We find new, simple cosmological solutions with flat, open, and closed spatial geometries, contrary to the previous wisdom that only the open model is allowed. The metric and the Stückelberg fields are given explicitly, showing nontrivial configurations of the Stückelberg in the usual FriedmannLemaître-Robertson-Walker coordinates. The solutions exhibit self-acceleration, while being free from ghost instabilities. Our solutions can accommodate inhomogeneous dust collapse represented by the Lemaître-Tolman-Bondi metric as well. Thus, our results can be used not only to describe homogeneous and isotropic cosmology but also to study gravitational collapse in massive gravity.It is very intriguing to explore whether or not the graviton can have a mass. The first attempt to add a mass term to the gravity action was made by Fierz and Pauli [1], who considered the quadratic action for the graviton h µν in flat space with the mass termThe linear theory with the Fierz-Pauli mass term is ghost-free. However, the theory does not reproduce general relativity in the massless limit m → 0. The extra three degrees of freedom in a massive spin 2 survive even in this limit, and therefore the prediction for light bending is away from that of general relativity, which clearly contradicts solar-system tests. This is called the vDVZ discontinuity [2]. As pointed out by Vainshtein [3], the discontinuity can in fact be cured by going beyond the linear theory. Massive gravity has a new length scale called the Vainshtein radius, below which the nonlinearities of the theory come in and the effect of the extra degrees of freedom is screened safely. The Vainshtein radius becomes larger as m gets smaller, and thereby a smooth massless limit is attained.However, the very nonlinearities turned out to cause another trouble. Boulware and Deser argued that there appears a sixth scalar degree of freedom at nonlinear order, which has a wrong sign kinetic term, i.e., the sixth mode is a ghost [4]. The ghost issue was emphasized in the effective field theory approach in Ref. [5]. The presence of the Boulware-Deser (BD) ghost has hindered us from constructing a consistent theory of massive gravity.Recently, a theoretical breakthrough in this field has been made. Adding higher-order self-interaction terms and tuning appropriately their coefficients, de Rham and collaborators successfully eliminated the dangerous scalar mode from the theory in the decoupling limit [6,7]. Then, Hassan and Rosen established a complete proof that the theory does not suffer from the BD ghost instability to all orders in perturbations and away from the decoupling limit [8]. Thus, there certainly exists a nonlinear theory of massive gravity that is free of the BD ghost.In addition to the theoretical interests described above, the mystery of the accelerated expansion of the Universe [9] motivates massive gravity theories as a possible alternative to dark energy. Since the attractive force mediated by a massive graviton is Yukawa-suppressed by a factor e −mr , massive gravity t...
The topology of event horizons is investigated. Considering the existence of the end point of the event horizon, the event horizon cannot be differentiable. Then there are new possibilities for the topology of the event horizon, excluded in smooth event horizons. The relation between the spatial topology of the event horizon and its end points is revealed. A toroidal event horizon is caused by two-dimensional end point sets. One-dimensional end point sets provide the coalescence of spherical event horizons. Moreover, these aspects can be removed by an appropriate time slicing. The result will be useful to discuss the stability and generality of the topology of the event horizon. ͓S0556-2821͑98͒07620-6͔PACS number͑s͒: 04.20.Gz, 02.40.Ma A. The Poincaré-Hopf theoremOur investigation is based on a well-known theorem regarding the relation between the topology of a manifold and *E-mail address: siino@yukawa.kyoto-u.ac.jp 1 The TOEH means the topology of the spatial section of the EH throughout the present article. Of course, it may depend on a time slicing.
We construct a simple expression for the spherical gravitational collapse in a single coordinate patch. To describe the dynamics of collapse, we use a generalized form of the Painlevé-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. We show the both cases corresponding to the gravitational collapse from infinity and from a finite radius.Subject Index: 420, 451 §1. IntroductionOne of the most important predictions of general relativity is black hole. Numerous studies have been done on the properties of the black hole and the formation by gravitational collapse. The theory of gravitational collapse was initiated in 1939 by Oppenheimer and Snyder 1) as a spherical contraction model of a uniformly distributed dust star. The standard method in Refs. 1) and 2) is making a physically reasonable junction of the two different spacetimes corresponding to the interior and exterior regions of the collapsing body. The interior dust solution is given by the Friedmann-Robertson-Walker metric in the synchronous comoving coordinates. We impose a junction condition at the surface of the star so that the solution connects smoothly to the exterior Schwarzschild solution. The interior and exterior solutions are described in different coordinate systems. Although it is nothing wrong to construct solutions in such a manner, one cannot describe the dynamics of the collapsing star in terms of the coordinates of the observer outside the event horizon. The main purpose of this paper is to analytically describe the both regions inside and outside the horizon by a single coordinate system in a physical way.The Painlevé-Gullstrand coordinates used in this paper is, in fact, the key to a simple physical picture of black hole and gravitational collapse. It was first introduced by Painlevé 3) and Gullstrand 4) in 1921. We leave the details of its history to Ref. 5), but mention here only its notable properties. Unlike the Schwarzschild form, the Painlevé-Gullstrand metric tensor has an off-diagonal element and spatially flat elements so that it is regular at the Schwarzschild radius and has a singularity only at the origin of the spherical coordinates. In other words, the surfaces t = constant traverse the event horizon to reach the singularity. Therefore, the Painlevé-Gullstrand coordinates are convenient for exploring the geometry of collapsing star and black hole both inside and outside the horizon altogether by a single coordinate patch. Moreover, the space given by the Painlevé-Gullstrand coordinates can be intuitively regarded as a river whose speed of current is the Newtonian escape velocity at each
We study the collision of black holes in the Kastor-Traschen space-time, at present the only such analytic solution. We investigate the dynamics of the event horizon in the case of the collision of two equal black holes, using the ray-tracing method. We confirm that the event horizon has trouser topology and show that its set of past end points ͑where the horizon is nonsmooth͒ is a spacelike curve resembling a seam of trousers. We show that this seam has a finite length and argue that twice this length be taken to define the minimal circumference C of the event horizon. Comparing with the asymptotic mass M , we find the inequality CϽ4M supposed by the hoop conjecture, with both sides being of similar order, Cϳ4M . This supports the hoop conjecture as a guide to general gravitational collapse, even in the extreme case of head-on black-hole collisions.
We investigate the quantum effects on the so-called critical phenomena in black hole formation. Quantum effects of a scalar field are treated semiclassically via a trace anomaly method. It is found that the demand of regularity at the origin implies the disappearance of the echo. It is also found that semiclassical equations of motion do not admit continuously self-similar solutions. The quantum effects would change the critical solution from a discretely self-similar one to a solution without critical phenomena.
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