Self-gravitating field configurations: The role of the energy–momentum trace

Abstract

Static spherically-symmetric matter distributions whose energy-momentum tensor is characterized by a non-negative trace are studied analytically within the framework of general relativity. We prove that such field configurations are necessarily highly relativistic objects. In particular, for matter fields with T ≥ α · ρ ≥ 0 (here T and ρ are respectively the trace of the energy-momentum tensor and the energy density of the fields, and α is a non-negative constant), we obtain the lower bound max r {2m(r)/r} > (2 + 2α)/(3 + 2α) on the compactness (mass-to-radius ratio) of regular field configurations. In addition, we prove that these compact objects necessarily possess (at least) two photon-spheres, one of which exhibits stable trapping of null geodesics. The presence of stable photon-spheres in the corresponding curved spacetimes indicates that these compact objects may be nonlinearly unstable. We therefore conjecture that a negative trace of the energy-momentum tensor is a necessary condition for the existence of stable, soliton-like (regular) field configurations in general relativity.

Upper bound on the gravitational masses of stable spatially regular charged compact objects

“…As we shall explicitly show below, the above stated question is directly related to the physically important theorem presented recently in [5] (see also [6,7]), according to which the innermost null circular geodesic of an horizonless compact object, if it exists, is stable [8]. In particular, combining this interesting physical property of the spatially regular self-gravitating compact objects that we consider in the present paper with the intriguing assertion made in [9] (see also [10,11]), according to which horizonless spacetimes which possess stable null circular geodesics (stable closed light rings) are expected to develop non-linear instabilities in response to the presence of time-dependent massless perturbation fields [12], one concludes that spatially regular compact objects that possess light rings in their exterior spacetime regions are dynamically unstable.…”

“…one can use the instability properties of spatially regular horizonless spacetimes which possess light rings [5][6][7]9], in order to derive an upper bound on the gravitational masses of physically realistic (stable) charged compact objects.…”

“…Taking cognizance of Eqs. (5), (6), (12), and (13), and using the conservation equation T µ r;µ = 0 [16], one finds the simple functional relation [6,7]…”

“…In particular, the innermost light ring of a spatially regular compact object, if it exists, is generally stable with the property V ′′ r (r = r innermost γ ) > 0 [see Eqs. (16) and (17)] [5][6][7]. The exterior spacetime regions (r ≥ R) of the spherically symmetric charged compact objects that we consider in the present paper are characterized by the relations [4] ρ = p = 0 (18) and [1]…”

No-go theorem for static boson stars

“…In addition, taking cognizance of Eqs. (2), (6), and (9), and using the fact that finite mass matter configurations in asymptotically flat spacetimes are characterized by the simple asymptotic behavior [24]…”