The equations of general relativity are nonlinear second-order partial differential equations, and, as a consequence, obtaining the exact solutions is a difficult problem. One of the solutions to this problem is to obtain models with a thin self-gravitating shell. This method is used to study most of the phenomena in the theory of gravity, where the reverse effect of matter on the geometry of space-time is a key factor. Another interesting problem that can be studied using the thin shell method is the «simulation» of a black hole. Consider a system consisting of a spherically symmetric Schwarzschild black hole and a thin shell surrounding it, located at a certain fixed distance from the black hole. From the viewpoint of gravitational physics, an observer at infinity is unable to distinguish a real black hole from a wormhole with a thin shell, in which the simulation condition is satisfied. Simulation of a black hole is possible only under sufficiently stringent conditions for the parameters of the model. In particular, the shell needs to be held at a fixed radius. In the general case, such a movement of the shell is non-geodesic, and external forces are required to hold it. The radius of the shell is also a parameter that determines the possibility / impossibility of simulation. In this paper, the radius is found for the case of a Schwarzschild black hole. In particular, the paper considers a model of a wormhole obtained as a result of gluing two space-times: a Schwarzschild black hole and a Damour-Solodukhin wormhole. The latter solution differs from the Schwarzschild black hole in the parameter of the dimensionless real deviation λ and is a twice asymptotically flat regular space-time. It is shown that they can be glued along a given radius. As a result, a thin shell is formed between two glued manifolds consisting of exotic matter. Cases are considered when the thin shell is stable. It turns out that zones corresponding to the «force» constraint are more restrictive than those corresponding to the «mass» constraint.
GEODESIC MOTION IN THE GLOBALLY REGULAR SPACE-TIME OF A SCHWARZSCHILD BLACK HOLE
