Exploring null geodesic of Finslerian hairy black hole

Abstract The study of hairy black holes within Finsler space-time is performed based on a fundamental set of criteria. These requirements include the presence of a clearly defined event horizon and compliance with the strong energy condition for the characteristics outside the horizon. This examination is conducted through the gravitational decoupling method to describe the deformation of a Finslerian Schwarzschild (FSch) black hole due to the inclusion of additional arbitrary sources (scalar field, tensor field, fluidlike dark matter, etc). So, it is characterized by the primary hair ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mrow> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> </mml:mrow> </mml:math> ) where the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is linked to gauge transformations of the seed FSch metric and α is the deformation factor apart from flag curvature ( λ ) and mass ( M ). This study focuses on determining the black hole’s temperature and heat capacity, which are crucial for understanding its thermodynamic properties. Additionally, we examine how the hairy parameters influence these thermodynamic characteristics, providing a deeper understanding of the interplay between the black hole’s structure and its thermal behavior. Moreover, this paper investigates the null geodesics around the Finslerian hairy Schwarzschild black hole, where we obtain the physical parameters associated with these geodesics, including the effective potential, the photon sphere radius, and the impact parameter and investigate the effects of Finslerian parameter ε and hairy parameters ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> </mml:mrow> </mml:math> ) on these values. This study aims to enhance our understanding of the structure of space-time and the behavior of light in proximity to black holes.