Photon orbits and phase transitions in Kiselev-AdS black holes from $$f(R,\; T)$$ gravity

Abstract The acceleration observed in cosmological expansion is often attributed to negative pressure, potentially arising from quintessence. We explore the relationship between the photon orbit radius and the phase transition of spherical AdS black holes in f ( R , T ) gravity influenced by quintessence dark energy, specifically Kiselev-AdS black holes in f ( R , T ) gravity. We are treating the negative cosmological constant $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> as the system’s pressure to examine the impact of the state parameter $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> and the f ( R , T ) gravity parameter $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> . Interestingly, Kiselev-AdS black holes within the f ( R , T ) gravity framework exhibit a van der Waals-like phase transition. In contrast, these black holes display a Hawking–Page-like phase transition in general relativity. We demonstrate that below the critical point, the black hole undergoes a first-order vdW-like phase transition, with $$r_{ps}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $$u_{ps}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> </mml:msub> </mml:math> serving as order parameters exhibiting a critical exponent of 1/2, similar to ordinary thermal systems. It suggests that $$r_{ps}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $$u_{ps}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> </mml:msub> </mml:math> can serve as order parameters in characterizing black hole phase transitions, hinting at a potentially universal gravitational relationship near critical points within black hole thermodynamic systems. Investigating the correlation between photon sphere radius and thermodynamic phase transitions provides a valuable means of distinguishing between different gravity theory models, ultimately shedding light on the nature of dark energy. Finally, as $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> tends towards zero, our results precisely align with those of Kiselev-AdS black holes.