Analytical investigations on the Maxwell electromagnetic invariant in the spinning and charged horizonless star background

We study properties of the Maxwell electromagnetic invariant in the external region of spinning and charged horizonless stars. We analytically find that the minimum negative value of the Maxwell electromagnetic invariant is obtained on the equator of the star surface. We are interested in scalar fields non-minimally coupled to the Maxwell electromagnetic invariant. The negative enough Maxwell electromagnetic invariant can lead to a negative effective mass term, which forms a binding potential well for the scalar field. It means that the scalar field coupled to the Maxwell electromagnetic invariant may mostly exist around the surface of the star on the equator.