Distinguishing Signature of Kerr-MOG Black Hole and Naked Singularity via Lense-Thirring Effect

We examine the geometrical difference between non-extremal black hole(NXBH), extremal black hole(XBH) and naked singularity(NS) via Lense-Thirring(LT) effect in spinning modified-gravity(MOG). For NXBH, we find that the LT frequency (ΩLT ) is proportional to the angular-momentum(J = a calM) parameter or spin parameter(a) i.e. ΩLT ∝ J or ΩLT ∝ a[where calM = M(1 + α) is ADM mass, α is MOG parameter and M is Komar mass] and is inversely proportional to the cubic value of radial parameter i.e. ΩLT ∝ 1 r3 . For XBH (a2 = G2 N calM2 1+α ), we find LT frequency is proportional to the angular-momentum parameter i.e. ΩLT ∝ 1 √ 1+α and is inversely proportional to the cubic value of radial parameter i.e. ΩLT ∝ 1 r3 . While for NS, we find ΩLT ∝ calM3 J3 and ΩLT ∝ [ r−( α 1+α )GN calM 2 ] √ 1+( α 1+α ) G2 N calM2 a2 in the limit θ = 0 and a = J calM >> r. It depends both on the angular momentum parameter and MOG parameter~α.