String effect on the relative time delay in the Kerr-Sen black hole

Abstract: A well known solution of heterotic string theory is the spinning Kerr-Sen black hole (KSBH) characterized by a string parameter $\xi$. Kerr black hole is recovered at $\xi=0$. The purpose of this paper is to investigate the effect of $\xi$ on a new diagnostic of relative time delay (RTD) to see how the latter deviates from that in general relativity. Assuming KSBH as the spinning lens partner in PSR-BH binary systems, which provide the best laboratory for testing the time delay predictions, we study here the RTD up to third PPN order in $\left(1/r\right)$ in the thin-lens approximation. We work out a useful generalization of the RTD formulas applicable to the experimentally viable \textit{finite} distance lens scales, while\ terms higher than the zeroth order are shown to contain the effect of $\xi$. We shall also relate RTD to the observable image magnification factor determined by $\beta/\theta_{E}$, where $\beta$ is the angular separation between the source and the observer and $\theta_{E}$ is the "Einstein angle" determined by an "effective" non-aligned static lens equivalent to the original aligned spinning lens. Numerical estimates for two typical binary lens systems show $\mu$sec level delay at the zeroth order consistent with predictions in the literature. However, the string effect at higher orders is too tiny to be measurable even in the far future leading to the conclusion that the stringy and general relativity BHs are yet observationally indistinguishable.