Ted Jacobson, Renaud Parentani
The surface gravity of any Killing horizon, in any spacetime dimension, can be interpreted as a local, two-dimensional expansion rate seen by freely falling observers when they cross the horizon. Any two-dimensional congruence of geodesics invariant under the Killing flow can be used to define this expansion, provided that the observers have unit Killing energy.
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We study the existence and stability of circular geodesics in a family of asymptotically AdS static black holes in New Massive Gravity. We show that the existence of such geodesics is determined by the sign of the hair parameter. For a positive hair parameter the stability regions follow the usual pattern, with an innermost unstable null geodesic separated from the horizon, followed by a region of unstable timelike geodesics and then stable timelike geodesics extending in the asymptotic region.
We study the null geodesics in a static circularly symmetric (SCS) black hole spacetime which is a solution in the $(2+1)$-dimensional massive gravity proposed by Bergshoeff, Hohm and Townsend (BHT massive gravity). We obtain analytic solutions for the null geodesic equation in the SCS black hole background and find the explicit form of deflection angles. We see that for various values of the impact parameter, the deflection angle can be positive, negative or even zero in this black hole spacetime. The negative deflection angle indicates the repulsive behavior of the gravity which comes from t…
We provide an exhaustive and illustrated classification of timelike and null geodesics in the near-horizon region of near-extremal Kerr black holes. The classification of polar motion extends to Kerr black holes of arbitrary spin. The classification of radial motion leads to a simple parametrization of the separatrix between bound and unbound motion. Furthermore, we prove that each timelike or null geodesic is related via conformal transformations and discrete symmetries to spherical orbits and we provide the explicit mappings. We detail the high-spin behavior of both the innermost stable and …
In order to classify and understand the spacetime structure, investigation of the geodesic motion of massive and massless particles is a key tool. So the geodesic equation is a central equation of gravitating systems and the subject of geodesics in the black hole dictionary attracted much attention. In this paper, we give a full description of geodesic motions in three-dimensional spacetime. We investigate the geodesics near charged BTZ black holes and then generalize our prescriptions to the case of massive gravity. We show that electric charge is a critical parameter for categorizing the geo…
The geometry of a spacetime containing a wormhole generated by a spherically symmetric electric field is investigated in detail. These solutions arise in high-energy extensions of General Relativity formulated within the Palatini approach and coupled to Maxwell electrodynamics. Even though curvature divergences generically arise at the wormhole throat, we find that these spacetimes are geodesically complete. This provides an explicit example where curvature divergences do not imply spacetime singularities.
We study the issue of length renormalization in the context of fully covariant gravity theories as well as non-relativistic ones such as Hořava-Lifshitz gravity. The difference of their symmetry groups implies a relation between the lengths of paths in spacetime in the two types of theory. Provided that certain asymptotic conditions hold, this relation allows us to transfer analytic criteria for the standard spacetime length to be finite to the Perelman length to be likewise finite, and therefore formulate conditions for geodesic incompleteness in partially covariant theories. We also discuss …
Causal radial geodesics with a positive interval in the Schwarzschild metric include a subset of trajectories completely confined under a horizon, which compose a thermal statistical ensemble with the Hawking-Gibbons temperature. The Bekenstein--Hawking entropy is given by an action at corresponding geodesics of particles with a summed mass equal to that of black hole in the limit of large mass.
Phase Space of SdS Geodesics and using the Cosmological Horizon to Observe a Black Hole
本文分析SdS时空的测地线相空间结构，提出利用宇宙学视界的蓝移效应抵消黑洞近域的强红移，明确了最优观测位置与运动策略，量化了宇宙学常数对黑洞信号观测的增强效果。
Near-horizon Geodesic Instabilities and Anomalous Decay of Quasinormal Modes in Weyl Black Holes
本文研究了韦伊尔黑洞的测地不稳定性及本征振荡的异常衰减，揭示了李雅普诺夫指数与本征振荡之间的关系。
In the framework of thermal quantization of radial geodesics completely confined behind the horizons we calculate the entropy of BTZ black hole in agreement with the Bekenstein--Hawking relation. Particles in the BTZ black hole occupy the only quantum ground level. The quantization allow us to find a linear dependence of black hole mass versus its orbital momentum.
The causal geodesics in the equatorial plane of a static extremal charged black holes in heterotic string theory are examined with regard to their geodesic stability, and compared with similar geodesics in the non-extremal situation. Extremization of the effective potential for time-like and null circular geodesics implies that in the extremal limit, the radius of ISCO(Inner-most Stable Circular Orbit) $(r_{ISCO})$, circular photon orbit (CPO) $(r_{ph})$ and marginally bound circular orbit (MBCO) $(r_{mb})$ are coincident with the event horizon $(r_{hor})$ i.e. $r_{ISCO}=r_{ph}=r_{mb}=r_{hor}=…
This article explores the characteristics of ergoregion, horizons and circular geodesics around a Kerr-Newman-Kasuya black hole. We investigate the effect of spin and dyonic charge parameters on ergoregion, event horizon and static limit surface of the said black hole. We observed that both electric, as well as magnetic charge parameters, results in decreasing the radii of event horizon and static limit, whereas increasing the area of ergoregion. The obtained results are compared with that acquired from Kerr and Schwarzschild black holes. Moreover, we figured out the photons orbit of circular …
The time-like and null-like geodesics around compact objects are some of the best tools to classify and understand the structure of a space-time. In this paper, we study the null geodesics around charged static dilaton black holes in Einstein-Maxwell-dilaton gravity. The physical parameters for non-radial geodesics including the effective potential, effective force, radius of the photon sphere and impact parameter are obtained and effects of the charge parameter and dilaton coupling constant on these quantities are studied. Possible photon motions for different values of the impact parameter a…
We study the derivation of the effective equation of motion for a pointlike particle in the framework of quantum gravity. Just like the geodesic motion of a classical particle is a consequence of classical field theory coupled to general relativity, we introduce the similar notion of an effective equation of motion, but starting from an abstract quantum gravity description. In the presence of entanglement between gravity and matter, quantum effects give rise to modifications of the geodesic trajectory, primarily as a consequence of the nonzero overlap between various coherent states of the gra…
In this work, we show that it is possible to study the notion of geodesic deviation equation in $f(T)$ gravity, in spite of the fact that in teleparallel gravity there is no notion of geodesics, and the torsion is responsible for the appearance of gravitational interaction. In this regard, we obtain the GR equivalent equations for $f(T)$ gravity which are in the modified gravity form such as $f(R)$ gravity. Then, we obtain the GDE within the context of this modified gravity. In this way, the obtained geodesic deviation equation will correspond to the $f(T)$ gravity. Eventually, we extend the c…
We study Born-Infeld gravity coupled to a static, nonrotating electric field in $2+1$ dimensions and find exact analytical solutions. Two families of such solutions represent geodesically complete, and hence nonsingular, spacetimes. Another family represents a point-like charge with a singularity at the center. Despite the absence of rotation, these solutions resemble the charged, rotating BTZ solution of General Relativity but with a richer structure in terms of horizons. The nonsingular character of the first two families turn out to be attached to the emergence of a wormhole structure on th…
Algebraic systems called the local geodesic loops and their tangent Akivis algebras are considered. Their possible role in theory of gravity is considered. Quantum conditions for the infinitesimal quantum events are proposed.
Free fall in modified symmetric teleparallel gravity
本文研究了修正的对称遥平行引力中的自由落体与等效原理，指出该理论虽不服从Weinberg意义上的等效原理，但自由下落测试粒子必然沿包含Levi-Civita联络的度规测地线运动。
Kerr Geodesics in horizon-penetrating Kerr coordinates: description in terms of Weierstrass functions
本文将 Biermann-Weierstrass 公式推广到穿透视界的克尔坐标中，给出在克尔视界上光滑的类时与类光测地线解，并实现最大解析延拓区域间的显式延拓与图形可视化。
Closed sections of totally geodesic null hypersurfaces are marginally outer trapped surfaces (MOTS), for which a well-defined notion of stability exists. In this paper we obtain the explicit form for the stability operator for such MOTS and analyze in detail its properties in the particular case of non-evolving horizons, which include both isolated and Killing horizons. We link these stability properties with the surface gravity of the horizon and/or to the existence of minimal sections. The results are used, in particular, to obtain an area-angular momentum inequality for sections of axially …
Stability criteria of nearly extreme black holes in the perturbations level is one of the interesting issues in gravitational systems. Considering the nearly extreme conformal-de Sitter black holes, in this paper, we obtain an exact relation for the quasinormal modes of scalar perturbations. As for stability criteria, we find a lower bound on the event horizon radius which is corresponding to an upper bound on the value of the quasinormal frequencies. In addition, we show that the asymptotic behavior of quasinormal modes gives highly damped modes which is important due to possible connection b…
In this paper we study the geodesic structure of the Schwarzschild black hole in rainbow gravity analyzing the behavior of null and time-like geodesic. We find that the structure of the geodesics essentially does not change when the semi-classical effects are included. However, we can distinguish different scenarios if we take into account the effects of rainbow gravity. Depending on the type of rainbow functions under consideration, inertial and external observers see very different situations in radial and non radial motion of a test particles.
In the present paper we study the Geodesic Deviation Equation (GDE) in the modified $f(Q)$-gravity theories. The formulation of GDE in General Relativity in the case of the homogeneous and isotropic Friedman-Lemaître-Robertson-Walker (FLRW) spacetime is briefly discussed and then extended in modified $f(Q)$-gravity using its covariant counterpart. The generalised Mattig relation is obtained. Finally, an equivalent expression to the Dyer-Roeder equation in General Relativity in the case of $f(Q)$-gravity is presented.
We derive the wave equation obeyed by electromagnetic fields in curved spacetime. We find that there are Riemann and Ricci curvature coupling terms to the photon polarisation which result in a polarisation dependent deviation of the photon trajectories from null geodesics. Photons are found to have an effective mass in an external gravitational field and their velocity in a local inertial frame is in general less than $c$. The effective photon mass in the Schwarzschild metric is $m_γ= ( 2GM/r^3 )^{1/2}$ and near the horizon it is larger than the Hawking temperature of the blackhole. Our result…
We investigate the geometry of a particular class of null surfaces in space-time called vacuum Non-Expanding Horizons (NEHs). Using the spin-coefficient equation, we provide a complete description of the horizon geometry, as well as fixing a canonical choice of null tetrad and coordinates on a NEH. By looking for particular classes of null geodesic congruences which live exterior to NEHs but have the special property that their shear vanishes at the intersection with the horizon, a good cut formalism for NEHs is developed which closely mirrors asymptotic theory. In particular, we show that suc…
We consider alternative theories of gravity with a direct coupling between matter and the Ricci scalar We study the relation between these theories and ordinary scalar-tensor gravity, or scalar-tensor theories which include non-standard couplings between the scalar and matter. We then analyze the motion of matter in such theories, its implications for the Equivalence Principle, and the recent claim that they can alleviate the dark matter problem in galaxies.
Kehagias-Sfetsos black hole in Hořava-Lifshitz gravity is probed through particle geodesics. Gravitational force of KS black hole becomes weaker than that of Schwarzschild around horizon and interior space. Particles can be always scattered or trapped in new closed orbits, unlike those falling forever in Schwarzschild black. The properties of null and timelike geodesics are classified with values of coupling constants. The precession rates of the orbits are evaluated. The time trajectories are also classified under different values of coupling constants for both null and timelike geodesics. Ph…
In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random trian…
The motion of a massive test particle around a Kehagias-Sfetsos black hole in deformed Hourava-Lifshitz gravity is studied. Employing the effective potential, the marginally bound orbits and the innermost stable circular orbits are analyzed. For the marginally bound orbits, their radius and angular momentum decrease with the parameter $ω$ of the gravity. For the innermost stable circular orbits, the energy and angular momentum also decrease with $ω$. Based on these results, we investigate the periodic orbits in the Kehagias-Sfetsos black holes. It is found that the apsidal angle parameter incr…
We write the equation of geodesic deviations in the spacetime of $pp$-waves in terms of the Newman-Penrose scalars and apply it to study gravitational waves in quadratic curvature gravity. We show that quadratic curvature gravity $pp$-waves can have a transverse helicity-0 polarization mode and two transverse helicity-2 general relativity-like wave polarizations. A concrete example is given in which we analyze the wave polarizations of an exact impulsive gravitational wave solution to quadratic curvature gravity.
We consider the black hole solutions of five-dimensional gravity with a Gauss-Bonnet term in the bulk and an induced gravity term on a 2-brane of codimension-2. Applying the Kerr-Schild method we derive additional solutions which include charge and angular momentum. To improve our understanding of this geometry we study the geodesic structure of such spacetimes.
Geodesic motions in extraordinary string geometry
该研究通过哈密顿-雅可比方法分析(4+1)维奇异弦解的测地线运动，发现自由落体粒子无法到达视界或奇点，存在稳定零圆轨道和反弹测地线，且进动角在不同几何中表现各异。
This note describes the behavior of null-geodesics near nondegenerate Killing horizons in language amenable to the application of a general framework, due to Vasy and Hintz, for the analysis of both linear and nonlinear wave equations. Throughout, the viewpoint of Melrose's b-geometry on a suitable compactification of spacetime at future infinity is adopted.
Towards black-hole horizons and geodesic focusing in causal sets
本文探讨如何在因果集中识别离散视界，通过局部诊断和测地聚焦方法，构建了1+1维黑洞时空的离散视界模型。
Study of Null Geodesics and their Stability in Horndeski Black Holes
本文研究Horndeski黑洞中零测地线结构及稳定性，通过太阳系经典检验约束耦合参数，并利用Lyapunov指数分析宇宙学常数对轨道稳定性的影响。
We compute the length of spacelike geodesics anchored at opposite sides of certain double-sided flow geometries in two dimensions. These geometries are asymptotically anti-de Sitter but they admit either a de Sitter or a black hole event horizon in the interior. While in the geometries with black hole horizons, the geodesic length always exhibit linear growth at late times, in the flow geometries with de Sitter horizons, geodesics with finite length only exist for short times of the order of the inverse temperature and they do not exhibit linear growth. We comment on the implications of these …
This work focuses on an unexplored aspect of non-symmetric geometry where {\it only} the off-diagonal metric components along the extra dimension, in a 5-dimensional spacetime, are non-symmetric. We show that the energy densities of the stationary non-symmetric models are similar to that of brane models thereby mimicking the thick-brane scenario. We find that the massive test particles are confined near the location of the brane for both growing and decaying warp factors. This feature is unique to the non-symmetric nature of our model. We have also studied the dynamical models where standard 4…
In the present paper we study the geodesic motion of test particles and light rays in the spacetime of a static charged black hole in $f(R)$ gravity. The complete set of analytic solutions of the geodesic equations in the spacetime of this black hole are presented. The geodesic equations are solved in terms of Weierstrass elliptic $\wp$ function and derivatives of Kleinian $σ$ function. With the help of parametric diagrams and effective potentials we analyze the geodesic motion and give a list of all possible orbit types. The different types of the resulting orbits are characterized in terms o…
Continuing past the inner horizon using WKB
通过WKB方法研究黑洞内部结构及内视界不稳定性
We show that the recipe for computing the expansions $θ_\ell$ and $θ_n$ of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar $κ(x)$, corresponding to the mapping $θ_\ell \to κθ_\ell$, $θ_n \to κ^{-1} θ_n$. Under this mapping, the product $θ_\ell θ_n$ is invariant, and thus the marginal surface computed from the vanishing of $θ_\ell$, which is used to define the apparent horizon, is invariant. This covariance group naturally appears in comparing the expansions computed with different choices of coordinate system.
Interpreting gravitational fields of Topologically Massive Gravity using geodesic deviation
本文利用测地偏离方程将三维拓扑大质量引力中的引力场分解为横向、纵向和牛顿分量，证明Cotton型N时空对应单偏振引力波，并分析了宇宙学常数、物质场及TN波和pp波的具体效应。
We investigate dynamics of probe particles moving in the near-horizon limit of extremal Myers-Perry black holes in arbitrary dimensions. Employing ellipsoidal coordinates we show that this problem is integrable and separable, extending the results of the odd dimensional case discussed in arXiv:1703.00713. We find the general solution of the Hamilton-Jacobi equations for these systems and present explicit expressions for the Liouville integrals, discuss Killing tensors and the associated constants of motion. We analyze special cases of the background near-horizon geometry were the system posses…
We analyze the horizon and geodesic structure of a class of 4D off--diagonal metrics with deformed spherical symmetries, which are exact solutions of the vacuum Einstein equations with anholonomic variables. The maximal analytic extension of the ellipsoid type metrics are constructed and the Penrose diagrams are analyzed with respect to adapted frames. We prove that for small deformations (small eccentricities) there are such metrics that the geodesic behaviour is similar to the Schwarzcshild one. We conclude that some vacuum static and stationary ellipsoid configurations may describe black el…
In this paper, we study the geodesic motion in the spacetime of a SU(2)-colored (A)dS black hole in conformal gravity, and also we investigate spacetime features, such as light spheres and horizons. Moreover, we derive the analytical solutions for the equation of motion of test particles and light rays using Weierstrass elliptic and Kleinian sigma functions. Depending on the particle energy levels and angular momentums, we classify the solutions of the geodesic equations. Furthermore, several examples of possible types of orbits illustrate the results
We study hidden symmetries, the symmetries associated with the Killing tensors, of the near horizon geometry of odd-dimensional Kerr-AdS-NUT black hole in two limits: generic extremal and extremal vanishing horizon (EVH) limits. Starting from Kerr-AdS-NUT black hole in ellipsoidal coordinates which admits integrable geodesic equations, we obtain the near horizon extremal/EVH geometries and their principal and Killing tensors by taking the near horizon limit. We explicitly demonstrate that geodesic equations are separable and integrable on theses near horizon geometries. We also compute the con…
Set of analytic solutions of the geodesic equation in a spherical conformal spacetime is presented. Solutions of this geodesics can be expressed in terms of the Weierstrass {\wp} function and the Kleinian σ function. Using conserved energy and angular momentum we can characterize the different orbits. Also, considering parametric diagrams and effective potentials, we plot some possible orbits. Moreover, with the help of analytical solutions, we investigate the light deflection for such an escape orbit.
We consider the motion of test particles in the spacetime of a black hole in f(R) gravity. The complete set of analytic solutions of the geodesic equation in the spacetime of this black hole are presented. The geodesic equations are solved in terms of Weierstrass elliptic functions and derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant $Λ$ and the real constant $β$.
Black Hole Topologies and Geodesic Structures in Symmetric Teleparallel f(Q) Gravity
本文研究了对称Teleparallel f(Q)引力中黑洞的拓扑与测地结构，推导出精确解并分析其热力学性质及奇点可达性。
Gravitational lensing in a space-time with cosmic string within the Eddington-inspired Born-Infeld gravity
本文在EiBI引力背景时空中引入宇宙弦，推导了类光与类时测地线的有效势及光子偏转角，揭示了宇宙弦对这些物理量的修正效应。
The geodesic deviation equation has been investigated in the framework of $f(T,\mathcal{T})$ gravity, where $T$ denotes the torsion and $\mathcal{T}$ is the trace of the energy-momentum tensor, respectively. The FRW metric is assumed and the geodesic deviation equation has been established following the General Relativity approach in the first hand and secondly, by a direct method using the modified Friedmann equations. Via fundamental observers and null vector fields with FRW background, we have generalized the Raychaudhuri equation and the Mattig relation in $f(T,\mathcal{T})$ gravity. Furth…
The study of the dynamics of a two-body system in modified gravity constitutes a more complex problem than in Newtonian gravity. Numerical methods are typically needed to solve the equations of geodesics. Despite the complexity of the problem, the study of a two-body system in $f(R)$ gravity leads to a new exciting perspective hinting the right strategy to adopt in order to probe modified gravity. Our results point out some differences between the {\em semiclassical} (Newtonian) approach, and the {\em relativistic} (geodesic) one thus suggesting that the latter represents the best strategy for…
Geodesic completeness from string T-duality
本文通过研究弦T对偶修正传播子的引力Raychaudhuri方程，发现短尺度引力存在屏蔽效应阻止曲率奇点，指出常规引力的吸引性仅是低能效应，为测地完备性分析提供新视角。
We study the zeroth law for Killing horizons in Lanczos-Lovelock gravity. We show that the surface gravity of a general Killing horizon in Lanczos-Lovelock gravity (except for general relativity) may not be constant even when the matter source satisfies dominant energy condition.
Quantum Gravity Fluctuations in the Timelike Raychaudhuri Equation
本文研究时空度规量子涨落对类时测地线汇的影响，推导出量子引力修正的Raychaudhuri方程，并发现量子引力普遍增强测地线汇的会聚和焦散线产生。
In this paper, we investigate the modified Geodesic Deviation Equation (GDE) in the framework of $f(R,T)$ theory of gravity where $R$ and $T$ are the curvature scalar and the trace of the energy-momentum tensor, respectively, using the FLRW background. In this way, we obtain the GR equivalent (GDE) in $f(R,T)$ metric formalism. We also extend our work to the generalization of the Matting relation and perform the numerical analysis with GDE for null vector.
We show that the Equivalence Principle (EP) is violated by Quantum Gravity (QG) effects. The predicted violations are compared to experimental observations for Gravitational Redshift, Law of Reciprocal Action and Universality of Free Fall. This allows us to derive explicit bounds for $β$ - the QG scale. In our approach, there appears a deviation in the geodesic motion of a particle. This deviation is induced by a non-commutative spacetime, consistent with a Generalized Uncertainty Principle (GUP). GUP admits the presence of a minimum length scale, that is advocated by QG theories. Remarkably, …
We present a systematic study of the geometric structure of non-singular spacetimes describing black holes in Lorentz-violating gravity. We start with a review of the definition of trapping horizons, and the associated notions of trapped and marginally trapped surfaces, and then study their significance in frameworks with modified dispersion relations. This leads us to introduce the notion of universally marginally trapped surfaces, as the direct generalization of marginally trapped surfaces for frameworks with infinite signal velocities (Hořava-like frameworks), which then allows us to define…
In the context of metric $f(R)$ gravity, the Geodesic Deviation Equation (GDE) was first studied in arXiv:1010.5279v3, giving a general expression and studying a particular case, the FLRW universe. In the paper arXiv:1312.2022v1 a similar analysis was made. However, there is a discrepancy in the expressions for the null vector field case due to an algebraic error in our original paper. Here, we make explicit the contribution of the different operators in the GDE, and we correct our previous results.
We generalize the Noether current method proposed by Majhi and Padmanabhan recently to $f(R)$ gravity and conformal gravity and analyze the near horizon symmetry of black holes in these models. It is shown that the entropy obtained with Cardy's formula agrees with Wald entropy.
The null geodesics of charged and non-charged black hole in mimetic gravity
本文在拟态引力框架下研究带电黑洞的零测地线，发现非线性电磁效应会阻止圆轨道形成，且拟态场可改变dS空间的排斥性质并调节黑洞引力强弱。
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