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In view of Ehlers-Pirani-Schild formalism, since 1972 Weyl geometries should be considered to be the most appropriate and complete framework to represent (relativistic) gravitational fields. We shall here show that in any given Lorentzian spacetime (M,g) that admits global timelike vector fields any such vector field u determines an essentially unique Weyl geometry ([g], Gamma) such that u is Gamma-geodesic (i.e. parallel with respect to Gamma).
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The linear stability of closed timelike geodesics (CTGs) is analyzed in two spacetimes with cylindrical sources, an infinite rotating dust cylinder, and a cylindrical cloud of static cosmic strings with a central spinning string. We also study the existence and linear stability of closed timelike curves in spacetimes that share some common features with the Godel universe (Godel-type spacetimes). In this case the existence of CTGs depends on the `background metric. The CTGs in a subclass of inhomogeneous stationary cosmological solutions of the Einstein-Maxwell equations with topology $ S^3times mathbb R$ are also examined.
In this paper, we derive the solutions of orbit equations for a class of naked singularity spacetimes, and compare these with timelike orbits, that is, particle trajectories in the Schwarzschild black hole spacetime. The Schwarzschild and naked singularity spacetimes considered here can be formed as end state of a spherically symmetric gravitational collapse of a matter cloud. We find and compare the perihelion precession of the particle orbits in the naked singularity spacetime with that of the Schwarzschild black hole. We then discuss different distinguishable physical properties of timelike orbits in the black hole and naked singularity spacetimes and implications are discussed. Several interesting differences follow from our results, including the conclusion that in naked singularity spacetimes, particle bound orbits can precess in the opposite direction of particle motion, which is not possible in Schwarzschild spacetime.
We derive here the orbit equations of particles in naked singularity spacetimes, namely the Bertrand (BST) and Janis-Newman-Winicour (JNW) geometries, and for the Schwarzschild black hole. We plot the orbit equations and find the Perihelion precession of the orbits of particles in the BST and JNW spacetimes and compare these with the Schwarzschild black hole spacetime. We find and discuss different distinguishing properties in the effective potentials and orbits of particle in BST, JNW and Schwarzschild spacetimes, and the particle trajectories are shown for the matching of BST with an external Schwarzschild spacetime. We show that the nature of perihelion precession of orbits in BST and Schwarzschild spacetimes are similar, while in the JNW case the nature of perihelion precession of orbits is opposite to that of the Schwarzschild and BST spacetimes. Other interesting and important features of these orbits are pointed out.
It is shown that the free motion of massive particles moving in static spacetimes are given by the geodesics of an energy-dependent Riemannian metric on the spatial sections analogous to Jacobis metric in classical dynamics. In the massless limit Jacobis metric coincides with the energy independent Fermat or optical metric. For stationary metrics, it is known that the motion of massless particles is given by the geodesics of an energy independent Finslerian metric of Randers type. The motion of massive particles is governed by neither a Riemannian nor a Finslerian metric. The properies of the Jacobi metric for massive particles moving outside the horizon of a Schwarschild black hole are described. By constrast with the massless case, the Gaussian curvature of the equatorial sections is not always negative.
The role of the wandering null geodesic is studied in a black hole spacetime. Based on the continuity of the solution of the geodesic equation, the wandering null geodesics commonly exist and explain the typical phenomena of the optical observation of event horizons. Moreover, a new concept of `black room is investigated to relate the wandering null geodesic to the black hole shadow more closely.
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