Accretion disks in Schwarzschild-MOG and Kerr-MOG backgrounds: MOG parameter in terms of observational quantities

Abstract

Abstract We apply a general relativistic framework to static and rotating black hole solutions in Scalar–Tensor–Vector Gravity or modified gravity (MOG). Our results yield exact analytic, closed-form relations expressing the mass M , the MOG coupling parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and the distance D of a Schwarzschild-MOG black hole in terms of a minimal set of directly measurable elements of the accretion disk: the total frequency shift, the telescope aperture angle, and the redshift rapidity . The resulting expressions are derived for particles close to the midline and line of sight, where the redshift rapidity is treated as a relativistic invariant encoding the evolution of the frequency shift with respect to the emitter’s proper time in MOG spacetime. We further extend the formalism to the rotating Kerr-MOG geometry and obtain corresponding relations that determine the rotation parameter a jointly with M , $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and D on the midline. In the rotating background, we introduced the redshift acceleration (general-relativistic version of jerk) to disentangle the spacetime parameters. Crucially, the explicit appearance of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> in these formulas enables direct empirical estimation of this parameter, thereby providing a means to test for departures from standard general relativity. The present study focuses on the geodesic motion of massive test particles revolving around Schwarzschild/Kerr-MOG black holes, and the previous results obtained in the standard Schwarzschild/Kerr backgrounds are recovered in the limit $$\alpha \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The derived expressions are concise and suitable for incorporation into black hole parameter-estimation pipelines.