Some perspective of thermodynamical and optical properties of black holes in Maxwell–dilaton–dRGT-like massive gravity

Abstract Motivated by integrating the dilaton field (as a UV correction) with dRGT-like massive gravity (as an IR correction) into Einstein gravity, we investigate the thermodynamic and optical properties of black holes within this gravitational framework. We begin by reviewing the black hole solutions in Maxwell–dilaton–dRGT-like massive gravity, followed by an analysis of how various parameters influence on the asymptotical behavior of the spacetime and the event horizon of these black holes. In the subsequent section, we examine the conserved and thermodynamic quantities associated with these black holes, paying particular attention to the effects of parameters like $$\beta ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$\alpha ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and the massive parameters $$(\eta _{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>η</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and $$\eta _{2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>η</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> on their local stability by simultaneously evaluating the heat capacity and temperature. We also adopt an alternative method to study phase transitions using geometrothermodynamics. Furthermore, we explore how the parameters of Maxwell–dilaton–dRGT-like massive gravity impacts the optical characteristics and radiative behavior of black holes. In particular, we analyze the effects of the dilaton coupling constant $$(\alpha ),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> charge ( q ), the massive gravity parameter $$(\eta _1),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>η</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and the graviton mass $$(m_g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> on the radius of the photon sphere and the resulting black hole shadow. Moreover, the theoretical shadow radius is compared to the observational data from $$Sgr A^*.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mi>r</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Additionally, we investigate the energy emission rate of these black holes, revealing that these parameters substantially influence the emission peak.