On the imbedding of the Schwarzschild space-time II.

Abstract

1. In the previous paper, we obtained the equation (4. 3) :( 1)z = y ( t , r)+ r(sin 0 sin cpe 4 + sin 0 cos cpe,+ cos 0e 6

Embeddings for the Schwarzschild metric: classification and new results

“…This embedding was first suggested in [16] (where γ = √ 2). It has also been studied in details in [27] (where γ = √ 2/R).…”

“…Using such parametrization the embedding function corresponding to V 1 and constructed following ( 19), (16), can be written for some of values r as y 0 = f (r) sinh(βt + w(r)) cos θ, y 3 = f (r) cosh(βt + w(r)) cos θ, y 1 = f (r) sinh(βt + w(r)) sin θ cos ϕ, y 4 = f (r) cosh(βt + w(r)) sin θ cos ϕ, y 2 = f (r) sinh(βt + w(r)) sin θ sin ϕ, y 5 = f (r) cosh(βt + w(r)) sin θ sin ϕ,…”

“…Consider the variant W 3 . According to (20), (21) in this case the initial vector y 0 is B 0 C (r) ⊕ D 2 F (r), resulting in the embedding function which, according to ( 18), ( 11), (19), (16) for some values of r can be written as…”

“…For both mentioned embeddings N = 6, i. e. the dimension of the ambient space is minimal. For N = 6 two more embeddings are known, they have been suggested in [16] and [17], see details in Sections 4.6 and 4.4 respectively. Note that in [14,6,17] the embeddings were demonstrated without any method of their obtaining, while in [16] some method is used, but, for example, it does not give the embedding [17].…”

Embeddings for solutions of Einstein equations

“…is a function that is smooth for r > R. This embedding was first proposed by Fujitani, Ikeda, and Matsumoto [27] (there γ = √ 2). The manifold defined by this embedding covers only the domain r > R and has a conical singularity at r = R. This embedding is not asymptotically flat and can be written in terms of elementary functions.…”