We consider the tidal forces between test particles falling along geodesics in the exterior spacetime generated by a static and axially symmetric compact matter source with non-vanishing mass quadrupole. Specifically, we analyze the radial and angular geodesic deviation, compare it with that of the Schwarzschild spacetime, and investigate the impact of the deformation parameter $q$, at different polar angles $θ$ with respect to the vertical symmetry axis. Furthermore, we examine the geodesic deviation for the case of non-constant $θ$ during the radial fall. It is shown that the presence of the deformation parameter affects the behavior of the geodesic deviation vectors, depending on its value. In particular, we observe that for arbitrary values of $q$ and $θ$ the behavior of the deviation vector differs as it approaches the singularity at $r = 2m$. Above all, we can witness either stretching or compressing of the deviation vector for various combinations of $q$ and $θ$. These findings provide insight into the effects of quadrupole deformation on the motion of test particles in the vicinity of the central object.
An important aspect of General Relativity is to study properties of geodesics. A useful tool for describing geodesic behavior is the geodesic deviation equation. It allows to describe the tidal properties of gravitating objects through the curvature of spacetime. This article focuses on the study of the axially symmetric Taub-NUT metric. We study tidal effects in this metric using the geodesic deviation equation. Radial geodesics along the symmetry axis of spacetime are considered. We show that all spatial components of tidal forces always change sign under the event horizon. We find a solutio…
The article generalizes the description of tidal forces to the case of geodesics with non-zero angular momentum in the metric of static spherically symmetric black holes. We show that the geodesic deviation equation can be diagonalized even with non-radial free motion of a test body in the gravitational field. We present expressions for the spatial components of the tidal force in a spherically symmetric metric. We also solve geodesic deviation equation in the Schwarzschild metric and demonstrate how the presence of angular momentum and its magnitude affect the solution.
A novel approximation method in studying the perihelion precession and planetary orbits in general relativity is to use geodesic deviation equations of first and high-orders, proposed by Kerner et.al. Using higher-order geodesic deviation approach, we generalize the calculation of orbital precession and the elliptical trajectory of neutral test particles to Kerr$-$Newman space-times. One of the advantage of this method is that, for small eccentricities, one obtains trajectories of planets without using Newtonian and post-Newtonian approximations for arbitrary values of quantity ${G M}/{R c^2}$…
The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally treated as the same order in smallness as the separation itself. This assumption is discussed in various papers, but is typically not recognized in textbooks. Relaxing this assumption leads to the generalized geodesic deviation equation (GGDE). We demonstrate the distinction between the GDE and the GGDE by computing the relative acceleration between timelike geod…
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.
Within the theory of General Relativity, we study the solution and range of applicability of the standard geodesic deviation equation in highly symmetric spacetimes. In the Schwarzschild spacetime, the solution is used to model satellite orbit constellations and their deviations around a spherically symmetric Earth model. We investigate the spatial shape and orbital elements of perturbations of circular reference curves. In particular, we reconsider the deviation equation in Newtonian gravity and then determine relativistic effects within the theory of General Relativity by comparison. The dev…
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…
While considering the chameleon scalar field model with the spatially flat FLRW background, we investigate the late-time acceleration phase of the universe, wherein we apply the typical potential usually used in this model. Through setting some constraints on the free parameters of the model, we indicate that the non-minimal coupling between the matter and the scalar field in such a model should be strongly coupled in order to have an accelerated expansion of the universe at the late-time. We also investigate the relative acceleration of the parallel geodesics by obtaining the geodesic deviati…
I point out a simple expression for the "Hubble" parameter $\mathscr{H}$, defined by Borde, Guth and Vilenkin (BGV) in their proof of past incompleteness of inflationary spacetimes. I show that the parameter $\mathscr{H}$ which an observer $O$ with four-velocity $\bf v$ will associate with a congruence $\bf u$ is equal to the fractional rate of change of the magnitude $ξ$ of the Jacobi field associated with $\bf u$, measured along the points of intersection of $O$ with $\bf u$, with its direction determined by $\bf v$. I then analyse the time dependence of $\mathscr{H}$ and $ξ$ using the geode…
One of the important and famous topics in general theory of relativity and gravitation is the problem of geodesic deviation and its related singularity theorems. An interesting subject is the investigation of these concepts when quantum effects are considered. Since, the definition of trajectory is not possible in the framework of standard quantum mechanics (SQM), we investigate the problem of geodesic equation and its related topics in the framework of Bohmian quantum mechanics in which the definition of trajectory is possible. We do this in a fixed background and we do not consider the back-…
We present a method to compute exact expressions for optical observables for static spherically symmetric spacetimes in the framework of the bilocal geodesic operator formalism. The expressions are obtained by solving the linear geodesic deviation equations for null geodesics, using the spacetime symmetries and the associated conserved quantities. We solve the equations in two different ways: by varying the geodesics with respect to their initial data and by directly integrating the equation for the geodesic deviation. The results are very general and can be applied to a variety of spacetime m…
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.
Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an specific class of local coordinate transformations is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit $k$-jet approximations and are invariant under arbitr…
By using the higher-order geodesic deviation equations for charged particles, we apply the method described by Kerner et.al. to calculate the perihelion advance and trajectory of charged test particles in the Riessner-Nordstrom spacetime. The effect of charge on the perihelion advance is studied and compared the results with those obtained earlier via the perturbation method. The advantage of this approximation method is to provide a way to calculate the perihelion advance and orbit of planets in the vicinity of massive and compact objects without considering Newtonian and post-Newtonian appro…
We consider an equation of the geodesic deviation appearing in the problem of gravitational wave detection in an environment of gravitons. We investigate a state-dependent graviton noise (as discussed in a recent paper of Parikh,Wilczek and Zahariade) from the point of view of the Feynman integral and stochastic differential equations. The evolution of the density matrix and the transition probability in an environment of gravitons is obtained. We express the time evolution by a solution of a stochastic geodesic deviation equation with a noise dependent on the quantum state of the gravitationa…
The quantum fluctuations of the geodesic deviation equation in a flat background spacetime are discussed. We calculate the resulting mean squared fluctuations in the relative velocity and separation of test particles. The effect of these quantum fluctuations of the spacetime geometry is given in terms of the Riemann tensor correlation function. Three different sources of the Riemann tensor fluctuations are considered: a thermal bath of gravitons, gravitons in a squeezed state, and the graviton vacuum state.
According to Penrose effect, particles with negative energy can exist in the ergospheres of rotating black holes. We analyze geodesics for such particles and show that there are no circular and elliptic orbits in the ergosphere of a rotating black hole. We also show that there are geodesics which begin and terminate at the singularity and present the conditions under which such geodesics do not begin and terminate at the singularity.
The tidal effects generated by a nonlinear gravitational wave are investigated in double-null v - u coordinates, as an exact solution of Einstein's field equations. The components $ξ^{v}$ and $ξ^{u}$ of the separation vector behave as in flat space but the transversal components $ξ^{x}$ and $ξ^{y}$ depend nonlinearly on $v$ through the Bessel and Neumann functions, far from the null surface $v = 0$. We show that the same results are obtained by means of the tetrad formalism.
