# Diamond-Shaped Regions as Microcosmoi

October 25, 2010

We give a geometrically intrinsic construction of a global time function for
relatively compact diamond-shaped regions in arbitrary spacetimes. In the case
of Minkowski spacetime, the flow of diffeomorphisms associated to a suitably
normalized gradient of this time function becomes the conformal isotropy
subgroup of the diamond. In full generality, this time function is elegantly
expressed in terms of the Lorentzian distance function, and it has an
asymptotic behavior at large absolute times similar to the one in Minkowski
spacetime.

open access link

Preliminary version, comments welcome

Keywords:

Lorentzian geometry, Cauchy time function