Analysis Of Geodesics Research Articles

Gravitational lensing in holonomy corrected spherically symmetric black holes with phantom global monopoles

In this paper, we address a theoretical investigation of the gravitational lensing phenomenon within the space-time framework of a holonomy-corrected spherically symmetric black hole (BH), incorporating both ordinary and phantom global monopoles. Our focus lies on the analysis of null geodesics within this BH background, examining the influence of ordinary and phantom global monopoles on the effective potential of null geodesics of the system. Afterwards, we derive analytical expressions for the deflection angle of photon light, considering weak field limit. The obtain expressions are presented up to the second order of the Loop Quantum Gravity parameter, enabling a thorough examination of the impact of ordinary and phantom global monopoles on the deflection angle.

Singular space-times with bounded algebraic curvature scalars

We show that the absence of unbounded algebraic curvature invariants constructed from polynomials of the Riemann tensor cannot guarantee the absence of strong singularities. As a consequence, it is not sufficient to rely solely on the analysis of such scalars to assess the regularity of a given space-time. This conclusion follows from the analysis of incomplete geodesics within the internal region of asymmetric wormholes supported by scalar matter which arise in two distinct metric-affine gravity theories. These wormholes have bounded algebraic curvature scalars everywhere, which highlights that their finiteness does not prevent the emergence of pathologies (singularities) in the geodesic structure of space-time. By analyzing the tidal forces in the internal wormhole region, we find that the angular components are unbounded along incomplete radial time-like geodesics. The strength of the singularity is determined by the evolution of Jacobi fields along such geodesics, finding that it is of strong type, as volume elements are torn apart as the singularity is approached. Lastly, and for completeness, we consider the wormhole of the quadratic Palatini theory and present an analysis of the tidal forces in the entire space-time.

Possible Wormholes in a Friedmann Universe

We study the properties of evolving wormholes able to exist in a closed Friedmann dust-filled universe and described by a particular branch of the well-known Lemaître–Tolman–Bondi solution to the Einstein equations and its generalization with a nonzero cosmological constant and an electromagnetic field. Most of the results are obtained with pure dust solutions. It is shown, in particular, that the lifetime of wormhole throats is much shorter than that of the whole wormhole region in the universe (which coincides with the lifetime of the universe as a whole), and that the density of matter near the boundary of the wormhole region is a few times smaller than the mean density of matter in the universe. Explicit examples of wormhole solutions and the corresponding numerical estimates are presented. The traversability of the wormhole under study is shown by a numerical analysis of radial null geodesics.

Gravitational-wave imprints of compact and galactic-scale environments in extreme-mass-ratio binaries

Circumambient and galactic-scale environments are intermittently present around black holes that reside in active galactic nuclei. As supermassive black holes impart energy on their host galaxy, so the galactic environment affects the dynamics of solar-mass objects around black holes and the gravitational waves emitted from non-vacuum asymmetric binaries. Only recently an exact general-relativistic solution has been found that describes a Schwarzschild black hole immersed in a dark matter halo of the Hernquist type. We perform an extensive analysis of generic geodesics delving in such non-vacuum spacetimes and compare our results with those obtained in Schwarzschild, as well as calculate their gravitational-wave emission. Our findings indicate that the radial and polar oscillation frequency ratios descend deeper into the strong gravity region as the compactness of the halo increases. This translates to a redshift of non-vacuum geodesics and their resulting waveforms with respect to the vacuum ones. We calculate the overlap between waveforms resulting from Schwarzschild and non-vacuum geometries and find that it decreases as the halo compactness grows, meaning that dark matter environments should be distinguishable by space-borne detectors. For compact environments, we find that the apsidal precession is strongly affected due to the gravitational pull of dark matter; the orbit's axis can rotate in the opposite direction as that of the orbital motion, leading to a retrograde precession drift that depends on the halo mass, as opposed to the typical prograde precession transpiring in galactic-scale environments. Gravitational waves in retrograde-to-prograde alterations demonstrate transient frequency phenomena around critical non-precessing turning points, thus they may serve as `smoking guns' for the presence of compact dark matter environments around supermassive black holes.

Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole

We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.

Schwarzschild–Finsler–Randers spacetime: geodesics, dynamical analysis and deflection angle

In this work, we extend the study of Schwarzschi ld–Finsler–Randers (SFR) spacetime previously investigated by a subset of the present authors (Triantafyllopoulos et al. in Eur Phys J C 80(12):1200, 2020; Kapsabelis et al. in Eur Phys J C 81(11):990, 2021). We will examine the dynamical analysis of geodesics which provides the derivation of the energy and the angular momentum of a particle moving along a geodesic of SFR spacetime. This study allows us to compare our model with the corresponding of general relativity (GR). In addition, the effective potential of SFR model is examined and it is compared with the effective potential of GR. The phase portraits generated by these effective potentials are also compared. Finally we deal with the derivation of the deflection angle of the SFR spacetime and we find that there is a small perturbation from the deflection angle of GR. We also derive an interesting relation between the deflection angles of the SFR model and the corresponding result in the work of Shapiro et al. (Phys Rev Lett 92(12):121101, 2004). These small differences are attributed to the anisotropic metric structure of the model and especially to a Randers term which provides a small deviation from GR.

Stability of circular geodesics in equatorial plane of Kerr spacetime

We analyze the stability of circular geodesics for timelike as well as null geodesics of the Kerr BH spacetime with rotation parameter on the equatorial plane by Lyapunov stability analysis. Also, we verify the results of stability by presenting the phase portrait for both timelike and null geodesics. Further, by reviewing the Kosambi-Cartan-Chern (KCC) theory, we analyze the Jacobi stability for Kerr spacetime and present a comparative study of the methods used for stability analysis of geodesics.

Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding

We endow the 8/3-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere S2. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric dQ on a countable dense subset of S2. Here, we show that dQ a.s. extends uniquely and continuously to a metric d‾Q on all of S2. Letting d denote the Euclidean metric on S2, we show that the identity map between (S2,d) and (S2,d‾Q) is a.s. Hölder continuous in both directions. We establish several other properties of (S2,d‾Q), culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of (S2,d‾Q) geodesics.

Stability analysis of geodesics and quasinormal modes of a dual stringy black hole via Lyapunov exponents

We investigate the stability of both timelike as well as null circular geodesics in the vicinity of a dual (3+1) dimensional stringy black hole (BH) spacetime by using an excellent tool so-called Lyapunov exponent. The proper time (\(\tau \)) Lyapunov exponent (\(\lambda _{p}\)) and coordinate time (t) Lyapunov exponent (\(\lambda _{c}\)) are explicitly derived to analyze the stability of equatorial circular geodesics for the stringy BH spacetime with electric charge parameter (\(\alpha \)) and magnetic charge parameter (Q). By computing these exponents for both the cases of BH spacetime, it is observed that the coordinate time Lyapunov exponent of magnetically charged stringy BH for both timelike and null geodesics are independent of magnetic charge parameter (Q). The variation of the ratio of Lyapunov exponents with radius of timelike circular orbits (\(r_{0}/M\)) for both the cases of stringy BH are presented. The behavior of instability exponent for null circular geodesics with respect to charge parameters (\(\alpha \) and Q) are also observed for both the cases of BH. Further, by establishing a relation between quasinormal modes (QNMs) and parameters related to null circular geodesics (like angular frequency and Lyapunov exponent), we deduced the QNMs (or QNM frequencies) for a massless scalar field perturbation around both the cases of stringy BH spacetime in the eikonal limit. The variation of scalar field potential with charge parameters and angular momentum of perturbation (l) are visually presented and discussed accordingly.

The tortoise and the hare: a causality puzzle in AdS/CFT

We pose and resolve a holographic puzzle regarding an apparent violation of causality in anti-de Sitter (AdS)/conformal field theory. If a point in the bulk of AdS moves at the speed of light, the boundary subregion that encodes it may need to move superluminally to keep up. With AdS3 as our main example, we prove that the finite extent of the encoding regions prevents a paradox. We show that the length of the minimal-size encoding interval gives rise to a tortoise coordinate on AdS that measures the nonlocality of the encoding. We use this coordinate to explore circular and radial motion in the bulk before passing to the analysis of bulk null geodesics. For these null geodesics, there is always a critical encoding where the possible violation of causality is barely avoided. We show that in any other encoding, the possible violation is subcritical.

Quantum fields, geometric fluctuations, and the structure of spacetime

Quantum fluctuations of the vacuum stress-energy tensor are highly non-Gaussian, and can have unexpectedly large effects on spacetime geometry. In this paper, we study a two-dimensional dilaton gravity model coupled to a conformal field, in which the distribution of vacuum fluctuations is well understood. In this model, the fluctuations of the matter field are responsible for the fluctuations of the geometry itself. By analyzing the geodesic deviation in this model, we show that a pencil of massive particles propagating on this fuzzy spacetime eventually converges and collapses. This is consistent with our earlier analysis of null geodesics in [Phys. Rev. Lett.\ 107, 021303 (2011)].

Dynamical analysis of null geodesics in brane-world spacetimes

In this work, we present an extensive dynamical analysis of the null geodesics in the spacetimes proposed by Casadio, Fabbri, and Mazzacurati (CFM), which model black holes and wormholes in a Randall-Sundrum brane. Geodesic stability is evaluated by Lyapunov and Jacobi criteria, with coinciding results. Semistable photon spheres are found in the geometries of interest. Bifurcations associated with the dynamical system are also investigated, characterizing a degenerated Bogdanov-Takens bifurcation. Our results suggest deep connections between the geometric characteristics of the CFM spacetimes and the dynamics of null geodesics in these backgrounds.

Null geodesics in conformal gravity

We present an analysis of the null geodesics of the static, spherically symmetric, vacuum solution to the equations of conformal (Weyl) gravity. We classify the full range of exotic spacetimes arising from the parameter space of the metric. The nature of various notable features of these spacetimes is investigated including light spheres, horizons and physical singularities.

Analysis of geodesics on different surfaces

It is widely known that some surfaces contain special curves as a geodesics, while a lots of geodesics on surface have complicated shapes. Goal of this research is to find on what surfaces are u and v-parameter curves geodesics. Developable surfaces that contain a given plane curve as a geodesic are studied in the article, whereas the plane curve is also an initial u-parameter curve on that surface. Parametric equations of the minimal surfaces that contain an epicycloid as a geodesic are also given. Visualization of geodesics was done in Mathematica.

Conformal geodesics in spherically symmetric vacuum spacetimes with cosmological constant

An analysis of conformal geodesics in the Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild–de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the case of the Schwarzschild–anti-de Sitter family, the natural parameter of the curves only covers a restricted time span so that these global conformal Gaussian systems do not exist.

Entanglement entropy in Galilean conformal field theories and flat holography.

We present the analytical calculation of entanglement entropy for a class of two-dimensional field theories governed by the symmetries of the Galilean conformal algebra, thus providing a rare example of such an exact computation. These field theories are the putative holographic duals to theories of gravity in three-dimensional asymptotically flat spacetimes. We provide a check of our field theory answers by an analysis of geodesics. We also exploit the Chern-Simons formulation of three-dimensional gravity and adapt recent proposals of calculating entanglement entropy by Wilson lines in this context to find an independent confirmation of our results from holography.

Who's Afraid of the Hill Boundary?

The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove the conjugate locus of any point near enough to the boundary is a hypersurface tangent to the boundary. Our method of proof is to reduce analysis of geodesics near the boundary to that of solutions to Newton's equations in the simplest model case: a constant force. This model case is equivalent to the beginning physics problem of throwing balls upward from a fixed point at fixed speeds and describing the resulting arcs, see Fig. 2.

Geodesic motion in the vacuumCmetric

Geodesic equations of the vacuum C-metric are derived and solved for various cases. The solutions describe the motion of timelike or null particles with conserved energy and angular momentum. Polar, nearly-circular orbits around weakly accelerated black holes may be regarded as a perturbation of circular Schwarzschild geodesics. Results indicate that circular Schwarzschild geodesics of radius $r_0>6m$ are stable under small uniform accelerations along the orbital plane. These stable orbits undergo small oscillations around $r_0$, behaving like a harmonic oscillator driven by a periodic force plus another constant force. Circular orbits with axis parallel to the direction of black hole acceleration are also considered. In this case an algebraic relation expressing the condition of stability is obtained. This refines the stability analysis done in previous literature. We also present an analysis of radial geodesics along the poles. There exist a solution where a particle remains at unstable equilibrium at a fixed distance directly behind the accelerating black hole. Examples of numerical solutions are presented for other more general cases.

Massive neutral particles on heterotic string theory

The motion of massive particles in the background of a charged black hole in heterotic string theory, which is characterized by a parameter α, is studied in detail in this paper. Since it is possible to write this space-time in the Einstein frame, we perform a quantitative analysis of the time-like geodesics by means of the standard Lagrange procedure. Thus, we obtain and solve a set of differential equations and then we describe the orbits in terms of the elliptic ℘-Weierstraß function. Also, by making an elementary derivation developed by Cornbleet (Am. J. Phys. 61(7):650–651, 1993) we obtain the correction to the angle of advance of perihelion to first order in α, and thus, by comparing with Mercury’s data we give an estimation for the value of this parameter, which yields an heterotic solar charge Q ⊙≃0.728 [Km]=0.493 M ⊙. Therefore, in addition to the study on null geodesics performed by Fernando (Phys. Rev. D 85:024033, 2012), this work completes the geodesic structure for this class of space-time.

Curvature invariants, geodesics, and the strength of singularities in Bianchi-I loop quantum cosmology

We investigate the effects of the underlying quantum geometry in loop quantum cosmology on spacetime curvature invariants and the extendibility of geodesics in the Bianchi-I model for matter with a vanishing anisotropic stress. Using the effective Hamiltonian approach, we find that even though quantum geometric effects bound the energy density and expansion and shear scalars, divergences of curvature invariants are potentially possible under special conditions. However, as in the isotropic models in LQC, these do not necessarily imply a physical singularity. Analysis of geodesics and strength of such singular events, point towards a general resolution of all known types of strong singularities. We illustrate these results for the case of a perfect fluid with an arbitrary finite equation of state $w > -1$, and show that curvature invariants turn out to be bounded, leading to the absence of strong singularities. Unlike classical theory, geodesic evolution does not break down. We also discuss possible generalizations of sudden singularities which may arise at a non-vanishing volume, causing a divergence in curvature invariants. Such finite volume singularities are shown to be weak and harmless.