Thermodynamic properties and shadows of black holes in $f(R,T)$ gravity

Abstract

We explored two f(R,T) gravity models and derived black hole solutions within these models. We focus on investigating how the f(R,T) model influences the thermodynamic characteristics of black holes by studying their thermodynamic topology and thermodynamic geometry. We considered five specific values of the thermodynamic parameter \omega, which signify five different classes of black hole solutions in general relativity (GR). We observed significant changes in the local topological properties of these black holes compared to GR, depending on the model parameters. Notably, we identified an additional topological class W=0 for some values of \omega that is absent in the GR framework. We also studied the thermodynamic geometry of the black hole using the Geometrothermodynamics (GTD) formalism. Our analysis demonstrates that the singular point, where the GTD scalar curvature diverges, corresponds exactly to the point where the heat capacity changes sign. Additionally, we constrained the model parameters of both models considered by utilizing black hole shadow data from the Sgr A* black hole, measured by the Event Horizon Telescope (EHT).

Graphical abstract

Keywords

${\color{khaki}{f(R,T)}} $ gravity / black hole thermodynamics / black hole shadows

References

[1] B. P. Abbott, . [LIGO Scientific Collaboration and Virgo Collaboration], Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116(6), 061102 (2016), arXiv: 1602.03837

[2] K. Akiyama, et al. [The Event Horizon Telescope Collaboration], . First M87 event horizon telescope results. I. The shadow of the supermassive black hole, Astrophys. J. 871, L1 (2019)

[3] K. Akiyama, et al. [The Event Horizon Telescope Collaboration], . First M87 event horizon telescope results. II. Array and instrumentation, Astrophys. J. Lett. 875(1), L2 (2019)

[4] K. Akiyama, et al. [The Event Horizon Telescope Collaboration], . First M87 event horizon telescope results. III. Data processing and calibration, Astrophys. J. Lett. 875(1), L3 (2019)

[5] K. Akiyama, et al. [The Event Horizon Telescope Collaboration], . First M87 event horizon telescope results. IV. Imaging the central supermassive black hole, Astrophys. J. 875, L4 (2019)

[6] K. Akiyama, et al. [The Event Horizon Telescope Collaboration], . First M87 event horizon telescope results. V. Physical origin of the asymmetric ring, Astrophys. J. Lett. 875(1), L5 (2019)

[7] K. Akiyama, et al. [The Event Horizon Telescope Collaboration], . First M87 event horizon telescope results. VI. The shadow and mass of the central black hole, Astrophys. J. Lett. 875(1), L6 (2019)

[8] A. G. Reiss, ., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116(3), 1009 (1998), arXiv: astro-ph/9805201

[9] S. Perlmutter, ., Measurements of Ω and Λ from 42 high-redshift supernovae, Astrophys. J. 517(2), 565 (1999), arXiv: astro-ph/9812133

[10] D. N. Spergel, ., Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for cosmology, Astrophys. J. Suppl. Ser. 170(2), 377 (2007), arXiv: astro-ph/0603449

[11] P. Astier, ., The Supernova Legacy Survey: Measurement of ΩM, ΩΛ and ω from the first year data set, Astron. Astrophys. 447, 31 (2006), arXiv: astro-ph/0510447

[12] P. D. Naselskii and A. G. Polnarev, Candidate missing mass carriers in an inflationary universe, Sov. Astron. 29, 487 (1985)

[13] T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, f(R, T) gravity, Phys. Rev. D 84(2), 024020 (2011)

[14] D. J Eisenstein, et al. [SDSS], . Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies, Astrophys. J. 633(2), 560 (2005)

[15] M. Sharif and M. Zubair, Thermodynamics in f(R, T) theory of gravity, J. Cosmol. Astropart. Phys. 2012, 028 (2012)

[16] M. Jamil,D. Momeni,M. Raza,R. Myrzakulov, Reconstruction of some cosmological models in f(R, T) gravity, Eur. Phys. J. C 72, 1999 (2012), arXiv: 1107.5807

[17] H. Shabani and M. Farhoudi, Cosmological and solar system consequences of f(R, T) gravity models, Phys. Rev. D 90(4), 044031 (2014)

[18] L. C. N. Santos, F. M. da Silva, C. E. Mota, I. P. Lobo, and V. B. Bezerra, Kiselev black holes in f(R, T) gravity, Gen. Relativ. Gravit. 55(8), 94 (2023)

[19] A. Sood, A. Kumar, J. K. Singh, and S. G. Ghosh, Photon orbits and phase transitions in Kiselev-AdS black holes from gravity, Eur. Phys. J. C 84(8), 876 (2024)

[20] G. Mohan,U. D. Goswami, Galactic rotation curves of spiral galaxies and dark matter in f(R, T) gravity theory, Int. J. Geom. Meth. Mod. Phys. 21(04), 2450082 (2024), arXiv: 2211.02948 [gr-qc]

[21] M. M. Gohain, C. Chetia, and K. Bhuyan, Emergent cosmology in magnetized Bianchi VI geometry within f(R, T) gravity, Int. J. Theor. Phys. 63(5), 133 (2024)

[22] T. Tangphati, M. Youk, and S. Ponglertsakul, Magnetically charged regular black holes in f(R, T) gravity coupled to nonlinear electrodynamics, J. High Energy Astrophys. 43, 66 (2024)

[23] P. Sarmah,U. D. Goswami, Observational constraints on cosmological parameters in the Bianchi type III universe with f(R, T) gravity theory, arXiv: 2410.02386 [gr-qc] (2024)

[24] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7(8), 2333 (1973)

[25] S. W. Hawking, Black hole explosions, Nature 248(5443), 30 (1974)

[26] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975) [Erratum: Commun. Math. Phys. 46, 206 (1976)]

[27] J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31(2), 161 (1973)

[28] R. M. Wald, Entropy and black-hole thermodynamics, Phys. Rev. D 20(6), 1271 (1979)

[29] J. D. Bekenstein, Black-hole thermodynamics, Phys. Today 33(1), 24 (1980)

[30] R. M. Wald, The thermodynamics of black holes, Living Rev. Rel. 4, 6 (2001), arXiv: gr-qc/9912119

[31] S. Carlip, Black hole thermodynamics, Int. J. Mod. Phys. D 23(11), 1430023 (2014)

[32] A. C. Wall, A survey of black hole thermodynamics, arXiv: 1804.10610 [gr-qc] (2018)

[33] P. Candelas and D. W. Sciama, Irreversible thermodynamics of black holes, Phys. Rev. Lett. 38(23), 1372 (1977)

[34] S. Mahapatra, P. Phukon, and T. Sarkar, Black hole entropy corrections in the grand canonical ensemble, Phys. Rev. D 84(4), 044041 (2011)

[35] G. P. Li, K. J. He, X. Y. Hu, and Q. Q. Jiang, Holographic images of an AdS black hole within the framework of f(R) gravity theory, Front. Phys. 19(5), 54202 (2024)

[36] P. C. W. Davies, Thermodynamic phase transitions of Kerr–Newman black holes in de Sitter space, Class. Quantum Gravity 6(12), 1909 (1989)

[37] S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87(4), 577 (1983)

[38] A. Curir, Rotating black holes as dissipative spin-thermodynamical systems, Gen. Relativ. Gravit. 13(5), 417 (1981)

[39] A. Curir, Black hole emissions and phase transitions, Gen. Relativ. Gravit. 13(12), 1177 (1981)

[40] D. Pavón and J. M. Rubi, Nonequilibrium thermodynamic fluctuations of black holes, Phys. Rev. D 37(8), 2052 (1988)

[41] D. Pavón, Phase transition in Reissner–Nordstrom black holes, Phys. Rev. D 43(8), 2495 (1991)

[42] O. Kaburaki, Critical behavior of extremal Kerr–Newman black holes, Gen. Relativ. Gravit. 28(7), 843 (1996)

[43] R. G. Cai,Z. J. Lu,Y. Z. Zhang, Critical behavior in (2+1)-dimensional black holes, Phys. Rev. D 55, 853 (1997), arXiv: gr-qc/9702032

[44] R. G. Cai,J. H. Cho, Thermodynamic curvature of the BTZ black hole, Phys. Rev. D 60, 067502 (1999), arXiv: hep-th/9803261

[45] Y. H. Wei, Thermodynamic critical and geometrical properties of charged BTZ black hole, Phys. Rev. D 80(2), 024029 (2009)

[46] K. Bhattacharya, S. Dey, B. R. Majhi, and S. Samanta, General framework to study the extremal phase transition of black holes, Phys. Rev. D 99(12), 124047 (2019)

[47] D. Kastor, S. Ray, and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quantum Gravity 26(19), 195011 (2009)

[48] B. P. Dolan, The cosmological constant and the black hole equation of state, Class. Quantum Gravity 28(12), 125020 (2011)

[49] B. P. Dolan, Pressure and volume in the first law of black hole thermodynamics, Class. Quantum Gravity 28(23), 235017 (2011)

[50] B. P. Dolan, Compressibility of rotating black holes, Phys. Rev. D 84(12), 127503 (2011)

[51] B. P. Dolan, Where is the PdV in the first law of black hole thermodynamics? arXiv: 1209.1272 [gr-qc] (2012)

[52] D. Kubizňák,R. B. Mann, P−V criticality of charged AdS black holes, J. High Energy Phys. 07, 033 (2012), arXiv: 1205.0559 [hep-th]

[53] D. Kubizňák, R. B. Mann, and M. Teo, Black hole chemistry: Thermodynamics with lambda, Class. Quantum Gravity 34(6), 063001 (2017)

[54] K. Bhattacharya, B. R. Majhi, and S. Samanta, Van der Waals criticality in AdS black holes: A phenomenological study, Phys. Rev. D 96(8), 084037 (2017)

[55] Z. M. Xu, Analytic phase structures and thermodynamic curvature for the charged AdS black hole in alternative phase space, Front. Phys. 16(2), 24502 (2021)

[56] P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light ring stability in ultra-compact objects, Phys. Rev. Lett. 119(25), 251102 (2017)

[57] P. V. P. Cunha and C. A. R. Herdeiro, Stationary black holes and light rings, Phys. Rev. Lett. 124(18), 181101 (2020)

[58] S. W. Wei, Topological charge and black hole photon spheres, Phys. Rev. D 102(6), 064039 (2020)

[59] M. Guo and S. Gao, Universal properties of light rings for stationary axisymmetric spacetimes, Phys. Rev. D 103(10), 104031 (2021)

[60] M. Guo, Z. Zhong, J. Wang, and S. Gao, Light rings and long-lived modes in quasiblack hole spacetimes, Phys. Rev. D 105(2), 024049 (2022)

[61] S. P. Wu and S. W. Wei, Topology of light rings for extremal and nonextremal Kerr–Newman–Taub-NUT black holes without Z2 symmetry, Phys. Rev. D 108(10), 104041 (2023)

[62] P. V. P. Cunha, C. A. R. Herdeiro, and J. P. A. Novo, Light rings on stationary axisymmetric spacetimes: Blind to the topology and able to coexist, Phys. Rev. D 109(6), 064050 (2024)

[63] S. W. Wei and Y. X. Liu, Topology of equatorial timelike circular orbits around stationary black holes, Phys. Rev. D 107(6), 064006 (2023)

[64] X. Ye,S. W. Wei, Topological study of equatorial timelike circular orbit for spherically symmetric (hairy) black holes, J. Cosmol. Astropart. Phys. 07, 049 (2023), arXiv: 2301.04786

[65] X. Ye,S. W. Wei, Novel topological phenomena of timelike circular orbits for charged test particles, arXiv: 2406.13270 (2024)

[66] S. W. Wei and Y. X. Liu, Topology of black hole thermodynamics, Phys. Rev. D 105(10), 104003 (2022)

[67] S. W. Wei, Y. X. Liu, and R. B. Mann, Black hole solutions as topological thermodynamic defects, Phys. Rev. Lett. 129(19), 191101 (2022)

[68] Y. S. Duan, The structure of the topological current, SLAC-PUB-3301, 1984

[69] Y. S. Duan and M. L. Ge, SU(2) gauge theory and electrodynamics with N magnetic monopoles, Sci. Sin. 9, 1072 (1979)

[70] P. K. Yerra and C. Bhamidipati, Topology of black hole thermodynamics in Gauss–Bonnet gravity, Phys. Rev. D 105(10), 104053 (2022)

[71] P. K. Yerra and C. Bhamidipati, Topology of Born–Infeld AdS black holes in 4D novel Einstein–Gauss–Bonnet gravity, Phys. Lett. B 835, 137591 (2022)

[72] M. B. Ahmed, D. Kubizňák, and R. B. Mann, Vortex/anti-vortex pair creation in black hole thermodynamics, Phys. Rev. D 107(4), 046013 (2023)

[73] N. J. Gogoi and P. Phukon, Topology of thermodynamics in R-charged black holes, Phys. Rev. D 107(10), 106009 (2023)

[74] M. Zhang and J. Jiang, Bulk-boundary thermodynamic equivalence: A topology viewpoint, J. High Energy Phys. 2023(6), 115 (2023)

[75] M. R. Alipour, M. A. S. Afshar, S. N. Gashti, and J. Sadeghi, Topological classification and black hole thermodynamics, Phys. Dark Univ. 42, 101361 (2023)

[76] Z. M. Xu,Y. S. Wang,B. Wu,W. L. Yang, Riemann surface, winding number and black hole thermodynamics, arXiv: 2305.05916 (2023)

[77] M. Y. Zhang, H. Chen, H. Hassanabadi, Z. W. Long, and H. Yang, Topology of nonlinearly charged black hole chemistry via massive gravity, Eur. Phys. J. C 83(8), 773 (2023)

[78] T. N. Hung and C. H. Nam, Topology in thermodynamics of regular black strings with Kaluza–Klein reduction, Eur. Phys. J. C 83(7), 582 (2023)

[79] J. Sadeghi,M. R. Alipour,S. N. Gashti,M. A. S. Afshar, Bulk-boundary and RPS thermodynamics from topology perspective, arXiv: 2306.16117 (2023)

[80] P. K. Yerra, C. Bhamidipati, and S. Mukherji, Topology of critical points and Hawking–Page transition, Phys. Rev. D 106(6), 064059 (2022)

[81] Z. Y. Fan, Topological interpretation for phase transitions of black holes, Phys. Rev. D 107(4), 044026 (2023)

[82] N. C. Bai, L. Li, and J. Tao, Topology of black hole thermodynamics in Lovelock gravity, Phys. Rev. D 107(6), 064015 (2023)

[83] N. C. Bai, L. Song, and J. Tao, Reentrant phase transition in holographic thermodynamics of Born–Infeld AdS black hole, Eur. Phys. J. C 84(1), 43 (2024)

[84] R. Li, C. H. Liu, K. Zhang, and J. Wang, Topology of the landscape and dominant kinetic path for the thermodynamic phase transition of the charged Gauss–Bonnet AdS black holes, Phys. Rev. D 108(4), 044003 (2023)

[85] P. K. Yerra,C. Bhamidipati,S. Mukherji, Topology of critical points in boundary matrix duals, arXiv: 2304.14988 (2023)

[86] C. X. Fang, J. Jiang, and M. Zhang, Revisiting thermodynamic topologies of black holes, J. High Energy Phys. 2023(1), 102 (2023)

[87] Y. Z. Du,H. F. Li,Y. B. Ma,Q. Gu, Topology and phase transition for EPYM AdS black hole in thermal potential, arXiv: 2309.00224 (2023)

[88] P. K. Yerra, C. Bhamidipati, and S. Mukherji, Topology of Hawking−Page transition in Born–Infeld AdS black holes, J. Phys. Conf. Ser. 2667(1), 012031 (2023)

[89] K. Bhattacharya, K. Bamba, and D. Singleton, Topological interpretation of extremal and Davies-type phase transitions of black holes, Phys. Lett. B 854, 138722 (2024)

[90] H. Chen,M. Y. Zhang,H. Hassanabadi,B. C. Lutfuoglu,Z. W. Long, Topology of dyonic AdS black holes with quasitopological electromagnetism in Einstein-Gauss-Bonnet gravity, arXiv: 2403.14730 (2024)

[91] B. Hazarika,N. J. Gogoi,P. Phukon, Revisiting thermodynamic topology of Hawking-Page and Davies type phase transitions, arXiv: 2404.02526 (2024)

[92] C. H. Liu and J. Wang, The topological natures of the Gauss–Bonnet black hole in AdS space, Phys. Rev. D 107(6), 064023 (2023)

[93] D. Wu, Topological classes of rotating black holes, Phys. Rev. D 107(2), 024024 (2023)

[94] D. Wu and S. Q. Wu, Topological classes of thermodynamics of rotating AdS black holes, Phys. Rev. D 107(8), 084002 (2023)

[95] N. Chatzifotis, P. Dorlis, N. E. Mavromatos, and E. Papantonopoulos, Thermal stability of hairy black holes, Phys. Rev. D 107(8), 084053 (2023)

[96] S. W. Wei, Y. P. Zhang, Y. X. Liu, and R. B. Mann, Static spheres around spherically symmetric black hole spacetime, Phys. Rev. Res. 5(4), 043050 (2023)

[97] Y. Du and X. Zhang, Topological classes of black holes in de Sitter spacetime, Eur. Phys. J. C 83(10), 927 (2023)

[98] C. Fairoos and T. Sharqui, Topological nature of black hole solutions in dRGT massive gravity, Int. J. Mod. Phys. A 38(25), 2350133 (2023)

[99] D. Chen, Y. He, and J. Tao, Topological classes of higher-dimensional black holes in massive gravity, Eur. Phys. J. C 83(9), 872 (2023)

[100] N. J. Gogoi and P. Phukon, Thermodynamic topology of 4D dyonic AdS black holes in different ensembles, Phys. Rev. D 108(6), 066016 (2023)

[101] J. Sadeghi, S. N. Gashti, M. R. Alipour, and M. A. S. Afshar, Bardeen black hole thermodynamics from topological perspective, Ann. Phys. 455, 169391 (2023)

[102] M. S. Ali, H. El Moumni, J. Khalloufi, and K. Masmar, Topology of Born–Infeld-AdS black hole phase transition, Ann. Phys. 465, 169679 (2024)

[103] D. Wu, Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT charged spacetimes, Eur. Phys. J. C 83(5), 365 (2023)

[104] D. Wu, Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes, Eur. Phys. J. C 83(7), 589 (2023)

[105] D. Wu, Topological classes of thermodynamics of the four-dimensional static accelerating black holes, Phys. Rev. D 108(8), 084041 (2023)

[106] J. Sadeghi, M. A. S. Afshar, S. N. Gashti, and M. R. Alipour, Thermodynamic topology and photon spheres in the hyperscaling violating black holes, Astropart. Phys. 156, 102920 (2024)

[107] F. Barzi, H. E. Moumni, and K. Masmar, Rényi topology of charged-flat black hole: Hawking–Page and van der Waals phase transitions, J. High Energy Astrophys. 42, 63 (2024)

[108] M. U. Shahzad, A. Mehmood, S. Sharif, and A. Övgün, Criticality and topological classes of neutral Gauss–Bonnet AdS black holes in 5D, Ann. Phys. 458, 169486 (2023)

[109] C. W. Tong,B. H. Wang,J. R. Sun, Topology of black hole thermodynamics via Rényi statistics, arXiv: 2310.09602 (2023)

[110] A. Mehmood,M. U. Shahzad, Thermodynamic topological classifications of well-known black holes, arXiv: 2310.09907 (2023)

[111] M. Rizwan and K. Jusufi, Topological classes of thermodynamics of black holes in perfect fluid dark matter background, Eur. Phys. J. C 83(10), 944 (2023)

[112] C. Fairoos, Topological interpretation of black hole phase transition in Gauss–Bonnet gravity, Int. J. Mod. Phys. A 39(05n06), 2450030 (2024)

[113] D. Chen, Y. He, J. Tao, and W. Yang, Topology of Horava–Lifshitz black holes in different ensembles, Eur. Phys. J. C 84(1), 96 (2024)

[114] J. Sadeghi, M. A. S. Afshar, S. N. Gashti, and M. R. Alipour, Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives, Ann. Phys. 460, 169569 (2024)

[115] B. Hazarika,P. Phukon, Thermodynamic topology of D = 4, 5 Horava Lifshitz black hole in two ensembles, arXiv: 2312.06324 (2023)

[116] N. J. Gogoi and P. Phukon, Thermodynamic topology of 4D Euler–Heisenberg-AdS black hole in different ensembles, Phys. Dark Univ. 44, 101456 (2024)

[117] M. Y. Zhang,H. Chen,H. Hassanabadi,Z. W. Long,H. Yang, Thermodynamic topology of Kerr–Sen black holes via Rényi statistics, arXiv: 2312.12814 (2023)

[118] J. Sadeghi, M. A. S. Afshar, S. Noori Gashti, and M. R. Alipour, Topology of Hayward-AdS black hole thermodynamics, Phys. Scr. 99(2), 025003 (2024)

[119] B. Hazarika and P. Phukon, Thermodynamic topology of black holes in f(R) gravity, Prog. Theor. Exp. Phys. 2024(4), 043E01 (2024)

[120] A. Malik, A. Mehmood, and M. U. Shahzad, Thermodynamic topological classification of higher dimensional and massive gravity black holes, Ann. Phys. 463, 169617 (2024)

[121] M. U. Shahzad, A. Mehmood, A. Malik, and A. Övgün, Topological behavior of 3D regular black hole with zero point length, Phys. Dark Univ. 44, 101437 (2024)

[122] S. P. Wu,S. W. Wei, Thermodynamical topology of quantum BTZ black hole, arXiv: 2403.14167 (2024)

[123] H. Chen,M. Y. Zhang,H. Hassanabadi,Z. W. Long, Thermodynamic topology of phantom AdS black holes in massive gravity, arXiv: 2404.08243 (2024)

[124] B. Hazarika,P. Phukon, Topology of restricted phase space thermodynamics in Kerr-Sen-AdS black holes, arXiv: 2405.02328 (2024)

[125] Z. Q. Chen,S. W. Wei, Thermodynamical topology with multiple defect curves for dyonic AdS black holes, arXiv: 2405.07525 (2024)

[126] B. E. Panah,B. Hazarika,P. Phukon, Thermodynamic topology of topological black hole in F(R)-ModMax gravity’s rainbow, arXiv: 2405.20022 (2024)

[127] H. Wang,Y. Z. Du, Topology of the charged AdS black hole in restricted phase space, arXiv: 2406.08793 (2024)

[128] A. S. Mohamed and E. E. Zotos, Motion of test particles and topological interpretation of generic rotating regular black holes coupled to non-linear electrodynamics, Astron. Comput. 48, 100853 (2024)

[129] D. Wu, S. Y. Gu, X. D. Zhu, Q. Q. Jiang, and S. Z. Yang, Topological classes of thermodynamics of the static multi-charge AdS black holes in gauged supergravities: Novel temperature-dependent thermodynamic topological phase transition, J. High Energy Phys. 2024(6), 213 (2024)

[130] X. D. Zhu, D. Wu, and D. Wen, Topological classes of thermodynamics of the rotating charged AdS black holes in gauged supergravities, Phys. Lett. B 856, 138919 (2024)

[131] S. W. Wei, Y. X. Liu, and R. B. Mann, Universal topological classifications of black hole thermodynamics, Phys. Rev. D 110(8), L081501 (2024)

[132] W. Liu,L. Zhang,D. Wu,J. Wang, Thermodynamic topological classes of the rotating, accelerating black holes, arXiv: 2409.11666 (2024)

[133] X. D. Zhu,W. Liu,D. Wu, Universal thermodynamic topological classes of rotating black holes, arXiv: 2409.12747 (2024)

[134] D. Wu,W. Liu,S. Q. Wu,R. B. Mann, Novel topological classes in black hole thermodynamics, arXiv: 2411.10102 (2024)

[135] F. Weinhold, Thermodynamics and geometry, Phys. Today 29(3), 23 (1976)

[136] G. Ruppeiner, Thermodynamics: A Riemannian geometric model, Phys. Rev. A 20(4), 1608 (1979)

[137] G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67(3), 605 (1995)

[138] R. Mrugała, On equivalence of two metrics in classical thermodynamics, Physica A 125(2−3), 631 (1984)

[139] J. E. Aman, I. Bengtsson, and N. Pidokrajt, Geometry of black hole thermodynamics, Gen. Relativ. Gravit. 35(10), 1733 (2003)

[140] H. Quevedo, Geometrothermodynamics, J. Math. Phys. 48, 013506 (2007)

[141] H. Quevedo, Geometrothermodynamics of black holes, Gen. Relativ. Gravit. 40(5), 971 (2008)

[142] S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh, P. S. Joshi, K. Jusufi, M. Khodadi, R. K. Walia, A. Övgün, and C. Bambi, Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A*, Class. Quantum Gravity 40(16), 165007 (2023)

[143] K. Jusufi, Quasinormal modes of black holes surrounded by dark matter and their connection with the shadow radius, Phys. Rev. D 101(8), 084055 (2020)

[144] T. Johannsen, A. E. Broderick, P. M. Plewa, S. Chatzopoulos, S. S. Doeleman, F. Eisenhauer, V. L. Fish, R. Genzel, O. Gerhard, and M. D. Johnson, Testing general relativity with the shadow size of SGR A*, Phys. Rev. Lett. 116(3), 031101 (2016)

[145] D. Psaltis, Testing general relativity with the Event Horizon Telescope, Gen. Relativ. Gravit. 51(10), 137 (2019)

[146] P. Kocherlakota, . Event Horizon Telescope, Phys. Rev. D 103(10), 104047 (2021)

[147] V. Prokopov, S. Alexeyev, and O. Zenin, Black hole shadows constrain extended gravity, J. Exp. Theor. Phys. 135(1), 91 (2022)

[148] K. Jafarzade, M. K. Zangeneh, and F. S. N. Lobo, Observational optical constraints of regular black holes, Ann. Phys. 446, 169126 (2022)

[149] I. Çimdiker, D. Demir, and A. Övgün, Black hole shadow in symmergent gravity, Phys. Dark Univ. 34, 100900 (2021)

[150] S. Haroon, K. Jusufi, and M. Jamil, Shadow images of a rotating dyonic black hole with a global monopole surrounded by perfect fluid, Universe 6(2), 23 (2020)

[151] M. Okyay and A. Övgün, Nonlinear electrodynamics effects on the black hole shadow. deflection angle.quasinormal modes and grey body factors, J. Cosmol. Astropart. Phys. 01, 009 (2022)

[152] A. Belhaj and Y. Sekhmani, Shadows of rotating quintessential black holes in Einstein–Gauss–Bonnet gravity with a cloud of strings, Gen. Relativ. Gravit. 54, 17 (2021)

[153] R. Roy, S. Vagnozzi, and L. Visinelli, Superradiance evolution of black hole shadows revisited, Phys. Rev. D 105(8), 083002 (2022)

[154] S. Vagnozzi, C. Bambi, and L. Visinelli, Concerns regarding the use of black hole shadows as standard rulers, Class. Quantum Gravity 37(8), 087001 (2020)

[155] K. Jusufi, M. Jamil, P. Salucci, T. Zhu, and S. Haroon, Black hole surrounded by a dark matter halo in the M87 galactic center and its identification with shadow images, Phys. Rev. D 100(4), 044012 (2019)

[156] K. Jusufi, Quasinormal modes of black holes surrounded by dark matter and their connection with the shadow radius, Phys. Rev. D 101(8), 084055 (2020)

[157] E. Berti, V. Cardoso, and C. M. Will, On gravitational- wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D 73(6), 064030 (2006)

[158] G. Franciolini, L. Hui, R. Penco, L. Santoni, and E. Trincherini, Effective field theory of black hole quasinormal modes in scalar-tensor theories, J. High Energy Phys. 2019(2), 127 (2019)

[159] R. Karmakar and U. D. Goswami, Quasinormal modes, thermodynamics and shadow of black holes in Hu–Sawicki f(R) gravity theory, Eur. Phys. J. C 84(9), 969 (2024)

[160] Y. Sekhmani,D. J. Gogoi,R. Myrzakulov,G. Gaetano Luciano,J. Rayimbaev, Four STU black holes shadows, Class. Quantum Gravity 41(18), 185002 (2024), arXiv: 2407.20621 [gr-qc]

[161] G. Lambiase,D. J. Gogoi,R. C. Pantig,A. Övgün, Shadow and quasinormal modes of the rotating Einstein-Euler-Heisenberg black holes, arXiv: 2406.18300 [gr-qc] (2024)

[162] Y. Sekhmani, D. J. Gogoi, M. Koussour, R. Myrzakulov, and J. Rayimbaev, Shadows of R-charged black holes in AdS5, Phys. Dark Univ. 44, 101442 (2024)

[163] D. J. Gogoi, Violation of Hod’s conjecture and probing it with optical properties of a 5-D black hole in Einstein Gauss–Bonnet bumblebee theory of gravity, Phys. Dark Univ. 45, 101535 (2024)

[164] D. J. Gogoi,S. Ponglertsakul, Constraints on quasinormal modes from black hole shadows in regular non-minimal Einstein Yang–Mills gravity, Eur. Phys. J. C 84(6), 652 (2024), arXiv: 2402.06186 [gr-qc]

[165] R. Karmakar, D. J. Gogoi, and U. D. Goswami, Thermodynamics and shadows of GUP-corrected black holes with topological defects in Bumblebee gravity, Phys. Dark Univ. 41, 101249 (2023)

[166] W. Liu,D. Wu,X. Fang,J. Jing,J. Wang, Kerr-MOG-(A)dS black hole and its shadow in scalar-tensorvector gravity theory, J. Cosmol. Astropart. Phys. 08, 035 (2024), arXiv: 2406.00579

[167] W. Liu,D. Wu,J. Wang, Shadow of slowly rotating Kalb−Ramond black holes, arXiv: 2407.07416 (2024)

[168] W. Liu, D. Wu, and J. Wang, Light rings and shadows of static black holes in effective quantum gravity, Phys. Lett. B 858, 139052 (2024)

[169] S. W. Wei, Y. C. Zou, Y. P. Zhang, and Y. X. Liu, Constraining black hole parameters with the precessing jet nozzle of M87*, Phys. Rev. D 110(6), 064006 (2024)

[170] M. X. Xu, T. Harko, and S. D. Liang, Quantum Cosmology of f(R, T) gravity, Eur. Phys. J. C 76(8), 449 (2016)

[171] J. Wu, G. Li, T. Harko, and S. D. Liang, Palatini formulation of f(R, T) gravity theory, and its cosmological implications, Eur. Phys. J. C 78(5), 430 (2018)

[172] T. Harko and P. H. R. S. Moraes, Comment on “Reexamining f(R, T) gravity”, Phys. Rev. D 101(10), 108501 (2020)

[173] M. A. S. Pinto, T. Harko, and F. S. N. Lobo, Gravitationally induced particle production in scalar-tensor f(R, T) gravity, Phys. Rev. D 106(4), 044043 (2022)

[174] M. A. S. Pinto, T. Harko, and F. S. N. Lobo, Challenging CDM with scalar-tensor f(R, T) gravity and thermodynamics of irreversible matter creation, Acta Phys. Pol. B Proc. Suppl. 16(6), A28 (2023)

[175] A. Bouali, H. Chaudhary, T. Harko, F. S. N. Lobo, T. Ouali, and M. A. S. Pinto, Observational constraints and cosmological implications of scalar–tensor f(R, T) gravity, Mon. Not. R. Astron. Soc. 526(3), 4192 (2023)

[176] G. I. Róis,J. T. S. S. Junior,F. S. N. Lobo,M. E. Rodrigues,T. Harko, Charged black hole solutions in f(R, T) gravity coupled to nonlinear electrodynamics, arXiv: 2412.00582 [gr-qc] (2024)

[177] F. G. Alvarenga, M. J. S. Houndjo, A. V. Monwanou, and J. B. C. Orou, Testing some f(R, T) gravity models from energy conditions, J. Mod. Phys. 4(1), 130 (2013)

[178] M. Jamil, D. Momeni, M. Raza, and R. Myrzakulov, Reconstruction of some cosmological models in f(R, T) gravity, Eur. Phys. J. C 72(4), 1999 (2012)

[179] H. Shabani and M. Farhoudi, f(R, T) cosmological models in phase space, Phys. Rev. D 88(4), 044048 (2013)

[180] J. W. York, Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33(8), 2092 (1986)

[181] E. A. Kontou and K. Sanders, Energy conditions in general relativity and quantum field theory, Class. Quantum Gravity 37(19), 193001 (2020)

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

B. H. would like to thank DST-INSPIRE, Ministry of Science and Technology fellowship program, Govt. of India for awarding the DST/INSPIRE Fellowship (IF220255) for financial support. Special thanks are extended to Pranjal Sarmah for his valuable suggestions regarding the fundamental concepts discussed in this manuscript. B. H. would also like to thank R. Karmakar and R. Bora for their fruitful discussion during the drafting of this manuscript.

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