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Register a new userGeodesics dynamics in the Linet-Tian spacetime with Lambda<0
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We investigate the geodesics kinematics and dynamics in the Linet-Tian metric with Lambda<0 and compare with the results for the Levi-Civita metric, when Lambda=0. This is used to derive new stability results about the geodesics dynamics in static vacuum cylindrically symmetric spacetimes with respect to the introduction of Lambda<0. In particular, we find that increasing |Lambda| always increases the minimum and maximum radial distances to the axis of any spatially confined planar null geodesic. Furthermore, we show that, in some cases, the inclusion of any Lambda<0 breaks the geodesics orbit confinement of the Lambda=0 metric, for both planar and non-planar null geodesics, which are therefore unstable. Using the full system of geodesics equations, we provide numerical examples which illustrate our results.
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We analyse the geodesics dynamics in cylindrically symmetric vacuum spacetimes with Lambda>0 and compare it to the Lambda=0 and Lambda<0 cases. When Lambda>0 there are two singularities in the metric which brings new qualitative features to the dynamics. We find that Lambda=0 planar timelike confined geodesics are unstable against the introduction of a sufficiently large Lambda, in the sense that the bounded orbits become unbounded. In turn, any non-planar radially bounded geodesics are stable for any positive Lambda. We construct global non-singular static vacuum spacetimes in cylindrical symmetry with Lambda>0 by matching the Linet-Tian metric with two appropriate sources.
We investigate the matching, across cylindrical surfaces, of static cylindrically symmetric conformally flat spacetimes with a cosmological constant $Lambda$, satisfying regularity conditions at the axis, to an exterior Linet-Tian spacetime. We prove that for $Lambdaleq 0$ such matching is impossible. On the other hand, we show through simple examples that the matching is possible for $Lambda>0$. We suggest a physical argument that might explain these results.
The static, apparently cylindrically symmetric vacuum solution of Linet and Tian for the case of a positive cosmological constant $Lambda$ is shown to have toroidal symmetry and, besides $Lambda$, to include three arbitrary parameters. It possesses two curvature singularities, of which one can be removed by matching it across a toroidal surface to a corresponding region of the dust-filled Einstein static universe. In four dimensions, this clarifies the geometrical properties, the coordinate ranges and the meaning of the parameters in this solution. Some other properties and limiting cases of this space-time are described. Its generalisation to any higher number of dimensions is also explicitly given.
Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the Godel spacetime appear to be elliptic-like if standard coordinates naturally adapted to the cylindrical symmetry are used. The orbit can then be suitably described through an eccentricity-semi-latus rectum parametrization, familiar from the Newtonian dynamics of a two-body system. However, changing coordinates such planar geodesics all become explicitly circular, as exhibited by Kundts form of the Godel metric. We derive here a one-to-one correspondence between the constants of the motion along these geodesics as well as between the parameter spaces of elliptic-like versus circular geodesics. We also show how to connect the two equivalent descriptions of particle motion by introducing a pair of complex coordinates in the 2-planes orthogonal to the symmetry axis, which brings the metric into a form which is invariant under Mobius transformations preserving the symmetries of the orbit, i.e., taking circles to circles.
In this research note we introduce a new approximation of photon geodesics in Schwarzschild spacetime which is especially useful to describe highly bent trajectories, for which the angle between the initial emission position and the line of sight to the observer approaches $pi$: this corresponds to the points behind the central mass of the Schwarzschild metric with respect to the observer. The approximation maintains very good accuracy overall, with deviations from the exact numerical results below $1%$ up to the innermost stable circular orbit (ISCO) located at $6~GM/c^2$.
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