We study gravitational perturbations of the Schwarzschild metric in the context of noncommutative gravity. rvarphi and rt noncommutativity are introduced through a Moyal twist of the Hopf algebra of diffeomorphisms. Differential geometric structures such as curvature tensors are also twisted. Noncommutative equations of motion are derived from the recently proposed NC vacuum Einstein equation. Here in addition to previously calculated axial NC potential we present the polar solution which generalizes the work done by Zerilli. Quasinormal mode frequencies of the two potentials are calculated using three methods WKB PschlTeller and RosenMorse. Notably we apply the WKB method up to the 13th order and determine the optimal order for each noncommutative parameter value individually. Additionally we provide comprehensive error estimations for the higherorder WKB calculations offering insights into the accuracy of our results. By comparing the spectra we conclude that the classical isospectrality of axial and polar modes is broken upon spacetime quantization. Isospectrality is restored in the eikonal limit.
['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']
Noncommutative quasinormal modes of Schwarzschild black hole
{'Noncommutative quasinormal modes of Schwarzschild black hole': "Nikola Herceg, ∗ Tajron Juri'c, † A. Naveena Kumara, ‡ and Andjelo Samsarov §  \nRudjer Boˇskovi'c Institute, Bijeniˇcka c.54, HR-10002 Zagreb, Croatia", "Ivica Smoli'c ¶": 'Department of Physics, Faculty of Science, University of Zagreb, Bijeniˇcka cesta 32, 10000 Zagreb, Croatia (Dated: September 17, 2024)  \nWe study gravitational perturbations of the Schwarzschild metric in the context of noncommutative gravity. r -ϕ and r -t noncommutativity are introduced through a Moyal twist of the Hopf algebra of diffeomorphisms. Differential geometric structures such as curvature tensors are also twisted. Noncommutative equations of motion are derived from the recently proposed NC vacuum Einstein equation. Here, in addition to previously calculated axial NC potential, we present the polar solution which generalizes the work done by Zerilli. Quasinormal mode frequencies of the two potentials are calculated using three methods: WKB, Poschl-Teller and Rosen-Morse. Notably, we apply the WKB method up to the 13th order and determine the optimal order for each noncommutative parameter value individually. Additionally, we provide comprehensive error estimations for the higher-order WKB calculations, offering insights into the accuracy of our results. By comparing the spectra, we conclude that the classical isospectrality of axial and polar modes is broken upon spacetime quantization. Isospectrality is restored in the eikonal limit.', 'I. INTRODUCTION': "Since its inception, general theory of relativity (GR) has proven to be the most successful theory of gravity. In a wide range of different physical circumstances in which it was tested, it has prevailed against all challenges. Indeed, the theory has passed many experimental tests involving diverse astrophysical phenomena ranging from gravitational lensing to compact binary coalescences. However, the ranges in which gravity was tested so far included mostly the weak-field, low-speed and linear gravity limits, while the extreme regimes like strong gravity and high curvature remained largely uninvestigated. The latter could presumably be accessed by studying the near-horizon region of black holes where energies as large as the Planck scale are expected to be present.  \nAt this energy scale, the gravitational effects become comparable to quantum effects and one might expect a modification of GR to be triggered there. This is the point where the gravitational waves come into play. Besides representing one of the most important predictions of general relativity, gravitational waves appear to play a crucial role in accessing the behaviour of GR in extreme conditions under which it is yet not so well studied. Namely, due to the fact that gravitational waves originate from regions of spacetime where gravity is extremely strong, just like the near-horizon region of black holes, they came out as the most suitable candidate for testing GR under such extreme conditions. In fact, since their recent experimental discovery, gravitational waves have been established as one of the most powerful tools for studying extreme regimes of relativistic speeds, large curvature, and strong gravity. In this way, gravitational waves provided a promising way for exploring spacetime dynamics in regions previously inaccessible to an experimental enquiry and helped bring them under a direct experimentalist's gaze.  \nGravitational waves as observed by the current ground-based gravitational wave detectors have been extensively used for testing various aspects of general relativity, but no discrepancies from the theory have been reported so far [1-3]. As it currently appears, possible deviations from GR, if anywhere found in the literature, may most likely be attributed to a false identification of GR violation arising from a series of technical issues such as detector noise, signal overlaps, gaps in the data, detector calibration, source misidentification or others [4]. This motivates an ongoing and  \nincreasingly enthusiastic search for potential deviations in the field.  \nAnother related observation involves GR itself and its validity and perseverance in the vicinity of a black hole. As already stated before, the region near a black hole horizon admits access to energies of the order of the Planck scale. As a result, GR may be potentially modified there, causing that some of its predictions, including gravitational quasinormal mode (QNM) spectrum of black holes, be altered too [5-9]. The latter scenario receives significant support within a scope of various quantum gravity theories, which suggest that strong gravity and high energy regimes like those found in a near-horizon region of black hole may create environment where the usual point-like structure of spacetime becomes replaced with certain noncommutative geometry constructs [10, 11] ultimately affecting the relaxation dynamics of black holes and their related ringdown spectra. Even more striking evidence of the discrete nature of spacetime and a possible necessity for modifying classical GR might be found in phenomena like gravitational wave echoes or tidal heating [12].  \nThe initial steps toward quantifying the black hole ringdown spectrum have been carried out on the example of the Schwarzschild black hole by Regge and Wheeler for the odd-parity (axial) perturbations [13] and subsequently by Zerilli for the even-parity (polar) perturbations [14]. In both cases the problem was shown to reduce to a scattering problem at a specially devised potential barrier, thus establishing the full analogy between the theory of scattering of gravitational waves on a background of a black hole and a scattering theory in quantum mechanics [15, 16]. The outcome of the analysis was a master equation in a form of a 1-dim Schrodinger-like equation in a radial coordinate with a particular effective potential that came out after separating the angular variables in the wave equation, using tensor spherical harmonics with angular indices ( /lscript, m ). Careful study of the master equation revealed that quasinormal modes in classical general relativity have the property of isospectrality. In particular, the axial and polar perturbations of the Schwarzschild black hole were shown by Chandrasekhar and Detweiler to have identical spectra [17].  \nIn our previous papers [18-20]. we have set up a noncommutative differential geometry framework sufficiently suited for studying metric perturbations in noncommutative gravity. This formalism is based on considering a set of deformed diffeomorphisms encompassing the symmetries of a noncommutative space which conveniently find their description within the structure of Hopf algebra. In addition, the formalism was applied to obtain noncommutative corrections to the Regge-Wheeler potential that governs the axial perturbations of the Schwarzschild black hole.  \nIn the present paper, we continue on our previous work by presenting noncommutative gravitational perturbations of the Schwarzschild black hole in their entirety, giving a description of metric perturbations in noncommutative gravity for both axial and polar cases. We do this by first recapitulating the results for axial perturbations obtained previously and then by extending the analysis to include polar perturbations. Curiously, contrary to the classical case, we find that these two types of perturbations do not share the same QNM spectra, yielding a conclusion that the presence of quantum spacetime violates isospectrality.  \nThe plan of the paper is the following. In Section II we briefly present a formalism of NC differential geometry and explain how it can be applied to linearized gravitational perturbation theory. In Section III we single out the NC corrections to the metric perturbations around the Schwarzschild black hole and calculate noncommutative corrections to the equation of motion for axial (i.e. odd-parity) and polar (i.e. even-parity) metric perturbations. In Section III A we discuss the axial sector. Rephrasing the problem in terms of a Schrodinger-type equation enables us to identify the NC correction to the Regge-Wheeler potential. Section III B is devoted to an extensive and in-detail analysis of the noncommutative polar perturbations of the Schwarzschild background. Utilising the classical infinitesimal diffeotransformations and a series of field redefinitions, as well as coordinate transformations we were able to determine the NC correction to the equation of motion governing the NC polar metric perturbations and to extract the related quantum correction to the Zerilli potential.  \nIn Section IV we gain the first insight into the spectra by applying the WKB, Poschl-Teller and Rosen-Morse methods. In Section V we demonstrate that the isospectrality between two types of perturbations no longer holds, in contrast to a well-established result in classical GR. We come to this conclusion by arguing that the two potentials may not be connected through a Darboux transformation, making the criterion for equivalence between the two potentials impossible to fulfill. This conclusion has been further corroborated by a detailed semi-analytic analysis of the QNM spectra in both cases, with the analysis being based on the WKB method [21] up to 13th order in the calculation. The paper ends with concluding remarks and 3 appendices where we present some technical details. In Appendix A we present the usual Zerilli gauge for the polar perturbation sector. In Appendix B we write out the coefficients of transformations used for finding the NC corrections to the polar sector. In Appendix C we give the QNM frequencies  \nobtained through the order-optimized WKB method together with error estimations up to the 13th order.", 'II. NONCOMMUTATIVE DIFFERENTIAL GEOMETRY': 'We consider a noncommutative gravity theory constructed using the mathematical framework of Hopf algebra theory [22-24]. First, we will provide a clear and concise introduction to Hopf algebras and their Drinfeld twist deformations. In the second subsection we outline the main building blocks of NC differential geometry - a theory that arises naturally by imposing covariance under the Hopf algebra of deformed diffeomorphisms. For a more detailed and pedagogical introduction to the topic, see [25].', 'A. Hopf algebra and deformed diffeomorphisms': 'The Lie algebra of vector fields (Ξ , [ · , · ]) plays a vital role in differential geometry. It describes the infinitesimal diffeomorphisms of the manifold M . It acts on tensor fields via the Lie derivative £ . From the Lie algebra (Ξ , [ · , · ]), we can create a universal enveloping algebra U Ξ, which we can upgrade to a Hopf algebra. Universal enveloping algebra is a free algebra on generators of the Lie algebra Ξ with associative product µ : U Ξ ⊗ U Ξ → U Ξ and unit η : C → U Ξ obtained by quotienting the former (the free algebra) by an ideal generated by relations ab -ba -[ a, b ] = 0.  \nTo encode the intuitive concepts of Leibniz rule, inverse, and normalization that we have on the Hopf algebra, we use C -linear maps coproduct ∆ : U Ξ → U Ξ ⊗ U Ξ, antipode S : U Ξ → U Ξ and counit /epsilon1 : U Ξ → C . While ∆ and /epsilon1 are multiplicative maps, the map S associated with inversion in the corresponding Lie group is antimultiplicative. The object of interest here is the structure H = ( U Ξ , µ, η, ∆ , /epsilon1, S ), known as a Hopf algebra.  \nA Hopf algebra can be understood as a generalization of the enveloping algebra that includes three important maps: coproduct ∆, counit /epsilon1 , and antipode S . These maps must satisfy certain compatibility criteria. To be considered a Hopf algebra, it must meet the following three conditions for all ξ ∈ H :  \n(∆ ⊗ id)∆( ξ ) = (id ⊗ ∆)∆( ξ ) , ( /epsilon1 ⊗ id)∆( ξ ) = ξ = (id ⊗ /epsilon1 )∆( ξ ) , µ (( S ⊗ id)∆( ξ ) = /epsilon1 ( ξ )1 = µ ((id ⊗ S )∆( ξ )) . (1)  \nIn summary, the Lie algebra of diffeomorphisms (Ξ , [ · , · ]) can be embedded into a Hopf algebra H = ( U Ξ , µ, η, ∆ , /epsilon1, S ). This Hopf algebra is known as the Hopf algebra of diffeomorphisms. It contains the relevant information on the Leibniz rule (through the coproduct ∆), the inverse of a diffeomorphism (through the antipode S ), and the normalization process (through the counit /epsilon1 ).  \nLet us present some properties of the Hopf algebra H . One important concept is that of a Drinfeld twist. A Drinfeld twist refers to an invertible element F ∈ H ⊗ H that satisfies the following two conditions:  \n( F ⊗ 1)(∆ ⊗ id) F = (1 ⊗F )(id ⊗ ∆) F , ( /epsilon1 ⊗ id) F = 1 = (id ⊗ /epsilon1 ) F . (2)  \nNotation F = f α ⊗ f α , F -1 = ¯ f α ⊗ ¯ f α , where summation over α is assumed, will often be employed. We further require that F = 1 ⊗ 1+ O ( a ), where a represents a deformation parameter, in order to keep the zeroth order (classical limit) unchanged. 1 If a Drinfeld twist F of the Hopf algebra H is provided, it is well-known that we can build a new Hopf algebra H F := ( U Ξ , µ, η, ∆ F , /epsilon1, S F ) by twisting the coproduct and the antipode as follows:  \n∆ F ( ξ ) := F ∆( ξ ) F -1 , S F ( ξ ) := χS ( ξ ) χ -1 . (3)  \nHere, χ := f α S ( f α ) and χ -1 := S ( ¯ f α ) ¯ f α . Note that f α , f α , ¯ f α , ¯ f α are all elements in H .  \nThe twisted Hopf algebra of diffeomorphisms, denoted by H F , raises the question of whether and in what sense it differs from the original Hopf algebra H . It should be noted that H is a cocommutative 2 Hopf algebra, i.e. ∆ cop = ∆. It is easy to show that the coopposite twisted coproduct (∆ F ) cop is related to the twisted coproduct ∆ F by performing a conjugation with the help of the element R = F 21 F -1 ∈ H F ⊗ H F for all ξ ∈ H F (where F 21 = f α ⊗ f α and F -1 21 = ¯ f α ⊗ ¯ f α ), as given by the equation:  \n(∆ F ) cop ( ξ ) = R ∆ F ( ξ ) R -1 . (4)  \nThis element R is called a universal R -matrix and satisfies the quantum Yang-Baxter equation, giving rise to braiding. We use the notation R = R α ⊗ R α and R -1 = ¯ R α ⊗ ¯ R α (sum over α assumed) for the R -matrix and its inverse, respectively.  \nThe twisted Hopf algebra of diffeomorphisms H F is generally not cocommutative. This means that H F differs structurally from the Hopf algebras generated by Lie algebras via the universal enveloping algebra construction. As a result of this noncocommutative behavior of H F , we will obtain a noncommutative structure on the spaces that the Hopf algebra acts on (modules), such as the algebra of functions on the spacetime manifold M .  \nIn the context of noncommutative manifolds, it is important to consider the construction of scalar, vector, and tensor transformations that are compatible with the twisted Hopf algebra of diffeomorphisms. Our focus is on the Hopf algebra H F which characterizes the deformed infinitesimal diffeomorphisms that correspond to the symmetry of a noncommutative manifold. To illustrate, let us consider the simplest type of tensor field: the smooth and complex functions C ∞ ( M ). This space can be equipped with an algebra structure by employing the pointwise multiplication ( hk )( x ) = h ( x ) k ( x ), for all h, k ∈ C ∞ ( M ). The algebra A := ( C ∞ ( M ) , · ) is covariant under the Hopf algebra H , meaning ξ /triangleright · ( f ⊗ g ) = · ∆ ξ /triangleright ( f ⊗ g ) for ξ ∈ H, f, g ∈ C ∞ ( M ). This is the usual Leibniz rule for the action of the Lie derivative, or more formally, left H -module algebra property of A .  \nHowever, when it comes to nontrivial deformations generated by F , ∆ in the preceding relation changes to ∆ F and algebra A fails to be covariant under H F . To resolve this issue, we must deform the product in A according to F , transforming A into A /star - the algebra covariant under H F . In other words, the twisted Hopf algebra H F describes the symmetries of a noncommutative manifold underlying the algebra A /star = ( C ∞ ( M ) , /star ), where the deformed multiplication, the so-called /star -product, is given by:  \nh /star k := · F -1 ( h ⊗ k ) = ¯ f α ( h ) ¯ f α ( k ) , (5)  \nfor all h, k ∈ C ∞ ( M ). A similar prescription can be used to deform the tensor product from ⊗ A to ⊗ A /star and space of tensors from T to T /star . It is important to note that as a vector space A is isomorphic to A /star and also T is isomorphic to T /star , therefore it is only the algebraic structure which we deform. Property (4) of the coproduct is reflected inside the algebra A /star as braided commutativity of A /star :  \nf /star g = ¯ R α ( g ) /star ¯ R α ( f ) . (6)  \nTo implement noncommutative gravity practically, we require a specific Hopf algebra of deformed diffeomorphisms. More precisely, for our study, we select a particular twist F , known as the Moyal-Weyl twist, and utilize it to twist H . If we consider M = R N and use x µ ( µ = 1 , ..., N ) as local coordinate functions on M , then the derivatives ∂ µ along x µ provide a local basis of Ξ. This means that any vector field v ∈ Ξ can be expressed as v = v µ ( x ) ∂ µ , where the coefficient functions v µ ∈ C ∞ ( M ). Notably, ∂ µ ∈ Ξ are locally defined vector fields for all µ = 1 , ..., N . The Moyal-Weyl twist F is an element in U Ξ ⊗ U Ξ, and is defined by the following expression:  \nF = exp( -i Θ µν ∂ µ ⊗ ∂ ν ) , (7)  \nwhere Θ µν is a constant and antisymmetric matrix.', 'B. Quantum Lie algebras and NC geometry': "We can associate a quantum Lie algebra (Ξ , [ · , · ] /star ) to H F , similar to how we can associate the Lie algebra (Ξ , [ · , · ]) of vector fields to H . However, this quantum Lie algebra fits naturally inside another Hopf algebra H /star , which is isomorphic to H F . Now we describe its construction in brief, with universal enveloping algebras as the starting point.  \nFirst note that ( U Ξ , µ ) is a left H -module algebra via the adjoint action. We deform this module algebra to ( U Ξ , µ /star ) by twisting the associative product µ . There is a remarkable relation between the algebras ( U Ξ , µ ) and ( U Ξ , µ /star ), as they are isomorphic. The algebra isomorphism D : ( U Ξ , µ ) → ( U Ξ , µ /star ) is given by the adjoint action D ( ξ ) := Ad ¯ f α ( ξ ) ¯ f α for all ξ ∈ U Ξ, where Ad ¯ f α ( ξ ) := µ (1 ⊗ S )∆ f α = ¯ f α 1 ξS ( ¯ f α 2 ). This algebra isomorphism can be lifted to the Hopf algebra level by introducing H /star = ( U Ξ , µ /star , η, ∆ /star , /epsilon1, S /star ), which is isomorphic to H F . The quantum Lie algebra and the R -matrix it carries are of central importance in our approach; details of the co-algebraic structures of H /star can be found in [24, 25].  \nSince H /star and H F are isomorphic Hopf algebras, any representation of H F is also a representation of H /star . We perform the H /star -action by the /star -Lie derivative, for all ξ ∈ U Ξ, as follows:  \n£ /star ξ := £ D ( ξ ) . (8)  \nWhen acting with the /star -Lie derivative on a /star -tensor product of tensor fields, we obtain  \n£ /star v ( τ ⊗ A /star σ ) = £ /star v ( τ ) ⊗ A /star σ + ¯ R α ( τ ) ⊗ A /star £ /star ¯ R α ( v ) ( σ ) , (9)  \nfor all v ∈ Ξ, and for all tensors τ, σ ∈ T /star . Furthermore, we can construct a quantum Lie algebra (Ξ , [ · , · ] /star ), where the /star -Lie bracket [ · , · ] /star satisfies the deformed antisymmetry property:  \n[ v, w ] /star = -[ ¯ R α ( w ) , ¯ R α ( v )] /star , (10)  \nand the deformed Jacobi identity:  \n[ v, [ w, z ] /star ] /star = [[ v, w ] /star , z ] /star +[ ¯ R α ( w ) , [ ¯ R α ( v ) , z ] /star ] /star , (11)  \nfor all v, w, z ∈ Ξ. The quantum Lie algebra (Ξ , [ · , · ] /star ) represents the infinitesimal deformed diffeomorphisms.  \nIn the framework of noncommutative differential geometry, we can introduce the covariant derivative, torsion, and curvature in a natural way. A /star -covariant derivative ˆ ∇ v is introduced along a vector field v ∈ Ξ. It is a C -linear map ˆ ∇ v : Ξ → Ξ that satisfies the following conditions:  \nˆ ∇ v + w z = ˆ ∇ v z + ˆ ∇ w z, ˆ ∇ h/starv z = h /star ˆ ∇ v z, ˆ ∇ v ( h /star z ) = £ /star v ( h ) /star z + ¯ R α ( h ) /star ˆ ∇ ¯ R α ( v ) z, (12)  \nfor all v, w, z ∈ Ξ and h ∈ C ∞ /star ( M ). When a /star -covariant derivative is given, its /star -torsion and /star -curvature are defined by C -linear maps ˆ T : Ξ ⊗ Ξ → Ξ and ˆ R : Ξ ⊗ Ξ ⊗ Ξ → Ξ, respectively.  \nˆ T ( v, w ) := ˆ ∇ v w -ˆ ∇ ¯ R α ( w ) ¯ R α ( v ) -[ v, w ] /star , ˆ R ( v, w, z ) := ˆ ∇ v ˆ ∇ w z -ˆ ∇ ¯ R α ( w ) ˆ ∇ ¯ R α ( v ) z -ˆ ∇ [ v,w ] /star z, (13)  \nfor all v, w, z ∈ Ξ. To construct the /star -Ricci tensor ˆ R : Ξ ⊗ Ξ → C ∞ /star ( M ), we need a local basis and a contraction covariant under H F . 3 In the /star -dual basis { dx ν } for which 〈 ∂ µ , dx ν 〉 /star = δ ν µ holds, the contraction defines the /star -Ricci tensor as follows:  \nˆ R ( v, w ) = 〈 dx µ , ˆ R ( ∂ µ , v, w ) 〉 /star , (14)  \nfor all v, w ∈ Ξ. It is independent of the choice of basis. The noncommutative Ricci tensor (14) is not symmetric, due to the fact that the noncommutative Riemann tensor defined in (13) is R -antisymmetric in the first two indices,  \nˆ R ( u, v, z ) = -ˆ R ( ¯ R c ( v ) , ¯ R c ( u ) , z ) , (15)  \nbut not in the last two. For the deformations induced by the Moyal-Weyl twist (7), the Christoffel symbols completely specify the /star -covariant derivative as shown below:  \nˆ ∇ ∂ µ ∂ ν = ˆ Γ ρ µν /star ∂ ρ = ˆ Γ ρ µν ∂ ρ . (16)  \nThe last equality follows from the fact that the twist acts trivially on all ∂ ρ . In the basis { ∂ ρ } , the /star -torsion, /star -curvature and /star -Ricci tensor can be expressed as:  \nˆ T ( ∂ µ , ∂ ν ) = ˆ T ρ µν ∂ ρ = ( ˆ Γ ρ µν -ˆ Γ ρ νµ ) ∂ ρ , ˆ R ( ∂ µ , ∂ ν , ∂ ρ ) = ˆ R σ µνρ = ( ∂ µ ˆ Γ σ νρ -∂ ν ˆ Γ σ µρ + ˆ Γ τ νρ /star ˆ Γ σ µτ -ˆ Γ τ µρ /star ˆ Γ σ ντ ) ∂ σ , ˆ R ( ∂ ν , ∂ ρ ) = ˆ R νρ = ∂ µ ˆ Γ µ νρ -∂ ν ˆ Γ µ µρ + ˆ Γ τ νρ /star ˆ Γ µ µτ -ˆ Γ τ µρ /star ˆ Γ µ ντ . (17)  \nIt's important to note that these equations hold in the case of Abelian twist in the so-called nice basis [25]. Abelian twist means that vector fields generating the twist commute and nice basis means that vector fields generating the twist commute with the basis vectors. For the twist (7) this trivially holds since it is composed of derivatives with respect to a coordinate chart.  \nTo formulate a noncommutative gravity theory, we also need to introduce a metric field g = g α ⊗ A /star g α ∈ Ω 1 ⊗ A /star Ω 1 . We can define the /star -inverse metric g -1 = g -1 α ⊗ A /star g -1 α ∈ Ξ ⊗ A /star Ξ by imposing the following conditions for all v ∈ Ξ and ω ∈ Ω 1 :  \n〈〈 v, g 〉 /star , g -1 〉 /star = 〈 v, g α 〉 /star /star 〈 g α , g -1 β 〉 /star /star g -1 β = v, 〈〈 ω, g -1 〉 /star , g 〉 /star = 〈 ω, g -1 β 〉 /star /star 〈 g -1 β , g α 〉 /star /star g α = ω. (18)  \nIn the context of the Moyal-Weyl twist (7), the metric field can be expressed as g = dx µ ⊗ A dx ν g µν = dx µ ⊗ A /star dx ν /star g µν , since the /star symbol reduces to its zeroth order commutative multiplication when acting on ∂ ρ or dx µ . The inverse metric field is given by g -1 = g /starµν /star ∂ µ ⊗ A /star ∂ ν , where g µν and g /starµν satisfy the conditions:  \ng µν /star g /starν ρ = δ ρ µ , g /starµν /star g νρ = δ µ ρ . (19)  \nSpecifically, we have [26]  \ng /starαβ = g αβ -g γβ i Θ µν ( ∂ µ g ασ )( ∂ ν g σγ ) + O (Θ 2 ) . (20)  \nThe unique torsion-free, ˆ T = 0, and metric compatible 4 , ˆ ∇ ( g ) = 0, NC Levi-Civita connection is given by [24],  \nˆ Γ ρ µν = 1 2 g /starρσ /star ( ∂ µ g νσ + ∂ ν g µσ -∂ σ g µν ) . (21)  \nIn the next section, we will use this formalism in the context of black hole perturbation theory.", 'III. NONCOMMUTATIVE PERTURBATIONS OF THE SCHWARZCHILD BLACK HOLE': "In this section, our investigation delves into the noncommutative deformation of black hole perturbations. The metric ˚ g µν characterizes the background spacetime of a black hole, and its perturbation is represented by the rank-2 symmetric tensor h µν . The complete metric is therefore  \ng µν =˚ g µν + h µν . (22)  \nIn our calculation, we adopt the Schwarzschild metric as the background metric ˚ g :  \nds 2 =˚ g µν dx µ dx ν = -( 1 -R r ) dt 2 + ( 1 -R r ) -1 dr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 ) , (23)  \nwhere R = 2 M denotes the horizon radius, and M is the mass of the black hole.  \nConventionally, the perturbation of a black hole in commutative theory is studied by imposing that the Ricci tensor is zero, R µν = 0, up to first order in h , following from the linearized Einstein equations 5 . Extending the notion of Einstein manifolds to noncommutative gravity requires careful consideration. Although one might intuitively consider ˆ R µν = 0 as a straightforward extension by merely accounting for the correct commutative limit, the relation ˆ R νµ = 0, in contrast to the commutative case, will not be satisfied automatically here. This is because the definition of ˆ R µν does not guarantee the symmetry property and therefore imposing ˆ R µν = 0 leads to contradictions. To navigate this challenge, a recently proposed approach involves postulating a noncommutative Einstein manifold [18, 19] as  \nˆ R µν = 0 . (24)  \nHere, ˆ R µν is the R -symmetrized Ricci tensor, defined as  \nˆ R µν ≡ 1 2 〈 dx α , ˆ R ( ∂ α , ∂ µ , ∂ ν ) + ˆ R ( ∂ α , ¯ R A ( ∂ ν ) , ¯ R A ( ∂ µ )) 〉 /star . (25)  \nNote that this proposal, first presented in [18, 19], differs from the one found in [22-25].  \nAs mentioned in the preceding section, we are in pursuit of noncommutative corrections to gravitational perturbations induced by the Moyal-Weyl twist (7). For this type of deformation, R -symmetrization reduces to the usual symmetrization, i.e.  \nˆ R µν = ˆ R ( µν ) ≡ 1 2 ( ˆ R µν + ˆ R νµ ) . (26)  \nIn addition, we consider an Abelian twist, a form of the Moyal-Weyl twist where globally defined vector fields { X µ } generating the twist commute. This twist has the form  \nF = exp( -i Θ µν X µ ⊗ X ν ) . (27)  \n/negationslash  \nIt is important to keep in mind that the selection of a specific twist is a matter of quantum-gravity phenomenology and ideally should be constrained by experiments. In the absence of such empirical inputs, one must rely on symmetry arguments to make a physically meaningful choice. The simplest approach in this regard might involve constructing a twist solely from a basis composed of Killing vectors K µ of the background ˚ g µν . However, in the unperturbed background, it becomes apparent that such a choice would not yield a nontrivial noncommutative contribution. This is because any solution to the commutative Einstein equation is also a solution to the NC Einstein equation in cases where the twist is constructed from the Killing vectors of the nonperturbed background metric since £ K ˚ g = 0. Similar arguments apply to the choice of a semi-Killing twist, constructed using a Killing vector K µ and an arbitrary basis vector V ν . For more details, see [18, 19, 24, 25]. In the realm of black hole perturbation studies, our focus is on the perturbed background black hole spacetime described by the metric ˚ g µν + h µν . In this scenario, opting for the mere Killing twist seems to result in non-vanishing NC corrections that are quadratic in the metric perturbation h . This comes from the fact that £ K ˚ g = 0 and £ K h = 0, resulting in only a quadratic leading-order term in noncommutative geometric quantities. In other words, if we want to respect the full symmetry of the background ˚ g , then the NC effects are inherently nonlinear in perturbation, and we are forced to study quadratic commutative corrections, which would go beyond the linearized metric perturbation theory.  \nHowever, the leading linearized noncommutative metric perturbation term persists if we opt for a semi-pseudoKilling twist constructed from the Killing field K µ of the background ˚ g and an arbitrary vector X ν . We consider a semi-pseudo-Killing twist of the form:  \nF = e -i a 2 ( K ⊗ X -X ⊗ K ) . (28)  \nIn the spherical coordinate basis, two Killing fields, namely ∂ t and ∂ ϕ are implemented as partial derivatives which makes them suitable for constructing the field K :  \nK = α∂ t + β∂ ϕ , X = ∂ r , α, β ∈ R . (29)  \nThis choice produces the following commutation relations between the coordinates:  \n[  \nt  \n/star  \n, r  \n] =  \niaα,  \n[ ϕ /star , r ] = iaβ.  \nBoth commutators display quantization of the radial coordinate as this is necessary in order to have noncommutative corrections to perturbations at the linear level. Furthermore, we introduce the parameter λ as an eigenvalue of the Killing field's action on the perturbation:  \nh µν ∝ e imϕ e -iωt = ⇒ £ K h µν = iλ h µν , (30)  \nfor K = α∂ t + β∂ ϕ , λ = -αω + βm. (31)  \nIn the calculation of the noncommutative entities, we can use the following linearized /star -product operation following (28) and (5):  \nh /star k = hk + i 2 a ( K ( h ) X ( k ) -X ( h ) K ( k ) ) + O ( a 2 ) , (32)  \nwhere h, k ∈ C ∞ /star ( M ) and V ( f ) := £ V f for generic vector field V µ and function f .  \nThe spherically symmetric nature of the Schwarzschild background allows us to decompose the perturbations h µν into tensor spherical harmonics Y n /lscriptm ( θ, ϕ ), which can all be expressed in terms of Y /lscriptm ( θ, ϕ ) and their derivatives with respect to θ and ϕ . Schwarzschild background allows us to decompose perturbations h µν into scalar, vectorial, and tensorial spherical harmonics, which are certain derivatives with respect to θ and ϕ of Y /lscriptm ( θ, ϕ ). These perturbations manifest as axial and polar modes, each displaying distinct behaviors under the parity transformation /vectorr →-/vectorr . Evenparity or polar modes, akin to scalar spherical harmonics Y /lscriptm ( θ, ϕ ), transform as ( -1) /lscript , while axial or odd-parity modes transform as ( -1) /lscript +1 . In the linear order, these modes are decoupled, facilitating separate treatment.  \nWe will calculate the equations of motion up to linear order in both perturbation h µν and the NC deformation a . Due to the semi-psuedo Killing form of the twist, the perturbation and deformation parts are inherently coupled, appearing in pairs.", 'A. Axial perturbations and noncommutative Regge-Wheeler potential': "Here we provide a brief overview of noncommutative axial perturbations [18-20]. The idea is to mimic the procedure of Regge and Wheeler [13] in the NC setting by deforming the geometric quantities using methods outlined in the previous section.  \nGenerally, axial perturbations are characterized by three families of radial functions h /lscriptm 0 ( r ), h /lscriptm 1 ( r ), and h /lscriptm 2 ( r ). However, in the Regge-Wheeler gauge (see Appendix A in [19] for details), one can choose h /lscriptm 2 = 0, resulting in only four non-zero components of the perturbation metric [27]:  \nh tθ = 1 sin θ ∑ /lscript,m h /lscriptm 0 ∂ φ Y /lscriptm ( θ, φ ) e -iωt , h tφ = -sin θ ∑ /lscript,m h /lscriptm 0 ∂ θ Y /lscriptm ( θ, φ ) e -iωt , h rθ = 1 sin θ ∑ /lscript,m h /lscriptm 1 ∂ φ Y /lscriptm ( θ, φ ) e -iωt , h rφ = -sin θ ∑ /lscript,m h /lscriptm 1 ∂ θ Y /lscriptm ( θ, φ ) e -iωt . (33)  \nSimilar to the commutative case [13, 28], we obtain only 7 non-zero components of the equation (24). These components are in separable form, resulting in only 3 distinguishable (mutually distinct) radial components.  \nFrom ˆ R rφ = 0 we get  \n4 ir 4 ( r -R ) ωh 0 +2 r 2 ( r -R ) ( r 3 ω 2 -( r -R )( /lscript ( /lscript +1) -2) ) h 1 -2 iωr 5 ( r -R ) h ' 0 + λa [ 2 ir 3 ω ( r -2 R ) h 0 + ( (2 /lscript ( /lscript +1) + 12) r ( r -R ) 2 -9( r -R ) 2 R -r 4 Rω 2 ) h 1 + ir 4 Rωh ' 0 +2 r ( r -R ) 3 h ' 1 ] = 0 , (34)  \nfrom ˆ R tφ = 0 we get  \n2 r (2 R -/lscript ( /lscript +1) r ) h 0 +4 ir 2 ω ( r -R ) h 1 +2 r 3 ( r -R )( iωh ' 1 + h '' 0 ) + λa [ (2 /lscript ( /lscript +1) r + R ) h 0 + irω (4 r -3 R ) h 1 + r (4 r -5 R ) h ' 0 + r 2 R ( iωh ' 1 + h '' 0 ) ] = 0 , (35)  \nand from ˆ R θφ = 0 we get  \nir 3 ω r -R h 0 + Rh 1 + r ( r -R ) h ' 1 -λa [ ir 2 Rω 2( r -R ) 2 h 0 -3 r -R r h 1 -1 2 Rh ' 1 ] = 0 . (36)  \nThe radial parts of the linearized equations ˆ R rθ = 0, ˆ R θθ = 0, ˆ R ϕϕ = 0, and ˆ R tθ = 0 are identical to the above equations. Among the 3 equations (34-36), only 2 are mutually independent. We are therefore in position to pick any two out of these three equations to find the solution of the whole system, the easiest choice being first-order equations (34) and (36). Combining these equations eliminates h 0 and yields a single second-order differential equation for h 1 , as demonstrated in [19],  \nr ( r -R ) ( /lscript ( /lscript +1) r ( R -r ) + 2 r 2 -6 rR +5 R 2 + ω 2 r 4 ) h 1 + r 2 ( r -R ) 2 ( (5 R -2 r ) h ' 1 + r ( r -R ) h '' 1 ) + λa [ ( /lscript ( /lscript +1) r ( r -R ) 2 -6 r 3 + R 2 (49 r 2 -64 rR +26 R 2 -ω 2 r 4 ) ) h 1 + r ( r -R ) 2 ( 3( r -2 R ) h ' 1 + 1 2 rRh '' 1 ) ] = 0 . (37)  \nThis can be re-expressed as a Schrodinger-like equation with a field redefinition  \nh 1 ( r ) = r 2 r -R [ 1 + λa 2 ( 3 r -1 r -R + 1 R log r r -R )] ψ ( r ) (38)  \nand a noncommutative tortoise variable transformation  \nˆ r ∗ = r + R log r -R R + λa 2 R r -R . (39)  \nFinally, we arrive at the differential equation,  \nd 2 ψ d ˆ r 2 ∗ + ( ω 2 -V ( r ) ) ψ = 0 , (40)  \nwhere V ( r ) = V RW + V NC represents the noncommutative Regge-Wheeler potential,  \nV ( r ) = ( r -R ) ( /lscript ( /lscript +1) r -3 R ) r 4 + λa /lscript ( /lscript +1)(3 R -2 r ) r + R (5 r -8 R ) 2 r 5 . (41)  \nIn the limit a → 0, only V RW remains, representing the commutative Regge-Wheeler potential [13, 28], while V NC denotes the noncommutative correction to the Regge-Wheeler potential. It is noteworthy that in case α = 0 , β = 1 in (31), V NC is dependent on /lscript and m , leading to a Zeeman-like splitting as depicted in Figure 1. From the figure, it is evident that two crucial aspects of the potential, namely the peak and the zero, are influenced by the noncommutative structure of spacetime. This behavior of the NC potential persists for higher /lscript modes as can be seen from Figure 2.  \nFIG. 1. The figure shows the behavior of the potential for R = 2 ( M = 1) and /lscript = 2. Both noncommutative and commutative cases exhibit similar behavior outside the horizon, particularly around the peak at r 0 ∼ = 1 . 5 R and for larger r . The right panel highlights the behavior of the potential near the peak.  \n<!-- image -->  \n<!-- image -->  \nIn the commutative case, the potential vanishes precisely at the event horizon r = R . However, in the noncommutative case, the potential vanishes slightly either inside or outside the horizon, depending on the sign of λa , i.e., r = R ± λa 2 + O ( a 2 ). This implies that each mode m effectively 'perceives' a somewhat different position of the horizon. Physically, this characteristic can be understood as the introduction of a level of fuzziness to the event horizon in particle scattering. Specifically, particle scattering in the potential (41) can be described as a summation of various modes ψ n/lscriptm , i.e., Ψ = ∑ ψ n/lscriptm e -iω n t . Since each mode experiences the position of the event horizon differently, the collective wave packet may perceive a slightly shifted event horizon instead of a sharply defined one.  \nThe second aspect altered by noncommutativity is the location, width and height of the peak of the potential. From an astrophysical perspective, this is intriguing as it will modify the photon sphere radius and the shadow of the black hole. We shall see in the next section that the noncommutative polar potential exhibits the same properties.", 'B. Polar perturbations and noncommutative Zerilli potential': "In this subsection we follow the derivation of Zerilli [14] to obtain NC corrections to the polar perturbations. The procedure outlined there can be adapted to the NC setting by deforming the geometry according to the rules of the previous section. Similar to the axial case, the solution of the NC vacuum Einstein equation (24) in the end turns out to be completely governed by the NC Zerilli potential in the Schrodinger equation.  \nPolar perturbations of the metric are characterized by seven families of functions. However, by exploiting the invariance of the theory under spacetime diffeomorphisms we can adopt a specific gauge, namely the Zerilli gauge (refer to Appendix A for details). In this gauge, the polar perturbations are parameterized by four families of radial functions: H /lscriptm 0 , H /lscriptm 1 , H /lscriptm 2 , and K /lscriptm as described below:  \nh tt = f ( r ) ∑ /lscript,m H /lscriptm 0 ( r ) Y /lscriptm ( θ, ϕ ) e -iωt , h tr = ∑ /lscript,m H /lscriptm 1 ( r ) Y /lscriptm ( θ, ϕ ) e -iωt , (42)  \nh rr = 1 f ( r ) ∑ /lscript,m H /lscriptm 2 ( r ) Y /lscriptm ( θ, ϕ ) e -iωt , h ab = ∑ /lscript,m K /lscriptm ( r )˚ g ab Y /lscriptm ( θ, ϕ ) e -iωt , (43)  \nwhere f ( r ) = 1 -R/r . Here, the indices a and b represent angular coordinates θ and ϕ . For this metric we find the /star -inverse metric, connection, curvature and Ricci tensor as defined in the previous section. Finally, the linearized Einstein equation obtained from the R -symmetrized Ricci tensor (24) gives rise to 10 coupled ordinary differential equations in r due to the separability of the angular parts. Out of these 10 equations, only 7 are distinguishable.  \nAfter separating the angular part, the ˆ R θϕ = 0 equation reduces to the following algebraic relation,  \nH 0 -H 2 + λa [ R 2 r ( r -R ) H 0 + R 2 r ( r -R ) H 2 ] = 0 . (44)  \nThis equation can be solved for H 2 in terms of H 0 , allowing for the elimination of H 2 in the remaining equations,  \nH 2 = H 0 -λa R r ( R -r ) H 0 . (45)  \nIn the commutative limit, this solution simplifies to H 2 = H 0 . The remaining 6 equations involve only three unknown functions K , H 0 , and H 1 .  \nEquation ˆ R tt = 0 reduces to  \n2( r -R ) r 2 H ' 0 + ( r -R ) 2 r 2 H '' 0 + ( /lscript ( /lscript +1)( R -r ) r 3 -ω 2 ) H 0 + 2 iω ( r -R ) r H ' 1 + iω (4 r -3 R ) r 2 H 1 + R ( R -r ) r 3 K ' -2 ω 2 K -λa [ ( r -R ) r 3 H ' 0 + R ( r -R ) 2 r 3 H '' 0 + ( ( r -R )( /lscript ( /lscript +1) r + R ) r 5 + Rω 2 2 r ( r -R ) ) H 0 + iωR r 2 H ' 1 + iω (2 r -R ) r 3 H 1 + R (4 r -5 R ) 2 r 4 K ' + ( 2 r 4 ω 2 + R ( r -R ) ) r 5 K ] = 0 . (46)  \nThe ˆ R tr = 0 equation reduces to  \n-H 0 r -i/lscript ( /lscript +1) 2 r 2 ω H 1 + K ' + 1 2 ( 1 R -r + 3 r ) K + λa [ ( 2 r 2 -6 rR +5 R 2 ) 4 r 2 ( r -R ) 2 H 0 + iR ( R -2 r ) 4 r 3 ω ( r -R ) H ' 1 + i 4 r 4 ω ( R 2 (3 R -4 r ) ( r -R ) 2 -2 /lscript ( /lscript +1) r ) H 1 + K ' r + ( 4 r 2 -12 rR +7 R 2 ) 4 r 2 ( r -R ) 2 K ] = 0 . (47)  \nThe ˆ R rr = 0 equation reduces to  \n2( r -R ) r 2 H ' 0 + ( r -R ) 2 r 2 H '' 0 + ( /lscript ( /lscript +1)( r -R ) r 3 -ω 2 ) H 0 + 2 iω ( r -R ) r H ' 1 + iωR r 2 H 1 -(4 r -3 R )( r -R ) r 3 K ' -2( r -R ) 2 r 2 K '' -λa [ ( r 2 -3 rR + R 2 ) r 4 H ' 0 + R ( R -r ) 2 r 3 H '' 0 + 2 /lscript ( /lscript +1) r 2 ( R -r ) + 3 ω 2 r 4 R +2 R (3 r -2 R ) 2 2 r 5 ( r -R ) H 0 -iωR r 2 H ' 1 + iωR r 3 H 1 + ( 8 r 2 -12 rR +5 R 2 ) 2 r 4 K ' + 2( r -R ) 2 r 3 K '' + R ( r -R ) r 5 K ] = 0 . (48)  \nThe ˆ R tθ = 0 equation reduces to  \niωH 0 + ( 1 -R r ) H ' 1 + R r 2 H 1 + iωK + λa [ iωR 2 r ( r -R ) H 0 -R 2 r 2 H ' 1 -( 4 r 2 -9 rR +6 R 2 ) 2 r 3 ( r -R ) H 1 + iω r K ] = 0 . (49)  \nThe ˆ R rθ = 0 equation reduces to  \n( 1 -R r ) H ' 0 + R r 2 H 0 + iωH 1 + ( R r -1 ) K ' + λa [ R 2 r 2 H ' 0 -( 4 r 2 -9 rR +3 R 2 ) 2 r 3 ( r -R ) H 0 + iωR 2 r ( r -R ) H 1 -( r -R ) r 2 K ' -2( R -r ) r 3 K ] = 0 . (50)  \nThe ˆ R θθ = 0 equation reduces to  \n2( R -r ) H ' 0 -2 H 0 -2 iωrH 1 +(4 r -3 R ) K ' + r ( r -R ) K '' + ( -/lscript ( /lscript +1) + r 3 ω 2 r -R +2 ) K -λa [ ( r + R ) r H ' 0 + ( 3 R -4 r r 2 + 2 r -R ) H 0 + iω ( r + R ) r -R H 1 + 1 2 RK '' +2 K ' -1 2 r 2 ( -2( /lscript ( /lscript +1) + 2) r + ω 2 r 4 R ( r -R ) 2 +6 R ) K ] = 0 . (51)  \nIn the limit a → 0, the linearized equations of motion derived from the condition ˆ R µν = 0 coincide with the equations obtained by Edelstein and Vishveshwara (Eq. (9 b ) -(9 g ) in [28]). The linearized equations ˆ R tϕ = 0, ˆ R rϕ = 0, and ˆ R ϕϕ = 0, in their radial part correspond identically to ˆ R tθ = 0, ˆ R rθ = 0 and ˆ R θθ = 0, respectively. The combination  \nˆ R rr -( 1 + λa R r ( r -R ) ) ˆ R tt -( -2( r -R ) r 3 -λa (2 r -R ) r 4 ) ˆ R θθ = 0 (52)  \nreduces the three second-order differential equations to a first-order relation  \n-4( r -R ) 2 r 3 H ' 0 + 2( /lscript ( /lscript +1) -2)( r -R ) r 3 H 0 -8 iω ( r -R ) r 2 H 1 + 2 ( 2 r 2 -3 rR + R 2 ) K ' r 3 + ( 4 ω 2 -2( /lscript ( /lscript +1) -2)( r -R ) r 3 ) K -λa [ ( 6 r 2 -6 rR + R 2 ) r 4 H ' 0 + ( -2 /lscript ( /lscript +1) r 3 +2( /lscript ( /lscript +1) + 10) r 2 R -28 rR 2 +9 R 3 ) r 5 ( r -R ) H 0 + iω ( 4 r 2 +2 rR -5 R 2 ) r 3 ( r -R ) H 1 + R ( R -2 r ) r 4 K ' + ( (4 r -3 R )( /lscript ( /lscript +1) r -2 R ) r 5 -4 ω 2 r -R ) K ] = 0 . (53)  \nThe first-order terms in the above expression can be removed by employing the first-order equations (47), (49) and (50), resulting in an algebraic relation  \n( -/lscript ( /lscript +1) -3 R r +2 ) H 0 + ( 2 irω -i/lscript ( /lscript +1) R 2 r 2 ω ) H 1 + ( -2( /lscript ( /lscript +1) -2) r 2 +2( /lscript ( /lscript +1) -3) rR +4 r 4 ω 2 +3 R 2 ) 2 r ( R -r ) K -λa [ ( /lscript ( /lscript +1) -1 r -R + 3 R 2 r 2 -8 r ) H 0 + i ( -/lscript ( /lscript +1) r (5 r -4 R )( r -R ) + 2 r 4 ω 2 ( r -3 R ) + R 3 ) 2 r 4 ω ( r -R ) H 1 + ( (18 -4 /lscript ( /lscript +1)) r 3 +7( /lscript ( /lscript +1) -7) r 2 R +(47 -3 /lscript ( /lscript +1)) rR 2 +4 r 5 ω 2 -15 R 3 ) 2 r 2 ( r -R ) 2 K ] = 0 . (54)  \nThe above derivation demonstrates that any of the second-order equations in (46)-(51) can be derived from the three first-order equations, provided the algebraic relationship (54) holds. Therefore, it suffices to concentrate on the first-order equations to obtain the complete solution for the system of equations (46)-(51). However, one can prove that one equation among the three is redundant - the algebraic equation eliminates H 0 from the first-order equations, leaving three equations in two functions K and H 1 . The dependence of the three equations can be proved in a similar way to appendix B of [19].  \nSolving the system therefore amounts to solving for K and H 1 from (47) and (49). Rearranging these equations, while eliminating H 0 with the aid of algebraic relation, yields two first-order equations (with redefinition 6 L = H 1 /ω ):  \nK ' = [ α 0 ( r ) + α 2 ( r ) ω 2 ] K + [ β 0 ( r ) + β 2 ( r ) ω 2 ] L, L ' = [ γ 0 ( r ) + γ 2 ( r ) ω 2 ] K + [ δ 0 ( r ) + δ 2 ( r ) ω 2 ] L, (55)  \nwhere α ( r ), β ( r ), γ ( r ), and δ ( r ), with the abbreviation /lscript ( /lscript +1) = 2Λ + 2 are given below:  \n,  \nα 0 ( r ) = R ((Λ -2) r +3 R ) r ( r -R )(2Λ r +3 R ) -λa [ ( 4(Λ -2)Λ r 4 +4 ( 4 -5Λ 2 ) r 3 R +(Λ(12Λ -17) -56) r 2 R 2 +15(Λ + 3) rR 3 -9 R 4 ) 2 r 2 ( r -R ) 2 (2Λ r +3 R ) 2 ] α 2 ( r ) = -2 r 3 ( r -R )(2Λ r +3 R ) -λa [ 2 r 2 ( (7 -3Λ) r 2 +(5Λ -11) rR +9 R 2 ) ( r -R ) 2 (2Λ r +3 R ) 2 ] ,  \nβ 0 ( r ) = 2 i (Λ + 1)(Λ r + R ) r 2 (2Λ r +3 R ) -λa [ i ( -10Λ(Λ + 1) r 4 +(Λ+1)(13Λ -8) r 3 R +(13 -5(Λ -2)Λ) r 2 R 2 -(8Λ + 3) rR 3 -3 R 4 ) r 4 ( r -R )(2Λ r +3 R ) 2 ] ,  \nβ 2 ( r ) = 2 ir 2Λ r +3 R -λa [ i ( 2(6Λ -7) r 2 +(31 -16Λ) rR -27 R 2 ) ( r -R )(2Λ r +3 R ) 2 ] ,  \nγ 0 ( r ) = -ir (8Λ r ( r -R ) + R (8 r -9 R )) 2( r -R ) 2 (2Λ r +3 R ) -λa [ i 4( r -R ) 3 (2Λ r +3 R ) 2 ( 32Λ 2 r 4 +16Λ r 4 -48Λ 2 r 3 R +96Λ r 3 R -32 r 3 R +16Λ 2 r 2 R 2 (56)  \nrR  \n3  \n+81  \nR  \n4  \n,  \n-188Λ r 2 R 2 +166 r 2 R 2 +72Λ rR 3 -225 ) γ 2 ( r ) = 2 ir 5 ( r -R ) 2 (2Λ r +3 R ) + λa [ ir 4 ( 14 r 2 +4Λ rR -13 rR +9 R 2 ) ( r -R ) 3 (2Λ r +3 R ) 2 ] ,  \nδ  \n0  \n(  \nr  \n) =  \n-  \nR  \n(3Λ  \nr  \n+  \nr  \n+3  \nR  \n)  \nr ( r -R )(2Λ r +3 R ) -λa [ -1 2 ( r 2 ( r -R ) 2 (2Λ r +3 R ) 2 ) ( 36Λ 2 r 4 +20Λ r 4 -68Λ 2 r 3 R +32Λ r 3 R +16 r 3 R  \n2  \n+32Λ 2 r 2 R 2 -131Λ r 2 R ) δ 2 ( r ) = 2 r 3 ( r -R )(2Λ r +3 R ) -λa [ 2 r 2 ( 3Λ r 2 -7 r 2 -5Λ rR +11 rR -9 R 2 ) ( r -R ) 2 (2Λ r +3 R ) 2 ] .  \n+  \nr  \n2  \nR  \n2  \n+73Λ  \nrR  \n3  \n-  \n66  \nrR  \n3  \n+42  \nR  \n4  \n]  \n,  \nTo cast the coupled differential equations (55) in a form resembling the Schrodinger equation, we introduce a field redefinition  \nK = ˆ f ( r ) ˆ K +ˆ g ( r ) ˆ L, L = ˆ h ( r ) ˆ K + ˆ l ( r ) ˆ L, (57)  \naccompanied by a coordinate transformation dr/d ˆ r ∗ = ˆ n ( r ). Our aim is to find the functions ˆ f, ˆ g, ˆ h , ˆ l and ˆ n such that the following conditions hold:  \nd ˆ K d ˆ r ∗ = ˆ L, d ˆ L d ˆ r ∗ = ( V -ω 2 ) ˆ K, (58)  \nwhich, when combined, form the Schrodinger equation for ˆ K ,  \nd 2 ˆ K d ˆ r 2 ∗ +( ω 2 -V ) ˆ K = 0 . (59)  \n]  \nThe system (55) expressed in terms of ˆ K and ˆ L is  \nd ˆ K d ˆ r ∗ = [ ˆ α 0 ( r ) + ˆ α 2 ( r ) ω 2 ] ˆ K + [ ˆ β 0 ( r ) + ˆ β 2 ( r ) ω 2 ] ˆ L, d ˆ L d ˆ r ∗ = [ ˆ γ 0 ( r ) + ˆ γ 2 ( r ) ω 2 ] ˆ K + [ ˆ δ 0 ( r ) + ˆ δ 2 ( r ) ω 2 ] ˆ L. (60)  \nThe requirement (58) imposes the following specific constraints on the coefficient functions:  \nˆ α 0 ( r ) = ˆ α 2 ( r ) = ˆ β 2 ( r ) = ˆ δ 0 ( r ) = ˆ δ 2 ( r ) = 0 , ˆ β 0 ( r ) = 1 , ˆ γ 2 ( r ) = -1 . (61)  \nOur requirement involves seven conditions for five unknowns: the coefficients ˆ f ( r ), ˆ g ( r ), ˆ h ( r ), ˆ l ( r ) and the tortoise coordinate ˆ n ( r ). Fortunately, the additional degrees of freedom coincide, allowing the system to admit a solution. Moreover, the solution reduces to the commutative result in the limit a → 0. The exact functional forms of the coefficients of the transformation equation (57) are complicated and are provided in Appendix B. The solutions we obtained have the form  \nˆ f ( r ) = f ( r ) -λa ˜ f ( r ) , ˆ g ( r ) = g ( r ) -λa ˜ g ( r ) , ˆ h ( r ) = h ( r ) -λa ˜ h ( r ) , ˆ l ( r ) = l ( r ) -λa ˜ l ( r ) , (62)  \nwhere { f ( r ) , g ( r ) , h ( r ) , l ( r ) } are the commutative components and { ˜ f ( r ) , ˜ g ( r ) , ˜ h ( r ) , ˜ l ( r ) } denote the noncommutative corrections. The NC tortoise coordinate of the transformation is given by  \nˆ r ∗ = r + R log r -R R -λa ( (2Λ + 7) R 2(2Λ + 3)( r -R ) -4(Λ + 3) log ( r R -1 ) (2Λ + 3) 2 -9 log ( 2Λ r R +3 ) Λ(2Λ + 3) 2 ) (63)  \nwhere we used the abbreviation /lscript ( /lscript +1) = 2Λ+2. Note that the noncommutative part of the tortoise coordinate differs from that of axial perturbation (see Ref. [19]) since it depends on both /lscript and m . The resulting Schrodinger-like form of the equation (59) features an effective potential expressed as V = V Z + V NC , where V Z denotes the commutative Zerilli potential, and V NC represents the noncommutative correction. The explicit forms of these potentials are  \nV Z = ( r -R ) ( 8Λ 2 (Λ + 1) r 3 +12Λ 2 r 2 R +18Λ rR 2 +9 R 3 ) r 4 (2Λ r +3 R ) 2 , V NC = λa 4 r 5 (2Λ r +3 R ) 3 [ 32Λ 2 ( 2Λ 2 +7 ) r 5 -8Λ 2 (2Λ(6Λ -13) + 121) r 4 R -12Λ(2Λ(15Λ -58) + 59) r 3 R 2 -2(Λ(440Λ -741) + 162) r 2 R 3 -3(316Λ -207) rR 4 -387 R 5 ] . (64)  \nThe potential V = V Z + V NC is illustrated in Figure 3(a). Similar to the axial case, the noncommutative structure of spacetime influences two significant characteristics of the potential: its peak and zero. In the commutative scenario, the polar potential precisely vanishes at the horizon. However, in the noncommutative case, the polar potential either vanishes slightly inside or outside the horizon, depending on the sign of the correction term. This phenomenon, as discussed in the axial case, can be attributed to the fuzziness introduced by noncommutativity around the black hole horizon. The behavior of the polar potential around its peak is depicted in Figure 3(b). Similar to the axial case, deviations observed around the peak of the potential, at approximately r 0 ≈ 1 . 5 R , can be interpreted as Zeeman-like splitting in case of pure r -ϕ noncommutativity ( α = 0 , β = 1 = ⇒ λ = m ). Contrasting Figure 1(b) with 3(b), we see that polar perturbations seem to alter the width of the potential by a larger margin. The same goes for the height, although the difference is less prominent.  \nIt is worth noting that a similar splitting of the potential and the corresponding QNM spectrum was observed in [29, 30] in the context of NC charged scalar field probing a background of the Reissner-Nordstrom black hole. This property reminiscent of Zeeman-like splitting, along with a distinct peak around r 0 ∼ = 1 . 5 R persists for higher multipoles as well, as illustrated in Figure 4. Additionally, Figure 4 demonstrates that the shifting of the potential near the horizon follows a consistent trend across all modes. Interestingly, zeros of the potential (and the position of the peak) are translated oppositely relative to the axial case. This is visible from the ordering of blue and red lines in Figures 1(b) and 3(b).  \nFIG. 3. Plot of the potential for R = 2 ( M = 1) and /lscript = 2 for a wider range of r and V ( r ) is on the left. Near-horizon region is shown on the right.  \n<!-- image -->  \n<!-- image -->", 'IV. NONCOMMUTATIVE CORRECTIONS TO QUASINORMAL MODES': 'In this section, we will employ semi-analytical methods to compute the quasinormal frequencies of both noncommutative axial and polar perturbations. By analyzing the results of these calculations, we aim to assess the stability of the black hole spacetime under perturbation and how the presence of noncommutativity affects the QNM spectra.', 'A. The WKB Method': "One of the most elegant methods for calculating quasinormal modes in black hole perturbation theory is the WKB (Jeffreys-Wentzel-Kramers-Brillouin) approximation. This method was initially introduced in the context of black hole scattering theory by Schutz and Will [21]. The key concept in this approach involves matching the solutions at spatial infinity and the black hole horizon using a Taylor expansion in the vicinity of the maximum of the potential barrier through the two turning points. Here, the Schrodinger-like equation is reformulated into the form,  \nd 2 Ψ dx 2 + Q ( x )Ψ( x ) = 0 , (65)  \nwhere x = r ∗ and Q ( x ) = ω 2 -V . Since -Q max ( x ) /lessmuch Q ( ±∞ ), the solution near the peak of the potential can be approximated by the Taylor series,  \nQ ( x ) = Q 0 + 1 2 Q '' 0 ( x -x 0 ) 2 + O (( x -x 0 ) 3 ) , (66)  \nwhere x 0 is the location of the maximum with the function value Q 0 = Q ( x 0 ) and second derivative Q '' 0 is with respect to x at the peak. The Q ' 0 term does not appear in the series expansion as it vanishes at the maximum. Substituting (66) into (65) results in a parabolic cylinder differential equation, known as the Weber equation. At first order, the solution to this equation reads (in terms of the effective potential),  \nω 2 = V 0 -i √ -2 V '' 0 ( n + 1 2 ) . (67)  \nHere, n denotes the harmonic or overtone number. While the analytical expression for the quasinormal mode frequency is derived, obtaining an analytical expression for x 0 or the tortoise coordinate at the peak of the potential is not always feasible. Thus, the WKB approach is considered semi-analytical in this regard.  \nTo enhance precision, the WKB method has been extended to the third order by Iyer and Will [31]. The third-order formula for QNM frequencies is given by  \nwhere  \nω 2 = [ V 0 + √ -2 V '' 0 Λ 1 ] -iν √ -2 V '' 0 [1 + Λ 2 ] , (68)  \nΛ 1 = 1 √ -2 V '' 0   1 8 ( V (4) 0 V '' 0 ) ( 1 4 + ν 2 ) -1 288 ( V (3) 0 V '' 0 ) 2 ( 7 + 60 ν 2 )   , (69a) Λ 2 = -1 2 V '' 0   5 6912 ( V (3) 0 V '' 0 ) 4 ( 77 + 188 ν 2 ) -1 384 V (3)2 0 V (4) 0 V '' 3 0 ( 51 + 100 ν 2 ) (69b) + 1 2304 ( V (4) 0 V '' 0 ) 2 ( 67 + 68 ν 2 ) + 1 288 V (3) 0 V (5) 0 V '' 2 0 ( 19 + 28 ν 2 ) -1 288 V (6) 0 V '' 0 ( 5 + 4 ν 2 ) ] ,  \nwith ν = n +1 / 2, and V ( j ) 0 denoting the j th derivative of the potential evaluated at its peak in tortoise coordinate. Subsequently, the method was extended to the sixth order by Konoplya [32], resulting in even greater accuracy. The expression for QNM frequencies is given by,  \ni ( ω 2 -V 0 ) √ -2 V '' 0 -6 ∑ i =2 ˜ Λ i = n + 1 2 . (70)  \nThe higher-order corrections ˜ Λ i involve complicated functions of derivatives of potential up to V (12) 0 , with explicit expressions available in [32]. Furthermore, the WKB method has been developed up to the thirteenth order by Matyjasek and Opala [33]. However, it is noteworthy that higher-order WKB does not always lead to superior approximations due to the fact that WKB expansion is an asymptotic series [34]. The optimal order for minimizing the error in QNM frequency calculation depends on the specific effective potential. The error is estimated by comparing the solution of two consecutive orders using the formula [34]  \n∆ k = | ω k +1 -ω k -1 | 2 , (71)  \nwhere for the sake of comparison, one usually takes fundamental mode frequency. To obtain the numerical values for the frequencies for the noncommutative potentials (41) and (64), we must specify the exact kind of twist. This amounts to specifying constants α, β and a in the equation (29). WKB analysis turns out to be simpler for the twist α = 0 , β = 1, to which we stick from now on. In that case, the only nontrivial commutator is the one between the radial and angular coordinate, [ ϕ /star , r ] = ia and all occurences of λa in the previous section can be replaced by am as per (31). This noncommutative space was analyzed in [18], where the NC-corrected axial potential was presented,  \nFIG. 5. The comparison between the noncommutative correction ∆ c and the relative error ∆ k in the optimal WKB order is illustrated. Left: Axial case. Right: Polar case.  \n<!-- image -->  \nand the axial QNM frequencies were calculated using the third-order WKB method. In that work, it was referred to as ¯ q -space, where ¯ q plays the role of noncommutativity parameter a in the case of pure r -ϕ twist.  \nFor commutative axial and polar potentials, the sixth-order WKB approximation usually yields the optimal result for the /lscript = 2 mode (see Table VI in appendix C). However, noncommutative QNM frequencies are optimized for different WKB orders (see Table VII -XXIII). We select the optimal frequencies for each am value (see Table I-II). In order to gauge the relative error inherent in the WKB method compared to the noncommutative (NC) correction, we introduce the absolute NC correction, denoted as ∆ c ,  \n∆ c = | ω NC -ω C | , (72)  \nwhere again ω refers to fundamental mode frequency. This absolute NC correction has been computed across a spectrum of am values for both axial and polar cases and is compared with the relative error ∆ k inherent in the optimized WKB order, as shown in Fig 5. It is evident that for /lscript = 3 (and similarly for higher /lscript values), the error is negligible compared to the /lscript = 2 case. The error ∆ k becomes comparable with the NC correction ∆ c after a certain value of am the positive/negative side of am axis for axial/polar modes is reached, with this claim being valid for both types of perturbation and all angular momentum channels analyzed. Within the domain of am where the WKB approximation proves effective, the impact of the noncommutative parameter on QNM frequencies surpasses the error in the WKB approximation. However, for positive am values where the WKB analysis breaks down, we can obtain the NC correction using other semi-analytic methods. We calculate the QNM frequencies using the Poschl-Teller and Rosen-Morse methods in the next section, thereby validating the corrections obtained from the WKB approach within the admissible domain of am .", 'B. Poschl-Teller and Rosen-Morse method': "The method of deriving quasinormal modes from the knowledge of bound states of the inverted potential has been widely utilized in black hole perturbation studies. The Poschl-Teller potential was originally introduced as a solvable potential for which the Schrodinger equation has exact solutions [35]. This method has been adapted for black holes by approximating the more complex black hole potentials with the Poschl-Teller potential [36-39]. This approach involves approximating the effective potential in the Schrodinger-like equation, derived from the radial perturbation equation, with the well-known Poschl-Teller potential. Consequently, the master wave equation can be rewritten as  \n∂ Ψ ∂r 2 ∗ + [ ω 2 -V 0 cosh 2 α ( r ∗ -¯ r ∗ ) ] Ψ = 0 , (73)  \nwhere  \nα 2 = -V '' 0 2 V 0 (74)  \nand ¯ r ∗ denotes the point where the effective potential reaches its maximum value in tortoise coordinate, and V 0 = V (¯ r ∗ ) is the value of the potential at this point. A semi-analytical form for the quasinormal mode (QNM) frequencies can  \nTABLE I. Table of NC axial QNMs for n = 0, M = 1 ( R = 2), and /lscript = 2 , 3 , 4, computed using the WKB higher-order method, Poschl-Teller (PT) method, and Rosen-Morse method. Order stands for the optimal WKB order. The estimated error of the WKB method is provided in parentheses.  \nbe obtained by transforming equation (73) into hypergeometric form and analyzing the asymptotic behavior of the solutions [40]. The formula for the frequencies of QNMs is given by  \nω = ± √ V 0 -α 2 4 -iα ( n + 1 2 ) . (75)  \nFrom the above expression, it is clear that both the real and imaginary parts depend on V 0 and V '' 0 , similar to the WKB method. However, only the imaginary part of the quasinormal mode (QNM) is influenced by the overtone number n . Thus, this method is not recommended for determining the real part of QNM, except in specific cases such as the eikonal limit ( /lscript →∞ ) or for the fundamental mode ( n = 0) [40].  \nThe effectiveness of this approach in approximating the effective potential can be enhanced by selecting a more accurate potential model to describe the black hole potential (see [41] for an extensive list of potential options for QNM calculations). Here, we opt for a specific potential known as the Rosen-Morse potential [9, 42], which introduces a correction term to add asymmetry to the Poschl-Teller potential, resulting in a better correlation with the black hole effective potential. The Rosen-Morse potential function is defined as [43]  \nV RM = V 0 cosh 2 α ( r ∗ -¯ r ∗ ) + V 1 tanh α ( r ∗ -¯ r ∗ ) , (76)  \nwhere V 1 adds asymmetry to the potential. By substituting this into the master wave equation and employing the appropriate boundary conditions, the resulting QNM frequencies are given by  \n√ ω 2 + V 1 + √ ω 2 -V 1 2 = ± √ V 0 -α 2 4 -iα ( n + 1 2 ) . (77)  \nTABLE II. Table of NC polar QNMs for n = 0, M = 1, and /lscript = 2 , 3 , 4, calculated using the WKB higher-order method, Poschl-Teller (PT) method, and Rosen-Morse method. Order stands for the optimal WKB order. The estimated error of the WKB method is provided in parentheses.  \nWhen V 1 = 0 this reduces to (75). The obtained QNM frequencies using the Poschl-Teller and Rosen-Morse potentials are presented alongside the WKB method in Table I and Table II. These methods supplement our findings on noncommutative corrections calculated using the WKB method.", 'V. AXIAL VS POLAR PERTURBATIONS - VIOLATION OF CLASSICAL ISOSPECTRALITY': "An important characteristic of quasinormal modes in classical general relativity is their isospectrality. For the commutative Schwarzschild black hole, axial and polar mode perturbations share identical spectra, despite being described by different master equations. This characteristic also extends to Reissner-Nordstrom, Kerr, and KerrNewman black holes (to linear order in rotation) in the commutative case [44, 45]. Chandrasekhar and Detweiler demonstrated for the commutative Schwarzschild case that the master equations of axial and polar perturbations are related by a specific transformation [17]. This transformation was later identified as a subclass of the Darboux transformation [46] (see Ref. [47-49] also). Isospectrality remains valid even in spacetimes including a cosmological constant for uncharged and non-rotating black holes [50]. However, the physical origin of this property remains unclear, and it is uncertain whether isospectrality is a generic feature or a mere coincidence for classical black holes. Efforts to understand isospectrality in black hole perturbation remain intriguing, particularly since this property holds only for very specific black hole spacetimes.  \nNotably, in classical general relativity, there are cases where isospectrality is broken in black hole perturbations [51-54]. Isospectral breaking is also observed in beyond general relativity theories [55-62]. In fact, parameterization approaches have shown that the isospectral property between even- and odd-parity quasinormal mode spectra is quite fragile [63]. Therefore, any observational indication of isospectrality breaking would strongly suggest new physics  \nbeyond general relativity [64]. From a quantum gravity perspective, it has even been suggested that isospectral breaking could potentially signal quantum gravity observables in current experiments [65].  \nClassically, the isospectrality can be understood as follows. The master equations are of the form  \nd 2 ψ dr ∗ + ( ω 2 -V ± ( r ) ) ψ = 0 ,  \nwhere + and -correspond to the polar (even) and axial (odd) perturbations respectively, governed by the commutative Zerilli and Regge-Wheeler potentials,  \nV + ( r ) = V Z ( r ) = ( r -R ) ( 8Λ 2 (Λ + 1) r 3 +12Λ 2 r 2 R +18Λ rR 2 +9 R 3 ) r 4 (2Λ r +3 R ) 2 , V -( r ) = V RW ( r ) = ( r -R )(2 r (1 + Λ) -3 R ) r 4 ,  \nwith the unique tortoise coordinate given by  \nr ∗ = r + R log r -R R .  \nThe criterion to test whether two potentials are equivalent via the Darboux transformation is given by [46]  \nd dr ∗ ( V + + V -) V + -V -= ∫ dr ∗ ( V + -V -) ,  \nwhere the potentials should be expressed using the standard tortoise coordinate r ∗ = r + R log r -R r . This condition is satisfied for the V + and V -in the commutative case. One of the requirements of the Darboux transformation is that initially, neither the polar nor the axial equations contain first derivative terms and that they have the same tortoise coordinate. In the noncommutative case, tortoise coordinates ˆ r ∗ differ for axial and polar modes as can be seen from equations (39) and (63). Trying to express one tortoise coordinate in terms of the other in any of the two equations (40) and (59) spoils their Schrodinger form by introducing a first derivative term in one of the equations. It is therefore impossible to relate them through the Darboux transformation since initial requirements cannot be satisfied, hinting at breaking of the classical isospectrality. The isospectrality violation is evident from the numerical values of QNM frequencies. In the commutative case ( am = 0), the QNM frequencies are identical for both the axial and polar cases (Table III).  \nTABLE III. QNM frequencies calculated using the WKB method for commutative case ( am = 0) with n = 0, M = 1 ( R = 2).  \nAs we introduce noncommutativity with a small value, such as am = -0 . 001, the breaking of isospectrality becomes apparent in higher /lscript modes (IV).  \nTABLE IV. QNM frequencies calculated using the WKB method for the noncommutative case am = -0 . 001 with n = 0, M = 1 ( R = 2).  \nTABLE V. QNM frequencies calculated using the WKB method for the noncommutative case am = -0 . 01 with n = 0, M = 1 ( R = 2) .  \nAs we further increase the strength of noncommutativity, for instance to am = -0 . 01, the breaking of isospectrality becomes more pronounced, being manifest even in the fundamental mode /lscript = 2 (see Table V).  \nThe increasing effect of isospectral breaking continues to appear for all negative values of am , and it also becomes evident on the positive side of am . We assess the degree of isospectral breaking as follows [65],  \n∆ ω R,I = 100 × ω axial R,I -ω polar R,I ω polar R,I . (78)  \nThe results are shown in Figure 6. Firstly, it is visible that the real part of the QNM frequency is affected similarly in both axial and polar cases, with deviations appearing at larger parameter values where the precision of the WKB method starts to decrease. Secondly, the negative imaginary part of QNM, ω I , is influenced by am in the reverse order compared to the real part. However, it should be noted that in the axial case, ω I is less influenced by noncommutativity. These observations imply that the effects of noncommutativity on oscillation frequencies and damping times differ in magnitude for axial and polar modes. The ∆ ω R,I depicted in the figure shows that the imaginary part experiences a more pronounced isospectral breaking effect than the real part.  \nThe fact that polar modes are significantly influenced by the noncommutativity stemming from the twist defined in (29) can be qualitatively analyzed as follows. Axial perturbations, which parameterize the off-diagonal components of the perturbation matrix, describe the twisting of spacetime around the black hole. These typically represent the gravitational waves that could be generated by the rotating or spiraling motion of matter in the vicinity of the black hole.  \nIn contrast, polar perturbations impact the scalar and spherical components of the metric, modifying distances while preserving the overall spherical symmetry. These perturbations include changes in the radial and temporal components of the metric, representing the expansion and contraction of spacetime, or the 'breathing-like' modes of the black hole.  \nGiven that polar perturbations predominantly relate to the radial direction, and considering that it is the radial coordinate that is quantized by the field X = ∂ r in the twist (29), it is understandable why polar perturbations are more sensitive to the coordinate quantization induced by the twist.  \nExpressions for the tortoise coordinate and effective potential become similar for noncommutative axial and polar modes when /lscript →∞ :  \nˆ r ∗ = r + R log r -R R ± λa 2 R r -R , ˆ V ( r ) = /lscript 2 ( r -R ) r 3 ± λa 2 /lscript 2 (3 R -2 r ) r 4 , (79)  \nwhere + corresponds to axial and -to polar perturbations. A natural question to ask is whether the spectrum corresponding to these potentials differs from the commutative one. Consider the limit a → 0 of the expressions above:  \nr ∗ = r + R log r -R R , V ( r ) = /lscript 2 ( r -R ) r 3 . (80)  \nReplacing r → r ± λa/ 2 here and expanding up to first order in a turns the RHS parts of these relations into RHS parts in the equations (79). Consequentially, solutions of (79) can be obtained as translated solutions of (80), provided boundary conditions are translated accordingly. Therefore the noncommutative QNM spectrum, up to the first order in a , becomes identical to the commutative one in the eikonal limit.  \nFIG. 6. The figures depict the breaking of the isospectrality caused by noncommutativity. The left panel corresponds to /lscript = 2, while the right panel corresponds to /lscript = 3. In the top row, the variation of the real part of the QNM ( ω R ) with am is shown. In the middle row, the variation of the imaginary part of the QNM ( ω I ) with am is depicted. In the bottom row, the relative deviation between the two modes, ∆ ω R,I , is illustrated. The parameters fixed in these plots are n = 0, and M = 1 ( R = 2). In each case, the QNM values corresponding to the optimal WKB order are chosen.  \n<!-- image -->", 'VI. OUTLOOK AND DISCUSSION': 'A generic black hole can be described completely by a small number of intrinsic parameters such as the mass, charge and spin of a black hole and in the case of somewhat more general theoretical framework such as that of beyond Einsteinian gravity, this description may include some additional parameters characterizing the theory.  \nWhen disturbed, a black hole tends to go back to its equilibrium characterized by a nonperturbed black hole solution. In turn, a response of a black hole to external perturbation is characterized by a discrete set of quasinormal mode frequencies which can also be described by a small set of intrinsic parameters of the theory. This turns QNMs into preferable instrument for accomplishing at least two important tasks: to reveal the information about parameters of the black hole in the ringdown phase resulting from a binary black hole merger [66] and to test theories of gravity  \nbeyond Einstein, as these vibration frequencies exhibit a universal part of the gravitational wave signals and above all can be measured [66, 67].  \nIn this paper, we have examined a specific type of Einstein gravity modification based on the framework of noncommutative geometry and quantization methods implemented by the Drinfeld twist. This approach takes into account that the geometry of spacetime acquires a granular structure at the Planck scale distances and is particularly well suited for implementing this feature into a formalism.  \nThe analysis has been made in a bottom-up approach, implying that the equation of motion is obtained by carefully revising the basic notions of differential geometry, such as the vector field, Lie derivative, covariant derivative, etc, rather than from some anticipated overruling/paramount action principle. In particular, a deformation quantization by a Drinfeld twist operator has been applied to a linearized gravitational perturbation theory. After developing a general perturbation theory up to linear order in both the deformation parameter and metric perturbation, we have applied it to a fixed Schwarzschild background in order to analyze the corresponding axial as well as polar type of gravitational perturbations. In this way, we have derived the noncommutative generalizations of the Regge-Wheeler and Zerilli potentials, respectively.  \nFor both potentials QNM spectrum has been calculated to a high precision in WKB, which has enabled us to investigate possible isospectrality breaking. Namely, isospectrality between even- and odd-parity perturbations of a Schwarzschild black hole and some more general geometries is a well-known property of classical general relativity. We have shown that this property becomes violated when the quantum structure of spacetime is assumed. This is consistent with other findings in the literature that were inferred for some effective models originating in loop quantum gravity [61, 68, 69]. Interestingly however, isospectrality appears to be restored in the eikonal limit of large angular momentum.  \nThe way how a black hole responds to external perturbation is of essential importance, as these response patterns influence the gravitational wave emission. As a consequence, they are generally measurable. In this paper, we have studied the linear response of a black hole to small perturbations, as described by the quasinormal mode frequencies, which describe how a black hole relaxes back to the Schwarzschild solution upon being disturbed.  \nAnother type of linear response that would be of interest to study is the linear response of a black hole to a long-wavelength tidal gravitational field. This type of response is expressed in terms of tidal Love numbers [70], which outline the feedback of a rigid body to an external deformation and thus represent a measure of the rigidity of the object being studied. In principle, the feedback of the black hole to an external field could also be studied in case the deformation is caused by the background electromagnetic and scalar field profiles.  \nAs the gravitational wave signal coming from an orbiting binary system depends on the tidal Love numbers 7 of the constituents, the latter may be inferred by measuring gravitational waves during the inspiral phase [71]. In our case it would be interesting to see how quantum spacetime deformation affects the tidal Love numbers of the Schwarschild black hole [72]. Another point of interest would be to extend the results for the QNM spectra obtained here to Kerr black holes, which we plan to address in an upcoming work.', 'Acknowledgment': 'This research was supported by the Croatian Science Foundation Project No. IP-2020-02-9614 Search for Quantum spacetime in Black Hole QNM spectrum and Gamma Ray Bursts . Part of the calculations were checked using the Mathematica package.', 'Appendix A: Zerilli gauge': 'The polar perturbations of a static spherically symmetric black hole are parameterized by seven families of functions: H /lscriptm 0 , H /lscriptm 1 , H /lscriptm 2 , α /lscriptm , β /lscriptm , K /lscriptm , and G /lscriptm , dependent on the variables ( r, t ) [13, 27]. The non-zero components of the metric perturbation h µν are expressed as follows:  \nh tt = A ( r ) ∑ /lscript,m H /lscriptm 0 ( t, r ) Y /lscriptm ( θ, ϕ ) , h tr = ∑ /lscript,m H /lscriptm 1 ( t, r ) Y /lscriptm ( θ, ϕ ) , (A1)  \nh ta = ∑ /lscript,m β /lscriptm ( t, r ) ∂ a Y /lscriptm ( θ, ϕ ) , h ra = ∑ /lscript,m α /lscriptm ( t, r ) ∂ a Y /lscriptm ( θ, ϕ ) , (A3)  \nh rr = 1 A ( r ) ∑ /lscript,m H /lscriptm 2 ( t, r ) Y /lscriptm ( θ, ϕ ) , (A2)  \nh ab = ∑ /lscript,m K /lscriptm ( t, r ) g ab Y /lscriptm ( θ, ϕ ) + ∑ /lscript,m G /lscriptm ( t, r ) D a D b Y /lscriptm ( θ, ϕ ) , (A4)  \nwhere A ( r ) = 1 -R/r . The indices a and b take values of angular coordinates θ and ϕ . Here, D a is the 2-dimensional covariant derivative associated with the metric of the 2-sphere dθ 2 +sin 2 θdϕ 2 . The explicit form of the angular part of the metric h µν is given by  \nh θθ = ∑ /lscript,m K /lscriptm ( t, r ) Y /lscriptm ( θ, ϕ ) + ∑ /lscript,m G /lscriptm ( t, r ) ∂ 2 θ Y /lscriptm ( θ, ϕ ) , (A5)  \nh ϕϕ = ∑ /lscript,m sin 2 θ K /lscriptm ( t, r ) Y /lscriptm ( θ, ϕ ) + ∑ /lscript,m G /lscriptm ( t, r ) ( ∂ 2 ϕ +sin θ cos θ ∂ θ ) Y /lscriptm ( θ, ϕ ) . (A7)  \nh θϕ = h ϕθ = -∑ /lscript,m G /lscriptm ( t, r ) cotan θ ∂ ϕ Y /lscriptm ( θ, ϕ ) , (A6)  \nDue to the spacetime diffeomorphism invariance of the theory, the parametrization of perturbations presented above is redundant and can be simplified through gauge fixing. At the linear level, an infinitesimal change of coordinates, x µ → x µ + ξ µ , induces the transformation:  \nh µν → h µν + ∇ µ ξ ν + ∇ ν ξ µ . (A8)  \nIn the polar sector, one can choose the gauge parameter ξ in such a way that three families of functions in (A1 - A4), namely G /lscriptm , α /lscriptm , and β /lscriptm , are set to zero. This particular gauge is known as the Zerilli gauge. In the Zerilli gauge, the polar perturbations can be parametrized by the remaining four families of functions, namely H /lscriptm 0 , H /lscriptm 1 , H /lscriptm 2 , and K /lscriptm , as follows:  \nh tt = A ( r ) ∑ /lscript,m H /lscriptm 0 ( t, r ) Y /lscriptm ( θ, ϕ ) , h tr = ∑ /lscript,m H /lscriptm 1 ( t, r ) Y /lscriptm ( θ, ϕ ) , (A9)  \nh rr = 1 A ( r ) ∑ /lscript,m H /lscriptm 2 ( t, r ) Y /lscriptm ( θ, ϕ ) , h ab = ∑ /lscript,m K /lscriptm ( t, r ) g ab Y /lscriptm ( θ, ϕ ) . (A10)', 'Appendix B: Coefficients of transformation': 'In this appendix we present the coefficients { f ( r ) , g ( r ) , h ( r ) , l ( r ) } and { ˜ f ( r ) , ˜ g ( r ) , ˜ h ( r ) , ˜ l ( r ) } that constitute the transformation equation (57). Together with the equation (63), the field redefinition satisfies the constraints outlined in (61). The functions appearing in Eq. (62) are given by  \ng ( r ) =1 , f ( r ) = 2Λ(Λ + 1) r 2 +3Λ rR +3 R 2 r 2 (2Λ r +3 R ) , l ( r ) = ir 2 R -r , h ( r ) = 3 iR 2Λ r +3 R -i (2 r -3 R ) 2( r -R ) , (B1)  \nwhich are the commutative results derived in Ref. [14] and the non commutative corrections are  \n˜ f ( r ) = 1 72 r 3 (2Λ r +3 R ) 2 { 36 r -R ( -4Λ(Λ(3Λ + 5) + 5) r 4 +8(Λ -1)Λ(2Λ + 1) r 3 R +(Λ(44Λ + 21) -36) r 2 R 2 +(32Λ + 57) rR 3 -6 R 4 ) -( r (2Λ r +3 R ) ( 4Λ(Λ + 1) r 2 +6Λ rR +6 R 2 ) ) × [ 9 ( 8Λ 2 +34Λ+9 ) log( r -R ) + 4Λ(Λ + 3)(10Λ -9) log(2Λ r +3 R ) (2Λ + 3) 2 R + 36 2Λ + 3 ( 1 r -R -3 2Λ r +3 R ) -(10Λ + 9) log( r ) R -123 r ] } , ˜ l ( r ) = -ir 36( r -R ) { -9 ( 8Λ 2 +34Λ+9 ) r log( r -R ) -4Λ(Λ + 3)(10Λ -9) r log(2Λ r +3 R ) (2Λ + 3) 2 R -36 r 2Λ + 3 ( 1 r -R -3 2Λ r +3 R ) + (10Λ + 9) r log( r ) R +69 } , ˜ h ( r ) = i 72(2Λ + 3) 2 rR ( r -R ) 2 (2Λ r +3 R ) 2 { ( 2 r ( r -R )(2Λ r +3 R ) ( 3 R (2Λ r + R ) -4Λ r 2 ) ) × [ Λ (( 20Λ 2 +78Λ+99 ) log( r ) -9(4Λ + 17) log( r -R ) -2(Λ + 3)(10Λ -9) log(2Λ r +3 R ) ) +81tanh -1 ( R 2 r -R )] +3(2Λ + 3) R [ 8Λ(Λ(74Λ + 183) + 72) r 4 -4Λ(Λ(202Λ + 69) -531) r 3 R +12(Λ(Λ(22Λ -189) -213) + 252) r 2 R 2 +9(2Λ(64Λ -89) -579) rR 3 +675(2Λ + 3) R 4 ]} , ˜ g ( r ) = -1 36 { 1 (2Λ + 3) 2 R [ 9 ( 8Λ 2 +34Λ+9 ) log( r -R ) + 4Λ(Λ + 3)(10Λ -9) log(2Λ r +3 R ) ] + 36 2Λ + 3 1 r R -3 2Λ r +3 R -(10Λ + 9) log( r ) R -123 r . (B2)  \n( -) }', 'Appendix C: Error estimation in WKB approximation method': "In this appendix, we provide the details of error estimation in the QNM calculation using the higher-order WKB method for both axial and polar modes. Error for the n -th order WKB is equal to | ω n +1 -ω n -1 | / 2 [34]. For the commutative case ( am = 0) with /lscript = 2 we obtain  \nTABLE VII. Error estimation in WKB method for the noncommutative case am = -0 . 2 with n = 0, M = 1 ( R = 2) and /lscript = 2. Orders for which the error exceeded the obtained value were omitted.  \nTABLE VI. Error estimation in WKB method for commutative case ( am = 0) with n = 0, M = 1( R = 2) and /lscript = 2.  \nFrom the table we conclude that the optimal WKB order for axial case is 6 whereas for the polar case, it is 5. The relative error between the WKB orders is of the order 10 -4 in both cases.  \nFor the noncommutative case am = -0 . 2 with /lscript = 2 we obtain  \nFrom the table we conclude that the optimal WKB order for the axial case is 6, whereas for the polar case it is 3. The relative error between the WKB orders is of the order of 10 -4 in axial and 10 -1 for the polar case. We observed that the higher-order WKB method breaks down after reaching certain orders in the polar case. This phenomenon appears to be a generic feature, particularly noticeable for higher (negative) values of the noncommutative parameter am in the polar case.  \nFor the noncommutative case am = -0 . 1 with /lscript = 2 we obtain  \nTABLE VIII. Error estimation in WKB method for the noncommutative case am = -0 . 1 with n = 0, M = 1 ( R = 2) and /lscript = 2. Orders for which the error exceeded the obtained value were omitted.  \nFrom the table we conclude that the optimal WKB order for axial case is 6, whereas for the polar case it is 4. The relative error between the WKB orders is of the order 10 -4 in axial and 10 -2 for the polar case.  \nFor the noncommutative case am = -0 . 01 with /lscript = 2 we obtain  \nTABLE IX. Error estimation in WKB method for the noncommutative case am = -0 . 01 with n = 0, M = 1 ( R = 2) and /lscript = 2.  \nFrom the table we conclude that the optimal WKB order for the axial case is 6, whereas for the polar case it is 5. The relative error between the WKB orders is of the order 10 -4 in both cases.  \nFor the noncommutative case am = -0 . 001 with /lscript = 2 we obtainTABLE X. Error estimation in WKB method for the noncommutative case am = -0 . 001 with n = 0, M = 1 ( R = 2) and /lscript = 2.  \nFrom the table we conclude that the optimal WKB order for the axial case is 6, whereas for the polar case it is 5. The relative error between the WKB orders is of the order 10 -4 in both cases.  \nFor the noncommutative case am = 0 . 001 with /lscript = 2 we obtain  \nTABLE XI. Error estimation in WKB method for the noncommutative case am = 0 . 001 with n = 0, M = 1 ( R = 2) and /lscript = 2.  \nFrom the table we conclude that the optimal WKB order for the axial case is 6, whereas for the polar case it is 5. The relative error between the WKB orders is of the order 10 -4 in both cases.  \nFor the noncommutative case am = 0 . 01 with /lscript = 2 we obtain  \nTABLE XII. Error estimation in WKB method for the noncommutative case am = 0 . 01 with n = 0, M = 1( R = 2) and /lscript = 2  \nFrom the table we conclude that the optimal WKB order for the axial case is 6, whereas for the polar case, it is 5. The relative error between the WKB orders is of the order 10 -4 in both cases.  \nFor the noncommutative case am = 0 . 1 with /lscript = 2 we obtain  \nTABLE XIII. Error estimation in WKB method for the noncommutative case am = 0 . 1 with n = 0, M = 1 ( R = 2) and /lscript = 2. Orders for which the error exceeded the obtained value were omitted.  \nFrom the table we conclude that the optimal WKB order for the axial case is 4, whereas for the polar case it is 6. The relative error between the WKB orders is of the order 10 -4 in both cases. We observe that the higher-order WKB method breaks down after reaching certain orders for axial case. This phenomenon appears to be a generic feature, particularly noticeable for higher (positive) values of the noncommutative parameter am for the axial case.  \nFor the noncommutative case am = 0 . 2 with /lscript = 2 we obtain  \nTABLE XIV. Error estimation in WKB method for the noncommutative case am = 0 . 2 with n = 0, M = 1 ( R = 2) and /lscript = 2. Orders for which the error exceeded the obtained value were omitted.  \nFrom the table we conclude that the optimal WKB order for the axial case is 4, whereas for the polar case it is 7. The relative error between the WKB orders is of the order 10 -2 in the axial case and 10 -4 in the polar case.  \nNow we consider the mode with /lscript = 3. For the commutative case ( am = 0) we obtain  \nTABLE XV. Error estimation in WKB method for the commutative case ( am = 0) with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order for the axial case is 12, whereas for the polar case it is 10. The relative error between the WKB orders is negligible, 10 -7 for the axial case and 10 -8 for the polar case.  \nFor the noncommutative case am = -0 . 2 with /lscript = 3 we obtain  \nTABLE XVI. Error estimation in WKB method for the noncommutative case am = -0 . 2 with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order is 8 for both the axial case and 3 for the polar case. The relative error between the WKB orders is of the order 10 -6 in axial case and 10 -2 in polar case.  \nFor the noncommutative case am = -0 . 1 with /lscript = 3 we obtain  \nTABLE XVII. Error estimation in WKB method for the noncommutative case am = -0 . 1 with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order for the axial case is 10, whereas for the polar case it is 4. The relative error between the WKB orders is of the order 10 -7 for the axial and 10 -3 for the polar case.  \nFor the noncommutative case am = -0 . 01 with /lscript = 3 we obtain  \nTABLE XVIII. Error estimation in WKB method for the noncommutative case am = -0 . 01 with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order for the axial case is 8, whereas for the polar case it is 9. The relative error between the WKB orders is of the order 10 -7 for both cases.  \nFor the noncommutative case am = -0 . 001 with /lscript = 3 we obtain  \nTABLE XIX. Error estimation in WKB method for the noncommutative case am = -0 . 001 with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order for the axial case is 8, whereas for the polar case it is 10. The relative error between the WKB orders is of the order 10 -7 for axial and 10 -8 for the polar case.  \nFor the noncommutative case am = 0 . 001 with /lscript = 3 we obtain  \nTABLE XX. Error estimation in WKB method for the noncommutative case am = 0 . 001 with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order for the axial case is 8, whereas for the polar case it is 10. The relative error between the WKB orders is of the order 10 -7 for the axial and 10 -8 for the polar case.  \nFor the noncommutative case am = 0 . 01 with /lscript = 3 we obtain  \nTABLE XXI. Error estimation in WKB method for the noncommutative case am = 0 . 01 with n = 0, M = 1 ( R = 2) and /lscript = 3.  \nFrom the table we conclude that the optimal WKB order for the axial case is 12, whereas for the polar case it is 10. The relative error between the WKB orders is of the order 10 -7 in the axial case and 10 -8 in the polar case.  \nTABLE XXII. Error estimation in WKB method for the noncommutative case am = 0 . 1 with n = 0, M = 1 ( R = 2) and /lscript = 3  \nFrom the table we conclude that the optimal WKB order is 4 for axial case and 9 for polar case. The relative error between the WKB orders is of the order 10 -4 in the axial and 10 -6 in the polar case.  \nFor the noncommutative case am = 0 . 2 with /lscript = 3 we obtainTABLE XXIII. Error estimation in WKB method for the noncommutative case am = 0 . 2 with n = 0, M = 1 ( R = 2) and /lscript = 3. Orders for which the error exceeded the obtained value were omitted.  \nFrom the table we conclude that the optimal WKB order for the axial case is 4, whereas for the polar case it is 8. The relative error between the WKB orders is of the order 10 -3 in the axial case and 10 -6 in the polar case. 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