2021/12/16 by Abbas M. Sherif, Peter K. S. Dunsby · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algorithm #Artificial intelligence #Black Holes and Theoretical Physics #Computer science #Geometric Analysis and Curvature Flows #Physics #gr-qc #math.DG
paper · pdf · open access · doi:10.1088/1361-6382/ac45db
published in Classical and Quantum Gravity 39(4), 045004 (IOP Publishing) · 21 pages, no figure, Accepted for publication in CQG
arxiv created 2021/12/16 · openalex publication_date 2021/12/22 · arxiv updated 2022/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this work, we study various properties of embedded hypersurfaces in 1+1+2 decomposed spacetimes with a preferred spatial direction, denoted eμ, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure a non-vanishing positivity scalar curvature of the induced metric, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor satisfies φ2+2φ>0, where ∗ denotes derivative along eμ. Firstly, it is demonstrated that such hypersurface is either Einstein or the twist vanishes on it, and that the scalar curvature of the induced metric is constant. It is then proved that if the hypersurface is compact and of Einstein, and admits a proper conformal transformation, then the hypersurface must be isomorphic to the 3-sphere, where we make use of well known results on Riemannian manifolds admitting conformal transformations. If the hypersurface is not Einstein and has nowhere vanishing sheet expansion, we show that this conclusion fails. However, with the additional conditions that the scalar curvatures of the induced metric and the conformal metric coincide, the associated conformal factor is strictly negative and the third and higher order derivatives of the conformal factor vanish, the conclusion follows. Furthermore, additional results are obtained under the conditions that the scalar curvature of a metric conformal to the induced metric is also constant. Finally, we consider some of our results in context of locally rotationally symmetric spacetimes and show that, if the hypersurfaces are compact and not of Einstein type, then under specified conditions the hypersurface is isomorphic to the 3-sphere, where we constructed explicit examples of proper conformal Killing vector fields along eμ.