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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. I. INTRODUCTIONThere are various studies of compact astrophysical objects in the interior of which the energy density ρ and the pressure p obey an equation of state typical of dark energy such as p = −ρ. Such objects have been variously named in the literature. We refer to them as "dark energy stars" for simplicity.Although dark energy stars could have spacetime singularities, one is commonly interested in dark energy stars that are nonsingular. Buchdahl's theorem [1] precludes the existence of nonsingular compact objects with radius smaller than 9/8 the Schwarzschild radius under the assumptions of spherical symmetry, isotropic stress, and nonnegative trace of the energy momentum tensor. Compact nonsingular dark energy stars are possible because the nonnegative trace condition does not apply for p < −ρ/3 dark energy (compact objects supported by anisotropy instead have also been studied [2]).In the mid 1960s the idea of objects with p = −ρ at their center was put forward by Gliner [3]. The first concrete solution was the Bardeen spacetime [4][5][6][7], which is a nonsingular, asymptotically flat, spherically symmetric spacetime that may have zero, one, or two event horizons depending on the value of a parameter. The Bardeen stress energy tensor features radial pressure p r = −ρ everywhere and tangential pressure p T = p r away from the center.In the 1980s the gravitational effects of false vacuum bubbles forming in true vacuum, and vice versa, were considered [8]. False-vacuum bubbles were studied as a possibility for wormholes [9] and localized inflation [10,11] when it was found that the null energy condition imposes that any spherically symmetric false-vacuum bubble that forms in an asymptotically flat space, and grows beyond a certain critical size, must have emerged from an initial singularity [11]. Smaller false vacuum bubbles may arise without initial singularities [12]. There were also attempts to replace the black hole singularity inside the horizon with a Planckian density vacuum bubble and a junction layer [13][14][15].Starting in the 1990s, compact objects with p = −ρ at their center, called vacuum nonsingular black holes, or lambda black holes [16][17][18][19], were studied within the class of "regular black-hole" solutions, i.e., asymptotically flat spacetimes that, like black holes, possess an event horizon but, unlike black holes, do not have a singularity (see, e.g., Ref.[20] for a review). Lambda black holes, and similar horizonless objects known as G lumps [21][22][23], are s...

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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. Usage PACS numbers Structure I. INTRODUCTIONNearly 100 years after its original discovery, the constant density interior Schwarzschild solution [1] was analyzed in more detail and shown to behave as a gravastar in the limit that the radius approaches the Schwarzschild radius R S = 2GM [2]. If the radius reaches 9/8R S the pressure at the center diverges and the convention was this implied a static solution no longer existed [1,3]. However, the static interior Schwarzschild solution may be maintained, without modification, if one accepts a region of negative pressure [4]. While the interior Schwarzschild solution strictly speaking does not avoid singularities (it is singular where the pressure diverges [5]) it is a simple and mathematically valid solution with potential for high compactness and negative pressures that can be interesting to study in its own right. For example, slowly rotating Schwarzschild stars in the compact gravastar limit behave almost exactly as non black hole extended Kerr sources [6]. Also, if one allows for a Dirac delta function in the transverse stress at the radius of the pressure divergence, the singularity has a well-defined contribution to the Komar integral [2].In this paper, we show that if one allows for a time-dependent radius and for anisotropic stress, the interior Schwarzschild solution generalizes into a new exact solution to the Einstein field equations. Its line element has the same form as the static interior Schwarzschild * † Electronic address:

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