The Gravitational Field of String Matter when the Dilaton is Massive

We study numerically the gravitational field of a star made of massive and neutral string states for the case in which the dilaton is massive. The solution exhibits very simple scaling properties in the dilaton mass. There is no horizon and the singularity is surrounded by a halo (the physical size of which is inversely proportional to the dilaton mass) where the scalar curvature is very large and proportional to the square of the dilaton mass. 1. Neutral and scalar string excitations which are massive already in ten dimensions [1] represent the simplest form of that exotic state of matter in which the constituents are string states not belonging to the massless sector of the string spectrum to which instead ordinary matter, like quarks and leptons, presumably belongs (their masses originating through some still unknown symmetry breaking mechanism). The physical relevance of such string matter stems from several studies [2] which indicate that, at high energy density, the most probable configuration in string theory is the one in which most of the massive states are excited. According to this view, an electrically neutral collapsing star of sufficiently large mass would start out as a celestial body made of ordinary matter but would eventually evolve into a string star in which most of its mass is now carried by neutral string excitations. It is therefore of some interest, we believe, to study the gravitational field around such a string star. The scalar, neutral and massive string excitations couple only to gravitons and dilatons; therefore, the gravitational field around a star made of a large number of such states is described by Einstein’s equations in which the energy-momentum tensor is the one for a scalar field. In the original string scenario, these equations arise as conditions for the vanishing of the one-loop beta functional required by Weyl invariance; the dilaton is massless and the field equations admit an exact solution [3, 4]. This solution, and its relationship to string theory, has been discussed recently in [5], to which this letter is closely related. The most significant feature of such a solution is the absence of horizon. The relevant elements grr and g00 of the space-time metric are shown in fig.1, where they are compared to the Schwarzschild’s ones. As it is possible to see from the figure, while the two solutions are equivalent at large distances, they are remarkably different closer to the gravitational radius: at the horizon Schwarzschild’s grr diverges and −g00, which is just its inverse, crosses into negative values, whereas the ones corresponding to the string star solution are never either negative or infinite. These results seem to be in agreement with previous work [6] in which the existence of a static and uncharged static solution with a scalar field has been shown to be incompatible with the presence of a nonsingular horizon (see, also, [4]). It is tempting to speculate that also in a realistic scenario, derived from string theory, in which ordinary matter, after obtaining a mass much lower than Planck mass by some yet unknown mechanism, continues to behave with respect to the