Observing black hole phase transitions in extended phase space and holographic thermodynamics approaches from optical features

Abstract The phase transitions of charged Anti-de Sitter (AdS) black holes are characterized by studying null geodesics in the vicinity of the critical curve of photon trajectories around black holes as well as their optical appearance as the black hole images. In the present work, the critical parameters including the orbital half-period $$\tau ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> the angular Lyapunov exponent $$\lambda _L,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and the temporal Lyapunov exponent $$\gamma _L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> are employed to characterize black hole phase transitions within both the extended phase space and holographic thermodynamics frameworks. Under certain conditions, we observe multi-valued function behaviors of these parameters as functions of bulk pressure and temperature in the respective approaches. We propose that $$\tau ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$\lambda _L,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$\gamma _L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> can serve as order parameters due to their discontinuous changes at first-order phase transitions. To validate this, we provide detailed analytical calculations demonstrating that these optical critical parameters follow scaling behavior near the critical phase transition point. Notably, the critical exponents for these parameters are found to be 1/2, consistent with those of the van der Waals fluid. Our findings suggest that static and distant observers can study black hole thermodynamics by analyzing the images of regions around the black holes.