| Summary: | The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>, magnetic field strength <i>H</i>, and magnetic permeability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>G</mi><mi>Q</mi><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In such a spacetime, we obtain an EoS with a constant scalar curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> in terms of the magnetic field strength <i>H</i> and magnetic permeability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi mathvariant="script">R</mi><mi>i</mi><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula> vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></semantics></math></inline-formula> with a Killing vector field. Furthermore, we demonstrate that the EoS <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></semantics></math></inline-formula> of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula> and obeying the equation of state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></semantics></math></inline-formula> is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively.
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