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Register a new userThe regularity of geodesics in impulsive pp-waves
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We consider the geodesic equation in impulsive pp-wave space-times in Rosen form, where the metric is of Lipschitz regularity. We prove that the geodesics (in the sense of Caratheodory) are actually continuously differentiable, thereby rigorously justifying the $C^1$-matching procedure which has been used in the literature to explicitly derive the geodesics in space-times of this form.
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We study geodesics in the complete family of nonexpanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we prove existence and uniqueness of continuously differentiable geodesics (in the sense of Filippov) and use a C^1-matching procedure to explicitly derive their form.
We investigate a class of gravitational pp-waves which represent the exterior vacuum field of spinning particles moving with the speed of light. Such exact spacetimes are described by the original Brinkmann form of the pp-wave metric including the often neglected off-diagonal terms. We put emphasis on a clear physical and geometrical interpretation of these off-diagonal metric components. We explicitly analyze several new properties of these spacetimes associated with the spinning character of the source, such as rotational dragging of frames, geodesic deviation, impulsive limits and the corresponding behavior of geodesics.
We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance $d_S$ from a spacelike hypersurface $S$ is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.
We take into account two further physical models which play an utmost importance in the framework of Analogue Gravity. We first consider Bose--Einstein condensates (BEC) and then surface gravity waves in water. Our approach is based on the use of the master equation we introduced in a previous work. A more complete analysis of the singular perturbation problem involved, with particular reference to the behavior in the neighbourhood of the (real) turning point and its connection with the WKB approximation, allows us to verify the thermal character of the particle production process. Furthermore, we can provide a simple scheme apt to calculate explicitly the greybody factors in the case of BEC and surface waves. This corroborates the improved approach we proposed for studying the analogue Hawking effect in the usual limit of small dispersive effects.
We generalize the classical junction conditions for constructing impulsive gravitational waves by the Penrose cut and paste method. Specifically, we study nonexpanding impulses which propagate in spaces of constant curvature with any value of the cosmological constant (that is Minkowski, de Sitter, or anti-de Sitter universes) when additional off-diagonal metric components are present. Such components encode a possible angular momentum of the ultra-relativistic source of the impulsive wave - the so called gyraton. We explicitly derive and analyze a specific transformation that relates the distributional form of the metric to a new form which is (Lipschitz) continuous. Such a transformation automatically implies an extended version of the Penrose junction conditions. It turns out that the conditions for identifying points of the background spacetime across the impulse are the same as in the original Penrose cut and paste construction, but their derivatives now directly represent the influence of the gyraton on the axial motion of test particles. Our results apply both for vacuum and nonvacuum solutions of Einsteins field equations, and can also be extended to other theories of gravity.
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