Bose-Einstein Condensate in Synchronous Coordinates

Analytical spherically symmetric static solution to the set of Einstein and Klein-Gordon equations in a synchronous reference frame is considered. In a synchronous reference frame, a static solution exists in the ultrarelativistic limit p=−ε/3 . Pressure p is negative when matter tends to contract. The solution pretends to describe a collapsed black hole. The balance at the boundary with dark matter ensures the static solution for a black hole. There is a spherical layer inside a black hole between two “gravitational” radii rg and rh>rg , where the solution exists, but it is not unique. In a synchronous reference frame, detgik and grr do not change signs. The non-uniqueness of solutions with boundary conditions at r=rg and r=rh makes it possible to find the gravitational field both inside and outside a black hole. The synchronous reference frame allows one to find the remaining mass of the condensate. In the model “ λ ψ 4 ”, total mass M=3c2/2k rh is three times that of what a distant observer sees. This gravitational mass defect is spent for bosons to be in the bound ground state, and for the balance between elasticity and density of the condensate.