Stability Conditions For a Noncommutative Scalar Field Coupled to Gravity

We consider a noncommutative scalar field with a covariantly constant noncommutative parameter in a curved space-time background. For a potential as a noncommutative polynomial it is shown that the stability conditions are unaffected by the noncommutativity, a result that is valid irrespective whether space-time has horizons or not.

However since covariant derivatives do not commute, the resulting Moyal product is not associative, i.e. (f ⋆ g) ⋆ h = f ⋆ (g ⋆ h). One could consider instead the Kontsevich product [29], but its covariant version is also nonassociative [28]. Since one usually implements the noncommutativity through a mapping, the Seiberg-Witten map [2] up to some order in θ µν , this procedure usually maintains, up to that order, the associativity, and hence, one chooses the simplest form of covariant deformed product as defined by Eq. (2).
One assumes that for curved space-times that i.e. the noncommutative tensor is covariantly constant (see discussion below). This condition was considered in Refs. [7,28] as it generalizes the condition that θ µν is constant. Following Ref. [8], one consider a scalar field whose commutative analytic potential V (Φ) = ∞ n=0 λn n! Φ n is defined by substituting the product between functions by the Moyal product Eq. (5) admits two classes of solutions. For det θ µν = 0, that is θ is invertible and then ∇ ν Φ = 0, a too strong condition for our problem. For det θ µν = 0, then ∇ µ Φ can be written as powers of the noncommutative parameter, a solution that does not trivialize our problem (cf. Eq. (25) and ensued discussion). One assumes that the gravity sector of the model is not affected by noncommutativity, therefore the space-time is still described by the usual Einstein equation with noncommutative sources where κ = 8πG and in the case under investigation the noncommutative energy-momentum tensor can be split into a scalar field and matter fields contributions:T µν =T Φ µν +T M µν . It is further assumed that matter fields satisfy the dominant energy condition 3 . The noncommutative action then readsS The noncommutative generalization of the energy-momentum tensors are given bỹ In order to discuss the stability conditions for the scalar field one considers the energymomentum density for the gravitational field so that the associated four-momentum vector p µ for a asymptotically flat space can be written as [31] 16πGp where V µ = ǫ 0 γ µ ǫ 0 , ǫ 0 represents a constant Dirac spinor, Σ is an arbitrary three-dimensional hypersurface and S its boundary ∂Σ at infinity. The two-form E σα is defined as 4 where ǫ is a Dirac spinor that at infinity behaves as ǫ → ǫ 0 +O 1 r . The total energy-momentum can be written with the use of spinor fields. Since one assumes that gravity is not affected by noncommutativity, the product between spinor fields and gamma matrices is actually the usual one. One further assumes that spinor fields commute with the noncommutative scalar field. 3 Physically this condition states that local energy density is positive, that is for any time-like vector W µ , T µν W µ W ν ≥ 0, and T µν W µ is not a space-like vector [30]. 4 Our conventions are the following: the metric signature is (

Generalized positive energy theorem
For supersymmetric theories the method used in Ref. [31] can be generalized by replacing Eq. (12) byÊ where∇ µ is the supercovariant derivative related to the change of the gravitino field ψ i µ under a supersymmetric transformation and i = 1, . . . , N is the number of supersymmetries. One can show that Eq. (11) is then generalized to where δχ a represents the change of spin-1 2 fields under a supersymmetric transformation. In the case of asymptotic Anti-de Sitter (AdS) space-time one requires another term on the L.H.S. of this equation in order to fix the four-momentum vector p µ . If T Mσ α satisfies the dominant energy condition, then since vector ǫ i 0 γ α ǫ i 0 is non-space-like the first term in the integrand of Eq. (14) is positive. Considering the time direction orthogonal to Σ, thus the last two terms of the R.H.S. of Eq. (14) can be expressed as 5 This term is positive definite if one chooses the Witten condition [22] γ n∇ n ǫ i = 0.
For supersymmetric theories the values of∇ n ǫ i and δχ a are automatically set by supersymmetry [22,23]. If a theory does not admit a supersymmetric extension this setup can be used as discussed in Ref. [23].

Stability conditions
In order to obtain the stability conditions, one must identify Eq. (19) with Eq. (14). Therefore the coefficients of the last five terms in Eq. (19) must vanish. One first notices that the resulting system of equations is quite difficult to solve, so one assumes that the conditions for indices i, j, a are single valued. This simplifies considerably the system of equations.
One needs now to examine each term at the R. H. S. of Eq. (19). The first term is positive definite given that the matter fields satisfy the dominant energy condition. Choosing "0" as the direction orthogonal to Σ, through Eq. (16) one gets that the second and the third terms can be written as As θ µν is covariantly constant, this will be positive definite if one chooses the conditions: One considers now the expansion of a noncommutative functionh(Φ) up to second order in the noncommutative parameter where h is a function of Φ, h µν is an antisymmetric function of Φ and its derivatives, and so on. One uses this expansion to compute terms at Eq. (19) that are functions of Φ.
One looks now to the term proportional to ǫΓ σαβ ǫ. After using that Γ σαβ is totally antisymmetric and Eq. (50) found in of the Appendix one obtains which vanishes if one chooses that Using Eqs. (51) and (52) in the Appendix, the term proportional to ǫǫ can be computed: which vanishes given condition (25). The term proportional to ǫγ α ǫ reads, after using Eqs. (25) and (50) 2κ Clearly, since coefficients of every order in the noncommutative parameter must vanish, one gets and thus thatf 2 (Φ) = √ κ.
The term proportional to ǫγ σ ǫ reads after using Eqs. (5), (6), (25) and (53) In order to proceed one assumes thatf (Φ) = a + bΦ ⋆ Φ, where constants a and b must be obtained by the boundary conditions of the system of equations; this condition generalizes the procedure of Ref. [24]. Using Eq. (25) one gets Eq. (31) yields substituting this into Eq. (32), it follows that Finally, the term proportional to ǫΓ σα ǫ is given by Using Eqs. Thus, the problem of stability consists in solving the system of equations However, this is precisely the set of equations for the commutative case for a quartic potential solved in Ref. [24]. Our result is then that the stability conditions for a scalar with a noncommutative potential are not affected by noncommutativity.
Let us now examine the consistency of Eqs. (21) and (22) after solving the stability conditions (28), (33) and (34). At first order in perturbation of the noncommutative parameter, one obtains The first two terms vanish by the assumption that spinors are not affected by noncommutativity. The last term vanishes on account of Eq. (5). Therefore, this equation is consistent with the results obtained above. One can also show that θ µν ∇ ν δχ = 0, using the assumption that spinors are not altered by noncommutativity and Eqs. (5) and (25).

Space-time with horizons
One considers now space-time configurations which admit horizons. In this situation, the divergence theorem must be modified so to include the horizon where H denotes the horizon. Clearly, if the second term in the L. H. S. of Eq. (40) vanishes the presence of horizons does not affect the stability conditions obtained in Section 4. Following Ref. [32] one introduces a orthonormal tetrad field at the horizon {eμ}, where e0 is normal to the hypersurface Σ, e1 is normal to the two-surface H and eÂ (A = 2, 3) are tangent to H. Using this coordinate system then one has only to evaluate the term HÊ01 dS01. (41) For simplicity one omits the hat on the indices. First one restricts the two-form to Σ, and thus through Witten's condition γ a∇ a ǫ = 0, one finds that Using the definition of the supercovariant derivative and ∇ b ǫ = (3) ∇ b ǫ + 1 2 K ab γ 0 γ a ǫ, where (3) ∇ b is the intrinsic three-dimensional covariant derivative and K ab is the second fundamental form of Σ, then the value of two-form on H is given by From Witten's condition: (3) where K = K a a . Substituting Eq. (44) into Eq. (43) and using that (3) where A further condition is required to restrict the spinor field on H. This has been put forward in Ref. [25], namely: γ 1 γ 0 ǫ = ǫ. Now Eq. (45) readŝ Notice that (J + K + K 11 ) = − √ 2ψ, where ψ is the expansion scalar [32], which is related to the rate of increase of the absolute value of the element of area. If two neighbouring geodesics are converging, then ψ < 0, if instead they diverge, then ψ > 0. This quantity vanishes if H is an apparent horizon. Given that γ 1 γ 0 anticommutes with γ 1 γ A D A and with γ 1 , then and Thus, choosing the boundary H to be an apparent horizon, from Eqs. (47) and (48) one finds that and therefore the presence of spaces with horizons does not affect the stability conditions found in Section 4.

Conclusions
In this work the stability conditions for a noncommutative scalar field coupled to gravity have been examined. Gravity is assumed not to be affected by noncommutativity and also that in the Moyal product usual derivatives are replaced by covariant derivatives. Associativity is ensured through an auxiliary condition, namely θ µν ∇ ν Φ = 0. It is then found that for a scalar field with a polynomial potential, the stability conditions are the very ones for the commutative case studied in Ref. [24]. 6(2) ∇ A is the intrinsic covariant derivative on H and J AB is the second fundamental form on H with J = J A A .
At first sight one might think that this result was already expected, given that no noncommutative corrections toṼ (Φ) andf (Φ) were considered up to the second order in θ. This is not quite the case as one encounters that we obtain a nontrivial condition for the term proportional to ǫΓ σαβ ǫ (Eq. (25)), which is actually absent in the commutative case. It is interesting to point out that the obtained conditions for the stability of a noncommutative scalar field, Eqs. (21), (22) and (25), are structurally related with the associativity condition, Eq. (5).
Finally, it has also been shown that the contribution of the surface integral HÊ σα dS σα on an apparent horizon vanishes. This means that stability results are not altered whether one considers space-time configurations with an apparent horizon.