Presenting Circular Gravitational Fields: A Numerical Exploration around Rotating Black Holes Second Revision
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We present a novel theoretical framework, Circular Gravitational Fields (CGF), which extends general relativity by introducing a geometric coupling between a U(1) gauge field and spacetime curvature through the Ricci tensor. This coupling preserves exact consistency with vacuum Einstein equations while predicting specific modifications in strong-field regimes. Using the Baumgarte- Shapiro-Shibata-Nakamura (BSSN) formalism, we perform detailed numerical simulations to explore CGF behavior around rotating black holes and derive testable predictions for gravitational wave signatures. Our approach maintains compatibility with current observational constraints from LIGO/Virgo data while suggesting deviations that could be probed by next-generation detectors such as the Einstein Telescope.