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

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 $T\geq\alpha\cdot\rho\geq0$ (here $T$ and $\rho$ are respectively the trace of the energy-momentum tensor and the energy density of the fields, and $\alpha$ is a non-negative constant), we obtain the lower bound $\text{max}_r\{2m(r)/r\}>(2+2\alpha)/(3+2\alpha)$ on the compactness (mass-to-radius ratio) of regular field configurations. In addition, we prove that these compact objects necessarily possess (at least) {\it two} photon-spheres, one of which exhibits {\it 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 {\it necessary} condition for the existence of stable, soliton-like (regular) field configurations in general relativity.


I. INTRODUCTION
Nonlinear solitons have a long and broad history in science. These regular particle-like configurations find applications in many areas of physics, such as: general relativity [1], string theory [2], condensed matter physics [3], nonlinear optics [4], and astrophysics [5].
Let us denote by T µ ν the energy-momentum tensor of the matter fields which compose a nonlinear static soliton. A simple argument [1,6] then reveals that, in flat space, the sum of the principal pressures, Σ i p i (here p i = T i i ), cannot have a fixed sign throughout the body volume. This can be seen from the conservation law ∂ j T j i = 0, which implies that the spatial components of the energy-momentum tensor satisfy [1,6] The volume integral (1) has a simple physical meaning: it states that the total stresses must balance in a static matter distribution [1,6]. The relation (1) then implies that the sum of the principal pressures must switch signs somewhere inside the volume of the extended body. In particular, no regular static matter distributions exist with Σ i p i > 0 throughout the entire space. Such systems are of a purely repulsive nature and thus the force balance is impossible [1,6].
Although for purely repulsive matter fields (with Σ i p i > 0 throughout the body volume) in flat space the force balance is impossible, the situation may change in curved spacetimes (that is, in the presence of gravity). This fact is nicely demonstrated by the existence of globally regular particle-like solutions of the coupled Einstein-Yang-Mills field equations [7].
These non-linear solitons describe extended objects in which the repulsive nature of the matter field [8] is balanced by the attractive nature of gravity.

II. THE TRACE OF THE ENERGY-MOMENTUM TENSOR
We have seen that any flat-space static matter distribution must be characterized by the relation Σ i p i < 0 in some part of it. Denoting by ρ > 0 [9] the energy-density of the matter fields, one concludes that the trace of the energy-momentum tensor, must also be negative in this part of the system volume. Thus, a negative trace of the energymomentum tensor, at least in some part of the system volume, is a necessary condition for the existence of static regular matter distributions in flat spacetimes.
However, it should be emphasized that this conclusion no longer holds true in curved spacetimes. In particular, static soliton-like field configurations which are characterized by a non-negative trace (throughout the entire configuration's volume) do exist. The gravitating Einstein-Yang-Mills solitons [7], which are characterized by the identity T = 0, are a wellknown example for such regular particle-like configurations.
The main goal of the present paper is to analyze, within the framework of general relativity, the physical properties of regular self-gravitating field configurations whose energymomentum tensor is characterized by a non-negative trace [10]. The rest of the paper is organized as follows: In Sec. III we shall describe our physical system. In particular, we shall formulate the Einstein field equations in terms of the trace of the energy-momentum tensor. In Sec. IV we shall prove that matter configurations which are characterized by a non-negative energy-momentum trace are necessarily highly relativistic objects. In particular, we shall derive a lower bound on the compactness (mass-to-radius ratio [11,12]) of these extended physical objects. In Sec. V we shall prove that the curved spacetime geometries which describe these self-gravitating objects necessarily possess (at least) two photon-spheres, compact hypersurfaces on which massless particles can follow null circular geodesics. We shall show that one of these photon-spheres exhibits stable trapping of the null circular geodesics. We conclude in Sec. VI with a summary of the main results.

III. DESCRIPTION OF THE SYSTEM
We study static spherically symmetric matter configurations in asymptotically flat spacetimes. The line element describing the spacetime geometry takes the following form in Schwarzschild coordinates [12][13][14][15] The metric functions δ(r) and µ(r) in (3) depend on the Schwarzschild areal coordinate r.
Regularity of the matter configurations at the center requires In addition, asymptotically flat spacetimes are characterized by The fields that compose the matter configurations are characterized by an energymomentum tensor T µ ν . The Einstein equations, G µ ν = 8πT µ ν , are given by [12,13,16] and where T t t = −ρ, T r r = p, and T θ θ = T φ φ = p T are respectively the energy density, the radial pressure, and the tangential pressure of the fields [11], and a prime denotes differentiation with respect to r.
The gravitational mass m(r) contained within a sphere of radius r is given by [17] For the total mass of the configuration to be finite, the energy density ρ should approach zero faster than r −3 at spatial infinity: Substituting the Einstein field equations (6) and (7) into the conservation equation one finds for the pressure gradient, where is the trace of the energy-momentum tensor, and Below we shall analyze the spatial behavior of the pressure function P (r) ≡ r 2 p(r), whose gradient is given by [see Eq. (11)] We shall assume that the matter fields satisfy the following conditions: (1) The dominant energy condition [18]. This means that the energy density bounds the pressures: (2) The trace of the energy-momentum tensor is non-negative. Specifically, we shall assume that the trace is bounded from below by where α ≥ 0 is a constant. Note that Eqs. (12) and (15) imply T ≤ 2ρ, an inequality which restricts the value of α to the regime [19,20]: From Eqs. (12), (15), and (16) one also finds (2) From Eqs. (5) and (14) one finds P ′ (r) ≃ 2rp T as r → ∞, which implies [22] In addition, from Eqs. (9) and (15) one learns that p(r) should approach zero faster than r −3 at spatial infinity, which implies [23] P (r → ∞) → 0 − .
Inspection of Eqs. (19) and (21) reveals that the pressure function P (r) must switch signs at some intermediate point, r = r 0 , such that: Note that the negativity of the pressure function P (r) in some part of the spacetime [see Eq.
(21)] implies that 0 ≤ T < ρ in this spacetime region, an inequality which further restricts the value of α to the regime 0 ≤ α < 1 .
Note that the larger is the value of α, the stronger is the lower bound (25) [24].

V. PHOTONSPHERES OF THE REGULAR COMPACT OBJECTS
One of the most remarkable features of black-hole spacetimes is the existence of photonspheres, closed hypersurfaces on which massless particles (null rays) can orbit the central black hole. In the present section we shall prove that compact regular objects whose energy- To that end, we shall first follow the analysis of [14,15,25] at the two boundaries of the spacetime.
Inspection of Eqs. (19) and (21) reveals that there must be a finite interval, (r a , r b ), in which the pressure function P (r) is negative and decreasing: P (r a < r < r b ) < 0 and P ′ (r a < r < r b ) < 0 .
Taking cognizance of Eqs. (14), (15), (16), and (30), one deduces that in this interval. From Eqs. (29) and (31) one concludes that there must be (at least) two radii, r γ1 and r γ2 , which are characterized by These two radii correspond to two photon-spheres which characterize our regular curved spacetimes.
It is worth emphasizing that the regular spacetime (3) may be characterized by more than two photon-spheres. We shall henceforth denote by r γ1 and r γ2 the innermost and the outermost radii of these photonspheres, respectively. Note that this implies [see Eqs. (29) and (31)] where the derivative N ′ (r = r γ ) can be expressed in the form [see Eqs. (6), (11), and (13)] The stability/instability properties of the characteristic circular geodesics are determined by the second spatial derivative of the effective radial potential [14,15]: unstable circular geodesics are characterized by V One therefore concludes that the inner photonsphere is stable, whereas the outer photonsphere is unstable.

VI. SUMMARY AND PHYSICAL IMPLICATIONS
In this paper we have analyzed the physical properties of self-gravitating field configurations whose energy-momentum tensor is characterized by a non-negative trace: T ≥ α·ρ ≥ 0.
The main results obtained in this paper and their physical implications are: (1) It has been shown that, regularity of the field configurations sets an upper bound on the magnitude of the energy-momentum trace [see Eq. (23)]: This bound excludes, for example, the existence of high density fluid matter configurations in which the inter-particle interactions are mediated by massive vector fields (such fluids are characterized by T → 2ρ [20]).
(2) It has been shown that these matter configurations are necessarily highly relativistic objects. In particular, we obtained the lower bound on the mass-to-radius ratio (compactness) of these regular field configurations.
(3) We have proved that regular objects whose energy-momentum tensor is characterized by a non-negative trace necessarily possess (at least) two photonspheres, one of which exhibits stable trapping of null circular geodesics [see Eq. (36)].
In this respect, it is interesting to mention an important recent result by Keir [26]: It was shown [26] that in spherically symmetric, asymptotically flat regular spacetimes which exhibit stable trapping of null geodesics, linear perturbation fields cannot decay faster than logarithmically. As emphasized in [26,27], this result indicates that the corresponding compact objects may be nonlinearly unstable.
Our results therefore suggest that, while gravity may allow for the existence of regular field configurations whose energy-momentum tensor is characterized by a non-negative trace [28], these highly compact objects are probably nonlinearly unstable [29]. It is therefore tempting to conjecture that, within the framework of classical general relativity, a negative trace of the energy-momentum tensor (at least in some part of the system volume) is a necessary condition for the existence of stable, particle-like (regular) field configurations.