Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry

It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation (massless scalar field). In the spherically symmetric case it is further shown that if the spacetime admits a maximal slice then there are crushing singularities both in the past and in the future. The essential properties of the matter models chosen are that their energy-momentum tensors satisfy certain inequalities and that they do not develop singularities in a given regular background spacetime.Comment: 19 page

2 Public Key Encryption Using the Squaring Function First Attempt

In the encryption we’ve been doing so far, the sender and the recipient needed to preagree on a key. This is traditionally called “symmetric ” or “secret-key ” encryption. The idea of public key encryption is that I can walk into a room, announce my key, and everybody in the room can send secret messages to me by simply shouting them out so that everyone can hear them. In other words, we can communicate secretly by publishing messages for everyone to see, even if we never pre-agreed on a shared secret value unknown to others. This sounds impossible, but can actually be done. The idea was first proposed by Diffie and Hellman [DH76]