An efficient ICD inducer based on Ir(III) cyclometalated complex for anticancer and anti-migration effects towards cisplatin-resistant cancer cells

Phys. Lett. B 866 (2025) 139581 Contents lists available at ScienceDirect Physics Letters B journal homepage: www.elsevier.com/locate/physletb Letter White dwarfs in regularized 4D Einstein-Gauss-Bonnet gravity Juan M.Z. Pretel a, , Takol Tangphati b,c, , İzzet Sakallı d, ,∗ , Ayan Banerjee e, a Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil b School of Science, Walailak University, Thasala, Nakhon Si Thammarat, 80160, Thailand c Research Center for Theoretical Simulation and Applied Research in Bioscience and Sensing, Walailak University, Thasala, Nakhon Si Thammarat 80160, Thailand d Physics Department, Eastern Mediterranean University, Famagusta 99628, North Cyprus via Mersin 10, Turkey e Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu–Natal, Private Bag X54001, Durban 4000, South Africa A R T I C L E Editor: N. Lambert I N F O A B S T R A C T White dwarfs (WDs), as the remnants of low to intermediate-mass stars, provide a unique opportunity to explore the interplay between quantum mechanical degeneracy pressure and gravitational forces under extreme conditions. In this study, we examine the structure and macroscopic properties of WDs within the framework of 4D Einstein-Gauss-Bonnet (4DEGB) gravity, a modified theory that incorporates higher-order curvature corrections through the Gauss-Bonnet coupling constant 𝛼 . Using the modified Tolman-Oppenheimer-Volkoff (TOV) equations tailored for 4DEGB gravity, we analyze the hydrostatic equilibrium of WDs modeled with a realistic equation of state (EoS). Our findings reveal that the inclusion of the Gauss-Bonnet (GB) term signiﬁcantly influences the mass-radius (𝑀 − 𝑅) relation, allowing for deviations from the Chandrasekhar mass limit. In particular, we observe that such stars become more compact and slightly smaller with the increase of the parameter 𝛼 . For WDs with |𝛼| ≤ 500 km2 , the impact of 4DEGB gravity appears to be negligible. However, a larger range for 𝛼 allows for appreciable changes in the 𝑀 − 𝑅 diagram, mainly in the high-central-density region. Furthermore, we explore the role of anisotropic pressures, quantified by the parameter 𝛽 , on such systems and demonstrate their impact on stability and compactness. For suﬃciently large values of |𝛽| keeping negative 𝛽 with a large and positive 𝛼 , there exists a second stable branch according to the classical stability criterion 𝑑𝑀∕𝑑𝜌𝑐 > 0. These results suggest that anisotropic WDs in 4DEGB gravity exhibit unique characteristics that distinguish them from their general relativistic counterparts, oﬀering a novel testing ground for modified gravity theories in astrophysical settings. 1. Introduction WDs [1--3] represent one of the most well-understood classes of com pact stellar remnants, forming as the final evolutionary state of low to intermediate-mass stars. These objects are supported against gravita tional collapse primarily by electron degeneracy pressure, as described by quantum mechanics [4]. The study of WDs has been pivotal in un derstanding stellar evolution, binary star systems, and supernova mech anisms [5--8]. Despite the empirical success of general relativity (GR) in modeling the macroscopic properties of compact objects, alterna tive theories of gravity oﬀer a fertile ground to test the robustness of GR under extreme conditions. One such theory, the regularized 4DEGB gravity, extends GR by incorporating higher-order curvature corrections in the gravitational action [9--11]. Modified theories of gravity [12--17] are motivated by several fun damental challenges in modern physics, including the quantization of gravity, the nature of singularities, and the cosmological constant prob lem. Among these, higher-curvature theories (HCTs) stand out as they modify the conventional Einstein-Hilbert action by introducing nonlin ear terms in the curvature tensor. These modifications are expected to play a signiﬁcant role in high-energy astrophysical scenarios, including those involving WDs [18]. Einstein-Gauss-Bonnet (EGB) gravity is a widely studied higher curvature theory [19--26], originally formulated in dimensions 𝐷 > 4. The Lagrangian in such gravitational theory includes the Gauss-Bonnet (GB) contribution which is quadratic in the Riemann tensor and is con sistent with the low-energy eﬀective action in heterotic string theory. The GB term in the action, * Corresponding author. E-mail addresses: juanzarate@cbpf.br (J.M.Z. Pretel), takoltang@gmail.com (T. Tangphati), izzet.sakalli@emu.edu.tr (İ. Sakallı), ayanbanerjeemath@gmail.com (A. Banerjee). https://doi.org/10.1016/j.physletb.2025.139581 Received 12 February 2025; Received in revised form 6 May 2025; Accepted 12 May 2025 Available online 16 May 2025 0370-2693/© 2025 The Author(s). Published by Elsevier B.V. Funded by SCOAP³. (http://creativecommons.org/licenses/by/4.0/). This is an open access article under the CC BY license Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al.  = 𝑅𝜇𝜈𝜌𝜏 𝑅𝜇𝜈𝜌𝜏 − 4𝑅𝜇𝜈 𝑅𝜇𝜈 + 𝑅2 , (1) for an anisotropic stellar fluid, which govern the hydrostatic equilibrium of spherically symmetric objects in 4DEGB gravity. These equations, de rived by incorporating the additional contributions from the GB term and its coupling with the scalar field, require careful numerical treat ment to explore a wide parameter space of 𝛼 values, while the degree of anisotropy is measured by the parameter 𝛽 . Since anisotropic pressure introduces substantial changes in the macroscopic properties of relativistic compact stars [44--52], such as QSs and NSs, then the interest of including a tangential pressure in the present study arises. Coupled with an appropriate EoS for WD matter and a phenomenological anisotropy profile, the solutions of the mod ified stellar structure equations enable us to analyze how the physical characteristics of (an)isotropic WDs, such as their radii, radial and trans verse pressures, and compactness, are modified under this gravitational framework. Such exploration paves the way for a more profound com prehension of the intricate relationship between gravitational forces and high-density matter. In summary, we aim to provide new insights into the astrophysical implications of 4DEGB gravity, oﬀering a compelling case for its consideration as an alternative to GR in modeling compact stellar remnants. Our work is motivated not only by mathematical curiosity but also by the physical insights oﬀered by 4DEGB gravity. Unlike simpler modified gravity theories such as 𝑓 (𝑅) or 𝑓 (𝑇 ), 4DEGB gravity emerges natu rally from string theory, providing a more fundamental basis to explore deviations from the Chandrasekhar mass limit [17]. In this study, we demonstrate that while standard values of the GB coupling constant 𝛼 yield minimal changes in WD properties, extending the range of 𝛼 sig niﬁcantly alters the mass-radius relation. In addition, the inclusion of an anisotropic pressure parameter 𝛽 introduces additional observable characteristics, such as the appearance of two stable branches, which can serve as distinctive signatures to discriminate between competing theories. The rest of this paper is organized as follows: In Section 2, we provide an overview of the theoretical framework, including the field equations and the modifications to the TOV equations in 4DEGB grav ity. Section 3 discusses the equations of state used to model WD matter as well as the anisotropy model. Section 4 presents the numerical re sults, including the mass-radius relationships and stability analysis for isotropic and anisotropic configurations. Finally, Section 5 concludes with a discussion of the implications of our findings and possible direc tions for future research. is a topological invariant in four dimensions, and thus does not con tribute to the field equations. However, recent developments have shown that a well-defined 𝐷 → 4 limit can be derived by rescaling the GB coupling constant 𝛼 as lim (𝐷 − 4)𝛼 → 𝛼, 𝐷→4 (2) leading to the 4DEGB theory [9,10,27]. This theory introduces a scalar ﬁeld 𝜙 coupled to the GB term, resulting in a scalar-tensor modification of GR. The action for 4DEGB gravity is given by [28]: 𝑆= √ [ ( )] 1 𝑑 4 𝑥 −𝑔 𝑅+𝛼 𝜙+4𝐺𝜇𝜈 ∇𝜇 𝜙∇𝜈 𝜙 − 4(∇𝜙)2 □𝜙+2(∇𝜙)4 ∫ 2𝜅 (3) + 𝑆𝑚 , where 𝜅 = 8𝜋 , 𝑅 is the Ricci scalar and 𝑆𝑚 is the matter action. The modification of GB provides a unique framework to explore the im pact of higher-curvature corrections on compact objects like quark stars (QSs) [28--30], electrically charged quark stars [31,32] and neutron stars (NSs) [29,33]. As will be seen later in our results, the inclusion of the GB term and its associated coupling constant 𝛼 introduces new phenomenological features that may alter the mass-radius relation and stability conditions [34--36] of WDs, mainly in the high central den sity region. For the first time, to the best of our knowledge, this work addresses the study of WDs in 4DEGB gravity under the presence of isotropic and anisotropic pressure. Observational astrophysics has increasingly unveiled deviations in the properties of compact objects that challenge the predictions of stan dard GR-based models. Among these are the discoveries of unusually light WDs and other compact remnants, whose physical characteristics strain the theoretical boundaries imposed by conventional EoSs. Such anomalies have sparked considerable interest in exploring alternative frameworks that could accommodate these deviations. For instance, the Chandrasekhar mass limit [1], which represents the maximum mass a non-rotating WD can achieve before collapsing into a neutron star or black hole, arises naturally within the context of GR due to the balance between gravitational forces and electron degeneracy pressure [37,38]. Nevertheless, this critical mass is fundamentally dependent on the un derlying gravitational theory [39--43], and modifications introduced by alternative gravity frameworks, such as higher-curvature theories, have the potential to shift or redefine this established limit. These modifica tions could result in more massive or more compact WDs than those allowed under GR, with signiﬁcant implications for their stability and observable properties. Additionally, deviations from GR may oﬀer ex planations for peculiar observational phenomena, such as compact rem nants that occupy the so-called mass gap between NSs and black holes or unusually small-radius WDs that defy the constraints imposed by GR. By introducing corrections to the gravitational interaction, alternative the ories like 4DEGB could provide a more comprehensive understanding of these anomalies. Investigating how such theories alter the structure and stability of WDs not only helps to explain these unusual observations but also serves as a critical test of GR’s validity in extreme environments, oﬀering potential avenues for discovering new physics beyond the stan dard model of gravity. In this paper, we investigate the structure and properties of WDs within the framework of 4DEGB gravity, focusing on how the modifi cations introduced by the GB coupling parameter 𝛼 aﬀect their funda mental characteristics. WDs, being ideal laboratories for exploring the interplay between quantum mechanical degeneracy pressure and gravi tational forces, provide an excellent testbed for assessing the predictions of modified gravity theories. By incorporating the GB corrections, we aim to evaluate their impact on key features such as the mass-radius relation, the maximum mass limit, and the overall stability of these com pact stars. Speciﬁcally, the theory’s deviations from GR may introduce novel astrophysical phenomena that could distinguish it from standard GR predictions. To achieve this, we solve the modified TOV equations 2. Theoretical framework 2.1. Field equations in 4DEGB gravity WDs, being highly dense stellar remnants, are excellent candidates for testing modifications to GR, particularly in the context of higher curvature theories like 4DEGB gravity. The incorporation of the GB term into the gravitational action provides a natural extension to GR by intro ducing corrections that become relevant in strong gravitational regimes. These corrections are encoded in the GB coupling constant 𝛼 , which quantifies the contribution of the higher-order curvature terms to the dynamics of spacetime. Considering the action of 4DEGB gravity (3), which combines the Einstein-Hilbert term with a regularized GB term, the variation with respect to the scalar field 𝜙 leads to its equation of motion: 𝜙 = −  + 8𝐺𝜇𝜈 ∇𝜈 ∇𝜇 𝜙 + 8𝑅𝜇𝜈 ∇𝜇 𝜙∇𝜈 𝜙 − 8(□𝜙)2 + 8(∇𝜙)2 □𝜙 + 16∇𝜇 𝜙∇𝜈 𝜙∇𝜈 ∇𝜇 𝜙 + 8∇𝜈 ∇𝜇 𝜙∇𝜈 ∇𝜇 𝜙 = 0. (4) The variation with respect to the metric 𝑔𝜇𝜈 yields the modified Einstein ﬁeld equations: { [( )( ) ] 𝜇𝜈 = 𝐺𝜇𝜈 + 𝛼 𝜙𝐻𝜇𝜈 − 2𝑅 ∇𝜇 𝜙 ∇𝜈 𝜙 + ∇𝜈 ∇𝜇 𝜙 + 8𝑅𝜎(𝜇 ∇𝜈) ∇𝜎 𝜙 )( ) ] ( [ +8𝑅𝜎(𝜇 ∇𝜈) 𝜙 ∇𝜎 𝜙 − 2𝐺𝜇𝜈 (∇𝜙)2 + 2□𝜙 [( )( ) ] −4 ∇𝜇 𝜙 ∇𝜈 𝜙 + ∇𝜈 ∇𝜇 𝜙 □𝜙 2 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. [ ( )( )] − 𝑔𝜇𝜈 (∇𝜙)2 − 4 ∇𝜇 𝜙 ∇𝜈 𝜙 (∇𝜙)2 ( )( ) ( )( )] [ +8 ∇(𝜇 𝜙 ∇𝜈) ∇𝜎 𝜙 ∇𝜎 𝜙 − 4𝑔𝜇𝜈 𝑅𝜎𝜌 ∇𝜎 ∇𝜌 𝜙 + ∇𝜎 𝜙 ∇𝜌 𝜙 ( ) + 2𝑔𝜇𝜈 (□𝜙)2 − 4𝑔𝜇𝜈 (∇𝜎 𝜙) (∇𝜌 𝜙) ∇𝜎 ∇𝜌 𝜙 )( ) ) ( ( +4 ∇𝜎 ∇𝜈 𝜙 ∇𝜎 ∇𝜇 𝜙 − 2𝑔𝜇𝜈 ∇𝜎 ∇𝜌 𝜙 (∇𝜎 ∇𝜌 𝜙) } ] [ (5) +4𝑅𝜇𝜈𝜎𝜌 (∇𝜎 𝜙) (∇𝜌 𝜙) + ∇𝜌 ∇𝜎 𝜙 = 𝜅𝑇𝜇𝜈 , lim 𝑒−2Ψ = 1 − 𝛼→0 𝑑Φ = 𝑑𝑟 (7) (8) and this can act as a useful consistency check to see whether prior solu tions generated via the Glavin/Lin method are even possible solutions to the gravity theory. The resulting equations form a scalar-tensor theory of gravity where the scalar field 𝜙 interacts with the spacetime curvature. These equations retain the second-order nature of GR while allowing for new phenomenological predictions in astrophysical contexts. (9) where Φ(𝑟) and Ψ(𝑟) are unknown metric functions to be determined. Furthermore, the anisotropic matter-energy distribution is described by (10) where 𝜌 is the energy density, 𝑝𝑟 the radial pressure and 𝑝⟂ is the trans verse pressure. Besides, 𝜎 ≡ 𝑝⟂ − 𝑝𝑟 is the anisotropic factor in the stellar source. Consequently, the 00 and 11 components of the field equations generate [ ] [ ( )] 2𝛼(1 − 𝑒−2Ψ ) 𝑑Ψ 𝛼(1 − 𝑒−2Ψ ) 2 1 − 𝑒−2Ψ = 𝑒2Ψ 8𝜋𝜌 − 1+ 1− , 𝑟 𝑑𝑟 𝑟2 𝑟2 𝑟2 (11) [ ] [ ( )] 2𝛼(1 − 𝑒−2Ψ ) 𝑑Φ 𝛼(1 − 𝑒−2Ψ ) 2 1 − 𝑒−2Ψ 1+ 1− . = 𝑒2Ψ 8𝜋𝑝𝑟 + 𝑟 𝑑𝑟 𝑟2 𝑟2 𝑟2 (13) The relation between the mass function 𝑚(𝑟) and metric Ψ(𝑟) is given by the usual way [9,28] 𝑒 [ ] √ 𝑟2 8𝛼𝑚 1− 1+ =1+ , 2𝛼 𝑟3 (17) (18) The EoS plays a crucial role in modeling the equilibrium structure of WDs by providing a functional relation between radial pressure and energy density. Such relation is essential for solving the modified TOV equations within the framework of 4DEGB gravity. WDs are primarily supported by electron degeneracy pressure, which can be eﬀectively described using the Chandrasekhar EoS [54]. Additionally, to capture potential anisotropic eﬀects in stellar matter, we employ the Quasi-Local (QL) model [55], which introduces a tangential pressure in addition to the radial pressure into the stellar system. It was shown that the main impact of GR with respect to the Newtonian context occurs on WDs with masses greater than 1.3 𝑀⊙ , indicating that general relativistic eﬀects become important for massive WDs [38]. This has motivated the study of WDs in modified gravity theories, such as 𝑅-squared gravity [40,41], linear 𝑓 (𝑅, 𝑇 ) gravity [39], 𝑓 (𝑅, 𝐿𝑚 ) theories [42] and Rastall-Rainbow gravity [43]. Furthermore, in standard Einstein gravity, several authors In addition, the covariant conservation of the energy-momentum tensor provides −2Ψ (16) 3. EoS for WDs and anisotropy model (12) 𝑑𝑝𝑟 𝑑Φ 2 = −(𝜌 + 𝑝𝑟 ) + 𝜎. 𝑑𝑟 𝑑𝑟 𝑟 , As in conventional Einstein gravity, the boundary conditions for the numerical integration are specified at the stellar center (𝑟 = 0) and the surface (𝑟 = 𝑅). At the center, the enclosed mass is zero, 𝑚(0) = 0, and the central mass density 𝜌(0) = 𝜌𝑐 is provided as an input parameter. The integration proceeds outward until the radial pressure vanishes, 𝑝𝑟 (𝑅) = 0, which defines the stellar radius 𝑅. The mass at this radius corresponds to the total mass of the compact star, i.e., 𝑀 = 𝑚(𝑅). The variation of the input parameter 𝜌𝑐 will allow us to obtain a family of anisotropic WDs represented in an 𝑀 − 𝑅 diagram. For anisotropic configurations, additional conditions are imposed on the diﬀerent physical quantities in the specific anisotropy model, ensur ing that the radial pressure is equal to the tangential pressure at the center of the fluid sphere. Other conditions are also required, such as gradients for mass density and radial pressure must be negative, the radial and tangential speed of sound should be less than the speed of light, etc. [53]. As we will see below, this model satisfies such physical acceptability conditions. It is worth emphasizing that the radial speed of sound inside WDs is always smaller than the speed of light, and since it depends exclusively on the EoS, then this causality condition is trivially satisfied in our study. The modified version of the TOV equations in the context of 4DEGB gravity are derived from the field equations, incorporating the addi tional contributions from the GB term. To model static, spherically sym metric compact stars, we adopt the usual line element 𝑇𝜇𝜈 = (𝜌 + 𝑝⟂ )𝑢𝜇 𝑢𝜈 + 𝑝⟂ 𝑔𝜇𝜈 − 𝜎𝜒𝜇 𝜒𝜈 , ( )√ 𝑟2 2𝛼 + 𝑟2 1 + 8𝛼𝑚 − 8𝛼𝑚𝑟 − 𝑟4 𝑟3 2.3. Boundary and initial conditions 2.2. Equilibrium configurations via modified TOV equations 𝑑𝑠2 = −𝑒2Φ(𝑟) 𝑑𝑡2 + 𝑒2Ψ(𝑟) 𝑑𝑟2 + 𝑟2 𝑑Ω2 , (√ ) 1 + 8𝛼𝑚 + 8𝜋𝛼𝑝 − 1 − 2𝛼𝑚 𝑟 𝑟3 The above diﬀerential equations (17) and (18) are known as modi ﬁed TOV equations in 4DEGB gravity and reduce to the isotropic case when 𝜎 = 0 [28]. To close the system of equations, an EoS that relates 𝑝𝑟 and 𝜌 must be supplied. We will also adopt an anisotropy profile that relates 𝜎 to radial pressure and mass function, so that we have only two unknown variables (i.e., 𝜌 and 𝑚) to determine. Therefore, we will em ploy both the Chandrasekhar EoS for WD matter and the QL model to account for anisotropic pressures. We will discuss this in more detail below. The above field equations satisfy the following relationship 𝛼 𝛼  = −𝑅 − , 2 𝜙 2 𝑟3 𝑑𝑚 = 4𝜋𝑟2 𝜌, 𝑑𝑟 [ ] 𝑑𝑝𝑟 (𝜌 + 𝑝𝑟 ) 2𝛼𝑚 + 𝑟3 (1 −  − 8𝜋𝛼𝑝𝑟 ) 2 + 𝜎, = ( ) 𝑑𝑟 𝑟 𝑟2  𝑟2 + 2𝛼 − 𝑟2  √ 8𝛼𝑚 where we have defined  ≡ 1 + 3 . 𝑟 Additionally, 𝐻𝜇𝜈 is the GB tensor, which is given by 𝜅𝑔 𝜇𝜈 𝑇𝜇𝜈 = 𝑔 𝜇𝜈 𝜇𝜈 + (15) so that Eqs. (11) and (13) become respectively (6) [ ] 1 𝐻𝜇𝜈 = 2 𝑅𝑅𝜇𝜈 − 2𝑅𝜇𝛼𝜈𝛽 𝑅𝛼𝛽 + 𝑅𝜇𝛼𝛽𝜎 𝑅𝛼𝛽𝜎 − 2𝑅𝜇𝛼 𝑅𝛼𝜈 − 𝑔𝜇𝜈  . 𝜈 4 2𝑚 + 𝛼 + ⋯. 𝑟 𝑟4 In view of Eqs. (12) and (14), we obtain where 𝑇𝜇𝜈 is the energy-momentum tensor of the matter content, de ﬁned by the usual form −2 𝛿𝑆𝑚 𝑇𝜇𝜈 = √ . −𝑔 𝛿𝑔 𝜇𝜈 4𝑚2 (14) namely, 𝑚(𝑟) is the enclosed gravitational mass within the radial coor dinate 𝑟. Note also that in the limit 𝛼 → 0, the last expression behaves asymptotically as 3 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. 𝜎 ≡ 𝑝⟂ − 𝑝𝑟 = 𝛽𝑝𝑟 𝜇, have shown that anisotropic pressure induces signiﬁcant changes in the global properties of massive compact stars [44--52]. These anisotropy eﬀects are therefore expected to be noticeable in massive WDs. In that regard, in the present study we will also consider tangential pressure in WDs within the framework of 4DEGB gravity. where 𝛽 is a constant that quantifies the extent of anisotropy, and 𝜇 = 2𝑚(𝑟) represents the local compactness of the star. The free parameter 𝛽 is 𝑟 limited within the range [−2, 2], see for example Refs. [45--49,51,64--68] for typical values. A positive 𝛽 corresponds to stellar systems where 𝑝⟂ > 𝑝𝑟 , while a negative 𝛽 indicates the opposite. The choice of anisotropic pressure becomes particularly signiﬁcant in dense regions of the star, where deviations from the isotropic context could influence the star’s stability and maximum mass. In the QL model, the anisotropy naturally vanishes as 𝑟 → 0, recovering the isotropic limit at the stellar center. Additionally, the model ensures that the radial and tangential pressures smoothly vanish at the stellar surface, satisfying the physical boundary conditions 3.1. Chandrasekhar EoS The Chandrasekhar EoS [54], a cornerstone of WD modeling, de scribes the pressure-density relationship for a relativistic degenerate electron gas. The radial pressure 𝑝𝑟 as a function of the Fermi momen tum 𝑘𝐹 is given by 𝑘𝐹 𝑝𝑟 (𝑘𝐹 ) = 𝑘4 1 𝑑𝑘, √ 2 3 3𝜋 ℏ ∫ 2 + 𝑚2 𝑘 0 𝑒 (19) 𝑝𝑟 (𝑟 → 𝑅) = 0, 𝜋𝑚4𝑒 [ 3ℎ3 ] √ 𝑥𝐹 (2𝑥2𝐹 − 3) 𝑥2𝐹 + 1 + 3 sinh−1 𝑥𝐹 , (20) with ℎ = 2𝜋ℏ being the Planck constant, and 𝑥𝐹 = 𝑝𝐹 ∕𝑚𝑒 𝑐 is the di mensionless Fermi momentum. The corresponding mass density 𝜌 is expressed as 𝜌= 𝑝⟂ (𝑟 → 𝑅) = 0. (23) We acknowledge that the specific physical origin of anisotropy in WDs requires further detailed investigation. Nonetheless, by includ ing this parameter in our analysis, we provide a more comprehensive framework for future studies that might incorporate specific physi cal mechanisms for anisotropy in WDs. Our study employs the QL anisotropy model as a phenomenological approach to explore how pres sure anisotropies might aﬀect WD structure in 4DEGB gravity. where ℏ is the reduced Planck constant, 𝑚𝑒 is the electron mass, and 𝑘 represents the momentum of the electrons. This integral can be evalu ated to yield 𝑝𝑟 (𝑘𝐹 ) = (22) 3.3. Physical implications of EoS and anisotropy 8𝜋𝜇𝑒 𝑚𝐻 𝑚3𝑒 The combined use of the Chandrasekhar EoS and the quasi-local ansatz provides a comprehensive framework for modeling WDs. The Chandrasekhar EoS captures the quantum mechanical eﬀects governing electron degeneracy pressure, while the QL anisotropy model accounts for anisotropic pressures that may arise due to strong gravitational fields or interactions in dense stellar matter. As we will see later, the inclusion of anisotropy is particularly relevant in the context of 4DEGB gravity, where higher-curvature corrections could enhance or suppress devia tions from the isotropic case. In subsequent sections, we will employ the above EoS along with Eq. (22) to solve the modified TOV equations numerically and inves tigate the impact of 4DEGB corrections and tangential pressure on the 𝑀 − 𝑅 relation and compactness (𝐶 ) of WDs. 𝑥3𝐹 , (21) 3ℎ3 where 𝜇𝑒 is the mean molecular weight per electron (we adopt 𝜇𝑒 = 2 for our analysis) and 𝑚𝐻 is the mass of a hydrogen atom. This EoS describes the essential physics of degenerate matter in WD stars, where pressure arises from the quantum mechanical eﬀects of Pauli exclusion rather than thermal motion. In the high-density regime, the relativistic corrections included in the Chandrasekhar EoS become crucial for accurately predicting the mass-radius diagram and stability of WDs. This model based on an ideal Fermi gas at zero temperature was recently used to examine the properties of anisotropic white dwarf stars within the framework of Rainbow gravity [56]. 3.2. Incorporating anisotropic pressure with the QL model 4. Numerical implementation and computational setup While isotropic pressure distributions are traditionally assumed in WD modeling, there are several physical mechanisms that could poten tially introduce anisotropy in these systems. As a matter of fact, the fluid pressures become anisotropic in the presence of strong magnetic fields [57,58]. In that regard, WDs as polytropes for anisotropic fluids were considered to estimate the maximum stable mass of magnetized WDs, which could be greater than 3 𝑀⊙ [59]. For further studies on the re lationship between strong magnetic fields and anisotropic pressure in WDs, we refer the reader to Refs. [60,61]. Accordingly, strong mag netic fields (which are commonly observed in WDs [62,63]) can create pressure anisotropies by aﬀecting the motion of charged particles dif ferently along and perpendicular to field lines. Similarly, rapid rotation can induce pressure diﬀerentials between diﬀerent directions. For in stance, strong magnetic fields (observed up to 109 𝐺 in some WDs), rapid rotation, or even crystallization eﬀects at high densities can create di rectional pressure diﬀerences. In this work, we adopt the QL anisotropy model as a phenomenological framework to capture these potential ef fects in 4DEGB gravity [17,45]. While the exact physical origins of anisotropy in WDs require further study, our approach oﬀers a versa tile starting point to explore its impact on stellar stability and structure. To extend the analysis to cases where the pressure may not be isotropic, we incorporate the QL model proposed by Horvat et al. [55]. Anisotropic pressure can arise in highly dense systems where the ra dial pressure 𝑝𝑟 diﬀers from the tangential pressure 𝑝⟂ . The degree of anisotropy is characterized by the parameter 𝜎 , defined as follows The GB coupling constant 𝛼 introduces a new degree of freedom in the gravitational dynamics, and here we will numerically examine its eﬀects on the relativistic structure of WDs. To investigate such as equilibrium configurations in the framework of 4DEGB gravity, we nu merically solve the modified TOV equations (17)-(18). This involves employing appropriate boundary conditions, EoS, and a robust numer ical scheme to explore the eﬀects of the GB coupling parameter 𝛼 on the mass-radius diagram and stability properties of WDs. The parame ter space explored includes both isotropic and anisotropic configurations of the stellar structure. 4.1. Numerical approach The stellar structure equations (17)-(18) are solved using a fourth order Runge-Kutta integration method. The equations are discretized on a fine grid, and care is taken to ensure numerical stability and conver gence. A parametric study is initially conducted for a given value of central density 𝜌𝑐 and GB coupling constant 𝛼 , spanning values in the range 𝛼 ∈ [−1.0, 1.0] × 104 km2 by considering the isotropic case (i.e., when 𝛽 = 0). Then the integration will be done over a wide range of central densities for both isotropic and anisotropic configurations. To visualize our numerical results, 𝑀 − 𝑅 diagrams and mass-central density (𝑀 − 𝜌𝑐 ) relations are constructed for various values of 𝛼 and 𝛽 . 4 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. Fig. 1. Radial behavior of the mass function (left plots) and energy density (right plots) inside a WD with central density 𝜌𝑐 = 2.8 × 1014 kg∕m3 in 4DEGB gravity. The upper panel corresponds to isotropic solutions (i.e., when 𝛽 = 0) for various values of 𝛼 , while the lower panel represents anisotropic solutions for a fixed 𝛼 = 5000 km2 . Fig. 2. Mass-radius relation (left panel) and mass-central density relation (right panel) for isotropic WDs in 4DEGB gravity, where we have varied the GB coupling parameter in the range 𝛼 ∈ [−500, 500] km2 , see the color scale on the right. The small plot embedded in the left panel is an enlargement of the 𝑀 − 𝑅 relation near the maximum-mass points. Although the eﬀect of 𝛼 on the mass function is appreciable, the mass density is slightly modified by 𝛼 . Now we will vary the central density 𝜌𝑐 to generate a family of equi librium WDs in 4DEGB gravity considering 𝛽 = 0, but before that it is necessary to comment on the values of 𝛼 . Very recently, Saavedra and collaborators [33] have shown that very signiﬁcant changes in the mass radius diagrams of NSs due to the GB term occur if 𝛼 is of the order of ∼ 300 km2 , where it is possible to obtain maximum NS masses of up to ∼ 12 𝑀⊙ . Taking this work as reference, in Fig. 2 we show our numeri cal results for the mass-radius (𝑀 − 𝑅) relationship (left panel) and the mass-central density (𝑀 − 𝜌𝑐 ) relation (right panel) for isotropic WDs by using 𝛼 ∈ [−500, 500] km2 , where we have varied the central density up to where the Chandrasekhar EoS is valid, i.e., the neutron drip den sity 𝜌drip ≈ 4.3 × 1011 g∕cm3 [69]. Above this density value, we would be describing NS interiors and a diﬀerent EoS would be required than the one established in Eqs. (20)-(21). Although for NSs this range of 𝛼 val These diagrams provide insight into the dependence of stellar proper ties on 𝛼 , the EoS and anisotropy profile. Both isotropic and anisotropic configurations are analyzed, highlighting the role of pressure anisotropy in modifying the compactness and stability of WDs. 4.2. Eﬀects of GB coupling constant Given a central density value 𝜌𝑐 = 2.8 × 1014 kg∕m3 , we begin our analysis by showing the numerical solution of the modified TOV equa tions for the isotropic case, where 𝛽 is null, see the top panel of Fig. 1. The mass 𝑚(𝑟) is an increasing function, while the density 𝜌(𝑟) decreases as we go from the center to the surface of the WD. We observe that pos itive (negative) values of the GB coupling constant increase (decrease) the mass distribution of the WD, mainly in the outermost regions of the star. Furthermore, the radius of the star 𝑅 decreases with increasing 𝛼 . 5 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. Fig. 3. 𝑀 − 𝑅 diagram (left panel) and 𝑀 − 𝜌𝑐 relation (right panel) for isotropic WDs when the GB parameter varies in the interval |𝛼| ≤ 1.0 × 104 km2 . For this range in 𝛼 , the gravitational mass of a WD undergoes appreciable changes for high central densities, namely, for 𝜌𝑐 ≳ 2.0 × 1013 kg∕m3 , while below this density value the modifications are irrelevant. The brown triangles on the left plot represent the observational data taken from the catalogue of isolated massive WDs [73]. Fig. 4. Radius (left) and compactness (right) as a function of central density for the isotropic WDs shown in Fig. 3. for the central density where the radii are ≳ 5000 km. However, the novelty of our findings lies in the high-mass region, beyond the usual Chandrasekhar limit. Remarkably, our results reveal that for suﬃciently large positive 𝛼 (i.e., 𝛼 ≳ 2000 km2 ), it is not possible to obtain a critical WD since the maximum mass cannot be found. Meanwhile, the critical point corresponding to the maximum-mass configuration can be found for negative 𝛼 (blue curves). This is an unprecedented result for WDs in 4DEGB gravity, something not manifested in GR. We will comment further on this matter in subsection 4.4. According to the plot on the left of Fig. 4, the compact star becomes slightly smaller with increasing 𝛼 , although its mass increases signif icantly for positive 𝛼 (i.e., for the smallest stars in the left panel of Fig. 3). As a consequence of these results, the compactness (given by 𝐶 = 𝑀∕𝑅) becomes larger with increasing GB coupling parameter. This can be clearly observed in the right plot of Fig. 4, where 𝐶 is of the or der of ∼ 10−3 . On the other hand, the lower 𝛼 , the less compact are the isotropic WDs in 4DEGB gravity. The plot on the left of Fig. 6 exhibits the compactness as a function of gravitational mass for the equilibrium configurations with 𝛽 = 0. As expected, 𝐶 changes noticeably due to the GB term only for high central densities, i.e. for 𝜌𝑐 ≳ 1014 kg∕m3 . Speciﬁcally, we see that the highest compactnesses are obtained for positive 𝛼 , which is consistent with the right plot of Fig. 4. In summary, the inclusion of the GB term signiﬁcantly aﬀects the maximum mass of WDs (although the radius barely changes) if the range for the coupling constant 𝛼 is large enough, potentially allowing them to exceed the Chandrasekhar limit. This phenomenon has important impli cations for understanding the observed properties of unusual compact objects, such as super-Chandrasekhar WDs or low-mass remnants. Nev ertheless, when 𝛼 is of the order of 500 km2 , the eﬀects of the GB term on the most basic properties of a WD are negligible (as already illustrated in Fig. 2). ues produces substantial eﬀects on the radius and mass, we observe that the impact of the GB term is irrelevant for the case of WDs in 4DEGB gravity. Speciﬁcally, for WDs with radii greater than 1000 km, the im pact of 4DEGB gravity appears to be negligible. Slight changes are more noticeable after the maximum mass, from where the WDs are unstable. Since the range of values for 𝛼 in Fig. 2 does not generate apprecia ble changes in the 𝑀 − 𝑅 diagram of WDs, we will now use a larger interval, taking advantage of the fact that the observational constraints on the GB coupling parameter have led to −10−36 km2 < 𝛼 < 104 km2 [70,71]. Although the literature also provides lower bounds [72], it is well known that these constraints depend heavily on the specific EoS used to describe dense stellar matter. Since our study deals with den sities lower than those corresponding to NSs or QSs, we can adopt a larger range for |𝛼|. In other words, since we are considering a diﬀer ent range of energy densities (and therefore diﬀerent stellar systems) than the one already examined in the literature, the GB parameter does not necessarily have to be the same here to appreciate substantial ef fects on the mass-radius relation due to 4DEGB gravity. Fig. 3 illustrates our results when the GB coupling parameter 𝛼 is varied in the range 𝛼 ∈ [−1.0, 1.0] × 104 km2 . Positive values of 𝛼 generally result in more massive compact configurations (see red curves), while negative 𝛼 de creases the gravitational mass of these stars (see blue curves). It should be noted that below a central density ∼ 2.0 × 1013 kg∕m3 , the parame ter 𝛼 has negligible eﬀects; however, the GB coupling constant plays an important role above this central density value. Unlike WDs in 𝑓 (𝑅, 𝑇 ) = 𝑅+2𝜆𝑇 gravity where the radius undergoes signiﬁcant modifications due to the parameter 𝜆 [39], in the present study we show that for small masses (or low central densities) the 𝑀 − 𝑅 curve is unaltered by the GB term, but the opposite occurs at high central densities. For comparison reasons, in Fig. 3 we have incorporated the observational data taken from the catalogue of isolated massive WDs [73]. Of course, measurements for several WDs within 100 pc from the Sloan Digital Sky Survey (SDSS) [74,75] lie in an even smaller range 6 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. Fig. 5. Anisotropic WDs in 4DEGB gravity for fixed 𝛼 = −5000 km2 (top panels) and 𝛼 = 5000 km2 (bottom panels). Here the anisotropy parameter has been varied in the interval |𝛽| ≤ 1.6, where the particular case 𝛽 = 0 corresponds to the isotropic solutions. It is observed that the anisotropic pressure introduces notable changes on the mass-radius results at high central densities, while the variations with respect to the isotropic case are irrelevant in the low-central-density branch. For suﬃciently large values of |𝛽| keeping negative 𝛽 with positive 𝛼 (see blue curves in the lower panel), there exists a second stable region where 𝑑𝑀∕𝑑𝜌𝑐 > 0. 4.3. Impact of anisotropic pressure 4.4. Comments on stability and observations To examine the eﬀects of anisotropy on WDs in 4DEGB gravity, we solve the modified stellar structure equations for a given central density 𝜌𝑐 = 2.8 × 1014 kg∕m3 while holding fixed 𝛼 = 5000 km2 as in the bottom panel of Fig. 1. For the range 𝛽 ∈ [−2.0, 2.0], we see that the gravita tional mass increases as the anisotropies grow (i.e., for positive values of 𝛽 ), while the opposite occurs for negative anisotropies. Nonetheless, the changes introduced by 𝛽 on the mass density are negligible with respect to the isotropic case. Fig. 5 demonstrates the influence of anisotropy on WD configura tions for fixed values of 𝛼 . The top panels correspond to 𝛼 = −5000 km2 , while the bottom panels represent 𝛼 = 5000 km2 . The anisotropy param eter 𝛽 is varied in the range 𝛽 ∈ [−1.6, 1.6], with 𝛽 = 0 representing the isotropic case. For both values of 𝛼 we see that the main conse quence of anisotropy is an increase (decrease) in the maximum masses for positive (negative) 𝛽 . However, for low central densities, the im pact of anisotropy is negligible, and the results converge to those of the isotropic solutions. Therefore, anisotropic pressure plays an important role for massive WDs (i.e. for stars with high central densities) in 4DEGB gravity. It is worth noting that this behavior is qualitatively similar to the case of NSs [45--47,49] and QSs [47,48,52] in GR, where the predomi nant role of anisotropy only occurs at high central densities. Of course, the range of central densities and EoSs here are diﬀerent from those of NSs, but the qualitative behavior regarding the impact of anisotropic pressure holds for WDs in 4DEGB gravity. The middle and right plots of Fig. 6 show the 𝐶 − 𝑀 relationship for 𝛼 = −5000 km2 and 5000 km2 , respectively. The eﬀect of anisotropy is irrelevant for low compactnesses, but above 𝐶 ∼ 0.001 again the anisotropic pressure plays a prominent role. It is well known in GR [76,77], even beyond Einstein gravity [78-80], that the turning point from stability to instability occurs when the mass is maximum, so that stable (unstable) compact stars correspond to 𝑑𝑀∕𝑑𝜌𝑐 > 0 (< 0) on the 𝑀(𝜌𝑐 )-curves. According to the right panel of Fig. 3, we can obtain a maximum mass point if 𝛼 ≲ 2000 km2 . However, above this value of 𝛼 it is not possible to find a critical WD indicating a maximum (see the red curves). This means that the GB term (with 𝛼 positive and suﬃciently large) is capable of generating always sta ble massive WDs, thus overcoming the Chandrasekhar limit. This is an unprecedented and peculiar outcome of 4DEGB gravity, which is not manifested within the gravitational theory of pure GR. In addition, the numerical results in Fig. 5 indicate that anisotropic pressures induce noticeable changes in the mass-radius profiles, partic ularly at high central densities. Positive values of 𝛽 (corresponding to 𝑝⟂ > 𝑝𝑟 ) tend to increase the maximum mass, whereas negative values of 𝛽 have the opposite eﬀect. Although for negative 𝛼 there is only one sta ble branch, it is observed that it is possible to obtain stable anisotropic WDs on two branches in the 𝑀(𝜌𝑐 )-curve when 𝛼 = 5000 km2 and for suﬃciently large values of |𝛽| keeping negative 𝛽 , see for example the curve for 𝛽 = −1.6, where there are two regions on the 𝑀(𝜌𝑐 )-curve with positive 𝑑𝑀∕𝑑𝜌𝑐 . The first critical central density 𝜌crit 𝑐 , where the ﬁrst stability branch ceases, becomes increasingly larger as 𝛽 increases from its negative values. On the other hand, the second critical density (where 𝑀(𝜌𝑐 ) is a minimum) decreases with increasing anisotropy pa rameter 𝛽 , so that the second stable branch starts at a lower 𝜌crit 𝑐 value as 𝛽 grows inside the WD. From this perspective, the anisotropic pressure in WDs has a remarkable and interesting eﬀect in the case of a posi tive GB coupling constant. Therefore, depending on the pair of values 7 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. Fig. 6. Compactness versus gravitational mass for the stellar configurations shown in Figs. 3 and 5. {𝛼, 𝛽}, it is possible to obtain two stable branches for anisotropic WDs in 4DEGB gravity. The computational results underline the importance of including both isotropic and anisotropic scenarios in the analysis of WDs under 4DEGB gravity. The variation of the parameters 𝛼 and 𝛽 leads to a di verse set of configurations, some of which could align with observed deviations from standard Chandrasekhar predictions. For instance, the presence of anisotropic pressures might explain certain anomalies in compact stellar remnants, oﬀering a theoretical basis for their unusual properties. These results also serve as a valuable testing ground for the validity of 4DEGB gravity in astrophysical contexts. WD stars would be in 4DEGB gravity. A suﬃcient condition for stability would be to determine the frequencies of the adiabatic radial vibration modes. In this case, the field equations have to be linearized around the equilibrium configuration, and we hope to address this more sophisti cated approach in a future study. In fact, for NSs in 4DEGB gravity, it has been shown very recently that the coincidence of the maximum-mass points with the transition to instability (where the squared fundamen tal eigenfrequencies are zero) still holds in this type of modified gravity [33]. Although the stellar systems addressed in such study are diﬀer ent from those considered in our work, we hope that this compatibility (or coincidence) will be maintained, so that the 𝑀(𝜌𝑐 ) method will still hold in the case of WDs. Furthermore, the analysis of anisotropic pressure, modeled using the QL profile [55], further highlights the importance of including anisotropy in the study of WDs. For positive anisotropy parameters 𝛽 (where 𝑝⟂ > 𝑝𝑟 ), we observed an increase in the maximum mass of the star, while negative 𝛽 values reduce this global quantity. These repercussions are particularly signiﬁcant at high central densities, where deviations from isotropic configurations are more pronounced. Accord ing to the classical stability criterion 𝑑𝑀∕𝑑𝜌𝑐 > 0 [45,55], our findings have shown that there exists a new branch of stable massive anisotropic WDs for some positive values of 𝛼 and suﬃciently large values of |𝛽| keeping negative 𝛽 . In particular, for the pair of values {𝛼, 𝛽} = {5000 km2 , −1.6}, we have shown the existence of two stability regions for anisotropic WDs in 4DEGB gravity. This result already leaves an in teresting avenue for future research and provides new insights into the astrophysical implications of modified gravity in dense stellar environ ments such as WDs. The inclusion of anisotropic pressure may also serve as a theoretical explanation for unusual stellar remnants that cannot be described by isotropic models. Indeed, the higher-curvature corrections introduced by the GB term oﬀer a new perspective on the physics of anisotropic WDs and may help reconcile theoretical predictions with observational data. For example, our results suggest that 4DEGB gravity could provide a framework for understanding the macroscopic proper ties of extreme compact objects, such as those that approach or exceed the Chandrasekhar limit. Looking ahead, there are several compelling avenues for future re search based on the findings of this study. The inclusion of rotation presents a natural and essential extension, as rapid rotation introduces signiﬁcant centrifugal support that can alter the equilibrium configura tions of WDs. Investigating the interplay between rotation, the GB cou pling, and anisotropic pressures would provide a more comprehensive understanding of these compact objects. Additionally, the stability of WDs under perturbations remains a critical aspect that warrants further 5. Concluding remarks and future perspectives In this work, we have explored the relativistic structure and basic macroscopic properties of WDs within the framework of 4DEGB grav ity, focusing on the impact of the GB coupling constant 𝛼 and the eﬀects of anisotropic pressures by means of an anisotropy parameter 𝛽 . By solv ing the modified TOV equations numerically for a wide range of central densities, coupling constants, and pressure anisotropy parameters, we have gained valuable insights into how higher-curvature corrections in ﬂuence these dense stellar remnants. We have shown that, when 𝛼 is of the order |𝛼| ≤ 500 km2 , the eﬀects of the GB term on the 𝑀 − 𝑅 relations of WDs are negligible. Although this range for 𝛼 has a quite appreciable impact on the global proper ties of NSs [33,72], our findings here reveal an irrelevant eﬀect for the isotropic WD case. Nevertheless, since we are dealing with another type of stellar system (i.e., a range of central densities and EoS diﬀerent from those of NSs), we have also considered a larger range for |𝛼| in order to appreciate notable changes in the 𝑀 − 𝑅 diagrams of WDs. To do so, we have adopted the range 𝛼 ∈ [−1.0, 1.0] × 104 km2 , which is within the observational constraints provided by Refs. [70,71]. Consequently, our results for this larger |𝛼| indicate that the GB coupling constant 𝛼 signif icantly aﬀects the equilibrium configurations of massive WDs. Positive values of 𝛼 tend to produce smaller and more compact configurations, while negative values lead to larger and less compact structures. Impor tantly, the inclusion of the GB extra term allows for deviations from the Chandrasekhar mass limit, potentially providing an explanation for ob served anomalies such as super-Chandrasekhar WDs [81]. The 𝑀 − 𝑅 diagrams and 𝑀 − 𝜌𝑐 relations obtained in this study demonstrate how 𝛼 modifies the maximum mass and stability of WDs, where its repercus sions become increasingly pronounced at high central densities. Although the 𝑀(𝜌𝑐 ) method is a necessary but not suﬃcient condi tion for stability, it already provides a hint of what the radial stability of 8 Physics Letters B 866 (2025) 139581 J.M.Z. Pretel, T. Tangphati, İ. Sakallı et al. exploration. Studying radial and non-radial oscillation modes in the con text of 4DEGB gravity could oﬀer new insights into stability criteria and help constrain the GB coupling constant. Another key direction involves comparing theoretical predictions with observed properties of WDs, par ticularly those in binary systems or acting as progenitors of Type Ia supernovae. Such comparisons, especially for systems that exceed the Chandrasekhar limit in modified gravity scenarios, could serve as vital tests of 4DEGB gravity. Lastly, the strong magnetic fields often observed in WDs present another layer of complexity, as they may interact with GB corrections in non-trivial ways. Incorporating magneticfield eﬀects into the modeling would enhance the realism and applicability of the results, providing a richer understanding of the dynamics and structure of these fascinating stellar remnants. It is worth emphasizing that this gravity theory needs to be examined within the context of solar-system tests, as was done for example in 𝑓 (𝑅) gravity [82,83], and we will also leave this for a future study. In conclusion, the results presented here contribute to the growing body of research on modified gravity and its implications for astro physics. WDs provide a robust platform for testing the consequences of higher-curvature corrections, and future studies that incorporate addi tional physical eﬀects and observational constraints will further enhance our understanding of these fascinating objects. 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Nandy, General relativistic calculations for white dwarfs, Res. As tron. Astrophys. 17 (2017) 061. [38] G.A. Carvalho, R.M. Marinho, M. Malheiro, General relativistic eﬀects in the struc ture of massive white dwarfs, Gen. Relativ. Gravit. 50 (2018) 38. [39] G.A. Carvalho, et al., Stellar equilibrium configurations of white dwarfs in the 𝑓 (𝑅, 𝑇 ) gravity, Eur. Phys. J. C 77 (2017) 871. [40] A.V. Astashenok, S.D. Odintsov, V.K. Oikonomou, Maximal masses of white dwarfs for polytropes in 𝑅2 gravity and theoretical constraints, Phys. Rev. D 106 (2022) 124010. Declaration of competing interest The authors declare the following financial interests/personal rela tionships which may be considered as potential competing interests: IZZET SAKALLI reports financial support was provided by The Scien tific and Technological Research Council of Türkiye. If there are other authors, they declare that they have no known competing financial in terests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We gratefully acknowledge the insightful comments and construc tive suggestions provided by the Editor and the anonymous referee. Their valuable feedback has signiﬁcantly improved both the clarity and the physical grounding of our manuscript. JMZP acknowledges support from ``Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro'' -- FAPERJ, Process SEI-260003/000308/2024. T.T. was supported by Walailak University under the New Researcher Devel opment scheme (Contract Number WU67268). He also acknowledges COST actions CA21106 and CA22113. İ.S. expresses gratitude to EMU, TÜBİTAK, ANKOS, and SCOAP3 for their academic and/or financial support. He also acknowledges COST Actions CA22113, CA21106, and CA23130 for their contributions to networking. 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