Four <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> </mml:mrow> </mml:math> black holes shadows

Abstract In this work, we examine the optical behaviors and thermodynamic phase structures using shadow analysis for four <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">U</mml:mi> </mml:mrow> </mml:mrow> </mml:math> black holes. The study is conducted for four cases of charge configurations on the parameter space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . As a matter of fact, both the electric charge as a parameter and the parameter space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> affect the geometry of the black hole shadow, particularly the size of the shadow. We also introduce a constraint on the charge of the black hole from the observational results of the M87 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow/> <mml:mo>⋆</mml:mo> </mml:msup> </mml:mrow> </mml:math> and Sgr A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow/> <mml:mo>⋆</mml:mo> </mml:msup> </mml:mrow> </mml:math> shadow. Furthermore, we show that the electric charge and the parameter space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> have a non-trivial impact on the variation of the energy emission rate. Interestingly enough, we find novel scenarios in which the evaporation is slower, which causes the lifetime of the black holes to be considerably elongated. On the other side, the phase structure of four <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">U</mml:mi> </mml:mrow> </mml:mrow> </mml:math> black holes is explored for two cases of electric charge configuration. The findings show a perfect correlation between the shadow and event horizon radii. This correlation is, in fact, helpful in discovering the phase transition in terms of the shadow radius. In addition, the microstructure is being analyzed in terms of shadow analysis, providing similar behavior to the ordinary situation of the Ruppeiner formalism.

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