abstract

This paper compares three criteria for a spacetime to be free of position drift: those by Hasse and Perlick (HP), Krasiński and Bolejko (KB), and Korzyński and Kopiński (KK). A spacetime having no position drift means that every observer sees all light sources in unchanging directions. The following is shown: (1) The HP criterion is a necessary condition for the KK criterion to apply. (2) If the spacetime metric obeys the Einstein equations with a perfect fluid source, then another necessary condition for the KK criterion is the Weyl tensor being zero. (3) Result (2) points to the Stephani metric, so it is shown that this metric obeys an equation which is still one more necessary condition for the KK criterion. (4) The general Szekeres metrics become drift-free by the KK criterion only in the Friedmann limit. (5) The HP and KB criteria coincide, and the HP zero-drift condition imposes on the Stephani metric the same restriction as found by Krasiński and Bolejko (KB). The relations between the three criteria are displayed and compared in a diagram.

abstract

The initial time-dependence of a state in circumstances where it makes transitions to, or decay to, a second state has been investigated. In classical stochastic processes, the observed time-dependence of transition or decay proportional to \(t^2\) is attributed to the noise with memory. In contrast to quantum mechanics, the quadratic form of initial decay is unable to decelerate the evolution of the system.

abstract

We study umbilically synchronized space-times \(M\). First, we show that \(M\) with vanishing electric part of the Weyl tensor is conformally flat if either \(\mathrm {dim}~M =4\) or spatial slices \({\mit \Sigma }\) are conformally flat. Next, for the vacuum case, we show that the scalar curvature of spatial slices \({\mit \Sigma }\) is a non-positive function of time \(t\) (this includes the case when \(M\) is Schwarzchild exterior space-time), and if, in addition, \(M\) is geodesic (acceleration-free) and electric part of the Weyl tensor vanishes, then \(M\) is a Lorentzian cone over a hyperbolic space which is, in dimension 4, an expanding hyperbolic cosmological model. Finally, we provide some characterizations of conformal (including inheriting conformal) vector fields of an umbilically synchronized space-time.