Volker Perlick

Gravitational Lensing from a Spacetime Perspective

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.

On Fermat's principle in general relativity. I, The general case

The following version of Fermats principle is proven to hold on an arbitrary Lorentzian manifold (i.e., without any kind of symmetry or causality condition being required): Among all lightlike curves connecting some event with some timelike curve, the geodesics are characterised by stationary arrival time. The arrival time is minimal if the geodesic is free of conjugate points, whereas it is a saddle point if there is a conjugate point in the interior.

Fermat principle in Finsler spacetimes

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point q (emission event) to a timelike curve γ (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here “arrival time” refers to a parametrization of the timelike curve γ. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.

Exact gravitational lens equation in spherically symmetric and static spacetimes

Lensing in a spherically symmetric and static spacetime is considered, based on the lightlike geodesic equation without approximations. After fixing two radius values r_O and r_S, lensing for an observation event somewhere at r_O and static light sources distributed at r_S is coded in a lens equation that is explicitly given in terms of integrals over the metric coefficients. The lens equation relates two angle variables and can be easily plotted if the metric coefficients have been specified; this allows to visualize in a convenient way all relevant lensing properties, giving image positions, apparent brightnesses, image distortions, etc. Two examples are treated: Lensing by a Barriola-Vilenkin monopole and lensing by an Ellis wormhole.

On Fermat's principle in general relativity. II. The conformally stationary case

For pt.I see ibid., vol.7, p.1319 (1990). On a conformally stationary spacetime, Fermats principle, and hence the equations of motion of light rays, can be formulated in three-dimensional (purely spatial) terms. This reduction from four-dimensional spacetime to three-dimensional space is very similar to the well known reduction formalism of Kaluza-Klein theory. The resulting equation of motion of light rays in 3-space is formally identical with the Lorentz force equation for charged particles of unit specific charge and fixed energy in a magnetostatic field on a Riemannian 3-manifold. This analogy, in which the magnetostatic field corresponds to the rotation of the timelike conformal Killing vector field, has the following consequence. To every magnetostatic electron lens there can be constructed mathematically an analogous gravitational lens.

Characterization of Standard Clocks by Means of Light Rays and Freely Falling Particles

A mathematical characterization of standard clocks (i.e., clocks measuring proper time) is presented, which yields an experimental method to test whether or not a given clock is a standard clock. The only tools needed are light rays and freely falling particles. For this reason our method fits very well in the framework of the axiomatic approach to space-time theory given by Ehlers, Pirani, and Schild [1], where just light rays and freely falling particles are used as primitive concepts. As the underlying space-time model we use a Weyl manifold (instead of a Lorentz manifold, which is the usual model of general relativity); this generalization is motivated by [1].

Rotating coordinates as tools for calculating circular geodesics and gyroscopic precession

If an axially symmetric stationary metric is given in standard form (i.e. in coordinates adapted to the symmetries) the transformationφ→φ′ =φ-ωt (ω=constant) of the azimuthal angle leads to another such standard form. The spatial latticeL′ corresponding to the latter rotates at angular velocityω relative to the latticeL of the former. For the standard form of a stationary metric there are simple formulae giving the four-acceleration of a given lattice point and the rotation of a gyroscope at a given lattice point. Applying these formulae toL′, we find the condition for circular paths about the axis inL to be 4-geodesic, and also the precession of gyroscopes along circular paths which are not necessarily geodesic. Among other examples we re-obtain the complete geodesic structure of the Gödel universe, and the gyroscopic precessions associated with the names of Thomas, Fokker and de Sitter, and Schiff.

Observer fields in Weylian spacetime models

Resting upon the Ehlers-Pirani-Schild axiomatic approach to spacetime theory, the author considers Weyl manifolds as reasonable spacetime models. He defines and discuss several concepts associated with a timelike vector field (=observer field=reference frame) on such a spacetime model. In particular, he derives propagation equations for rotation, expansion and shear of V, prove some kinematical vorticity theorems and present a new method how to measure length curvature operationally. As important mathematical tools, he uses the orthogonal decomposition of the Lie derivative and of the covariant derivative; the latter is the Weylian generalization of the Fermi-Walker derivative.

Observable effects in a class of spherically symmetric static Finsler spacetimes

After some introductory discussion of the definition of Finsler spacetimes and their symmetries, we consider a class of spherically symmetric and static Finsler spacetimes which are small perturbations of the Schwarzschild spacetime. The deviations from the Schwarzschild spacetime are encoded in three perturbation functions

Influence of a plasma on the shadow of a spherically symmetric black hole

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