Null Trajectories Research Articles

Exploring the possibility of testing the no-hair theorem with Minkowski-deformed regular hairy black holes via photon rings

In this paper, we investigate the optical appearance of a regular static spherically symmetric hairy black hole within the context of Minkowski deformation governed by the parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. This black hole describes a hairy black hole with geometric deformations in the radial and temporal metric components, parameterized by α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. The optical appearance of the black hole, illuminated by a static thin accretion disk in three toy emission function models, exhibits distinctive shadows and photon rings. Our findings reveal that for static spherically symmetric hairy Minkowski-deformed regular black holes, the event horizon radius rh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_h$$\\end{document}, photon sphere radius rph\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_{ph}$$\\end{document}, critical impact parameter bph\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b_{ph}$$\\end{document}, and innermost stable circular orbit radius risco\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_{isco}$$\\end{document} all have a positive correlation with α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. These parameters affect the null geodesic trajectories, shadows, and the optical appearance of the photon rings illuminated by a static thin accretion disk around the black hole. Utilizing this, we obtained the optical appearance of this black hole when observed in the forward direction of the optical and geometric thin accretion disk models. In all three models, the radius of the photon ring images also shows a positive correlation with the parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Therefore, under the static spherically symmetric hairy Minkowski-deformed regular black hole, there is no degeneracy of photon rings dependent on the parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Theoretically, this allows for the distinction of different spacetime metrics of hairy Minkowski-deformed regular black holes, providing a potential method for testing the no-hair theorem through future observations of black hole photon rings.

Abnormal light propagation and the underdetermination of theory by evidence in astrophysics

We investigate the propagation of certain non-plane wave solutions to Maxwell’s equations in both flat and curved spacetimes. We find that such solutions (or rather parts of them) exhibit accelerative behaviour, and in particular do not propagate on straight lines. Having established these results, we then turn to their conceptual significance—which, in brief, we take to be the following: (i) one should not assume that the part of electromagnetic waves from outer space that is subject to detection is localised onto null trajectories; therefore (ii) astrophysicists and cosmologists should at least be wary about making such assumptions in their inferences from obtained data, for to do so may lead to incorrect inferences regarding the nature of our universe.

If you want to cross singularity, wrap it!

In two-dimensional string theory, a probe D0-brane does not see the black hole singularity due to a cancellation between its metric coupling and the dilaton coupling. A similar mechanism may work in the Schwarzschild black hole in large D dimensions by considering a suitable wrapped membrane. From the asymptotic observer, the wrapped membrane looks disappearing into nothing while the continuation of the time-like trajectory beyond the singularity suggests that it would reappear as an instantaneous space-like string stretching from the singularity. A null trajectory can be extended to a null trajectory beyond the singularity. Not only the effective particle but an effective string from the wrapped membrane can exhibit the same feature.

Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions

The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.

Entanglement harvesting from conformal vacuums between two Unruh-DeWitt detectors moving along null paths

It is well-known that the (1 + 1) dimensional Schwarzschild and spatially flat FLRW spacetimes are conformally flat. This work examines entanglement harvesting from the conformal field vacuums in these spacetimes between two Unruh-DeWitt detectors, moving along outgoing null trajectories. In (1 + 1) dimensional Schwarzschild spacetime, we considered the Boulware and Unruh vacuums for our investigations. In this analysis, one observes that while entanglement harvesting is possible in (1+1) dimensional Schwarzschild and (1 + 3) dimensional de Sitter spacetimes, it is not possible in the (1 + 1) dimensional de Sitter background for the same set of parameters when the detectors move along the same outgoing null trajectory. The qualitative results from the Boulware and the Unruh vacuums are alike. Furthermore, we observed that the concurrence depends on the distance d between the two null paths of the detectors periodically, and depending on the parameter values, there could be entanglement harvesting shadow points or regions. We also observe that the mutual information does not depend on d in (1 + 1) dimensional Schwarzschild and de Sitter spacetimes but periodically depends on it in (1 + 3) dimensional de Sitter background. We also provide elucidation on the origin of the harvested entanglement.

Parametrized black holes: scattering investigation

We study the scattering of light-like geodesics and massless scalar waves by a static Konoplya–Zhidenko black hole, considering the case that the parametrized black hole solution contains a single deformation parameter. By performing a geodesic analysis, we compute the classical differential scattering cross section and probe the influence of the deformation parameter on null trajectories. Moreover, we investigate the propagation of a massless scalar field in the vicinity of the static Konoplya–Zhidenko black hole and use the plane waves formalism to compute the differential scattering cross section. We confront our numerical results in the backward direction with the glory approximation, finding excellent agreement. We compare the results for the deformed black hole with the Schwarzschild case, finding that the additional parameter has an important role in the behavior of the scattering process for moderate-to-high scattering angles.

Null geodesics and shadow of 4D Einstein–Gauss–Bonnet black holes surrounded by quintessence

We study the properties of the null geodesics of four-dimensional Einstein–Gauss–Bonnet (4D EGB) black holes surrounded by quintessence matter, recently derived by Shah et al. [Phys. Lett. B 818 (2021) 136383]. Due to the quintessence correction, we discuss the radial and nonradial geodesics. For the case of nonradial geodesics we obtain the effective potential, photon sphere and impact parameter associated to the null geodesics in detail. We analyze the different possible orbits of massless particles (i.e. photons) such as unstable circular orbits and unbounded orbits and investigate the role of the Gauss–Bonnet coupling [Formula: see text] and the quintessence parameter q on the null geodesic trajectories. Moreover, we study the effects of the model parameters on the shadow cast by such black holes. The results are compared to that of Schwarzschild black hole with and without quintessence matter and 4D EGB black holes without quintessence. As a physical application of null geodesics, the deflection angle is calculated and the effect of the quintessence matter on it is investigated.

Möbius mirrors

An accelerating boundary (mirror) acts as a horizon and black hole analog, radiating energy with some particle spectrum. We demonstrate that a Möbius transformation on the null coordinate advanced time mirror trajectory uniquely keeps invariant not only the energy flux but the particle spectrum. We clarify how the geometric entanglement entropy is also invariant. The transform allows generation of families of dynamically distinct trajectories, including -symmetric ones, mapping from the eternally thermal mirror to the de Sitter horizon, and different boundary motions corresponding to Kerr or Schwarzschild black holes.

Shock wave quantum memory in shocked detector

Usual uniformly accelerated frame, in Dray–’t Hooft spacetime, does not see the any quantum imprint on Unruh effect due to localized shock wave in Minkowski spacetime. Here, we argue that such non-appearance of quantum memory is specific to those particular observers which do not incorporate the presence of wave in their trajectory. In fact, a detector, associated with a frame which is affected by the shock, cannot be trivial in terms of its response. We explicitly show that the later type of frame detects particle in the shock wave Minkowski vacuum which is the effect of shock. Therefore, this quantum memory is very special to specific class of observers as far as Dray–’t Hooft spacetime is concerned. We analyze for a null like trajectory along which the detector is moving.

Page curves for a family of exactly solvable evaporating black holes

We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space. The islands and Page curves of eternal, evaporating and bath-removed black holes are investigated. The different theories in this parameter class are identified through field redefinitions which leave the island invariant. The Page transition is found to occur at the first a third of the black hole life time in the evaporating case for this family of solutions. In addition, we consider gluing the equilibrium black hole and the evaporating one along a null trajectory and study the effect of gluing on the islands and Page curves. In the glued space, the island jumps across two different geometries at a certain retarded time. As a result, the Page transition is stretched and split into two separate ones -- the first transition happens when the net entropy generation stops and the second one occurs as the early radiation effectively starts to become purified. Finally, we discuss the issues concerning the inconsistent rates of purification and the paradox related to the state of the radiation.

Mass parameter and the bounds on redshifts and blueshifts of photons emitted from geodesic particle orbiting in the vicinity of regular black holes

We obtain the mass parameter for a class of static and spherically symmetric regular black holes (BHs) (namely Bardeen, Hayward and Ay\'{o}n-Beato-Garc\'{i}a BHs) which are solutions of Einstein's field equations coupled to nonlinear electrodynamics (NED) in terms of redshifts and blueshifts of photons emitted by geodesic particles (for instance, stars) orbiting around these BHs. The motion of photons is not governed by null geodesics for these type of spacetime geometries which reflects the direct effects of the electrodynamic nonlinearities in the photon motion; hence, an effective geometry needs to be constructed to study null trajectories [Phys. Rev. D61, 045001 (2000)]. To achieve the above, we first study the constants of motion from the analysis of the motion of both geodesic particles moving in stable circular orbits and photons ejected from them and reaching a distant observer (or detector) in the equatorial plane for the above mentioned regular BHs. The relationship between red/blueshifts of photons and the regular BH observables is presented. We also numerically find the bounds on the photon shifts for these regular BH cases.

Null Cartan curve variations in 3D semi-Riemannian manifold

The aim of this study is to investigate the variations of the Bishop frame curvatures for null Cartan curves in semi-Riemannian manifolds. The Killing equations in terms of the variations of the Bishop curvatures along the null Cartan curve is especially derived. Killing equations are used to interpret the movement of the charged particles in a magnetic field. The charged particle motion along a null Cartan curve through the Killing equations is examined as an application in the study. It is found that the charged particle traces a trajectory in the form of the null cubic during its movement in the Killing magnetic vector field. According to the results obtained, an example of the null Cartan magnetic trajectories is presented.

Near-horizon local instability and quantum thermality

We revisit our previous proposed conjecture -- horizon creates a local instability which acts as the source of quantum temperature of black hole. It is found that a chargesless massless particle moving along the null trajectory in Eddington-Finkelstein (EF) coordinates feels instability in the vicinity of the horizon. Such instability is observer independent for this particle motion. Moreover, an observer associated to EF coordinates finds the local Hamiltonian as $xp$ where $p$ is the canonical momentum corresponding the coordinate $x$. Finally, using this Hamiltonian we notice that at the quantum level this class of observers feel the horizon as thermal object with temperature is given by the Hawking expression. We provide this by using various techniques in quantum mechanics and thereby bolstered our earlier claim -- the automatic local instability can be a mechanism for emerging horizon as a thermal object. In this process, the present analysis provides another set of coordinates (namely EF frame), in addition to our earlier Painleve ones, in which the null trajectory of the massless particle is governed by $xp$ type Hamiltonian in near the horizon regime.

Fuzzy Fixed-Time Learning Control with Saturated Input, Nonlinear Switching Surface and Switching Gain to Achieve Null Tracking Error

A class of generalized nonlinear dynamic systems is first approximated by $N$ fuzzy-based linear subsystems using the identification of input–output data or the linearizing system with respect to suitable operating points. To obtain null trajectory tracking error in fixed time, a fuzzy fixed-time control (FFTC) with nonlinear switching surface and switching gain is first designed. It can be said that the FFTC is based on a class of passive and distributive models with uncertainties. To compensate enormous uncertainties, a fuzzy fixed-time learning control (FFTLC) by learning two unknown coefficients for the upper bound of uncertainties in each subsystem is designed. As compared with radial basis function neural network, the computational complexity for the compensated uncertainties is much simple. It can be said that an FFTLC is based on a class of online active and distributive uncertain models. Due to the fixed-time control design, the transients often occur, particularly for the larger uncertainties or initial tracking error. Hence, the saturated input of nonlinear dynamic system is addressed and online compensated. Finally, the compared simulations and application to two-link robot manipulator confirm the effectiveness, robustness, and less computation as compared with previous studies.

Transition to light-like trajectories in thin shell dynamics

It was recently shown that a massive thin shell that is sandwiched between a flat interior and an exterior geometry given by the outgoing Vaidya metric becomes null in a finite proper time. We investigate this transition for a general spherically-symmetric metric outside the shell and find that it occurs generically. Once the shell is null its persistence on a null trajectory can be ensured by several mechanisms that we describe. Using the outgoing Vaidya metric as an example we show that if a dust shell acquires surface pressure on its transition to a null trajectory it can evade the Schwarzschild radius through its collapse. Alternatively, the pressureless collapse may continue if the exterior geometry acquires a more general form.

Developmental trajectories of callous-unemotional traits, anxiety and oppositionality in 3–7 year-old children in the general population

There is increasing interest in describing different variants and subtypes that characterize the heterogeneity of behavior problems with the aim of supporting early detection and prevention, as well as facilitating research into etiological differences. This work examines the course of co-occurrence of callous-unemotional traits (CU), anxiety (ANX) and oppositionality (ODD-s) levels in a longitudinal community sample. A sample of 622 3-year-old preschoolers, followed up until age 7, was assessed annually with dimensional measures of callous-unemotional traits, oppositional defiant disorder and anxiety and related psychological constructs answered by parents, teachers and performed by children. Growth-Mixture-Modeling yielded six trajectories (null 58.9%, ANX-increasing 4.9%, CU+ANX+ODD 2.4%, CU+ODD-decreasing 8.1%, ODD-increasing 16.4%, and ANX-decreasing 9.2%) that represent the variants previously described with older subjects. The specifier ODD “with limited prosocial emotion” (CU+ODD decreasing and increasing) showed deficits in executive functioning, attention, aggressive behavior and social cognition in comparison with null trajectory. The secondary variant showed a more severe clinical picture and presented more difficulties in executive functioning, worse environmental characteristics, and worse outcomes at age 7. It is possible to identify the heterogeneity of disruptive behavior problems from preschool age. The identification of homogeneous groups in this category of disorders may help to design more suitable treatments with specific components for specific difficulties, and to progress in the etiological research of each class.

Exact gravitational lensing in conformal gravity and Schwarzschild–de Sitter spacetime

An exact solution is obtained for the gravitational bending of light in static, spherically symmetric metrics which includes the Schwarzschild-de Sitter spacetime and also the Mannheim-Kazanas metric of conformal Weyl gravity. From the exact solution, we obtain a small bending-angle-approximation for a lens system where the source, lens and observer are co-aligned. This expansion improves previous calculations where we systematically avoid parameter ranges that correspond to non-existent null trajectories. The linear coefficient $\gamma$ characteristic to conformal gravity is shown to contribute enhanced deflection compared to the angle predicted by General Relativity for small $\gamma$.

On a nonlinear gravitational wave. Geodesics

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and all pressures are finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, the $z$-axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

Images and spectral properties of two-component advective flows around black holes: effects of photon bending

Two component advective flow (TCAF) successfully explains spectral and timing properties of black hole candidates. We study the nature of photon trajectories in the vicinity of a Schwarzschild black hole and incorporate this in predicting images of TCAF with a black hole at the Centre. We also compute the emitted spectra. We employ a Monte-Carlo simulation technique to achieve our goal. For accurate prediction of the image and the spectra, null trajectories are generated without constraining the motion to any specific plane. Red shift, bolometric flux and corresponding temperature have been calculated with appropriate relativistic consideration. The centrifugal barrier dominated boundary layer or CENBOL near the inner region of the disk which acts as the Compton cloud is appropriately modelled as a thick accretion disk in Schwarzschild geometry for the purpose of imaging and computing spectra. The variations of spectra and image with physical parameters such as the accretion rate ($\dot{m}_d$) and inclination angle are presented. We show that the gravitational bending effects of photons do change the spectral shape to some extent.

On a particular Morris–Thorne wormhole

The properties of a particular Misner–Thorne wormhole (WH) are investigated. The ‘exotic stress–energy’ needed to maintain the WH open corresponds to a massless scalar field whose Lagrangean density contains a negative kinetic term. While the Komar energy of the spacetime is vanishing due to the negative energy density and radial pressure, the ADM energy is (minus) the Planck energy. The timelike geodesics are hyperbolae and any static observer is inertial. The null radial trajectories are also hyperbolae and Lorentz invariant as Coleman–de Luccia expanding bubble or Ipser–Sikivie domain wall. Using a different equation of state for the fluid on the dynamic WH throat of Redmount and Suen, we reached an equation of motion for the throat (a hyperbola) that leads to a negative surface energy density and the throat expands with the same acceleration as the Ipser–Sikivie domain wall.