Axially symmetric solutions in Ricci-inverse modified gravity

Abstract We investigate a family of axial symmetry solution constructed in general relativity (GR) within the framework of Ricci-inverse (RI) gravity theory. In GR, these solutions admitted closed time-like curves at an instant of time from an initial spacelike hypersurface in a causally well-behaved manner, thus, violates the causality condition. Our aim is to examine these axial symmetry solutions within the context of Ricci-inverse gravity theory to determine whether closed time-like curves still appear in this new gravity theory. We consider two Classes of RI-gravity models: (i) Class-II models defined by a function $$f=f({\mathcal {R}}, A^{\mu \nu }\,A_{\mu \nu })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msup> <mml:mspace/> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity and (ii) Class-III models defined by $$f=f({\mathcal {R}},{\mathcal {A}}, A^{\mu \nu }\,A_{\mu \nu })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msup> <mml:mspace/> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$A^{\mu \nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msup> </mml:math> is the anti-curvature tensor, $${\mathcal {A}}=g_{\mu \nu }\,A^{\mu \nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:mspace/> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> as its scalar, and $$R^{\mu \nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msup> </mml:math> is the Ricci tensor. We are able solved the modified field equations considering these axial symmetry solutions as background in RI-gravity with null radiation as the matter content and the cosmological constant. This confirms that the chosen family of axial symmetry solutions are valid solutions in RI-gravity theory and, consequently, closed time-like curves is still form, analogous to their formation in GR.