Do naked singularities generically occur in generalized theories of gravity?

A new mechanism for causing naked singularities is found in an effective superstring theory. We investigate the gravitational collapse in a spherically symmetric Einstein-Maxwell-dilaton system in the presence of a pure cosmological constant “potential”, where the system has no static black hole solution. We show that once gravitational collapse occurs in the system, naked singularities necessarily appear in the sense that the field equations break down in the domain of outer communications. This suggests that in generalized theories of gravity, the non-minimally coupled fields generically cause naked singularities in the process of gravitational collapse if the system has no static or stationary black hole solution. The singularity theorem [1] states that the occurrence of singularities is inevitable under some physical conditions in general relativity. There are two notable scenarios where singularities may appear in our universe. One is the initial singularity at the birth of the universe and the other is the final stage of gravitational collapse. In the latter case we believe that an event horizon is formed which encloses all occurring singularities as the collapse proceeds, following the cosmic censorship hypothesis (CCH) [2]. CCH is classified into two types, the weak cosmic censorship hypothesis (WCCH) and the strong one (SCCH). WCCH says that observers at an infinity should not see singularities while SCCH says that no observer should see them and the whole region of a space-time can be uniquely determined by initial regular data. Mathematically, SCCH is equivalent to the statement that no Cauchy horizon can be formed in a physical gravitational collapse. Recently many elegant results [3] suggest that SCCH holds for charged and/or rotating black holes due to the destruction of the Cauchy horizon by the mass inflation phenomenon. The general proof of CCH is, however, far from complete and many counter examples have been found in the framework of general relativity [4]. A new approach has been considered in the context of generalized theories of gravity. Gibbons and Maeda [5] discovered the static black hole solutions in the EinsteinElectronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] Maxwell-dilaton system, which comes from an effective superstring theory, and later Garfinkel, Horowitz and Strominger [6] showed that the inner (Cauchy) horizon in the Reissner-Nordström solution is replaced by a spacelike singularity. This suggests that the occurrence of the inner horizon is not generic and hence SCCH holds if we take the effect of string theory into account. On the other hand Horne and Horowitz [7] obtained the opposite result that extremal electrically charged non-static black hole solutions in the presence of a central charge have timelike singularities. It seems, however, not to be a counter example of SCCH in the sense that such solutions have no regular initial spacelike hypersurface because of the central singularities. Thus, the following question naturally arises. Does CCH really hold in generalized theories of gravity? There is not much evidence deciding the matter yet because the above results do not take any physical process of the gravitational collapse from initial regular data into account. Although a large number of studies have been made on finding static/stationary black hole solution and investigating their thermodynamical properties and geometrical stability and so on in generalized theories of gravity, only few studies have so far been made on investigating gravitational collapse. We should not overlook that the gravitational collapse in generalized theories of gravity will be much different from that in general relativity. In general relativity, the black hole no-hair conjecture states that the matter fields are swallowed by a black hole and a space-time asymptotically approaches the electrovacuum Kerr-Newman solution after gravitational collapse. On the other hand, the above aspects are not necessarily satisfied in generalized theories of gravity because the dilaton field couples to the curvature and/or other matter fields. In this letter we investigate the gravitational collapse in the spherically symmetric Einstein-Maxwell-dilaton system analytically in the presence of a pure positive cosmological constant, which corresponds to g1 = 0 in the Liouville-type potential [8]. This system is interesting because it has been proved that no static black hole solution exists for spherically symmetric space-times [9]. Such property seems to be general in the sense that there exists no asymptotically flat, asymptotically de Sitter, or asymptotically anti-de Sitter solution in the arbitrary exponential dilaton potential [8]. Furthermore, it is known that there is a critical mass below which no asymptotically flat black hole solution exists in the system which