Fatimah Shojai, Ali Shojai
We shall investigate the properties of a congruence of geodesics in the framework of Palatini f(R) theories. We shall evaluate the modified geodesic deviation equation and the Raychaudhuri's equation and show that f(R) Palatini theories do not necessarily lead to attractive forces. Also we shall study energy condition for f(R) Palatini gravity via a perturbative analysis of the Raychaudhuri's equation.
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Geodesic deviation equation in generalized hybrid Metric-Palatini gravity
本文研究了广义混合度规-Palatini引力中的测地偏离方程，分析了宇宙测地线结构及偏离向量随红移的数值行为，并推导了Mattig关系的修正形式。
We investigate a particular regularization of big bang singularity, which remains within the domain of 4-dimensional general relativity but allowing for degenerate metrics. We study the geodesics and geodesic congruences in the modified Friedmann-Lemaître-Robertson-Walker universe. In particular, we calculate the expansion of timelike and null geodesic congruences. Based on these results, we also briefly discuss the cosmological singularity theorems.
We investigate the geodesic deviation equation in the context of quantum improved spacetimes. The improved Raychaudhuri equation is derived, and it is shown that the classical strong energy condition does not necessarily lead to the convergence of geodesics in a congruence in the quantum improved spacetime.
The purpose of this paper is to check the impact of observer and Palatini $f(R)$ terms in the formulations of inhomogeneity factors of spherical relativistic systems. We consider Lemaître-Tolman-Bondi dynamical model as a compact object and studied its evolution with both tilted and non-tilted observers. We performed our analysis for particular cases of fluid distribution in tilted frame and found some energy density irregularity variables. We found that these variables are drastically different from those observed by non-tilted observer. The conformal flat dust and perfect matter contents are…
We perform a complete rho-integration of the GHP equations for all spacetimes that admit a geodesic shear-free expanding null congruence, whose Ricci spinor is aligned to the congruence and whose Ricci scalar is constant. We also deduce the system of GHP equations after the integration is completed, and discuss a few applications.
We show that, though they are rare, there are asymptotically flat space-times that possess null geodesic congruences that are both asymptotically shear- free and twist-free (surface forming). In particular, we display the class of space-times that possess this property and demonstrate how these congruences can be found. A special case within this class are the Robinson- Trautman space-times. In addition, we show that in each case the congruence is isolated in the sense that there are no other neighboring congruences with this dual property.
We study the timelike geodesic congruences in the generalized Ellis-Bronnikov spacetime (4D-GEB) and in recently proposed 5D model where a 4D-GEB is embedded in a warped geometry (5D-WGEB) and conduct a comparative study. Analytical expressions of ESR variables (for 4D geometries) are found which reveal the role of the wormhole parameter. In more general 4D and 5D scenarios geodesic equation, geodesic deviation equation and Raychaudhury equations are solved numerically. The evolution of cross-sectional area of the congruences of timelike geodesics (orthogonal to the geodesic flow lines) projec…
In this article, we look into geodesics in the Schwarzschild-Anti-de Sitter metric in (3+1) spacetime dimensions. We investigate the class of marginally bound geodesics (timelike and null), while comparing their behavior with the normal Schwarzschild metric. Using $\textit{Mathematica}$, we calculate the shear and rotation tensors, along with other components of the Raychaudhuri equation in this metric and we argue that marginally bound timelike geodesics, in the equatorial plane, always have a turning point, while their null analogues have at least one family of geodesics that are unbound. We…
Space-times admitting a shear-free, irrotational, geodesic null congruence are studied. Attention is focused on those space-times in which the gravitational field is a combination of a perfect fluid and null radiation.
In the stringy cosmology, we investigate singularities in geodesic surface congruences for the time-like and null strings to yield the Raychaudhuri type equations possessing correction terms associated with the novel features owing to the strings. Assuming the stringy strong energy condition, we have a Hawking-Penrose type inequality equation. If the initial expansion is negative so that the congruence is converging, we show that the expansion must pass through the singularity within a proper time. We observe that the stringy strong energy conditions of both the time-like and null string congr…
Displacement and velocity memory effects in the exact, vacuum, plane gravitational wave line element have been studied recently by looking at the behaviour of pairs of geodesics or via geodesic deviation. Instead, one may investigate the evolution of geodesic congruences. In our work here, we obtain the evolution of the kinematic variables which characterise timelike geodesic congruences, using chosen pulse profiles (square and sech-squared) in the exact, plane gravitational wave line element. We also analyse the behaviour of geodesic congruences in possible physical scenarios describable usin…
Principal equatorial null geodesic congruences in the Kerr metric, and their quantum propagators
该研究利用Raychaudhuri方程为Kerr度规中的赤道主向内、向外零测地线汇构造量子传播子，发现尽管膨胀标量在环形奇点发散，传播子仍有限，从而暗示量子效应可能消除或缓和奇点。
Power law coupling Higgs-Palatini inflation with a congruence between physical and geometrical symmetries
本文研究了幂律耦合 Higgs-Palatini 暴胀模型，通过施加物理与几何对称性的一致性条件，在强耦合极限下导出了标量-张量型不变作用量，并获得了符合观测数据的标度不变功率谱与张量标量比。
The evolution of timelike geodesic congruences in a spherically symmetric, nonstatic, inhomogeneous spacetime representing gravitational collapse of a massless scalar field is studied. We delineate how initial values of the expansion, rotation and shear of a congruence, as well as the spacetime curvature, influence the global behavior and focusing properties of a family of trajectories. Under specific conditions, the expansion scalar is shown to exhibit a finite jump (from negative to positive value) before focusing eventually occurs. This nonmonotonic behavior of the expansion, observed in ou…
Geodesic congruences in acoustic spacetimes and the role of Raychaudhuri equation
本研究利用Raychaudhuri方程，结合程函近似，在黑洞与宇宙学时空的声学类比模型中探究对应类光测地线汇的声音传播，明确了该方程的鲁棒性及声学类比的局限。
In this article, we explore the kinematics of timelike geodesic congruences in warped five dimensional bulk spacetimes, with and without thick or thin branes. Beginning with geodesic flows in the Randall--Sundrum AdS (Anti de Sitter) geometry without and with branes we find analytical expressions for the expansion scalar and comment on the effects of including thin branes on its evolution. Later, we move on to congruences in more general warped bulk geometries with a cosmological thick brane and a time-dependent extra dimensional scale. Using analytical expressions for the velocity field, we i…
Charged black holes in gravity theories in the Palatini formalism present a number of unique properties. Their innermost structure is topologically nontrivial, representing a wormhole supported by a sourceless electric flux. For certain values of their effective mass and charge curvature divergences may be absent, and their event horizon may also disappear yielding a remnant. We give an overview of the mathematical derivation of these solutions and discuss their geodesic structure and other geometric properties.
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…
By the method of rho-integration we obtain all Lanczos potentials L_{ABCA'} of the Weyl spinor that, in a certain sense, are aligned to a geodesic shear-free expanding null congruence. We also obtain all spinors H_{ABA'B'}=Q_{AB}o_{A'}o_{B'}, Q_{AB}=Q_{(AB)} satisfying nabla_{(A}{}^{B'}H_{BC)A'B'}=L_{ABCA'}. We go on to prove that H_{ABA'B'} can be chosen so that Gamma_{ABCA'}=nabla_{(A}{}^{B'} H_{B)CA'B'} defines a metric asymmetric curvature-free connection such that L_{ABCA'}=Gamma_{(ABC)A'} is a Lanczos potential that is aligned to the geodesic shear-free expanding congruence. These result…
Using the Raychaudhuri equation, we show that a quantum probability amplitude (Feynman propagator) can be univocally associated to any timelike or null affinely parametrized geodesic congruence.
We studied the singularity of the geodesic surface congruence for timelike and null strings using the expansion of the universe in the string theory. We had Raychaudhuri type equation for the expansion. Assuming the stringy strong energy condition and initial convergence, we induced the existence of singularity and got the same inequality equation of the string strong energy condition for both timelike and null stringy congruence. In this paper we want to study the twist and shear aspects of the stingy geodesic surface congruence. Under some natural conditions we derive the equations of the tw…
We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element $U$ of the time-like 2--plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of $U$ and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of $U$. The type D case is analyze…
We derive a set of equations describing the linear response of the convergence properties of a geodesic congruence to arbitrary geometry variations. It is a combination of equations describing the deviations from the standard Raychaudhuri-type equations due to the geodesic shifts and an equation describing the geodesic shifts due to the geometry variations. In this framework, the geometry variations, which can be chosen arbitrarily, serve as probes to investigate the gravitational contraction processes from various angles. We apply the obtained framework to the case of conformal geometry varia…
A comprehensive analysis of general relativistic spacetimes which admit a shear-free, irrotational and geodesic timelike congruence is presented. The equations governing the models for a general energy-momentum tensor are written down. Coordinates in which the metric of such spacetimes takes on a simplified form are established. The general subcases of `zero anisotropic stress', `zero heat flux vector' and `two component fluids' are investigated. In particular, perfect fluid Friedmann-Robertson-Walker models and spatially homogeneous models are discussed. Models with a variety of physically re…
The Kerr solution is defined by a null congruence which is geodesic and shear free and has a singular line contained in a bounded region of space. A generalization of the Kerr congruence for nonstationary case is obtained. We find a nonstationary shear free geodesic null congruence which is generated by a given analytical complex world line. Solutions of the Einstein equations are analyzed. It is shown that there exists complex radiative solution which is generalization of the Kerr solution and the Kinnersley accelerating solution for "photon rocket".
In this paper we introduce a new approach to the study of the effects that an impulsive wave, containing a mixture of material sources and gravitational waves, has on a geodesic congruence that traverses it. We find that the effect of the wave on the congruence is a discontinuity in the B-tensor of the congruence. Our results thus provide a detector independent and covariant characterization of gravitational memory.
Teleparallel Gravity (TG) describes gravitation as a torsional- rather than curvature-based effect. As in curvature-based constructions of gravity, several different formulations can be proposed, one of which is the Teleparallel equivalent of General Relativity (TEGR) which is dynamically equivalent to GR. In this work, we explore the evolution of a spatially homogeneous collapsing stellar body in the context of two important modifications to TEGR, namely f (T) gravity which is the TG analogue of f (R) gravity, and a nonminimal coupling with a scalar field which has become popular in TG for it…
Palatini (or metric-affine) theories of gravity are characterized by having {\it a priori} independent metric and affine structures. The theories built in this framework have their field equations obtained as independent variations of the action with respect to the metric, the affine connection, and the other fields. In this work we consider the issue of singularity-removal in several members of a family of theoretically consistent and observationally viable subclass of them, built as contractions of the Ricci tensor with the metric (Ricci-based gravities or RBGs, for short). Several types of …
The effective potential in universal like coordinates$(U, V, θ, ϕ)$, which are smooth across the event horizon is derived and investigated the ISCO(Innermost Stable Circular Orbits) explicitly in these coordinates for extremal Kerr spacetime. Extremization of the effective potential for timelike circular orbit shows that the existence of a stable circular geodesics in the extremal spacetime for direct orbit, precisely {\it on} the event horizon in terms of the radial coordinate which coincides with the \emph{principal null geodesic congruences} of the event horizon. These null geodesic congrue…
We study and report on the class of vacuum Maxwell fields in Minkowski space that possess a non-degenerate, diverging, principle null vector field (null eigenvector field of the Maxwell tensor) that is tangent to a shear-free null geodesics congruence. These congruences can be either surface forming (the tangent vectors proportional to gradients) or not, i.e., the twisting congruences. In the non-twisting case, the associated Maxwell fields are precisely the Lienard-Wiechert fields, i.e., those Maxwell fields arising from an electric monopole moving on an arbitrary worldline. The null geodesic…
A recent attempt to arrive at a quantum version of Raychaudhuri's equation is looked at critically. It is shown that the method, and even the idea, has some inherent problems. The issues are pointed out here. We have also shown that it is possible to salvage the method in some limited domain of applicability. Although no generality can be claimed, a quantum version of the equation should be useful in the context of ascertaining the existence of a singularity in the quantum regime. The equation presented in the present work holds for arbitrary $n + 1$ dimensions. An important feature of the Ham…
Non-Commutativity in Geodesic Congruences
该研究通过弗罗贝尼乌斯定理推广测地汇向量对易关系，提出时空非对易性的几何实现，引入NCEC修正聚焦相关定理，为暗能量演化提供纯几何起源。
The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat space-time with a given Bondi shear, one can find (by integrating a partial differential equation) a class of asymptotically shear-free (but, in general, twistiing) null geodesic congruences. The class is uniquely given up to the arbitrary choice of a complex analytic world-line in a four-parameter complex space. Surprisingly this parameter space turns out to…
Gravitational wave memory and geodesic congruence response in generalised Ellis-Bronnikov wormholes
研究广义Ellis-Bronnikov虫洞中引力波脉冲对测地线及测地线汇的响应，发现记忆效应依赖喉半径、陡度参数和脉冲位置，喉处脉冲产生零膨胀但有限剪切响应。
Reconstruction of $f(Q)$ Gravity from Geodesic Congruence Dynamics: Raychaudhuri Equation
本研究利用Raychaudhuri方程作为几何工具，在对称遥平行引力框架下重建$f(Q)$引力函数形式，所得模型无需暗能量即可解释宇宙加速膨胀，并可作为逆重建的中心方程。
We study the energy conditions and geodesic deformations in Bertrand space-times. We show that these can be thought of as interesting physical space-times in certain regions of the underlying parameter space, where the weak and strong energy conditions hold. We further compute the ESR parameters and analyze them numerically. The focusing of radial time-like and radial null geodesics is shown explicitly, which verifies the Raychaudhuri equation.
We study the accelerated expansion of the Universe through its consequences on a congruence of geodesics. We make use of the Raychaudhuri equation which describes the evolution of the expansion rate for a congruence of timelike or null geodesics. In particular, we focus on the space-time geometry contribution to this equation. By straightforward calculation from the metric of a Robertson-Walker cosmological model, it follows that in an accelerated expanding Universe the space-time contribution to the Raychaudhuri equation is positive for the fundamental congruence, favoring a non-focusing of t…
It is shown that small elements of perfect fluid in adiabatic processes move along geodesic lines of a Riemannian space-time.
We study null geodesic congruences (NGCs) in the presence of spacetime torsion, recovering and extending results in the literature. Only the highest spin irreducible component of torsion gives a proper acceleration with respect to metric NGCs, but at the same time obstructs abreastness of the geodesics. This means that it is necessary to follow the evolution of the drift term in the optical equations, and not just shear, twist and expansion. We show how the optical equations depend on the non-Riemannian components of the curvature, and how they reduce to the metric ones when the highest spin c…
We investigate the evolution of timelike geodesic congruences, in the background of a charged black hole spacetime surrounded with quintessence. The Raychaudhuri equations for three kinematical quantities namely the expansion scalar, shear and rotation along the geodesic flows in such spacetime are obtained and solved numerically. We have also analysed both the weak and the strong energy conditions for the focusing of timelike geodesic congruences. The effect of the normalisation constant and equation of state parameter on the evolution of the expansion scalar is discussed, for the congruences…
In classical electromagnetic theory, one formally defines the complex dipole moment (the electric plus 'i' magnetic dipole) and then computes (and defines) the complex center of charge by transforming to a complex frame where the complex dipole moment vanishes. Analogously in asymptotically flat space-times it has been shown that one can determine the complex center of mass by transforming the complex gravitational dipole (mass dipole plus 'i' angular momentum) (via an asymptotic tetrad trasnformation) to a frame where the complex dipole vanishes. We apply this procedure to such space-times wh…
An introduction to extended theories of gravity formulated in metric-affine (or Palatini) spaces is presented. Focusing on spherically symmetric configurations with electric fields, we will see that in these theories the central singularity present in General Relativity is generically replaced by a wormhole structure. The resulting space-time becomes geodesically complete and, therefore, can be regarded as non-singular. We illustrate these properties considering two different models, namely, a quadratic $f(R)$ theory and a Born-Infeld like gravity theory.
We present the complete family of higher dimensional spacetimes that admit a geodesic, shearfree, twistfree and expanding null congruence, thus extending the well-known D=4 class of Robinson-Trautman solutions. Einstein's equations are solved for empty space with an arbitrary cosmological constant and for aligned pure radiation. Main differences with respect to the D=4 case (such as the absence of type III/N solutions, related to ``violations'' of the Goldberg-Sachs theorem in D>4) are pointed out, also in connection with other recent works. A formal analogy with electromagnetic fields is brie…
We describe here what appears to be a new structure that is hidden in all asymptotically vanishing Maxwell fields possessing a non-vanishing total charge. Though we are dealing with real Maxwell fields on real Minkowski space nevertheless, directly from the asymptotic field one can extract a complex analytic world-line defined in complex Minkowski space that gives a unified Lorentz invariant meaning to both the electric and magnetic dipole moments. In some sense the world-line defines a `complex center of charge' around which both electric and magnetic dipole moments vanish. The question of ho…
It is known that the Cosmic Microwave Background (CMB) temperature fluctuations are modified by gravitational lensing since the angular positions of photons are altered by the metric perturbations. We reconsider this effect in the context of geodesic deviation on a perturbed Friedmann-Robertson-Walker (FRW) spacetime with scalar and tensor modes. We first give an alternative derivation of the Sachs-Wolfe effect by using the solutions of the null geodesic equation. Then, we determine the two-dimensional induced metric on the transverse cross section of the null geodesic congruence corresponding…
We argue that the well-known geodesic completeness property of pp-waves, can be disregarded once the geodesics are extracted as being extended along sets of Brinkmann coordinates. This issue is investigated in the more general context of congruence convergence and we show that the problem leads to diverse issues for non-geodesic congruences. The discussion is mostly based on the null congruence expansion and a generalized Raychaudhuri equation is also provided.
We study geometric structures associated with shear-free null geodesic congruences in Minkowski space-time and asymptotically shear-free null geodesic congruences in asymptotically flat space-times. We show how in both the flat and asymptotically flat settings, complexified future null infinity acts as a "holographic screen," interpolating between two dual descriptions of the null geodesic congruence. One description constructs a complex null geodesic congruence in a complex space-time whose source is a complex world-line; a virtual source as viewed from the holographic screen. This complex nu…
We reconsider the problem of $f(R)$ theories of gravity coupled to Born-Infeld theory of electrodynamics formulated in a Palatini approach, where metric and connection are independent fields. By studying electrovacuum configurations in a static and spherically symmetric space-time, we find solutions which reduce to their Reissner-Nordström counterparts at large distances but undergo important non-perturbative modifications close to the center. Our new analysis reveals that the point-like singularity is replaced by a finite-size wormhole structure, which provides a geodesically complete and thu…
Kerr isolated horizon revisited: Caustic-free congruence and adapted tetrad
本文重新审视克尔时空的近视界描述，采用非扭的null测地线正交网，消除固定卡特常数导致的坐标和测地路径病态。通过依赖极角的卡特常数选择，构建了适配于孤立视界的纽曼-彭罗斯四元组及适配坐标，并计算了曲率标量和初始数据。
The purpose of the present work is to extend the earlier results for asymptotically flat vacuum space-times to asymptotically flat solutions of the Einstein-Maxwell equations. Once again, in this case, we get a class of asymptotically shear-free null geodesic congruences depending on a complex world-line in the same four-dimensional complex space. However in this case there will be, in general, two distinct but uniquely chosen world-lines. One of which can be assigned as the complex center-of- charge while the other could be called the complex center of mass. Rather than investigating the situ…
A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension $n$. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in $n-4$ spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric …
The Raychaudhuri equation for a geodesic congruence in the presence of a zero-point length has been investigated. This is directly related to the small-scale structure of spacetime and possibly captures some quantum gravity effects. The existence of such a minimum distance between spacetime events modifies the associated metric structure and hence the expansion as well as its rate of change deviates from standard expectations. This holds true for any kind of geodesic congruences, including time-like and null geodesics. Interestingly, this construction works with generic spacetime geometry with…
Shear-free or asymptotically shear-free null geodesic congruences possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant affects. It is the purpose of this paper to develop these issues and find applications in GR. The applications center around the problem of extracting interior physical properties of an asymptotically flat space-time directly from the asymptotic gravitational (and Maxwell) field itself in analogy with the determination of total charge by an integral over the Maxwell fi…
Second-Order Effects in Gravitational Wave Spacetime
本文研究了引力波的二阶效应，通过构造一般时空度规并推导二阶测线偏离方程，发现测试粒子在波传播方向上存在微小的二阶位移。
Graviton noise on tidal forces and geodesic congruences
本文利用Feynman-Vernon影响泛函与Schwinger-Keldysh形式研究引力子量子噪声对点质量运动的影响，将其视为随机潮汐力，推导并估计Minkowski真空与压缩真空中的表达式，进而通过Raychaudhuri方程分析其对测地线汇、剪切和旋转张量的效应。
The structural properties of geodesic currents in an ambient Kerr background is studied from an analytical point of view. The geodesics in the congruence correspond to charged particles that carry energy and angular momentum from the black hole through the Blandford-Znajek mechanism. It is shown that the resulting magnetosphere naturally satisfies the Znajek regularity condition. Particular attention is paid here to the energy extracted by matter currents rather than by electromagnetic Poynting fluxes.
We show that, within modified gravity, the non-linear nature of the field equations implies that the usual naive averaging procedure (replacing the microscopic energy-momentum by its cosmological average) is invalid. We discuss then how the averaging should be performed correctly and show that, as a consequence, at classical level the physical masses and geodesics of particles, cosmology and astrophysics in Palatini modified gravity theories are all indistinguishable from the results of general relativity plus a cosmological constant. Palatini gravity is however a different theory from general…
We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, $SL(2,\mathbb C)$ Yang-Mills and complex Maxwell fields, the latter produced by integer-valued electric charges ("elementary" for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the …
A study of Raychaudhuri equation and geodesic focusing in Fractal Universe
本文在分形宇宙中研究修正的Raychaudhuri方程，分析三种分形函数下类时测地线聚焦，并依据收敛标量符号探讨大爆炸奇点的存在与规避。
A new systematic approach extending the notion of frames to the Palatini scalar-tensor theories of gravity in various dimensions n>2 is proposed. We impose frame transformation induced by the group action which includes almost-geodesic and conformal transformations. We characterize theories invariant with respect to these transformations dividing them up into solution-equivalent subclasses (group orbits). To this end, invariant characteristics have been introduced. Unlike in the metric case, it turns out that the dimension four admitting the largest transformation group is rather special for s…
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