2 where and µ +( − ) nlm are the usual creation (annihilation) opera- tors for a particle with angular momentum l , azimuthal momentum m and energy E$_{n}$ . These operators sat- isfy the usual commutation relations [ µ + $_{nlm}$, µ + n ′ l ′ m ′ ] = [ µ − $_{nlm}$, µ − n ′ l ′ m ′ ] = 0 and [ µ + $_{nlm}$, µ − n ′ l ′ m ′ ] = δ$_{nn}$ ′ δ$_{ll}$ ′ δ$_{mm}$ ′ . With the operator ˆ Φ, it is now possible to construct the energy-momentum tensor operator ˆ T$_{µν}$ just by inserting the operator ˆ φ into the classical expression for the energy- momentum tensor ˆ Φ + = ∑ nlm µ + $_{nlm}$R$_{nl}$ ( r ) Y l $_{m}$( θ, ψ ) e − iE n t ˆ Φ − = ∑ nlm µ − $_{nlm}$R$_{nl}$ ( r ) Y l ∗ m ( θ, ψ ) e + iE n t (5) 0 0.02 0.04 0.06 0.08 0.1 0.12 (0) 0.1 0.2 0.3 0.4 0.5 M[m p 2 /m a ]  = 20  = 40  = 60  = 80 T µ ν = g $^{µσ}$∂$_{σ}$ φ∂$_{ν}$φ − 1 2 δ µ $_{ν}$g$^{λσ}$∂$_{λ}$ φ∂$_{σ}$φ − δ µ $_{ν}$V ( φ ) . (6) FIG. 1. Gravitational mass as a function of the central value of the scalar field σ (0) for different values of Λ. Dotted lines include only an expansion of the axion potential up to the term Φ 4 The average 〈 Q | ˆ T$_{µν}$ | Q 〉 is done by considering a state | Q 〉 for which all N particles are in the ground state ( l = m = 0, n = 1). The ground state satisfies µ − 100 | Q 〉 = 0. This procedure, as pointed out in [16], cancels all time dependence on the vacuum expectation value 〈 Q | ˆ T$_{µν}$ | Q 〉 and, for the case of a free scalar field ( V (Φ) = m 2 2 Φ $^{2}$), the real quantized scalar field yields the same field equations as those obtained by using a classi- cal complex scalar field. At this level, the self-gravitating system for a real quantized scalar field does not differ from a complex classical scalar field, hence, a real quan- tized scalar field does not produce the so called “oscilla- tons” [18], which are time-dependent. In our case we are interested in the axion potential (1). In order to compute 〈 ˆ T $^{µν}$〉 , we perform a Taylor expansion of (1), i.e. Since axions are real scalar particles, it is very use- ful to remember how self-gravitating systems made of spin zero particles are constructed. We will follow the method developed by Ruffini and Bonazzola [16]. The self-gravitating system arise as a solution of the EKG equations: I. EINSTEIN-KLEIN-GORDON WITH AN AXION-LIKE POTENTIAL V (Φ) ∼ m 2 2 Φ 2 − 1 4! m 2 f 2 a Φ 4 + 1 6! m 2 f 4 a Φ 6 − ... (7) G$_{µν}$ = 8 πG < ˆ T$_{µν}$ > , (2) ( □ − dV (Φ) d Φ 2 ) Φ = 0 , (3) We will show that the final results do not depend strongly on the number of terms considered in the Taylor expan- sion of (1). The relevant term that should be consid- ered is the self-interacting term Φ 4 and the sign it carries with itself, which differs from the one considered in Bo- son Stars (BS) [8]. With the potential (7), it is possible to compute 〈 ˆ T µ ν 〉 by performing the quantization and av- eraging procedure previously discussed. The computed average of the stress energy tensor is where □ = (1 / √ − g ) ∂$_{µ}$ [ √ − gg $^{µν}$∂$_{ν}$ ] and V ( φ ) is the scalar field potential. Here < · · · > denotes an average over the ground state of a system of many particles. Its presence refers to the fact that we are working in the semi-classical limit of the Einstein’s equations. We will work with units where c = ℏ = 1. In the case of a spherically symmetric, static space-time described by it has been shown that such self-gravitating systems are fully characterized by the scalar field properties, i.e. the mass m of the scalar field and its energy density potential V (Φ) [10, 17]. The total mass of the resulting object and the typical radius depend mainly on these two properties of the scalar field. The axion is no exception. To deal with the quantum nature of the axion field, we have to compute the average 〈 ˆ T $^{µν}$〉 in eq. (2). What is usually done is to quantize the scalar field Φ → ˆ Φ = ˆ Φ + + ˆ Φ − We have dropped all sub-indexes since, as we have al- ready pointed out, we will assume that the axion is in 〈 T 0 〉 = − E $^{2}$R2 2 B − R ′ 2 2 A − m $^{2}$R2 2 + m $^{2}$R4 12 f 2 a − m $^{2}$R6 144 f 4 a + . . . , 〈 T 1 〉 = E $^{2}$R2 2 B + R ′ 2 2 A − m $^{2}$R2 2 + m $^{2}$R4 12 f 2 a − m $^{2}$R6 144 f 4 a + . . . , 〈 T 2 〉 = E $^{2}$R2 2 B − R ′ 2 2 A − m $^{2}$R2 2 + m $^{2}$R4 12 f 2 a − m $^{2}$R6 144 f 4 a + . . . . (8) ds 2 = B ( r ) dt 2 − A ( r ) dr 2 − r $^{2}$(sin$^{2}$θdφ2 + dθ ) , (4)