Exact solutions for isometric embeddings of pseudo-Riemannian manifolds

7th International Conference on Gravitation and Cosmology

Amery, G.^1 ; Moodley, J.^1

Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa^1

关键词: Embedding problems;    Explicit solutions;    Gauss-bonnet gravities;    General Relativity;    Higher dimensions;    Higher-dimensional;    Isometric embeddings;    Riemannian manifold;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/484/1/012009/pdf
DOI  :  10.1088/1742-6596/484/1/012009

来源: IOP

【 摘 要 】

Embeddings into higher dimensions are of direct importance in the study of higher dimensional theories of our Universe, in high energy physics and in classical general relativity. Theorems have been established that guarantee the existence of local and global codimension-1 embeddings between pseudo-Riemannian manifolds, particularly for Einstein embedding spaces. A technique has been provided to determine solutions to such embeddings. However, general solutions have not yet been found and most known explicit solutions are for embedded spaces with relatively simple Ricci curvature. Motivated by this, we have considered isometric embeddings of 4-dimensional pseudo-Riemannian spacetimes into 5-dimensional Einstein manifolds. We have applied the technique to treat specific 4-dimensional cases of interest in astrophysics and cosmology (including the global monopole exterior and Vaidya-de Sitter-class solutions), and provided novel physical insights into, for example, Einstein-Gauss-Bonnet gravity. Since difficulties arise in solving the 5-dimensional equations for given 4-dimensional spaces, we have also investigated embedded spaces, which admit bulks with a particular metric form. These analyses help to provide insight to the general embedding problem.