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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mo>−</mo><mi>ε</mi><mo>/</mo><mn>3</mn></mrow></semantics></math></inline-formula>. Pressure <i>p</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mi>g</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mi>h</mi></msub><mo>></mo><msub><mi>r</mi><mi>g</mi></msub></mrow></semantics></math></inline-formula>, where the solution exists, but it is not unique. In a synchronous reference frame, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><msub><mi>g</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mrow><mi>r</mi><mi>r</mi></mrow></msup></mrow></semantics></math></inline-formula> do not change signs. The non-uniqueness of solutions with boundary conditions at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>=</mo><msub><mi>r</mi><mi>g</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>=</mo><msub><mi>r</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula> 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 “<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><msup><mrow><mfenced close="|" open="|"><mi>ψ</mi></mfenced></mrow><mn>4</mn></msup></mrow></semantics></math></inline-formula>”, total mass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mn>3</mn><mfenced><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn><mi>k</mi></mrow></mfenced><mo> </mo><msub><mi>r</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula> 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.