Overview
This research investigates the fundamental group of specific classes of spacetimes, specifically focusing on compact decent 4-dimensional spacetimes that possess lightlike hypersurface curvature. The study identifies properties of the fundamental group under certain geometric conditions, including the nature of foliations within these spacetimes and characteristics of pp-wave spacetimes.
Approach
The methodology employed involves the study of spacetimes foliated by totally geodesic null hypersurfaces. A specific condition for these foliations is considered: when the leaves are compact. Additionally, the research examines non-totally vicious 4-dimensional compact pp-wave spacetimes.
Findings
- For compact decent 4-dimensional spacetimes exhibiting lightlike hypersurface curvature, if these spacetimes are foliated by totally geodesic null hypersurfaces with compact leaves, then their fundamental group is proven to be polycyclic.
- The fundamental group of a non-totally vicious 4-dimensional compact pp-wave spacetime is also proven to be polycyclic.
Why This Matters
The paper contributes to the understanding of the topological structure, specifically the fundamental group, of certain spacetimes. These findings provide insight into the algebraic properties of the fundamental group in relation to specific geometric characteristics, such as lightlike hypersurface curvature and foliation structures, within 4-dimensional spacetimes.