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In a talk given in 2013, S. Hawking conjectured that the event horizon ofblack holes does not exist and suggested redefining black holes as bound statesof the gravitational field. Inspired by this idea, we investigated the couplingof the Bardeen action and a complex scalar field model. Numerically, weobtained a class of boson stars solutions with the magnetic monopole charge$q$. When the constant $q$ exceeds a certain threshold, we observed that as thefrequency approaches zero, a critical position $r_c$ emerges where the scalarfield concentrates within its interior. Outside this critical position, theseboson star solutions tend to infinitely approach what is known as an extremeblack hole. However, there is no event horizon present. While our results aremodel-dependent and their generality remains uncertain, they align well withHawking's conjecture that real, regular black holes do not have an eventhorizon and provided valuable insights into the understanding and developmentof concepts such as fuzzballs, firewalls and black hole soft hairs.

From Bardeen-boson stars to black holes without event horizon