Noncommutative fields in Reissner–Nordström black hole background

Since the discovery of the black hole (BH) solution of Einstein’s equation by Schwarzschild [1] more than one century ago, BH physics has been a very active and intriguing field of research. Although BHs are fully described just by their mass $M$, electric and/or magnetic charge $Q/P$, and angular momentum $J$, their properties are far from being fully understood. In parallel to the theoretical research, there have been groundbreaking observations of BHs and their mergers recently: LIGO and Virgo collaborations detected gravitational waves arising from the mergers of binary black holes [2], while the Event Horizon Telescope (EHT) produced images of the supermassive BH in the center of the M87 galaxy [3].

Much of the insight regarding gravitational radiation and properties of BHs may be gained by studying BH perturbations. Once perturbed, a BH responds by going through a ringdown phase, during which it starts emitting gravitational waves. The most important part of the ringdown phase is a long period of damped oscillations, dominated by the quasinormal modes (QNMs). Besides carrying the intrinsic information about the BH itself, QNMs may also carry information about the properties of the underlying spacetime structure. In particular, if the underlying spacetime structure is deformed in such a way that it departs from the usual notion of a smooth manifold, then this deviation should leave its trace in the QNMs’ spectrum too.

To study QNM spectra in the nonommutative (NC) geometry setting, it is necessary to first derive equations of motion for NC fields in a fixed gravitational background. We will focus on the twist formalism approach to NC geometry and to the Reissner–Nordström (RN) black hole gravitational background. RN BH is a solution of the Einstein-Maxwell equations in the presence of a point electric charge $Q$ and magnetic charge $P$ in the origin $r=0$. Although astrophysical BHs are expected to be nearly neutral due to charge neutralization by surrounding plasma, charged BHs can serve as valuable laboratories to probe the nature of horizons, singularities and some aspects of quantum gravity. They are also important in holography and AdS/CFT, where charged black holes in anti-de Sitter space correspond to finite-temperature states in strongly coupled quantum field theories with a chemical potential [4]. Recently, there have been suggestions that a charged BH can arise in the presence of dark matter halos [5].

In this short paper we discuss dynamics of NC matter fields in the RN black hole background. The NC deformation is introduced via the angular twist [6]. We will review the dynamics of NC scalar and spinor fields in the RN black hole background and then discuss the dynamics of the NC electromagnetic field and the NC gravitational field. In the next section, we briefly review the angular twist deformation and introduce the notation. Then we discuss dynamics of the NC matter fields: scalar, spinor and vector fields in Section 3. Finally, in Section 4 we present equations of motion for the NC gravitational field obtained in two different models. We postpone a detailed discussion on the propagation of the NC gravitational perturbations for future work.

The NC deformation is introduced via the twist operator of the form

with the small NC parameter $a$. Vector fields $X_{1}=\partial_{t}$, $X_{2}=\partial_{\varphi}$ are commuting vector fields, $[X_{1},X_{2}]=0$, therefore the twist (1) is an Abelian twist [7]. It is dubbed as "angular twist" becuase the vector field $X_{2}=\partial_{\varphi}$ is a generator of rotations around $z$-axis. The twist (1) defines the $\star$-product of functions,

The algebra of differential forms is deformed in the similar way [6]. Noncommutativity of $\star$-products is controled by the triangular $R$-matrix ${\cal R}={\cal F}_{21}{\cal F}^{-1}$, with ${\cal R}_{21}=\mathsf{R}_{\alpha}\otimes\mathsf{R}^{\alpha}={\cal R}^{-1}$. In particular:

Let us fix the background geometry to that of the RN black hole with the electric charge $Q\neq 0$ and no magnetic charge. The metric tensor is then given by

where $M$ and $Q$ are the mass and charge of the RN black hole, respectively. The vector fields $X_{1}$ and $X_{2}$ are two Killing vectors for this metric, thus the twist (1) does not act on the RN metric. In this way we ensure that the background geometry remains intact by the deformation.

The NC actions for the charged scalar field $\hat{\phi}$, the charged spinor field $\hat{\psi}$ and the massless vector field $\hat{A}_{\mu}$ are given by [6, 8]

The NC scalar field $\hat{\phi}$ has the mass $\mu$ and charge $q$ and transforms in the fundamental representation of NC $U(1)_{\star}$. The NC spinor field $\hat{\psi}$ has the mass $m$ and charge $q$ and it also transforms in the fundamental representation of NC $U(1)_{\star}$. The NC $U(1)$ gauge field $\hat{A}_{\mu}$ is introduced in the model through a minimal coupling and the NC field-strength tensor is defined as

The covariant derivatives are defined as

with the spin connection $\hat{\omega}_{\mu}=\frac{i}{2}\hat{\omega}_{\mu}^{~~ab}\Sigma_{ab}$ and $\Sigma^{ab}$ are the generators of the local Lorentz transforamtions.

The actions (4)-(6) are invariant under the infinitesimal $U(1)_{\star}$ gauge transformations:

where $\hat{\Lambda}$ is the NC gauge parameter. The last two transformations in (8) make clear that the models studied here are semiclassical. By this we mean that only scalar, spinor and gauge field are subject to a NC deformation, while gravitational field remains unaffected. A more general situation dealing with a NC deformation of gravitational field becomes increasingly more involved. For more details see [7], [9], [10] and references therein.

In order to expand the actions (4)-(6) up to first order in the deforamtion parameter $a$, we systematically expand all $\star$-products and use the Seiberg-Witten (SW) map. SW map enables to express NC variables as functions of the corresponding commutative variables [11]. In this way, the problem of charge quantization in $U(1)_{\star}$ gauge theory does not exist. In the case of NC Yang-Mills gauge theories, SW map guarantees that the number of degrees of freedom in the NC theory is the same as in the corresponding commutative theory. That is, no new degrees of freedom are introduced.

Using the SW map NC fields can be expressed as function of corresponding commutative fields and can be expanded in orders of the deformation parameter $a$. Expansions for an arbitrary Abelian twist deformation are known to all orders [9]. Applying these results to the twist (1), expansions of fields up to first order in the deformation parameter $a$ follow. They are given by¹¹1In the spherical coordinate system $x^{\mu}=(t,r,\theta,\varphi)$ the twist (1) reduces to the Moyal-like twist $${\cal F}=e^{-\frac{i}{2}\theta^{\alpha\beta}\partial_{\alpha}\otimes\partial_{\beta}}\,.$$ The matrix $\theta^{\alpha\beta}$ is related to the NC paramter $a$ as: $$\begin{pmatrix}0&0&0&a\\ 0&0&0&0\\ 0&0&0&0\\ -a&0&0&0\\ \end{pmatrix}\,.$$ (9) :

In the following we discuss equations of motion for each of the three matter fields.

There are different approaches to the construction of a NC deformation of gravity, see [14] and references there in. Here we will focus on two different models: $SO(2,3)_{\star}$ NC gravity [10] based on the $\star$-NC gauge symmetry and the braided NC gravity [15] based on the braided NC gauge symmetry.

Gravity models based on the $SO(2,3)$ gauge symmetry appeared in the 1970ies [16]. The original motivation for those models was to study properties of supergravity. The $SO(2,3)$ algebra reduces via the Inonü-Wigner contraction to the usual Poincaré algebra in $4D$ spacetime. This is used to obtain General Relativity from (A)dS gauge theory.

The NC generalization of the action proposed in [16] is given by

with

We introduced here the $SO(2,3)_{\star}$ field strength tensor $\hat{F}$ and the scalar field $\hat{\phi}$ transforming in the adjoint representation of $SO(2,3)_{\star}$

with the $SO(2,3)$ algebra generators, $M_{AB}=-M_{BA}$ ($A,B=0,1,2,3,5$) and the $5$D gamma-matrices $\Gamma^{A}$. We choose to work in the spherical coordinate system that is adapted to the twist (1), therefore the $\star$-products between the basis one forms and basis one forms and functions reduce to the ordinary commutative multiplication.

To obtain a NC $D=4$ gravity action, we proceed as follows: Firstly, we expand $\star$-products and insert the SW map solutions for the NC fields $\hat{F}_{\mu\nu}$ and $\hat{\phi}$ into the action (28). As expected, the NC correction linear in the NC parameter $\theta^{\mu\nu}$ disappears and the first nontrivial NC correction is quadratic in $\theta^{\mu\nu}$. The SW expansion guarantees that the NC action is invariant unded the commutative $SO(2,3)$ gauge symmetry. Next, we break the $SO(2,3)$ symmetry to the $SO(1,3)$ by fixing the value of the background field to $\phi=l\gamma_{5}$.

The obtained result is very cumbersome [10], and we will not write the full expression here. However, one can still analyze the model in different regimes of parameters. If we are interested in the low-energy sector of the theory, we should keep only terms that have at most two derivatives on vierbeins. Therefore, we include only terms linear in curvature and linear and quadratic in torsion (we assumed that the spin connection and the first-order derivatives of vierbeins are of the same order). The equations of motion are obtained by varying the action over vierbein and spin connection, independently. If we consider only the class of NC solutions with classically vanishing torsion $T_{\mu\nu}^{a(0)}=0$ (where $T_{\mu\nu}^{a(0)}=0$ is the zeroth order in $\theta^{\mu\nu}$ expansion of the full NC torsion), in the low-energy limit, equations of motion can be written as

with

Expanding the NC metric as $g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu}$, equation (31) can be used to study gravitational perturbations in the RN background. Developing further ideas from [17], we plan to study gravitational perturbations in our future work.

Recently, another approach to NC gauge theories was introduced in [15]. Unlike the $\star$-gauge transformations (8), braided NC gauge transformations close the Lie algebra. Therefore, no new degrees of freedom appear and the SW map is not necessary. The NC braided gravity action in $D=4$ dimensions is given by [15]

The curvature $R^{\star}$ and the torsion $T_{\mbox{\tiny L,R}}$ are defined as

The action (32) is invariant under the braided diffeomorphism symmetry and under the braided gauge $SO(1,3)$ symmetry. In particular, the curvature and the torsion are covariant under the braided gauge $SO(1,3)$ transformations

with the NC gauge parameter $\rho=\frac{1}{2}\rho^{ab}\Sigma_{ab}$.

The equations of motion are obtained by varying the action (32) with respect to the NC spin connection and the NC vierbein and they are exact in the NC parameter $\theta^{\mu\nu}$ [15]. In order to compare these equations with the equations following form the $SO(2,3)_{\star}$ model, in particular with (31), we now expand the exact equations of motion up to the 2nd order in the NC parameter $\theta^{\mu\nu}$.

The expanded equation of motion for the NC vierbein $e^{a}_{\mu}$ is given by

with