Explicit isometric embeddings of collapsing dust ball

Abstract

The work is devoted to the search for explicit isometric embeddings of a metric corresponding to the collapse of spherically symmetric matter with the formation of a black hole. Two approaches are considered: in the first, the embedding is constructed for the whole manifold at once; in the second, the idea of a junction of solutions, obtained separately for areas inside and outside the dust ball, is used. In the framework of the first approach, a global smooth embedding in 7D space with a signature (2 + 5) was constructed. It corresponds to the formation of the horizon as a result of matter falling from infinity. The second approach generally leads to an embedding in 7D space with the signature (1 + 6). This embedding corresponds to the case when matter flies out of a white hole with the disappearance of its horizon, after which the radius of the dust ball reaches its maximum, and then a collapse occurs with the formation of the horizon of a black hole. The embedding obtained is not smooth everywhere --- it contains a kink on the edge of the dust ball, and {also, it is} not quite global. In the particular case, when the maximum radius of the dust ball coincides with the radius of the horizon, it is possible to construct a global smooth embedding in a flat 6D space with a signature (1 + 5).

arXiv commentLaTeX, 18 pages

References in corpus (18)

• Study on anisotropic strange stars in $f(R,T)$ gravity: An embedding approach under simplest linear functional of matter-geometry coupling
• Mapping Hawking into Unruh Thermal Properties
• Generating physically realizable stellar structures via embedding
• A new class of relativistic model of compact stars of embedding class I
• Geodetic Brane Gravity
• Equivalence of Hawking and Unruh Temperatures and Entropies Through Flat Space Embeddings
• Canonical formulation of the embedded theory of gravity equivalent to Einstein's General Relativity
• Ostrogradski approach for the Regge-Teitelboim type cosmology
• Gravity as a field theory in flat space-time
• Embeddings for Schwarzschild metric: classification and new results
• Embedding theory as new geometrical mimetic gravity
• Hawking into Unruh mapping for embeddings of hyperbolic type
• When does the Hawking into Unruh mapping for global embeddings work?
• Global Embedding of the Reissner-Nordström Metric in the Flat Ambient Space
• GEMS embeddings and freely falling temperatures of Schwarzschild(-AdS) black holes in massive gravity
• Conformally isometric embeddings and Hawking temperature
• Classification of global minimal embeddings for nonrotating black holes
• Embedding into flat spacetime and black hole thermodynamics

Cited by in corpus (3)

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