Abstract
The possibility of an equilibrium state of a gravitating scalar field (describing ordinary matter) inside a black hole, compressed to the state of boson condensate, in balance with a longitudinal vector field (describing dark matter) from the outside, is considered. Analytical analysis, confirmed numerically, shows that there are regular static solutions to the Einstein equations with no limitation on the mass of a black hole. The metric tensor component grr(r) changes sign twice. The behavior of the gravitational field and material fields in the vicinity of these two Schwarzschild radii were studied in detail. The equality of the energy–momentum tensors of the scalar and longitudinal vector fields at the interface supports the phase equilibrium of a black hole and dark matter. Considering the gravitating scalar field as an example, a possible internal structure of a black hole and its influence on the dark matter at the periphery of a galaxy are clarified. In particular, the speed on the plateau of a galaxy rotation curve as a function of a black hole’s mass is determined.
Highlights
Astrophysical observations indicate the existence of super-massive objects at the centers of galaxies.In our Milky Way, there is an invisible object located in the same place as the radio source “SagittariusA”
A black hole is a process of unlimited compression of matter under the action of dominant forces from its own gravitational field
The Lagrangian (15) of a scalar field does not depend on the derivatives of the metric tensor
Summary
Astrophysical observations indicate the existence of super-massive objects at the centers of galaxies. Turning grr (r) to zero, followed by the change of the metric signature, can take place inside a body, or on its surface, or on the interface of two gravitating objects As it follows from the identity in Equation (12), in the presence of a layer with a different signature, the radius rh is not connected with the whole mass of a gravitating object, but only with its part inside the layer. The presence of dark matter opens up a possibility for existence of static solutions to the Einstein equations describing the gravitational field of spherically symmetric matter with no limitation of mass An example of such a solution, in which the metric component grr (r) changes sign twice (firstly, inside a black hole and, secondly, on the interface with dark matter), is investigated below. Let us demonstrate this considering a gravitating scalar field as an example of ordinary (not dark) matter inside a black hole
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