Exterior solutions of ultra-compact object candidate from semi-classical gravity

In a recent paper by Carballo-Rubio [Phys. Rev. Lett. 120, 061102 (2018)], the author proposed an ultra-compact object model, i.e., a combination of black stars and gravastars, from semi-classical gravity to obtain a generalized Tolman-Oppenheimer-Volkoff (TOV) equation with a new coupling constant lp. The resulting TOV equations have two different forms differentiated by the sign in metric function equation dgtt/dr. In the limit lp 0, the second (respectively, the first) form of the equations from the negative (positive) sign is (is not) going back to the TOV equation. By defining a suitably new constant parameter , the author has found a solution from the first form of obeying boundary conditions. In this work, we investigate the model in a vacuum to obtain exterior solutions for the model. We calculate its exterior solutions using the perturbation method by a small parameter =lp/rs where rs=2GM is the Schwarzschild radius and obtain the Schwarzschild metric as its leading terms. We also investigate its geodesic equations, whose effective potential from our exterior solutions has similar qualitative features as the geodesic of the Schwarzschild metric, i.e., it contains stable and unstable circular orbits.