, vector and fermionic `hair' as objects fall quasi- statically towards the horizon. In the process we find the Lienard-Wiechert potential for hyperbolic space and calculate the force between electrons mediated by neutrinos, extending the flat space result of Feinberg and Sucher. We use the enhanced conformal symmetry of the Schwarzschild and Reissner-Nordstrom backgrounds to re-derive the electrostatic field due to a point charge in a simple fashion.
We interpret the well known fact that the equations for light rays in the Kottler or Schwarzschild-de Sitter metric are independent of the cosmological constant in terms of the projective equivalence of the optical metric for any value of \Lambda. We explain why this does not imply that lensing phenomena are independent of \Lambda. Motivated by this example, we find a large collection of one-parameter families of projectively equivalent metrics including both the Kottler optical geometry and the constant curvature metrics as special cases. Using standard constructions for geodesically equivalent metrics we find classical and quantum conserved quantities and relate these to known quantities.
In this geometrical approach to gravitational lensing theory, we apply the Gauss-Bonnet theorem to the optical metric of a lens, modelled as a static, spherically symmetric, perfect non-relativistic fluid, in the weak deflection limit. We find that the focusing of the light rays emerges here as a topological effect, and we introduce a new method to calculate the deflection angle from the Gaussian curvature of the optical metric. As examples, the Schwarzschild lens, the Plummer sphere and the singular isothermal sphere are discussed within this framework.
Horndeski gravities are theories of gravity coupled to a scalar field, in
which the action contains an additional non-minimal quadratic coupling of the
scalar, through its first derivative, to the Einstein tensor or the analogous
higher-derivative tensors coming from the variation of Gauss-Bonnet or Lovelock
terms. In this paper we study the thermodynamics of the static black hole
solutions in n dimensions, in the simplest case of a Horndeski coupling to
the Einstein tensor. We apply the Wald formalism to calculate the entropy of
the black holes, and show that there is an additional contribution over and
above those that come from the standard Wald entropy formula. The extra
contribution can be attributed to unusual features in the behaviour of the
scalar field. We also show that a conventional regularisation to calculate the
Euclidean action leads to an expression for the entropy that disagrees with the
Wald results. This seems likely to be due to ambiguities in the subtraction
procedure. We also calculate the viscosity in the dual CFT, and show that the
viscosity/entropy ratio can violate the bound for
appropriate choices of the parameters.
Within effective heterotic superstring theory compactified on a six-torus we derive minimum energy (supersymmetric), static, spherically symmetric solutions, which are manifestly invariant under the target space O(6,22) and the strong-weak coupling SL(2) duality symmetries with 28 electric and 28 magnetic charges subject to one constraint. The class of solutions with a constant axion corresponds to dyonic configurations subject to two charge constraints, with purely electric (or purely magnetic) and dyonic configurations preserving 1/2 and 1/4 of N = 4 supersymmetry, respectively. General dyonic configurations in this class have a space-time of extreme Reissner-Nordström black holes while configurations with more constrained charges have a null or a naked singularity.
Light deflection by non-axisymmetric cosmic strings is studied. Various properties of the geodesics are obtained including the non-existence of closed or almost closed geodesics. It is shown that the behaviour of the geodesics resembles the case of the axisymmetric cosmic strings usually studied. In particular no exotic behaviour such as glory scattering is possible. The usual formula for the deflection of distant geodesics is obtained. An application to the slow motion of SU(3) monopoles, considered as geodesic motion on a moduli space, is also given.
We obtain spherically-symmetric and -symmetric dyonic black holes that
are asymptotic to anti-de Sitter space-time (AdS), which are solutions in
maximal gauged four-dimensional supergravity, with just one of the U(1) fields
carrying both the electric and magnetic charges (Q,P). We study the
thermodynamics, and find that the usually-expected first law does not hold
unless P=0, Q=0 or P=Q. For general values of the charges, we find that the
first law requires a modification with a new pair of thermodynamic conjugate
variables. We show that they describe the scalar hair that breaks some of the
asymptotic AdS symmetries.
An upper limit of 10-42 cm-2 is placed on the absolute value of the cosmological constant by comparing with the prediction of the perihelion shift of Mercury. It is shown that the bending of starlight near the sun gives no limit on the cosmological constant since the equation for a null geodesic takes the same form with or without the cosmological constant.
We consider scale invariant theories which couple gravity to Maxwell
fields and antisymmetric tensor fields with a dilaton field. We exhibit
in a unified way solutions representing black hole, space-time membrane,
vortex and cosmological solutions. Their physical properties depend
sensitively on the coupling constant of the dilaton field, there being
critical value separating qualitatively different types of behaviour,
e.g. the temperature of a charged black hole in the extreme limit. It is
also shown that compactification into the 4-dimensional Minkowski space
in terms of a membrane solution is possible in 10-dimensional
supergravity model.
Present address.
We present new anti-de Sitter black hole solutions of gauged N = 8, SO(8) supergravity, which is the massless sector of the AdS4 × S70 vacuum of M-theory. By focusing on the U(1)4 Cartan subgroup, we find non-extremal 1, 2, 3 and 4 charge solutions. In the extremal limit, they may preserve up to of the supersymmetry, respectively. In the limit of vanishing SO(8) coupling constant, the solutions reduce to the familiar black holes of the M4 × T7 vacuum, but have very different interpretation since there are no winding states on S7 and no U-duality. In contrast to the T7 compactification, moreover, we find no static multi-center solutions. Also in contrast, the S7 fields appear “already dualized” so that the 4 charges may be all electric or all magnetic rather than 2 electric and 2 magnetic. Curiously, however, the magnetic solutions preserve no supersymmetries. We conjecture that a subset of the extreme electric black holes preserving the supersymmetry may be identified with the S7 Kaluza-Klein spectrum, with the non-abelian SO(8) quantum numbers provided by the fermionic zero-modes.
It is well known that the low energy electron excitations of the curved
graphene sheet are solutions of the massless Dirac equation on a 2+1
dimensional ultra-static metric on . An externally
applied electric field on the graphene sheet induces a gauge potential which
could be mimicked by considering a stationary optical metric of the Zermelo
form, which is conformal to the BTZ black hole when the sheet has a constant
negative curvature. The Randers form of the metric can model a magnetic field,
which is related by a boost to an electric one in the Zermelo frame. We also
show that there is fundamental geometric obstacle to obtaining a model that
extends all the way to the black hole horizon.
Black-hole spacetimes with a "photonsphere", a hypersurface on which massless
particles can orbit the black hole on circular null geodesics, are studied. We
prove that among all possible trajectories (both geodesic and non-geodesic)
which circle the central black hole, the null circular geodesic is
characterized by the {\it shortest} possible orbital period as measured by
asymptotic observers. Thus, null circular geodesics provide the fastest way to
circle black holes. In addition, we conjecture the existence of a universal
lower bound for orbital periods around compact objects (as measured by
flat-space asymptotic observers): , where M is the
mass of the central object. This bound is saturated by the null circular
geodesic of the maximally rotating Kerr black hole.
Einstein-matter theories in which hairy black-hole configurations have been
found are studied. We prove that the nontrivial behavior of the hair must
extend beyond the null circular orbit (the photonsphere) of the corresponding
spacetime. We further conjecture that the region above the photonsphere
contains at least 50% of the total hair's mass. We support this conjecture with
analytical and numerical results.
We consider the scattering and lens rigidity of compact surfaces with
boundary that have a trapped geodesic. In particular we show that the flat
cylinder and the flat M\"obius strip are determined by their lens data. We also
see by example that the flat M\"obius strip is not determined by it's
scattering data. We then consider the case of negatively curved cylinders with
convex boundary and show that they are lens rigid.
We prove that the flat product metric on is scattering rigid
where is the unit ball in and . The scattering data
(loosely speaking) of a Riemannian manifold with boundary is map from unit vectors V at the boundary that point inward to
unit vectors at the boundary that point outwards. The map (where defined) takes
V to where is the unit speed geodesic determined
by V and is the first positive value of t (when it exists) such that
again lies in the boundary. We show that any other Riemannian
manifold with boundary isometric to
and with the same scattering data must be isometric
to . This is the first scattering rigidity result for a manifold
that has a trapped geodesic. The main issue is to show that the unit vectors
tangent to trapped geodesics in have measure 0 in the unit
tangent bundle.
Nurowski [arXiv:1003.1503] has recently suggested a link between the observation of Dark Energy in cosmology and the projective equivalence of certain Friedman-Lemaitre-Robertson-Walker (FLRW) metrics. Specifically, he points out that two FLRW metrics with the same unparameterized geodesics have their energy densities differing by a constant. From this he queries whether the existence of dark energy is meaningful. We point out that physical observables in cosmology are not projectively invariant and we relate the projective symmetry uncovered by Nurowski to some previous work on projective equivalence in cosmology.
It is shown that in classical general relativity, if space-time is nonempty at one time, it will be nonempty at all times provided that the energy momentum tensor of the matter satisfies a physically reasonable condition. The apparent contradiction with the quantum predictions for the creation and annihilation of matter particles by gravitons is discussed and is shown to arise from the lack of a good energy momentum operator for the matter in an unquantised curved space-time metric.
We study the thermodynamic stability of charged black holes in gauged supergravity theories in D=5, D=4 and D=7. We find explicitly the location of the Hawking-Page phase transition between charged black holes and the pure anti-de Sitter space-time, both in the grand-canonical ensemble, where electric potentials are held fixed, and in the canonical ensemble, where total charges are held fixed. We also find the explicit local thermodynamic stability constraints for black holes with one non-zero charge. In the grand-canonical ensemble, there is in general a region of phase space where neither the anti-de Sitter space-time is dynamically preferred, nor are the charged black holes thermodynamically stable. But in the canonical ensemble, anti-de Sitter space-time is always dynamically preferred in the domain where black holes are unstable. We demonstrate the equivalence of large R-charged black holes in D=5, D=4 and D=7 with spinning near-extreme D3-, M2- and M5-branes, respectively. The mass, the charges and the entropy of such black holes can be mapped into the energy above extremality, the angular momenta and the entropy of the corresponding branes. We also note a peculiar numerological sense in which the grand-canonical stability constraints for large charge black holes in D=4 and D=7 are dual, and in which the D=5 constraints are self-dual. Comment: 30 pages, 6 figures
We derive and analyse the full set of equations of motion for non-extreme static black holes (including examples with the spatial curvatures k=-1 and k=0) in D=5 N=2 gauged supergravity by employing the techniques of "very special geometry". These solutions turn out to differ from those in the ungauged supergravity only in the non-extremality function, which has an additional term (proportional to the gauge coupling g), responsible for the appearance of naked singularities in the BPS-saturated limit. We derive an explicit solution for the STU model of gauged supergravity which is incidentally also a solution of D=5 N=4 and N=8 gauged supergravity. This solution is specified by three charges, the asymptotic negative cosmological constant (minimum of the potential) and a non-extremality parameter. While its BPS-saturated limit has a naked singularity, we find a lower bound on the non-extremality parameter (or equivalently on the ADM mass) for which the non-extreme solutions are regular. When this bound is saturated the extreme (non-supersymmetric) solution has zero Hawking temperature and finite entropy. Analogous qualitative features are expected to emerge for black hole solutions in D=4 gauged supergravity as well. Comment: 16 pages ; references added, minor changes
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 match with the chiral operators of super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the theory has a large N phase transition related to the thermodynamics of AdS black holes. Comment: 40 pp.; additional references and assorted corrections
We summarize results for all four-dimensional Bogomol'nyi-Sommerfield-Prasat (BPS) saturated and non-extreme solutions of the (4+n)-dimensional Abelian Kaluza-Klein theory. Within effective N=4 supersymmetric string vacua, parameterized in terms of fields of the heterotic string on a six-torus, we then present a class of BPS saturated states and the corresponding non-extreme solutions, specified by O(6,22,Z) and SL(2,Z) orbits of general dyonic charge configurations with zero axion. The BPS saturated states with non-negative O(6,22,Z) norms for electric and magnetic charge vectors, along with the corresponding set of non-extreme solutions, are regular with non-zero masses. BPS saturated states with the negative charge norms are singular, unaccompanied by non-extreme solutions and become massless at particular points of the moduli space. The role that such massless states may play in the enhancement of non-Abelian gauge symmetry as well as local supersymmetry is addressed.
Within effective heterotic superstring theory compactified on a six-torus we derive minimum energy (supersymmetric), static, spherically symmetric solutions, which are manifestly invariant under the target space O(6,22) and the strong-weak coupling SL(2) duality symmetries with 28 electric and 28 magnetic charges subject to one constraint. The class of solutions with a constant axion corresponds to dyonic configurations subject to two charge constraints, with purely electric [or purely magnetic] and dyonic configurations preserving and of N=4 supersymmetry, respectively. General dyonic configurations in this class have a space-time of extreme Reissner-Nordstr\" om black holes while configurations with more constrained charges have a null or a naked singularity. Comment: 10 pages, uses RevTex, improved version to appear in Rapid Communication of PRD
We review boundary rigidity theorems assessing that, under appropriate conditions, Riemannian manifolds with the same spectrum of boundary geodesics are isometric. We show how to apply these theorems to the problem of reconstructing a d+1 dimensional, negative curvature space-time from boundary data associated to two-point functions of high-dimension local operators in a conformal field theory. We also show simple, physically relevant examples of negative-curvature spaces that fail to satisfy in a subtle way some of the assumptions of rigidity theorems. In those examples, we explicitly show that the spectrum of boundary geodesics is not sufficient to reconstruct the metric in the bulk. We also survey other reconstruction procedures and comment on their possible implementation in the context of the holographic AdS/CFT duality. Comment: 26 pages, 4 figures
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T. Jacobson, When is g tt g rr = −1?, Class. Quant. Grav. 24 (2007) 5717,
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