Ultracompact vector stars

We analytically investigate a new family of horizonless compact objects in vector-tensor theories of gravity, called ultracompact vector stars. They are sourced by a vector condensate, induced by a nonminimal coupling with gravity. They can be as compact as black holes, thanks to their internal anisotropic stress. In the spherically symmetric case their interior resembles an isothermal sphere, with a singularity that can be resolved by tuning the available integration constants. The star interior smoothly matches to an exterior Schwarzschild geometry, with no need of extra energy momentum tensor at the star surface. We analyze the behavior of geodesics within the star interior, where stable circular orbits are allowed, as well as trajectories crossing in both ways the star surface. We analytically study stationary deformations of the vector field and of the geometry, which break spherical symmetry, and whose features depend on the vector-tensor theory we consider. We introduce and determine the vector magnetic susceptibility as a probe of the star properties, and we analyze how the rate of rotation of the star is affected by the vector charges.

Figure 1

The time-time metric component $A(r)$ (see our metric Ansatz in Eq. (2.5)) both in the interior and exterior of the star. We make the following choices (from left-to-right and top-to-bottom) of the exponent $\gamma$: $\gamma =10.2$, 1.2, 0.2, 0.02. Red dashed line: the Schwarzschild radius $r=2M$. Black dashed line: the position of the star surface $r=R$.

Figure 2

The time-time metric profile $A(r)$ of Eq. (2.34) evaluated nearby the origin, choosing $\gamma =2$ and $\sigma =0.2$ (left panel) and $\sigma =0.8$ (right panel). We represent only the region $r\ge 0$ since the configurations are symmetric with respect to the axis $r=0$ and can be continuously connected to negative values of $r$. Black dashed line: the position of the star surface $r=R$.

Figure 3

Particle potential for timelike geodesics. Left panel: $\gamma =\ell =1/8$. Right panel $\gamma =2$, $\ell =1.4$. Black dotted line: star surface. Red dashed line: Schwarzschild radius. On the right we include in magenta dot-dashed line the innermost stable circular orbit position calculated within GR, and located at $y=6MR$. Both panels makes manifest that there are extrema of the geodesic potential located within the star radius.

Figure 4

Plots of the vector magnetic susceptibility of Eq. (4.12), as function of $\gamma$ (left panel) and $Q_E$ (right panel). Left: $Q_E=1/10$ (black); $Q_E=1$ (red dashed); $Q_E=10$ (blue dotted). Right: $\gamma =1/10$ (black); $\gamma =1$ (red dashed); $\gamma =10$ (blue dotted).