Geodesic motion and confinement in Gödel's universe (original) (raw)

Geodesic time travel in Gödel's universe

Revista Mexicana De Fisica E, 2016

This work is an introduction at a beginning graduate or advanced undergraduate level to Kurt Godel's foray into cosmology. After an elementary introduction to the basics of Einstein's theory of gravitation, we simply present the Godel's solution and the geodesic equations associated with it. This equations are then explicitly solved obtaining its full set of temporal geodesics. Armed with such explicit expressions, the geodesic time-travelling possibilities of Godel's universe are discussed. We search for their time-like closed geodesics that, following Godel's analysis, other people has imagined as possible routes for time-travel. We next exhibit that such time-travelling possibility do not exist in his model universe. This is done in the most straightforward way possible, framing the discussion as to serve as a simple example for students of General Relativity

A Simple Geodesic Equation for Gravity, Electromagnetism and all Sources of Energy

Advanced Studies in Theoretical Physics, 2024

Riadh Al Rabeh final motion of a particle in a space and this is the only cause of motion. Our work might then help to understand the function of the space-time curvature in GR and could encourage some new simplified methods to implement GR in practical situations. Other questions of whether GR is complete, permit singular solutions, or an expansion of spacetime in astronomy can be understood in a more enlightened fashion. For simplicity and accessibility, we use only single scalar variables and leave a generalization to tensor variables for anyone wishing to take the next step.

J ul 2 00 1 On the Motion of Matter in Spacetime

2003

We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a re-formulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.

Stability of Closed Timelike Geodesics in different Spacetimes

Arxiv preprint arXiv:0706.3212, 2007

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 Gödel universe (Gödel-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 3 × R are also examined. *

On the Status of the Geodesic Principle in Newtonian and Relativistic Physics

Studies in History and Philosophy of Modern Physics, 2011

A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1), (1975)] provides a sense in which the geodesic principle has the status of a theorem in General Relativity (GR). I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton-Cartan theory) [Weatherall, J. O. "The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52(3), (2011)]. Here I compare the interpretations of these two theorems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case.

Stability of closed timelike curves in the Gödel universe

General Relativity and Gravitation, 2007

We study, in some detail, the linear stability of closed timelike curves in the Gödel universe. We show that these curves are stable. We present a simple extension (deformation) of the Gödel metric that contains a class of closed timelike curves similar to the ones associated to the original metric. This extension correspond to the addition of matter whose energy-momentum tensor is analyzed. We find the conditions to have matter that satisfies the usual energy conditions. We study the stability of closed timelike curves in the presence of usual matter as well as in the presence of exotic matter (matter that does satisfy the above mentioned conditions). We find that the closed timelike curves in the Gödel universe with or without the inclusion of regular or exotic matter are stable under linear perturbations. We also find a sort of structural stability.

On the motion of matter in spacetime

2001

We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a reformulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.

Constructing the Gödel universe

By a suitable transformation, we derive the rotating Gödel universe from a static one and we show, how rotation may be implemented geometrically. The rotation law turns out to be a differential one. By increasing distance from the rotation axis the velocity of a rotating point will exceed the velocity of light and the cosmos has a cut-off radius. Thus, closed time-like curves do not occur in the Gödel universe.

The geodesics structure of Schwarzschild black hole immersed in an electromagnetic universe

International Journal of Modern Physics D, 2017

The geodesic equations are considered in a spacetime that represents a Schwarzschild metric coupled to a uniform external electromagnetic (em) field. Due to the em field horizon shrinks and geodesics are modified. By analyzing the behavior of the effective potentials for the massless and massive particle we study the radial and circular trajectories. Radial geodesics for both photons and particles are solved exactly. It is shown that a particle that falls toward the horizon in a finite proper time slows down so that the particle reaches the singularity slower than Schwarzschild case. Timelike and null circular geodesics are investigated. We have shown that, there are no stable circular orbits for photons, however stable and unstable second-kind orbits exist for the massive particle. An exact analytical solution for the innermost stable circular orbits (ISCO) has been obtained. It has been shown that the radius of the ISCO shrinks due to the presence of the em field.