Relative time delay in a spinning black hole as a diagnostic for no-hair theorem

The spinning regular black hole (spin a ) metric proposed by Johannsen shares the Kerr horizon but contains independent dimensionless parameters marking deviation from the Kerr metric. Non-zero value of any of the parameters would indicate violation of the no-hair theorem. We shall find the influence of these parameters on the relative time delay (not Shapiro time delay) treated here as a diagnostic for no-hair theorem using aligned, finite, thin-lens approximation in realistic spinning astrophysical configurations. Precise measurement of this delay would then help us determine, from observational perspective, whether or not any of the parameters is really non-zero. We shall also point out that the aligned spinning lens is completely equivalent to a “static” lens with a fictitious lens geometry, which would enable us to re-express the relative time delay components in terms of the spin a . Numerical values are tabulated for three astrophysical lens systems. The advantage of the present treatment is that it can accommodate a variety of spinning lens systems that are likely to be detected in the near future.