Thomas Foertsch

Inertial forces and photon surfaces in arbitrary spacetimes

Given, in an arbitrary spacetime (M, g), a two-dimensional timelike submanifold Σ and an observer field n on Σ, we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline λ in Σ with respect to n. We prove that centrifugal and Coriolis forces vanish, for all λ in Σ with respect to any n, if and only if Σ is a photon 2-surface, i.e., generated by two families of lightlike geodesics. We further demonstrate that a photon 2-surface can be characterized in terms of gyroscope transport and we give several mathematical criteria for the existence of photon 2-surfaces. Finally, examples of photon 2-surfaces in conformally flat spacetimes, in Schwarzschild and Reissner–Nordstrom spacetimes, and in Godel spacetime are worked out.

GROUP ACTIONS ON GEODESIC PTOLEMY SPACES

In this paper we study geodesic Ptolemy metric spaces

Bilipschitz embeddings of negative sectional curvature in products of warped product manifolds

X

Hyperbolic rank of products

which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that

Characterizing complete

X

\operatorname{CAT}(\kappa)

is equivariantly roughly isometric to a Euclidean space.

-spaces,

Generalizing results due to Brady and Farb (1998) we prove the existence of bilipschitz embedded manifolds of negative sectional curvature in Riemannian products of certain types of warped products.

\kappa<0

Generalizing a result of Brady and Farb (1998), we prove the existence of a bilipschitz embedded manifold of negative curvature bounded away from zero and dimension m 1 + m 2 -1 in the product X := X m1 1 x X m2 2 of two Hadamard manifolds X mi i of dimension m i with negative curvature bounded away from zero. Combining this result with a result of Buyalo and Schroeder (2002), we prove the additivity of the hyperbolic rank for products of manifolds with negative curvature bounded away from zero.

, with geodesic boundary

We investigate the Bourdon and Hamenstadt boundaries of complete CAT(κ)-spaces, κ < 0, and characterize those with geodesic Hamenstadt boundary up to isometry.