On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.Comment: 38 pages, 8 figures, late

General Relativistic Gravity Gradiometry

Gravity gradiometry within the framework of the general theory of relativity involves the measurement of the elements of the relativistic tidal matrix, which is theoretically obtained via the projection of the spacetime curvature tensor upon the nonrotating orthonormal tetrad frame of a geodesic observer. The behavior of the measured components of the curvature tensor under Lorentz boosts is briefly described in connection with the existence of certain special tidal directions. Relativistic gravity gradiometry in the exterior gravitational field of a rotating mass is discussed and a gravitomagnetic beat effect along an inclined spherical geodesic orbit is elucidated.Comment: 18 pages, invited contribution to appear in "Relativistic Geodesy: Foundations and Applications", D. Puetzfeld et al. (eds.), 2018; v2: matches version published in: D. Puetzfeld and C. L\"ammerzahl (eds.) "Relativistic Geodesy" (Springer, Cham, 2019), pp. 143-15

Cauchy Horizon Endpoints and Differentiability

Cauchy horizons are shown to be differentiable at endpoints where only a single null generator leaves the horizon. A Cauchy horizon fails to have any null generator endpoints on a given open subset iff it is differentiable on the open subset and also iff the horizon is (at least) of class C 1 on the open subset. Given the null convergence condition, a compact horizon which is of class C 2 almost everywhere has no endpoints and is (at least) of class C 1 at all points. I. Introduction Cauchy horizons and black hole event horizons have been extensively studied and used in relativity [2 -- 6, 10 -- 14, 16, 17]. For general spacetimes, horizons may fail to be stable under small metric perturbations, however, some sufficiency conditions for various stability questions have been obtained [1], [7]. In the present paper, we will consider some differentiability questions for Cauchy horizons. Let (M; g) be a spacetime with a partial Cauchy surface S. The future Cauchy development D + (S..