Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Jacobi-metric for timelike geodesics in static spacetimes
116
0
0.0
(
0
)
Authors
G. W. Gibbons
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwells equations in the curved spacetime. In particular, we see in very concrete terms how the magnetolectric susceptibility enters the Jacobi-Maupertuis-Randers-Finsler function.
We propose a new concept, the transversely trapping surface (TTS), as an extension of the static photon surface characterizing the strong gravity region of a static/stationary spacetime in terms of photon behavior. The TTS is defined as a static/stationary timelike surface $S$ whose spatial section is a closed two-surface, such that arbitrary photons emitted tangentially to $S$ from arbitrary points on $S$ propagate on or toward the inside of $S$. We study the properties of TTSs for static spacetimes and axisymmetric stationary spacetimes. In particular, the area $A_0$ of a TTS is proved to be bounded as $A_0le 4pi(3GM)^2$ under certain conditions, where $G$ is the Newton constant and $M$ is the total mass. The connection between the TTS and the loosely trapped surface proposed by us [arXiv:1701.00564] is also examined.
A perturbative method to compute the total travel time of both null and lightlike rays in arbitrary static spherically symmetric spacetimes in the weak field limit is proposed. The resultant total time takes a quasi-series form of the impact parameter. The coefficient of this series at a certain order $n$ is shown to be determined by the asymptotic expansion of the metric functions to the order $n+1$. To the leading order(s), the time delay, as well as the difference between the time delays of two kinds of relativistic signals, is then shown to take a universal form for all SSS spacetimes. This universal form depends on the mass $M$ and a post-Newtonian parameter $gamma$ of the spacetime. The analytical result is numerically verified using the central black hole of M87 as the gravitational lensing center.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
