Non-minimally coupled dark fluid in Schwarzschild spacetime

If one assumes a particular form of non-minimal coupling, called the conformal coupling, of a perfect fluid with gravity in the fluid-gravity Lagrangian then one gets modified Einstein field equation. In the modified Einstein equation, the effect of the non-minimal coupling does not vanish if one works with spacetimes for which the Ricci scalar vanishes. In the present work we use the Schwarzschild metric in the modified Einstein equation, in presence of non-minimal coupling with a fluid, and find out the energy-density and pressure of the fluid. In the present case the perfect fluid is part of the solution of the modified Einstein equation. We also solve the modified Einstein equation, using the flat spacetime metric and show that in presence of non-minimal coupling one can accommodate a perfect fluid of uniform energy-density and pressure in the flat spacetime. In both the cases the fluid pressure turns out to be negative. Except these non-trivial solutions it must be noted that the vacuum solutions also remain as trivial valid solutions of the modified Einstein equation in presence of non-minimal coupling.