Weak Deflection Angle by the Einstein–Cartan Traversable Wormhole Using Gauss–Bonnet Theorem with Time Delay

In this article, we estimate the gravitational deflection angles of light in the spacetime of Einstein–Cartan wormholes supported by normal matter or phantom energy utilizing the Gauss–Bonnet theorem. The obtained deflection angles are examined in relation to the wormhole throat radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mn>0</mn></msub></semantics></math></inline-formula> and the equation of state parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> across four scenarios, and it has been seen that the larger throat radii <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mn>0</mn></msub></semantics></math></inline-formula> result in higher deflection angles. Moreover, the wormholes filled with phantom energy exhibit greater deflection angles compared to those filled with normal matter. The reported deflection angles are influenced by dark matter and Maxwell’s fish eye matter: Dark matter, as well as Maxwell’s fish eye matter, increases the deflection angles. The deflection angle is also estimated using the Keeton and Petters method, which is proportional to wormhole throat <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mn>0</mn></msub></semantics></math></inline-formula> and inversely proportional to the impact parameter <i>b</i>. Additionally, a comparative study is performed on the deflection angles obtained from four different scenarios. Finally, analytical results for time delay due to Einstein–Cartan wormholes are estimated for the four <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> cases which are decreasing for increasing values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>c</mi></msub></semantics></math></inline-formula>.