Dark energy stars are finite size astrophysical objects with an interior equation of state typical of dark energy. Examples are self-gravitating false vacuum bubbles, vacuum nonsingular black holes, and gravastars. We present a time-dependent solution of Einstein’s field equations that describes the collapse of a spherical system from an initial state of positive pressure to a final state with a dark energy core. Our solution has no singularities, no event horizons, and does not violate the weak or null energy conditions.

We solve Einstein’s equations for slowly rotating gravitational condensate stars (gravastars) up to second order in the rotation by expanding about the spherically symmetric gravastar with de Sitter interior and Schwarzschild exterior matched at their common horizon. Requiring that the perturbations are finite on the null surface reduces the exterior geometry to that of a Kerr black hole, implying that a slowly rotating gravastar cannot be distinguished from a Kerr black hole by any measurement or observation restricted to the macroscopic spacetime exterior to the horizon. We determine the interior solution, the surface stress tensor, and the Komar mass and angular momentum localized on the slowly rotating horizon surface. With the interior equation of state fixed at $p=-\rho$, finite junction conditions on the null horizon surface necessarily lead to an interior solution with a singular core, where the perturbative expansion breaks down. Comparison to other models and implications for more rapidly rotating gravastars are briefly discussed.

The Newman-Janis algorithm and its generalizations can be used mathematically to generate rotating solutions from nonrotating spherically-symmetric solutions within general relativity. The energy-momentum tensors of these solutions may or may not represent the same physical system, in the sense of both being a perfect fluid, or an electromagnetic field, or a $\mathrm{\Lambda }$-term, and so on. In a series of two papers, we compare the structure of the eigenvalues and eigenvectors of the rotating and nonrotating energy-momentum tensors (their Segre types) and look for the existence of equations of state relating the energy density and the principal pressures. Part I covers Kerr-Schild systems, Part II more general systems. We find that there is a unique family of stationary axisymmetric Kerr-Schild systems that obey the same equation of state in both the rotating and nonrotating configurations. This family includes the Kerr and Kerr-Newman black holes, as well as rotating spacetimes whose mass function in the nonrotating limit contains a constrained superposition of a cloud of strings term, a Reissner-Nordstrom term, a cosmological constant term, and a Schwarzschild term. We describe the common equation of state relating energy density and pressure in this family of spacetimes and discuss some of its properties.

The general form of the surface stress tensor of an infinitesimally thin shell located on a rotating null horizon is derived, when different interior and exterior geometries are joined there. Although the induced metric on the surface must be the same approached from either side, the first derivatives of the metric need not be. Such discontinuities lead to a Dirac $\delta$-distribution in the Einstein tensor localized on the horizon. For a general stationary axisymmetric geometry the surface stress tensor can be expressed in terms of two geometric invariants that characterize the surface, namely the discontinuities [$\kappa$] and $[\mathcal{J}]$ of the surface gravity $\kappa$ and angular momentum density $\mathcal{J}$. The Komar energy and angular momentum are given in coordinates adapted to the Killing symmetries, and the surface contributions to each determined in terms of [$\kappa$] and $[\mathcal{J}]$. Guided by these, a simple modification of the original Israel junction conditions is verified directly from the Einstein tensor density to give the correct finite result for the surface stress, when the normal $\mathit{n}$ to the surface is allowed to tend continuously to a null vector. The relation to Israel’s original junction conditions, which fail on null surfaces, is given. The modified junction conditions are suitable to the matching of a rotating “black hole” exterior to any interior geometry joined at the Kerr null horizon surface, even when the surface normal is itself discontinuous and the Barrabès-Israel formalism is also inapplicable. This joining on a rotating null horizon is purely of the matter shell type and does not contain a propagating gravitational shock wave.

The present paper is devoted to a study of the equilibrium configurations of slowly rotating anisotropic stars in the framework of general relativity. For that purpose, we provide the equations of structure where the rotation is treated to second order in the angular velocity. These equations extend those first derived by Hartle for slowly rotating isotropic stars. As an application of the new formalism, we study the rotational properties of Bowers-Liang fluid spheres. A result of particular interest is that the ellipticity and mass quadrupole moment are negative for certain highly anisotropic configurations; thus, such systems are prolate rather than oblate. Furthermore, for configurations with high anisotropy and compactness close to their critical value, quantities like the moment of inertia, change of mass, and mass quadrupole moment approach to the corresponding Kerr black hole values, similar to other ultracompact systems like sub-Buchdahl Schwarzschild stars and analytic rotating gravastars.

We present a time-dependent uniform-density interior Schwarzschild solution, an exact solution to the Einstein field equations. Our solution describes the collapse (or the time-reversed expansion) of an object from an infinite radius to an intermediate radius of $9/8$ of the Schwarzschild radius, at which time a curvature singularity appears at the origin, and then continues beyond the singularity to a gravastar with radius equal to the Schwarzschild radius.