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The Kerr family of solutions of the Einstein and Einstein-Maxwell equations is the most general class of solutions known at present which could represent the field of a rotating neutral or electrically charged body in asymptotically flat space. When the charge and specific angular momentum are small compared with the mass, the part of the manifold which is stationary in the strict sense is incomplete at a Killing horizon. Analytically extended manifolds are constructed in order to remove this incompleteness. Some general methods for the analysis of causal behavior are described and applied. It is shown that in all except the spherically symmetric cases there is nontrivial causality violation, i.e., there are closed timelike lines which are not removable by taking a covering space; moreover, when the charge or angular momentum is so large that there are no Killing horizons, this causality violation is of the most flagrant possible kind in that it is possible to connect any event to any other by a future-directed timelike line. Although the symmetries provide only three constants of the motion, a fourth one turns out to be obtainable from the unexpected separability of the Hamilton-Jacobi equation, with the result that the equations, not only of geodesics but also of charged-particle orbits, can be integrated completely in terms of explicit quadratures. This makes it possible to prove that in the extended manifolds all geodesics which do not reach the central ring singularities are complete, and also that those timelike or null geodesics which do reach the singularities are entirely confined to the equator, with the further restriction, in the charged case, that they be null with a certain uniquely determined direction. The physical significance of these results is briefly discussed.

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... Meanwhile, there is a third area of research that has already been discussed over fifty years, and was recently again reactivated. This is the treatment of the elementary quantum particle as an overrotating the Kerr-Newman (KN) solution [1][2][3][4][5][6][7][8][9][10][11][12][13]. Formation of black holes is related with classical gravitational effect of frame-dragging [14], which has not previously been considered in particle physics and, when applied to the problem of interaction between quantum theory and gravity, it leads to two important new consequences: ...

... In 1968, Carter noticed that KN solution has gyromagnetic ratio g = 2, just the same as that of the Dirac electron [1], which gave rise to two lines in the study of the electron model based on the KN solution: the disk-like and bubble models [3,4,8,15], and the string-like models [5,6,11], based on the analogue of the Kerr singular ring with a string, similar to the Nielsen-Olesen string model in superconductor [16]. These lines were united in the subsequent series of the works [9,10], where the source of the KN electron model was considered as a superconducting bag model with a string formed on the sharp border of the bag-like core. ...

... The KN model of electron is consistent with gravity by nature [2], and these works show, how the known insuperable contradictions between gravity and quantum physics can actually be resolved. 1 As opposed to gravitational radius of the Schwarzschild solution l s = Gm c 2 , effective zone of gravitational interaction in the KN solution is determined by radius of the Kerr singular ring ...

We consider a consistent with gravity electron based on the overrotating Kerr-Newman (KH) solution and show that the earlier KH electron models proposed by Carter, Israel and López in 1970–1990 should be modified by the Landau-Ginzburg theory, leading to a superconducting electron model consistent with gravity and quantum theory. Truncated by Israel and López, the second sheet of the KN solution is rearranged and represented in a mirror form as a sheet of the positron, so that the modified KN system forms a quantum electron-positron vacuum interacting with gravity. Regularization of the KN black hole solution creates two new important effects leading to a strong gravitational interaction that acts on the Compton scale contrary to the usual Planck scale of Schwarzschild gravity: (A)—gravitational frame-dragging creates two Wilson loops acting at two boundaries of the modified KN solution, and (B)—formation of the flat superconducting core of the regularized KN solution creates a superconducting electron-positron vacuum state. The Landau-Ginzburg model shows that Wilson loops determine phases of two Higgs fields forming superconducting vacuum state of the modified KN solution, quantum vacuum of the electron-positron pairs. The phases of these Higgs fields correspond to two light-like modes of a classical relativistic ring string. We come to the conclusion that the electron models considered by Israel and López are not complete and must be supplemented by a mirror structure that forms a quantum system consistent with QED.

... After recalling about Kerr-Newman (charged rotating), and Reissner-Nordström (charged) black holes. we revisit our proposal of particles as microscopic black holes as initially proposed in [1], inspired by the work of A. Burinskii [4,[39][40][41][42][43][44][45]49], the subtle connections already encountered among spacetime, particles, blackholes, and their thermodynamics, and between the properties and scattering of particles and black holes [27][28][29][30][31][32][33]38,[64][65][66][67][68]. We then recast the model, using the latest developments from Burinskii, to argue that black holes are (extremal) solitons of microscopic Kerr Newman black holes, induced by space time matter and microscopically resulting from the condensation of the massless Higgs bosons. ...

... Indeed, the solutions include a closed ring of naked singularities that can be analytically extended by bifurcated double manifold solutions. It's a foliated (pseudo-)Riemann manifold with one inside solution and one solution outside the singularity ring [27,30]. Such situations present physical interpretation challenges. ...

... All these considerations only reinforce the proposal for multi-fold spacetime (re)construction as proposed in [1] and the role of the Higgs boson [6,9] and the electroweak symmetry breaking [9,25,26,27,55] in the concretization of spacetime. ...

In a multi-fold universe, gravity emerges from Entanglement through the multi-fold mechanisms. As a result, gravity-like effects appear in between entangled particles that they be real or virtual. Long range, massless gravity results from entanglement of massless virtual particles. Entanglement of massive virtual particles leads to massive gravity contributions at very smalls scales. Multi-folds mechanisms also result into a spacetime that is discrete, with a random walk fractal structure, and non-commutative geometry that is Lorentz invariant and where spacetime nodes and particles can be modeled with microscopic black holes. All these recover General relativity at large scales, and semi-classical model remain valid till smaller scale than usually expected. Gravity can therefore be added to the Standard Model (SM) resulting into what we defined as SM G. This can contribute to resolving several open issues with the Standard Model without new Physics other than gravity. These considerations hint at an even stronger relationship between gravity and the Standard Model. This paper investigates the details of modeling particles as microscopic black holes previously proposed during multi-fold spacetime reconstruction. We reuse work done on Kerr Newman regularization, by modeling the region inside the singularity ring as a Dirac soliton in Kerr-Newman metric within a kind of Q-ball where the Higgs field condense, due to its symmetry being broken. The Q-ball edge appears superconductive with an oblateness that for the electron is given by the fine structure constant. It recovers charged particle scatterings, spin quantization, magnetic momentum. Massless particles and concretized spacetime are modeled by Schwarzschild black holes. Our analysis is the result of combining different results obtained by others, but re-interpreted when put together in the context of multi-fold mechanisms. The analysis also clarifies and reinforces our proposals for the role of Higgs boson, the Higgs field and the Ultimate Unification (UU) in multifold universes, in term of random walk, spacetime point concretization, and inflation. We also confirm the possible relationship between supersymmetry, superstrings, 2D gravity and multi-fold random walks in multifold spacetime reconstructions. Indeed the formers can approximate the random walks, something we already concluded in recent papers. At large scale, all these models seem to converge even if the challenges related to asymptotic safety and SM remain a problem for superstrings and supersymmetry.

... To start with, new black hole solutions in the context of modified theories of gravity have long been known to exist [7][8][9][10][11][12][13][14][15] by evading the * georgios.antoniou@nottingham.ac.uk † papageo@ibs.re.kr ‡ pkanti@uoi.gr no-hair theorems of GR [16][17][18][19][20][21][22][23][24][25][26][27][28] with a plethora of additional solutions having emerged during the last few years . ...

... For l obs l 0 , l ph , asymptotic flatness demands that v → 0. The wormhole shadow is retrieved again in the limit l c → l ph , for which b → b crit according to Eq. (19). Thus, for a faraway observer, we obtain a sh ≈ b crit /l obs . ...

... Then, Eq. (17) gives l ph = 0, and thus there is only one circular photon orbit located around the throat. Then, in the limit l c → l ph , Eq. (19) yields that b crit = l 0 , and Eq. (20) takes the simplified form ...

We study a number of well-motivated theories of modified gravity with the common overarching theme that they predict the existence of compact objects such as black holes and wormholes endowed with scalar hair. We compute the shadow radius of the resulting compact objects and demonstrate that black hole images such as that of M87$^*$ or the more recent SgrA$^*$ by the Einstein Horizon Telescope (EHT) collaboration may provide a powerful way to constrain deviations of the metric functions from what is expected from general relativity (GR) solutions. We focus our attention on Einstein-scalar-Gauss-Bonnet (EsGB) theory with three well motivated couplings, including the dilatonic and $Z_2$ symmetric cases. We then analyze the shadow radius of black holes in the contest of the spontaneous scalarization scenario within EsGB theory with an additional coupling to the Ricci scalar (EsRGB). Finally, we turn our attention to spontaneous scalarization in the Einstein-Maxwell-Scalar (EMS) theory and demonstrate the impact of the parameters on the black hole shadow. Our results show that black hole imaging is an important tool for constraining black holes with scalar hair and for some part of the parameter space, black holes solutions with scalar hair may be marginally favoured compared to solutions of GR.

... The Hamilton−Jacobi equation determines the photon motion in the spacetime (1) (Carter 1968): ...

... The metric (1) is time translational and rotational symmetry, which leads to two conserved quantities, namely energy p t =and axial angular momentum p = f , where p μ is the photon's four-momentum. The Petrov-type D character of metric (1) guarantees the separable constant (Carter 1968), and then we get a set of null geodesics equations in the first-order differential form (Carter 1968;Chandrasekhar 1985): ...

... The metric (1) is time translational and rotational symmetry, which leads to two conserved quantities, namely energy p t =and axial angular momentum p = f , where p μ is the photon's four-momentum. The Petrov-type D character of metric (1) guarantees the separable constant (Carter 1968), and then we get a set of null geodesics equations in the first-order differential form (Carter 1968;Chandrasekhar 1985): ...

The Event Horizon Telescope (EHT) observation unveiled the first image of supermassive black hole Sgr A* showing a shadow of diameter θ sh = 48.7 ± 7 μ as with fractional deviation from the Schwarzschild black hole shadow diameter δ = − 0.08 − 0.09 + 0.09 ( VLTI ) , − 0.04 − 0.10 + 0.09 ( Keck ) . The Sgr A* shadow size is within 10% of the Kerr predictions, providing us with another tool to investigate the nature of strong-field gravity. We use the Sgr A* shadow observables to constrain metrics of four independent and well-motivated, parametrically different from Kerr spacetime, rotating regular spacetimes and the corresponding no-horizon spacetimes. We present constraints on the deviation parameter g of rotating regular black holes. The shadow angular diameter θ sh within the 1 σ region places bounds on the parameters a and g . Together with EHT bounds on the θ sh and δ of Sgr A*, our analysis concludes that the three rotating regular black holes, i.e., Bardeen, Hayward, and Simpson–Visser black holes, and corresponding no-horizon spacetimes agree with the EHT results of Sgr A*. Thus, these three rotating regular spacetimes and Kerr black holes are indiscernible in some parameter space, and one cannot rule out the possibility of the former being strong candidates to be astrophysical black holes.

... We start with the Hamilton-Jacobi equation to determine the null geodesics followed by photons in the LMRBH spacetime (3) (Carter 1968;Chandrasekhar 1985) ...

... S r (r) and S θ (θ), respectively, are functions only of the r and θ coordinates. Therefore, the four integrals of motions, the Lagrangian, energy E, axial angular momentum L and the Carter constant associated to the latitudinal motion of the photon, are sufficient to determine the geodesics equations of motion in the first-order differential form (Carter 1968;Chandrasekhar 1985), as follows ...

... where the separability constant K is related to the Carter constant Q through Q = K + (aE − L) 2 (Carter 1968;Chandrasekhar 1985), which, in essence, represents the isometry of metric (3) along the second-order Killing tensor field. The photon's θ-motion is influenced by the Q, but restricted to an equatorial plane when Q = 0. ...

A mathematically consistent rotating black hole model in Loop Quantum Gravity (LQG) is yet lacking. The scarcity of rotating black hole solutions in LQG substantially hampers the development of testing LQG from observations, e.g., from the Event Horizon Telescope (EHT) observations. The EHT observation revealed event horizon-scale images of the supermassive black holes Sgr A* and M87*. The EHT results are consistent with the shadow of a Kerr black hole of general relativity. We present LQG-motivated rotating black hole spacetimes (LMRBH), which are regular everywhere and asymptotically encompass the Kerr black hole as a particular case. LMRBH metric describes a multi-horizon black hole in the sense that it can admit up to three horizons, such that an extremal LMRBH, unlike Kerr black hole, refers to a black hole with angular momentum $a>M$. The metric, depending on the parameters, describes (1) black holes with only one horizon (BH-I), (2) black holes with an event and Cauchy horizons (BH-II), (3) black holes with three horizons (BH-III) or (4) no-horizon spacetime (NH) which, we show, is almost ruled out by the EHT observations. We constrain the LQG parameter with an aid of the EHT shadow observational results of M87* and Sgr A*,respectively, for an inclination angle of $17^0$ and $50^0$. In particular, the VLTI bound for the Sgr A*, $\delta\in (-0.17,0.01)$, constrains the parameters ($a,l$) such that for $0< l\leq 0.347851M\; (l\leq 2\times 10^6$ Km), the allowed range of $a$ is $(0,1.0307M)$. Together with EHT bounds of Sgr A$^*$ and M87$^*$ observables, our analysis concludes that the substantial part of BH-I and BH-II parameter space agrees with the EHT results of M87* and Sgr A*. While EHT M87* results totally rules out the BH-III, but not that by Sgr A*.

... The fact that the silhouette of black hole shadows encode in them, the strong-field properties of the spacetime, suggests that, we can use them for performing strongfield gravitational tests (Johannsen & Psaltis 2010;Cunha & Herdeiro 2018;Baker et al. 2015). The lightlike geodesics in the LQG spacetime (1), just as in the Kerr spacetime, follow the Hamilton-Jacobi equation (Carter 1968), ...

... t , axial angular momentum L z = p.∂ φ , arise due to the translational and rotational symmetry of the Kerr-like metric (1). Further, the axially symmetric metric (1) insinuates a fourth conserved quantity, the Carter's constant Q , which ensures the decoupling of r and θ equations (Carter 1968). Following (Tsukamoto 2018;Brahma et al. 2021;Liu et al. 2020), we obtain null geodesics in the firstorder differential form ...

The Event Horizon Telescope (EHT) collaboration's image of the compact object at the galactic centre is the first direct evidence of the supermassive black hole Sgr A$^*$. The shadow of Sgr A$^*$ has an angular diameter $d_{sh}= 48.7 \pm 7\,\mu$as with fractional deviation from the Schwarzschild black hole shadow diameter $\delta= -0.08^{+0.09}_{-0.09}\,,-0.04^{+0.09}_{-0.10}$ (for the VLTI and Keck mass-to-distance ratios). Sgr A$^*$'s shadow size is within $~10\%$ of the Kerr predictions, equipping us with yet another tool to analyse the gravity in the strong-field regime, including testing loop quantum gravity (LQG). We use Sgr A$^*$'s shadow to constrain the metrics of two well-motivated LQG-inspired rotating black holes (LIRBHs) models characterized by an additional deviation parameter $L_q$, which recover the Kerr spacetime in the absence of quantum effects ($L_q \to 0$). We use the astrophysical observables shadow area $A$ and oblateness $D$ to estimate the black hole parameters. When increasing the size of the quantum effects through $L_q$, the black hole shadow size increases monotonically, while the shape gets more distorted, allowing us to constrain the fundamental parameter $L_q$. While the EHT observational results completely rule out the wormhole region in the second LIRBH, a substantial parameter region of the generic black holes in both models agree with the EHT results. We find upper bounds on $L_q$ from the shadow of Sgr A$^*$: $L_q \lesssim 0.0423$ and $L_q \lesssim 0.0821$ for the two LIRBHs respectively, both more stringent than those obtained with the EHT image of M87$^*$.

... In this section, we will investigate the null geodesics and focus on the photons with circular orbits in the background of (29). The geodesics of photon orbiting in spacetime can be obtained using the Hamilton-Jacobi approach [74], and the associated equation is ∂S ∂λ ...

... where Σ(r) = G F H + a 2 sin 2 θ = k(r) + a 2 and K is the Cater constant of the motion related to the Killing-Yano tensor [74] due to the integrability of the system. Furthermore, by combining the null geodesic condition p µ p µ = 0 with the results (36) and (37), we can obtain four equations of motion which control the photon orbits. ...

In this paper, we investigate the shadow cast by non-rotating and rotating charged black holes with scalar Q-hair. We find that in addition to the spin parameter of black hole and inclination angle of the observer, the charge parameter and the self-interaction parameters of the scalar hair also influence the shape of the black hole shadow. Our studies show that the charged black holes with scalar Q-hair always have smaller shadow size compared to those without hair. Moreover, it is found that the parameters significantly affect the shadow observables. In particular, for the fixed spin parameter and inclination angle, a larger charge parameter will increase the shadow size but decrease the shadow distortion, whilst stronger self-interaction parameters have the opposite influence. In short, the shadow of the charged black hole with scalar Q-hair can be distinguished from the Reissner–Nordstro¨m (RN) black hole or Kerr–Newmann (KN) black hole, and they indeed generate new templates with large deviations from general relativity those are invariably smaller in size.

... However, to separate the equations of motion in Kerr black holes, an additional constant must be introduced. In contrast to the separation constants from isometries, the additional constant originates from a hidden symmetry related to an irreducible rank-two Killing tensor [70,71]. The separability of the equations of motion depends on the black holes and matter in the background; therefore, each case has been studied separately for Kerr black holes and other black holes [70][71][72][73][74][75][76][77][78][79][80][81][82][83]. ...

... In contrast to the separation constants from isometries, the additional constant originates from a hidden symmetry related to an irreducible rank-two Killing tensor [70,71]. The separability of the equations of motion depends on the black holes and matter in the background; therefore, each case has been studied separately for Kerr black holes and other black holes [70][71][72][73][74][75][76][77][78][79][80][81][82][83]. Based on hidden symmetries to spacetime, the equations of motion are solvable and understandable; therefore, this is an important property for studying the dynamics of test particles and fields. ...

We investigate the weak cosmic censorship conjecture in Myers-Perry black holes with arbitrary rotations in general dimensions based on the scattering of a massless scalar field. From the fluxes of the scalar field flowing into the black hole, the changes in mass and angular momenta of the black hole are obtained. However, the extremal and near-extremal black holes with the aforementioned changes are still black holes in the final state. Hence, the conjecture is valid for our investigation. Furthermore, we analyze the changes in the black hole from a thermodynamic perspective to highlight that the laws of thermodynamics support the conjecture.

... By taking trace of this equation (26) we obtained: ...

... In equation (54) it is observed that the partial differential equation does not contain any cross terms of variables u and θ and hence this can be solved by separation of variables as calculating in ( [24] pp. 344-345) and [26]. So we considered, ...

In this paper, we solved the Einstein's field equation and obtained a line element for static, ellipsoidal objects characterized by the linear eccentricity ($\eta$) instead of quadrupole parameter ($q$). This line element recovers the Schwarzschild line element when $\eta$ is zero. In addition to that it also reduces to the Schwarzschild line element, if we neglect terms of the order of $r^{-2}$ or higher which are present within the expressions for metric elements for large distances. Furthermore, as the ellipsoidal character of the derived line element is maintained by the linear eccentricity ($\eta$), which is an easily measurable parameter, this line element could be more suitable for various analytical as well as observational studies.

... The special integrability properties of the Kerr spacetime reduce its geodesic equation to a problem of quadratures [46], resulting in elliptic integrals that are expressible in Legendre normal form [39,47,48]. Modern computers can evaluate the Legendre elliptic integrals very fast and to arbitrary precision, reducing the computational cost of ray tracing in Kerr. In this section, we review the exact solution of the Kerr null geodesic equation in Legendre form [5,39] as well as the approximate solution derived in Schwarzschild by Beloborodov [43], and we use them to plot the transfer functions mapping Bardeen's coordinates in the observer sky [14] to the equatorial plane. ...

... a quantity that is also conserved along null geodesics-thanks to the Petrov type D nature of the Kerr metric [53]-and can be used to algebraically solve the parallel transport problem for a linear polarization vector f by evaluating κ at the source. The separability of Kerr geodesic motion allows the r and θ trajectories to be decoupled [46], These independent motions are then controlled by radial and angular geodesic potentials ...

Recent interferometric observations by the Event Horizon Telescope have resolved the horizon-scale emission from sources in the vicinity of nearby supermassive black holes. Future space-based interferometers promise to measure the ''photon ring''--a narrow, ring-shaped, lensed feature predicted by general relativity, but not yet observed--and thereby open a new window into strong gravity. Here we present AART: an Adaptive Analytical Ray-Tracing code that exploits the integrability of light propagation in the Kerr spacetime to rapidly compute high-resolution simulated black hole images, together with the corresponding radio visibility accessible on very long space-ground baselines. The code samples images on a nonuniform adaptive grid that is specially tailored to the lensing behavior of the Kerr geometry and therefore particularly well-suited to studying photon rings. This numerical approach guarantees that interferometric signatures are correctly computed on long baselines, and the modularity of the code allows for detailed studies of equatorial sources with complex emission profiles and time variability. To demonstrate its capabilities, we use AART to simulate a black hole movie of a stochastic, non-stationary, non-axisymmetric equatorial source; by time-averaging the visibility amplitude of each snapshot, we are able to extract the projected diameter of the photon ring and recover the shape predicted by general relativity.

... • In addition to the symmetries generated by T, Z, K(a, m) possesses also a non-trivial Killing tensor 4 , i.e. a symmetric 2-tensor C αβ verifying the property D (γ C αβ) = 0. The tensor carries the name of its discoverer B. Carter, see [15], who made use of it to show that the geodesic flow in Kerr is integrable. Its presence, in addition to T and Z, as a higher order symmetry, is at the heart of what Chandrasekhar, see [17], called the most striking feature of Kerr, "the separability of all the standard equations of mathematical physics in Kerr geometry". ...

... 14 In the case of Kerr, both cases are present as we shall see below. 15 Orbital stability can be established directly (i.e. without establishing the stronger version) only in rare occasions, such as for hamiltonian equations with weak nonlinearities. ...

This a brief introduction to the sequence of works \cite{KS:Kerr}, \cite{GKS-2022}, \cite{KS-GCM1}, \cite{KS-GCM2} and \cite{Shen} which establish the nonlinear stability of Kerr black holes with small angular momentum. We are delighted to dedicate this article to Demetrios Christodoulou for whom we both have great admiration. The first author would also like to thank Demetrios for the magic moments of friendship, discussions and collaboration he enjoyed together with him.

... Factors like velocity of baryonic density relocation and appearance from gravitational singularity in measurable Spacetime are currently measured in C. This advances for explanations to current challenges inherent in Dark energy dynamics and entropy rate of Relativistic speeds of light in the purported Inflationary vaccum [8] [9]. ...

... To derive this we need to consider the Bardeen tetrads [99][100][101][102] which are associated with observers to whom the black hole appears non-rotating. These tetrads are given by, Figure 1: Variation of the shape and size of the BH shadow with charge parameter k, spin parameter a and inclination angle θ. ...

With the recent release of the black hole image of Sgr A* alongside the earlier image of M87*, one can now really hope to acquire a better understanding of the gravitational physics at the horizon scale. In this paper, we investigate the prospect of the regular black hole scenario with a Minkowski core in explaining the observed shadow of M87* and Sgr A*. Regular black holes generally appear in Einstein gravity coupled to non-linear electrodynamics and are interesting as they can evade the r = 0 curvature singularity arising in general relativity. Using the previously determined mass and distance we compute the observables associated with the black hole shadow. These when compared with the observed angular diameter reveals that the shadow of M87* and Sgr A* favor the regular black hole scenario with a small but non-zero charge. The implications are discussed.

... Picture taken from Ref. [23] non-spinning black hole spacetime (Schwarzschild): in Kerr, to describe a generic orbit requires the knowledge of its radial, polar, and azimuthal frequencies [48]. The three fundamental frequencies correspond to three conserved quantities of Kerr geodesics: energy, angular momentum, and the Carter constant [46] 1 . ...

Gravitational-wave astronomy has been a burgeoning field of research since the first detection of a merging black hole binary in 2015. As gravitational-wave detector sensitivity improves, our models must keep pace. The planned space-based detector LISA will be sensitive to new gravitational wave sources, such as extreme-mass-ratio inspirals (EMRIs). Precise extraction of EMRI parameters from LISA data will require highly accurate waveform templates. These templates need models which include, among other things, the dissipative piece of the second-order self-force in Kerr. This thesis formulates methods to help calculate the second-order self-force in Kerr. In the first part of the thesis, I develop a general framework for second-order calculations by deriving a new form of the second-order Teukolsky equation. I show that the source of this equation is well defined (in a highly regular gauge) for second-order self-force calculations. Additionally, I present methods for calculating second-order gauge-invariants. I produce an algebraic method for calculating a gauge-invariant. I also provide a formalism for calculating a gauge-invariant associated with the Bondi--Sachs gauge (with a fixed BMS frame). The asymptotically flat property of the Bondi--Sachs gauge is shown to circumvent infrared divergences that arise in generic second-order calculations. Next, I calculate a general formula for the second-order source, decomposed into spherical harmonics, in Schwarzschild. Using this formula, I help to implement a framework for quasi-circular inspirals in Schwarzschild. I transform the source to a near-Bondi--Sachs gauge, increasing the asymptotic falloff by two orders in r. My collaborator Ben Leather integrates the resulting source. From the resulting quantity, we will extract fluxes and evolve inspirals to first post-adiabatic accuracy. In the final part, I take a step toward implementation in Kerr by developing a new method of constructing a more regular first-order perturbation. To help formulate this method, I implement Green--Hollands--Zimmerman metric construction for a stationary point-mass in flat spacetime.

... In general these are described in terms of Killing spinors. An important example is a second order symmetry operator related to the Carter constant [13] used by Andersson and Blue in [7]. This symmetry operator can not be built from Killing vectors. ...

Employing the covariant language of two-spinors, we find what conditions a curved Lorentzian spacetime must satisfy for existence of a second order symmetry operator for the massive Dirac equation. The conditions are formulated as existence of a set of Killing spinors satisfying a set of covariant linear differential equations. Using these Killing spinors, we then state the most general form of such an operator. Partial results for the zeroth and first order are presented and interpreted as well. Computer algebra tools from the Mathematica package suite xAct were used for the calculations.

... In general relativity, the geodesic motion of particles in a generic Kerr-Newman black hole spacetime [4] is integrable and there is no chaos in this system. In order to study chaotic motion in general relativity and to ensure that the dynamical system describing the motion of the mass is integrable, it is necessary to resort to some spacetime with a complex geometry or to introduce some additional interactions. ...

In this paper, we have studied the variation of the chaos bound in two regions of the torus-like black hole, i.e., the region close to the black hole horizon and the region at a certain distance from the black hole horizon. The angular momentum of the particle affects the effective potential and influences the magnitude of the chaotic behavior of the particle. Therefore, the angular momentum of particle is important in the study. The angular momentum of a particle not only affects the particle equilibrium orbital position, but also affects the Lyapunov exponent. As the angular momentum of the particle increases, the particle equilibrium position gradually moves away from the black hole horizon. In the near black hole horizon region, the chaos bound is not violated, however, at the far black hole horizon region, the chaos bound is violated. In addition unlike the charged AdS black hole which has a spherical topology of the horizon, the torus-like black hole has a toroidal topology of the horizon.

... The cosmological black holes of the Vaidya-Bonnor-monopole-de Sitter solution (3.5) are described by the value of r , when Δ = 0 which haves four roots r ++ , r +− , r −+ and r −− where the first two represent the cosmological horizon r ++ and the event horizon r +− . The surface gravity [24][25][26][27] for the metric (3.5) is given by ...

In this paper, we propose a class of embedded solutions of Einstein's field equations with the de Sitter cosmological function Λ(u). This class of solutions describes Vaidya-Bonnor-monopole-de Sitter space-time with variable Λ(u). It may also be interpreted as Vaidya-Bonnor black hole in monopole-de Sitter space with variable Λ(u). It is shown the natural modification of Einstein's field equations with the de Sitter cos-mological function Λ(u). In the energy-momentum tensor of the gravitational field in the embedded solution it is also seen the interaction of electromagnetic field with monopole and de Sitter fields having different equations of state parameters. It is also established that the time like vector field of the matter distribution in the embedded space-time geometry is expanding, accelerating and shearing, but non-rotating. We have also discussed the areas, entropies, surface gravities and temperatures for the different horizons for the solution. From the embedded solution we may also recover possible solutions with variable Λ(u) such as (i) Vaidya-Bonnor-de Sitter, (ii) Vaidya-Bonnor-monopole, (iii) Vaidya-monopole, (iv) charged monopole-de Sitter and (v) uncharged monopole-de Sitter. From the study of these exact solutions, it is found that the physical properties of an embedded black hole are depended on the nature of the background spaces. It is also true in the case of the cosmological de Sitter space with variable Λ(u) that the charged monopole-de Sitter and the uncharged monopole-de Sitter spaces have different properties depending upon the charged spaces.

... The results of Mino [19] were surprising, since they also contained an expression for the orbit-averaged change in the third constant of motion in the Kerr spacetime, the Carter constant K [20]. While the conserved currents used by [17,18] for arbitrary Killing vectors were constructed from the stress-energy tensor of the theory in question (or effective stress-energy tensor, in the case of gravity), it was not known if conserved currents associated with Killing tensors (from which the Carter constant can be constructed [21]) could be constructed in a similar way. ...

The motion of a radiating point particle can be represented by a series of geodesics whose "constants" of motion evolve slowly with time. The evolution of these constants of motion can be determined directly from the self-force equations of motion. In the presence of spacetime symmetries, the situation simplifies: there exist not only constants of motion conjugate to these symmetries, but also conserved currents whose fluxes can be used to determine their evolution. Such a relationship between point-particle motion and fluxes of conserved currents is a flux-balance law. However, there exist constants of motion that are not related to spacetime symmetries, the most notable example of which is the Carter constant in the Kerr spacetime. In this paper, we first present a new approach to flux-balance laws for spacetime symmetries, using the techniques of symplectic currents and symmetry operators, which can also generate more general conserved currents. We then derive flux-balance laws for all constants of motion in the Kerr spacetime, using the fact that the background, geodesic motion is integrable. For simplicity, we restrict derivations in this paper to the scalar self-force problem, although the generalization to the gravitational case is straightforward.

... To facilitate our manipulation as well as choose an observer, we put the test particle into the orthogonal nor-malized Carter tetrad e (a) µ [36,37], with ...

We explore how the internal structure of a test particle affects its equatorial stable circular orbits around the Kerr black hole with or without a cosmological constant. To this end, we first explicitly write equations of motion for a test particle in the pole-dipole-quadrupole approximation specifying the quadrupole momentum tensor to a spin-induced model. Then we calculate characteristic quantities -- radius, angular momentum, energy, angular velocity, and impact parameter -- for the particles on the stable circular orbits. Once the pole-dipole-quadrupole approximation is taken, we find that for a particle on an innermost stable circular orbit, all characteristic quantities, except the angular velocity, become greater relative to the pole-dipole case. In contrast, for a particle on an outermost stable circular orbit, which only exists in the case of the spacetime background being asymptotically de Sitter, it is the radius that becomes smaller while all other quantities become greater.

... The plots show the horizons structure of the KNKL black hole for different values of the parameters geometry of the KNKL black hole. We adopt the Hamilton-Jacobi formalism to study the null geodesic structure of the KNKL black hole spacetime, as follows[105] ...

We investigate the null geodesics and the shadow cast by the Kerr–Newman–Kiselev–Letelier (KNKL) black hole for the equation of state parameter ωq=-2/3 and for different values of the spacetime parameters, including the quintessence parameter γ, the cloud of string (CS) parameter b, the spin parameter a and the charge Q of the black hole. We notice that for the increasing values of the parameters γ and b the size of the shadow of the KNKL black hole increases and consequently the strength of the gravitational field of the black hole increases. On the other hand with increase in the charge Q of the black hole the size of the shadow of the black hole decreases. Further with the increase in the values of the spin parameter a of the KNKL black hole, we see that the distortion of the shadow of the black hole becomes more prominent. Moreover we use the data released by the Event Horizon Telescope (EHT) collaboration, to restrict the parameters b and γ for the KNKL black hole, using the shadow cast by the KNKL black hole. To this end, we also explore the relation between the typical shadow radius and the equatorial and polar quasinormal mods (QNMs) for the KNKL black hole and extend this correspondence to non-asymptotically flat spacetimes. We also study the emission energy rate from the KNKL black hole for the various spacetime parameters, and observe that it increases for the increasing values of both the parameters γ and b for fixed charge-to-mass and spin-to-mass ratios of the KNKL black hole. Finally, we investigate the effects of plasma on the photon motion, size and shape of the shadow cast by the KNKL black hole. While keeping the spacetime parameters fixed, we notice that with increase in the strength of the plasma medium the size of the shadow of the KNKL black hole decreases and therefore the intensity of the gravitational field of the KNKL black hole decreases in the presence of plasma.

... Furthermore D F is a connection on the space of forms which depends on the fluxes F of the supergravity theory that it is not necessarily form degree preserving. A consequence of the TCFH is that the form bilinears Ω satisfy a generalisation of the CKY equation with respect to D F connection as one can easily verify by skew-symmetrising and taking the contraction with respect to the spacetime metric of (1). This raises the question on whether the form bilinears generate symmetries for appropriate probes propagating on supersymmetric backgrounds. ...

We present the TCFH of 11-dimensional supergravity and so demonstrate that the form bilinears of supersymmetric solutions satisfy a generalisation of the conformal Killing-Yano equation with resepct to the TCFH connection. We also compute the Killing-Stäckel, Killing-Yano and closed conformal Killing-Yano tensors of all spherically symmetric M-branes that include the M2-brane, M5-brane, KK-monopole and pp-wave and demonstrate that their geodesic ﬂows are completely integrable by giving all independent conserved charges in involution. We then ﬁnd that all form bilinears of pp-wave and KK-monopole solutions generate (hidden) symmetries for spinning particle probes propagating on these backgrounds. Moreover, there are Killing spinors such that some of the 1-, 2- and 3-form bilinears of the M2-brane solution also generate symmetries for spinning particle probes. We also explore the question on whether the form bilinears are suﬃcient to prove the integrability of particle probe dynamics on 11-dimensional supersymmetric backgrounds.

... Ray tracing.-We use the public ray-tracing code GEOKERR [30] to perform geodesic integration of synchrotron photons from the emission point to a distant observer's screen, in postprocessing (see the Supplemental Material [31], which includes Ref. [32]). When analyzing the 2D simulation, we focus on the time dependence on the image. ...

Accreting supermassive black holes can now be observed at the event-horizon scale at millimeter wavelengths. Current predictions for the image rely on hypotheses (fluid modeling, thermal electrons) which might not always hold in the vicinity of the black hole, so that a full kinetic treatment is in order. In this Letter, we describe the first 3D global general-relativistic particle-in-cell simulation of a black-hole magnetosphere. The system displays a persistent equatorial current sheet. Synthetic radio images are computed by ray-tracing synchrotron emission from nonthermal particles accelerated in this current sheet by magnetic reconnection. We identify several time-dependent features of the image at moderate viewing angles: a variable radius of the ring, and hot spots moving along it. In this regime, our model predicts that most of the flux of the image lies inside the critical curve. These results could help promote understanding of future observations of black-hole magnetospheres at improved temporal and spatial resolution.

... The Kerr geometry has a number of interesting properties, we will here briefly refer to the "integrability" of its geodesics, as it will be important for following considerations. This property was demonstrated in a celebrated paper by Brandon Carter back in 1968 21 . But what is the precise meaning of integrability here? ...

We briefly discuss explicit compact object solutions in higher-order scalar-tensor theories. We start by so-called stealth solutions, whose metric are General Relativity (GR) solutions, but accompanied by a non-trivial scalar field, in both spherically-symmetric and rotating cases. The latter then enables to construct an analytic stationary solution of scalar tensor theory which is called disformed Kerr metric. This solution constitutes a measurable departure from the usual Kerr geometry of GR. We finally consider a scalar-tensor theory stemming from a Kaluza-Klein reduction of a higher-dimensional Lovelock theory, and which enables to obtain non-stealth black holes, highly compact neutron stars and finally wormhole solutions.

... Numerically, for every pixel of our observed image, the position of the emission source in the accretion disk can be traced back. For simplicity, assuming that the neutrinos are emitted isotropically at each radius from the equatorial plane, i.e., i em = π/2, and that the disk is in Keplerian rotation; at last, by ignoring the shading effect caused by the thickness of the disk, from which the trajectory of these emitted neutrinos should satisfy the geodesic equation (Carter 1968), i.e., The energy shift of neutrinos can be calculated by considering the corresponding velocity and the gravitational potential of the emission location. The total observed spectrum is obtained by integrating all the pixels. ...

In the coalescence events of binary neutron star (NS) or a black hole (BH) and an NS, a BH hyperaccretion disk might be eventually formed. At very high mass accretion rates, MeV neutrinos will be emitted from this disk, which is called a neutrino-dominated accretion flow (NDAF). Neutrino annihilation in the space out of the disk is energetic enough to launch ultrarelativistic jets to power gamma-ray bursts. Moreover, vertical advection might exist in NDAFs, which can generate the magnetic buoyancy bubbles to release gamma-ray photons. In this paper, we visit the effects of the vertical advection in NDAFs on the disk structure and gamma-ray and neutrino luminosities for different accretion rates. Then we study the anisotropic emission of kilonovae and the following gravitational waves (GWs) driven by the gamma-ray photons and neutrinos from NDAFs. Comparing NDAFs without vertical advection, the neutrino luminosity and GW strains slightly decrease for the case with vertical advection, and the kilonovae will be brightened by the injected gamma-ray photons. The future joint multimessenger observations might distinguish whether the vertical advection exists in NDAFs or not after compact binary coalescences.

... The constants E and L z are the energy and the angular momentum of the photon, respectively, and D is commonly referred to as the Carter separation constant [30,32,33]. ...

The correspondence between the shadow radius and the real part of the quasinormal modes (QNMs) of a Kerr–Sen black hole is studied. By using the equation of the shadow radius of Kerr–Sen black hole and the angular separation constant of the QNMs, the expression of QNMs related to shadow radius is established in the eikonal limit. We found that, our formula can reduce to the previous result of Kerr black hole when Kerr-Sen parameter b sets to zero.

... The starting point of the computation regarding black hole shadow starts from the geodesic equations of a photon, which are separable in the above spacetime, due to the presence of a Killing tensor. From the three constants of motion, the energy E, angular momentum L and the Carter constant K [48], one may introduce two impact parameters ξ ≡ (L/E) and η ≡ (K/E 2 ), these denote the distance of the photon from the axis of rotation and the equatorial plane, respectively. Finally, the construction of the shadow in the observer's sky, with the observer located at a large distance r 0 from the ultra-compact object, with an inclination angle θ 0 from the rotation axis, requires defining two celestial coordinates α and β, such that [49][50][51], ...

We show that the observed angular diameter of the shadow of the ultracompact object Sgr A*, favors the existence of an extra spatial dimension. This holds irrespective of the nature of the ultracompact object, i.e., whether it is a wormhole or a black hole mimicker, but with the common feature that both of them have an extra dimensional origin. This result holds true for the mass and the distance measurements of Sgr A* using both Keck and the Gravity collaborations and whether we use the observed image or, the observed shadow diameter. In particular, the central value of the observed shadow or the observed image diameter predicts nonzero hairs inherited from the extra dimensions.

... We will use the Hamilton-Jacobi equation to calculate. In this method, Carter et al. introduced a new integral constant and obtained analytical solutions of the geodesic equations by using variable separation approach [43,44]. Now, let's introduce the main process. ...

The successful observation of M87 supermassive black hole by the Black Hole Event Horizon Telescope(EHT) provides a very good opportunity to study the theory of gravity. In this work, we obtain the exact solution for the short hair black hole (BH) in the rotation situation, and calculate in detail how hairs affect the BH shadow. For the exact solution part, using the Newman-Janis algorithm, we generalize the spherically symmetric short-hair black hole metric to the rotation case (space-time lie element (2.25)). For the BH shadow part, we study two hairy BH models. In model 1, the properties of scalar hair are determined by the parameters $\alpha_{0}$ and $L$. In model 2, the scalar hair of the BH is short hair. In this model, the shape of the BH shadow is determined by scalar charge $Q_{m}$ and $k$. In general, various BH hairs have different effects on the shadows, such as non-monotonic properties and intersection phenomena mentioned in this work. Using these characteristics, it is possible to test the no-hair theorem in future EHT observations, so as to have a deeper understanding of the quantum effect of BHs. In future work, we will use numerical simulations to study the effects of various hairs on BHs and their observed properties.

We calculate the scalar self-force experienced by a scalar point-charge orbiting a Kerr black hole along rθ-resonant geodesics. We use the self-force to calculate the averaged rate of change of the charge’s orbital energy ⟨E˙⟩, angular momentum ⟨L˙z⟩, and Carter constant ⟨Q˙⟩, which together capture the leading-order adiabatic, secular evolution of the point-charge. Away from resonances, only the dissipative (time antisymmetric) components of the self-force contribute to ⟨E˙⟩, ⟨L˙z⟩, and ⟨Q˙⟩. We demonstrate, using a new numerical code, that during rθ resonances conservative (time symmetric) scalar perturbations also contribute to ⟨Q˙⟩ and, thus, help drive the adiabatic evolution of the orbit. Furthermore, we observe that the relative impact of these conservative contributions to ⟨Q˙⟩ is particularly strong for eccentric 2∶3 resonances. These results provide the first conclusive numerical evidence that conservative scalar perturbations of Kerr spacetime are nonintegrable during rθ resonances.

Orbits of Finslerian Schwarzschild black hole have been investigated in this paper. The Finslerian Schwarzschild black hole is a warp product spacetime where its two dimensional subspace possesses constant Finslerian curvature. Due to this special geometrical structure, four constants of motion have been obtained from geodesic equations of Finslerian Schwarzschild spacetime. Finslerian parameter ε which describes deviation between Finslerian Schwarzschild black hole and Schwarzschild black hole affects both orbital precession and orbital plane precession. Orbit of Finslerian Schwarzschild black hole will ergodically fill a toruslike region. It is shown that the orbital precession of Finslerian Schwarzschild black hole is a constant if we calculate the two nearby perihelion of orbit by arc length of two dimensional subspace of Finslerian Schwarzschild black hole. However, present astronomical observations calculate the two nearby perihelion of orbit by arc length of Riemannian 2-sphere. Under this viewpoint, the orbital precession of Finslerian Schwarzschild black hole will depend on the Finslerian parameter and orbital elements. Observations of orbits of several stars around Sagittarius A* by GRAVITY collaboration provide an approach to falsify Finslerian Schwarzschild black hole. We have used data of orbital precession of the star S2 as an anchor to constrain the Finslerian parameter. The value of ε is range from −1.42×10−4 to 1.66×10−4. Then, we used the constrained ε to give prediction of orbital precession of other stars around Sagittarius A*. Due to the parity symmetry violation, it is shown in numerical results that both the direction of motion and the sign of ε will affect the orbital precession.

The properties of light rays around compact objects surrounded by a plasma are affected by both strong gravitational fields described by a general-relativistic spacetime and by a dispersive and refractive medium, characterized by the density distribution of the plasma. We study these effects employing the relativistic Hamiltonian formalism under the assumption of stationarity and axisymmetry. The necessary and sufficient conditions on the metric and on the plasma frequency are formulated such that the rays can be analytically determined from a fully separated Hamilton–Jacobi equation. We demonstrate how these results allow us to analytically calculate the photon region and the shadow if they exist. Several specific examples are discussed in detail: the “hairy” Kerr black holes, the Hartle–Thorne spacetime metrics, the Melvin universe, and the Teo rotating traversable wormhole. In all of these cases, a plasma medium is present as well.

We use the Relativistic Precession Model (RPM) and quasi-periodic oscillation (QPO) observations from the Rossi X-ray Timing Explorer to derive constraints on the properties of the black holes that power these sources and to test General Relativity (GR) in the strong field regime. We build upon past techniques by using pairs of simultaneously measured QPOs, rather than triplets, and by including characteristic frequencies from the broad noise components of the power spectra in our fits. We find the inclusion of these broad noise components causes an overestimate in masses and underestimate in spins compared to values derived independently from optical spectra. We extend the underlying spacetime metric to constrain potential deviations from the predictions of GR for astrophysical black holes. To do this, we modify the RPM model to a Kerr-Newman-deSitter spacetime and model changes in the radial, ecliptic, and vertical frequencies. We compare our models with X-ray data of XTE J1550-564 and GRO J1655-40 using robust statistical techniques to constrain the parameters of the black holes and the deviations from GR. For both sources, using QPO and characteristic frequency data, we constrain particular deviations from GR to be less than one part per thousand.

The gravitation lensing features are different for gravities with the extra dimensions, hence, there is an opportunity to find the sign of extra dimension in the supermassive black hole. In new of then, we investigate the shadow cast by the higher-dimensional rotating black hole and demonstrate the higher-dimensional effect on rotating black hole shadow. The shape and size of the higher-dimensional black hole shadow is mainly characterized by its mass parameter (M), spin parameter (a) and spacetime dimensions (D). For the increasing values of the spin parameter a and spacetime dimensions D, the effective size of the black hole shadow decreases, and the shape gets more deformed. Our study includes the shadow observables in higher dimensions with the data of M87 black hole. We also calculate the energy emission rate for the higher-dimensional rotating black hole which increases with the spacetime dimensions D.

We study several classes of exterior and interior axially symmetric spacetimes, such as wormholes, accelerating black holes, and binary black hole systems, from the point of view of light surfaces related to the generators of Killing horizons. We show that light surfaces constitute a useful framework for the study of the more diverse axially symmetric geometries. In particular, we point out the existence of common properties of the light surfaces in different spacetimes. We introduce a deformation of the Kerr–Newman metric and apply the light surfaces framework to analyze several generalizations in a compact form. As particular examples, we analyze static and spinning wormhole solutions, black holes immersed in external (perfect fluid) dark matter, spacetimes with (Taub) NUT charge, acceleration, magnetic charge, and cosmological constant, binary Reissner–Nordström black holes, a solution of a (low-energy effective) heterotic string theory, and the $$(1+2)$$ ( 1 + 2 ) dimensional BTZ geometry.

We investigate the quasibound states of charged massive scalar fields in the Kerr–Newman black hole spacetime by using a new approach recently developed, which uses the polynomial conditions of the Heun functions. We calculate the resonant frequencies related to the spectrum of quasibound states, as well as its corresponding angular and radial wave eigenfunctions. We also analyze the instability of the system. These results are particularized to the cases of Schwarzschild and Kerr black holes. Additionally, we compare our analytical results with the numerical ones known in the literature. Finally, we apply the obtained results to compute the characteristic times of growth and decay of bosonic particles around a supermassive black hole situated at the center of the M87 galaxy.

We consider a pseudoscalar axionlike field coupled to a Chern-Simons gravitational anomaly term. The axion field backreacts on a rotating Kerr black hole background, resulting in modifications in the spacetime. In an attempt to determine potentially observable signatures, we study the angular momentum of the system of the modified Kerr-like black hole and the axionic matter outside the horizon of the black hole. As the strength of the coupling of the axion field to the Chern-Simons term is increasing, the requirement that the total angular momentum of the system remains constant forces the black hole angular momentum to decrease. There exists a critical value of this coupling beyond which the black hole starts to rotate in the opposite direction, with an increasing magnitude of its angular momentum. We interpret this effect as a consequence of the exchange of energy between the axionic matter and the gravitational anomaly, which is sourced by the rotating black hole.

We develop the 3-dimensional general relativistic radiative transfer code: CARTOON (Calculation code of Authentic Radiative Transfer based On phOton Number conservation in curved space-time) which is improved from the 2-dimensional code: ARTIST developed by Takahashi & Umemura (2017). In CARTOON, the frequency-integrated general relativistic radiative transfer equation is solved in a photon number-conserving manner, and the isotropic and coherent scattering in the zero angular momentum observers (ZAMO) frame and the fluid rest frame is incorporated. By calculating the average energy of photons, energy conservation of the radiation is also guaranteed. With the test calculations in 2-dimensional and 3-dimensional space, we have demonstrated that the wavefront propagation in black hole space-time can be correctly solved in CARTOON conserving photon numbers. The position of the wavefront coincides with the analytical solution and the number of photons remains constant until the wavefront reaches the event horizon. We also solve the radiative transfer equation on the geodesic reaching the observer's screen. The time variation of the intensity map on the observer's screen can be simultaneously and consistently calculated with the time variation of the radiation field around the black hole. In addition, the black hole shadow can be reproduced in moderately optically thin situations.

We explore how the internal structure of a test particle affects its equatorial stable circular orbits around the Kerr black hole with or without a cosmological constant. To this end, we first explicitly write equations of motion for a test particle in the pole-dipole-quadrupole approximation specifying the quadrupole momentum tensor to a spin-induced model. Then we calculate characteristic quantities – radius, angular momentum, energy, angular velocity, and impact parameter – for the particles on the stable circular orbits. Once the pole-dipole-quadrupole approximation is taken, we find that for a particle on an innermost stable circular orbit, all characteristic quantities, except the angular velocity, become greater relative to the pole-dipole case. In contrast, for a particle on an outermost stable circular orbit, which only exists in the case of the spacetime background being asymptotically de Sitter, it is the radius that becomes smaller while all other quantities become greater.

In this paper, we study the shadow of a 4D Einstein–Gauss–Bonnet black hole as photons couple to the Weyl tensor and find that the propagation of light depends on its polarization which leads to the existence of a double shadow. Then, we discuss the effect of the coupling parameter [Formula: see text], the polarization of light and the Gauss–Bonnet coupling constant [Formula: see text] on the shadow. Further, we explore the influence of the Gauss–Bonnet coupling constant [Formula: see text] on the quasinormal modes (QNMs) of massless scalar field and investigate the connection between the real part of QNMs in the eikonal limit and the shadow radius of black holes. We find that in the eikonal limit, the real part of QNMs is inversely proportional to the shadow radius under the case of the photons uncoupled to the Weyl tensor.

We develop the 3-dimensional general relativistic radiative transfer code: CARTOON (Calculation code of Authentic Radiative Transfer based On phOton Number conservation in curved space–time) which is improved from the 2-dimensional code: ARTIST developed by Takahashi & Umemura (2017). In CARTOON, the frequency-integrated general relativistic radiative transfer equation is solved in a photon number-conserving manner, and the isotropic and coherent scattering in the zero angular momentum observers (ZAMO) frame and the fluid rest frame is incorporated. By calculating the average energy of photons, energy conservation of the radiation is also guaranteed. With the test calculations in 2-dimensional and 3-dimensional space, we have demonstrated that the wavefront propagation in black hole space–time can be correctly solved in CARTOON conserving photon numbers. The position of the wavefront coincides with the analytical solution and the number of photons remains constant until the wavefront reaches the event horizon. We also solve the radiative transfer equation on the geodesic reaching the observer’s screen. The time variation of the intensity map on the observer’s screen can be simultaneously and consistently calculated with the time variation of the radiation field around the black hole. In addition, the black hole shadow can be reproduced in moderately optically thin situations.

In this paper, we investigate the optical properties by a charged rotating braneworld black hole, in the Randall–Sundrum scenario. We study the horizon, the photon region, the shadow of the black hole and other observables. The results show that in addition to the black hole spin parameter [Formula: see text], the other two parameters, tidal charge [Formula: see text] and electric charge [Formula: see text], are found to affect the horizon, the photon region and the black hole shadow. We also have researched different observables and found that with the increase of the three parameters, the area and perimeter of the black hole shadow decrease, while the deformation of the shadow intensifies. Finally, through the observations of the oblateness [Formula: see text], the circularity deviation [Formula: see text] and the angular diameter [Formula: see text] and the latest M87[Formula: see text] and SgrA[Formula: see text] black hole shadows, the three parameters are analyzed to turn out that those observations give different constraints due to the three parameters.

The paper discusses novel solutions of Einstein-Maxwell field equations by describing the static spherically symmetric isotropic matter distributions and electrifying a well-known uncharged model in general relativity. We start by selecting a Tolman−IV type potential for the gravitational potential gtt and a physically reasonable choice for the electric field intensity E in view to exploiting isotropic sources of matter which act as a basis for generating confined compact stars. These solutions are constructed to match interior space–time geometries with exterior Reissner-Nordström solution at the pressure-free boundary. The viability of the models is compared with observational data of some heavy pulsars coming from the Neutron Star Interior Composition Explorer (NICER). In particular, Her X-1, SAX J1808.4-3658, and 4U 1538-52 are considered. We stretch our findings by measuring the mass–radius relationship for our developed model, which reveals that the predicted radii via the M−R curve are lying in the range of R≤14–16km. Moreover, the generated model satisfies all of the required major physical properties for realistic compact stars. We can also say that this approach can play a vital role in encouraging the study of the electrification of well-known neutral stellar systems.

Motivated by (i) more and more interest in strong gravitational lensing by supermassive black holes due to the achievement of EHT observations, (ii) the ongoing popular topic on the possibility of Lorentz symmetry being broken in gravitation and its consequences, we will apply the Einstein bumblebee gravity with Lorentz violation (LV) to the study of strong gravitational lensing effect and the black hole shadow of slowly rotating Kerr-like black hole. In the strong gravitational lensing sector, we first calculate the deflection angel; then treating the slowly rotating Kerr-like black hole as supermassive M87* black hole, we evaluate the gravitational lensing observables (position, separation and magnification) and the time delays between the relativistic images. In the black hole shadow sector, we show the effect of LV parameter on the luminosity of the black hole shadow and photon sphere using the infalling spherical accretion. Moreover, we explore the dependence of various shadow observables on the LV parameter, and then give the possible constraint on the LV parameter by M87* black hole of EHT observations. We find that the LV parameter show significant effect on the strong gravitational lensing effect, the black hole shadow and photon sphere luminosity by accretion material. Our results point out that the future generations of EHT observation may help to distinguish the Einstein bumblebee gravity from GR, and also give a possible constrain on the LV parameter.

In this article, we considered the strong field approximation of nonlinear electrodynamics black hole and constructed its rotating counterpart by applying the modified Newman–Janis algorithm. The corresponding metric function in the strong field limit of the static black hole is identified in order to study the radius of photon sphere. However, the metric function for the rotating counterpart in the strong field limit is considered in order to study the horizon radius w.r.t spin parameter. We considered the Hamilton–Jacobi method to derive the geodesic equations for photon and constructed an orthonormal tetrad for deriving the equations for celestial coordinates in the observer’s sky. Shadows, distortions and energy emission rates are investigated and the results are compared for different values of nonlinear electrodynamics parameter, charge and spin. It is found that the presence of the nonlinear electrodynamics parameter affects the shape and size of the shadows and thus the distortion in the case of rotation. It is also found that the nonlinearity of electrodynamics diminishes the flatness in the shadow due to the effect of spin and other parameters.

This thesis is devoted to a new proof of the nonlinear stability of the slowly-rotating Kerr-de Sitter family as a black hole family of solutions to Einstein's vacuum equations with a positive cosmological constant. The proof uses harmonic coordinates to define a quasinormal spectrum for the linearized Einstein operator in the spirit of the regularity quasinormal spectrum constructed by Warnick on asymptotically anti-de Sitter spaces. Vectorfield methods are then used to produce a series of almost-Killing, redshift, and Morawetz energy estimates. These in turn imply resolvent estimates, giving both a Fredholm alternative for the linearized Einstein operator as well as the existence of a high-frequency spectral gap. Nonlinear stability is then a direct application of a classical bootstrap method. The proof in this thesis varies from the original proof by Hintz-Vasy, which relied on microlocal analysis of the meromorphic extension of the resolvent, constraint damping, and a Nash-Moser iteration to prove nonlinear stability. The difference in the nonlinear method in particular allows for a lower initial data regularity requirement.

Gravitational Faraday Rotation (GFR) is a frame-dragging effect induced by rotating massive objects, which is one of the important, yet studied characteristics of lensed gravitational waves (GWs). In this work, we calculate the GFR angle χ g of GWs in the weak deflection limit, assuming it is lensed by a Kerr black hole (BH). We find that the GFR effect changes the initial polarization state of the lensed GW. Compared with the Einstein deflection angle, the dominant term of the rotation angle χ g is a second-order correction to the polarization angle, which depends on the light-of-sight component of BH angular momentum. Such a rotation is tiny and degenerates with the initial polarization angle. In some critical cases, the GFR angle is close to the detection capability of the third-generation GW detector network, although the degeneracy has to be broken.

Extreme mass ratio inspirals (EMRIs)—systems with a compact object orbiting a much more massive (e.g., Galactic Center) black hole—are of interest as a new probe of the environments of galactic nuclei, and their waveforms are a precision test of the Kerr metric. This work focuses on the effects of an external perturbation due to a third body around an EMRI system. This perturbation will affect the orbit most significantly when the inner body crosses a resonance with the outer body, and results in a change of the conserved quantities (energy, angular momentum, and Carter constant) or equivalently of the actions, which results in a subsequent phase shift of the waveform that builds up over time. We present a general method for calculating the changes in action during a resonance crossing, valid for generic orbits in the Kerr spacetime. We show that these changes are related to the gravitational waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl scalar ψ4 at the horizon and at ∞) at the frequency corresponding to the resonance. This allows us to compute changes in the action variables for each body, without directly computing the explicit metric perturbations, and therefore we can carry out the computation by calling an existing black hole perturbation theory code. We show that our calculation can probe resonant interactions in both the static and dynamical limit. We plan to use this technique for future investigations of third-body effects in EMRIs and their potential impact on waveforms for LISA.

The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zero- G limit (z GKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the z GKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.

We study the null and timelike geodesics of light and the neutral particles, respectively, in the exterior of Kerr-Newman black holes. The geodesic equations are known to be written as a set of first-order differential equations in Mino time from which the angular and radial potentials can be defined. We classify the roots for both potentials, and mainly focus on those of the radial potential with an emphasis on the effect from the charge of the black holes. We then obtain the solutions of the trajectories in terms of the elliptical integrals and the Jacobian elliptic functions for both null and timelike geodesics, which are manifestly real functions of the Mino time that the initial conditions can be explicitly specified. We also describe the details of how to reduce those solutions into the cases of the spherical orbits. The effect of the black hole’s charge decreases the radii of the spherical motion of the light and the particle for both direct and retrograde motions. In particular, we focus on the light/particle boomerang of the spherical orbits due to the frame dragging from the back hole’s spin with the effect from the charge of the black hole. To sustain the change of the azimuthal angle of the light rays, say for example Δϕ=π during the whole trip, the presence of the black hole’s charge decreases the radius of the orbit and consequently reduces the needed values of the black hole’s spin. As for the particle boomerang, the particle’s inertia renders smaller change of the angle Δϕ as compared with the light boomerang. Moreover, the black hole’s charge also results in the smaller angle change Δϕ of the particle than that in the Kerr case. The implications of the obtained results to observations are discussed.

Algebraically special solutions of Einstein's empty-space field ; equations that are characterized by the existence of a geodesic and shear-free ; ray congruence are considered. A class of solutions is presented for which the ; congruence is diverging and is not necessarily hypersurface orthogonal. (C.E.S.);

With the introduction of multiply connected topologies into physics, a question of causality arises. There are alternative routes between two points in a multiply connected space. Therefore, one may ask if a signal traveling at the speed of light along one route could be outpaced by a signal which has traveled a much shorter path through a handle or "wormhole." This paper examines one such situation and shows that in this example causality is preserved. It proves essential in the analysis to distinguish between those regions of space-time which are catastrophic and those which are not. A catastrophic region is composed of catastrophic points. A catastrophic point in space-time is so located with respect to eventual singularities in the intrinsic geometry that every time-like geodesic through it necessarily runs into a region of infinite curvature at some time in the future-or was born out of a region of infinite curvature at some time in the past-or both. If a classical analysis of nature were possible-which it is not-then it would be natural to postulate that laboratory physics is carried out in noncatastrophic regions of space-time. Two such regions are shown to exist in the example considered in the paper. It is shown that no signal can ever be sent from one to the other. The key point in preventing any violation of causality is simple: The (Schwarzschild) throat of the wormhole pinches off in a finite time and traps the signal in a region of infinite curvature. This investigation also displays some of the unusual geometric features of the Schwarzschild solution of Einstein's equations for a spherically symmetrical center of attraction. Radial spacelike geodesics passing through the throat are calculated and it is shown that there exist regions of space-time unreachable by any radial geodesics that issue from a given point. Also, there exist points in space-time from which light signals can never be received no matter how long one waits.

It is shown that singularities of space-time are inevitable if the Einstein equations hold, if matter has normal properties and if the universe satisfies certain reasonable global conditions. The singularities would be in the past and would, in principle, be observable. Observation to determine whether such singularities actually occurred would provide a powerful test of the Einstein equations in strong fields. The singularity would not necessarily constitute a beginning of the universe.

The following theorem is established. Among all static, asymptotically flat vacuum space-times with closed simply connected equipotential surfaces ${g}_{00}=\mathrm{constant}$, the Schwarzschild solution is the only one which has a nonsingular infinite-red-shift surface ${g}_{00}=0$. Thus there exists no static asymmetric perturbation of the Schwarzschild manifold due to internal sources (e.g., a quadrupole moment) which will preserve a regular event horizon. Possible implications of this result for asymmetric gravitational collapse are briefly discussed.

It is shown that singularities of space-time are inevitable if the Einstein equations hold, if matter has normal properties and if the universe satisfies certain reasonable global conditions. The singularities would be in the past and would, in principle, be observable. Observation to determine whether such singularities actually occurred would provide a powerful test of the Einstein equations in strong fields. The singularity would not necessarily constitute a beginning of the universe.

DOI:https://doi.org/10.1103/PhysRevLett.14.57

Kruskal's transformation of the Schwarzschild metric is generalized to apply to the stationary, axially symmetric vacuum solution of Kerr, and is used to construct a maximal analytic extension of the latter. In the low angular momentum case, a2 < m2, this extension consists of an infinite sequence Einstein‐Rosen bridges joined in time by successive pairs of horizons. The number of distinct asymptotically flat sheets in the extended space can be reduced to four by making suitable identifications. Several properties of the Kerr metric, including the behavior of geodesics lying in the equatorial plane, are examined in some detail. Completeness is demonstrated explicitly for a special class of geodesics, and inferred for all those that do not strike the ring singularity.

The 2-dimensional metric on the symmetry axis of the Kerr solution is examined and it is shown that in the form usually given it is incomplete when ${a}^{2}<~{m}^{2}$. The method developed by Kruskal for completing the Schwarzschild solution is adapted to the distinct cases ${a}^{2}<{m}^{2}$ and ${a}^{2}={m}^{2}$. In each case a singularity-free metric is obtained which is periodic with respect to a timelike coordinate, and which is shown to be a complete analytic extension. The generalization to the full 4-dimensional Kerr solution is discussed, and finally the questions of uniqueness and causality are considered.

It is shown that by means of a complex coordinate transformation performed on the monopole or Schwarzschild metric one obtains a new metric (first discovered by Kerr). It has been suggested that this metric be interpreted as that arising from a spinning particle. We wish to suggest a more complicated interpretation, namely that the metric has certain characteristics that correspond to a ring of mass that is rotating about its axis of symmetry. The argument for this interpretation comes from three separate places: (1) the metric appears to have the appropriate multipole structure when analyzed in the manner discussed in the previous paper, (2) in a covariantly defined flat space associated with the metric, the Riemann tensor has a circular singularity, (3) there exists a closely analogous solution of Maxwell's equations that has characteristics of a field due to a rotating ring of charge.

A transformation is presented to remove coordinate ("pseudo") singularities from metrics of a certain class, a special case of which is the transformation of Kruskal, extending the Schwarzschild metric beyond its pseudosingularity. The transformation is applied to the Reissner-Nordström metric, which describes a concentration of charge and mass in general relativity. On an initial surface this metric shows the same general behavior as the Schwarzschild metric, describing a "wormhole," or bridge, between two asymptotically flat spaces, but with electric flux flowing through the wormhole. It is found that the region of minimum radius, the so-called "throat" of the wormhole, begins to contract, but reaches a minimum and re-expands after a finite proper time, rather than pinching off as in the Schwarzschild-Kruskal case: the raduis of the throat pulsates periodically in time, "cushioned" by Maxwell pressure of the electric field through the throat. The motion of charged particles in this metric is investigated, and it is shown that no particle can hit the geometric singularity at r=0; (1) quite in general, provided only that the mass of the test particle exceeds the value associated in general relativity with its charge, and (2) in particular when the test particle has no charge at all, but (3) such collisions are not avoided when the throat itself is not endowed with any electric flux.

The questions about singularities that remain to be answered are discussed. It is shown, without any assumption about causality, that there are fully general solutions which evolve from a non-singular state to an inevitable singularity. An observationally testable condition is given which would imply the existence of a singularity if a reasonable assumption about causality were made. This condition would also be satisfied in an approximately spherical collapsing star and so would enable one to prove the occurrence of a singularity in such circumstances without assuming that space-time admits a Cauchy surface. If the assumption about causality held, the singularity could not be of the Misner type but would presumably involve infinite curvature. This would probably indicate that the Einstein theory broke down but only in very strong fields.

A number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.

Kerr's metric is often said to describe the geometry exterior to a body whose mass and rotation are measured by Kerr's parameters m and a, respectively, even though no interior solution is known. In this paper we give an interior solution valid in the limit when the rotation parameter a is sufficiently small so that terms of higher power than the first are negligible, but the mass parameter m is allowed to be large. This is accomplished by bringing Kerr's exterior metric into the form of the metric for a slowly rotating mass shell. Also, the connection is found between Kerr's parameters and the physical parameters characterizing the rotating body.

A new solution of the Einstein-Maxwell equations is presented. This solution has certain characteristics that correspond to a rotating ring of mass and charge.

The field equations governing the gravitational field of a uniformly
rotating axially symmetric source are reformulated in terms of a simple
variational principle. The new formalism affords a concise unified
derivation of the solutions discovered by Weyl and Papapetrou, and
permits a simple derivation of the Kerr metric in terms of prolate
spheroidal coordinates. More complex solutions are identified by
applying perturbation theory.

In what follows we shall derive some properties of the gravitational field of an isolated, axially symmetric, uniformly rotating mass of perfect fluid in a steady state, according to the general theory of relativity. Several exact models describing rotating fluids are known in Newtonian mechanics, the Maclaurin and Jacobi ellipsoids ((6)) being perhaps the most interesting. In general relativity, no such exact solution is known in its entirety, although Kerr ((4)) has exhibited a certain vacuum solution possessing features that one might expect of a space-time exterior to some rotating body. Throughout this paper we shall have Kerr's solution in mind. The question that we shall keep before us is whether a perfect fluid interior can be matched to any given exterior field. Our main results exhibit the class of all possible fluid boundaries, given the exterior field, and some relations between the pressure, density, 4-velocity, and interior metric tensor.(Received August 12 1964)

The preceding paper ((1)) dealt with some general properties of the gravitational field of a rotating fluid mass. An interesting example of a vacuum solution that might be the exterior field of some rotating body was recently found by Kerr ((4)). It was natural to apply the preceding theory to the Kerr solution. This paper deals with other aspects of that solution, particularly the behaviour of its bounded geodesics (planetary orbits). It would seem desirable to know what sort of rotating body could be a source of the Kerr field. It will appear that one of the parameters in Kerr's solution can plausibly be related to the angular momentum per unit mass of a uniformly rotating sphere, the other parameter being a measure of the mass of the sphere.(Received August 12 1964)

Es wird eine neue exakte rotationssymmetrische Lösung der Feldgleichungen der allgemeinen Relativitätstheorie abgeleitet und deren physikalische Bedeutung untersucht.

A method originally developed for the Kerr solution is adapted in order to obtain the complete analytic continuation of the Reissner-Nordström metric in the special case e2 = m2. This case is the exterior field of a static spherical charged dust cloud.

- S. C. Darwin