Time Delay in Black Hole Gravitational Lensing as a Distance Estimator

Abstract

We calculate the time delay between different relativistic images formed by black hole gravitational lensing in the strong field limit. For spherically symmetric black holes, it turns out that the time delay between the first two images is proportional to the minimum impact angle. Their ratio gives a very interesting and precise measure of the distance of the black hole. Moreover, using also the separation between the images and their luminosity ratio, it is possible to extract the mass of the black hole. The time delay for the black hole at the center of our Galaxy is just few minutes, but for supermassive black holes with M = 10 8 ÷ 10 9 M⊙ in the neighbourhood of the Local Group the time delay amounts to few days, thus being measurable with a good accuracy.

Time delay between relativistic images as a probe of cosmic censorship

“…Schwarzschild black hole time delay between successive relativistic ER then becomes roughly 2π × 3 √ 3M and for JNW it becomes 2π × 1+2ν with previously studied examples for relativistic time delay in the literature which, to the best of our knowledge, have all been either black holes or WNS [6,27,37]. We have confirmed that there are important differences between SNS and WNS for time delays between relativistic images which we discuss below.…”

“…The last two solutions have been shown to be obtainable as collapse end states [23,28]. Time delay between relativistic images have previously been studied for Schwarzschild black hole in [6] and in general for spacetimes with photons spheres in [27] (which includes black holes and Weakly Naked Singularities, as we will see later). The present study complements these by computing time delay between relativistic images for Strongly Naked Singularities, ie., naked singularities not covered by photon sphere.…”

“…This is intuitive to understand because in the presence of a photon sphere the light trajectories corresponding to successive rings differ by one extra nearly circular loop at the photon sphere radius as can be easily seen from figure 2. Indeed the time delay between ERs is well approximated by ∆t m,n = 2π(m − n)r ph g(r ph ) = 2π(m − n)J ph (29) where r ph is radius of photon sphere and J ph is corresponding impact parameter [36]. For…”

“…Time delay between far field images is a probe of the so-called 'effective distance' of the system and can be a probe of cosmological parameters [26]. Time delay between relativistic images for black hole has been suggested as a distance estimator [27]. In this paper we focus on time delay as a discriminator between black holes and naked singularities.…”

Analytical discussion on strong gravitational lensing for a massive source with af(R)global monopole

“…The measurement of time delay provides a probe of the Hubble constant [3,21]. The general approximative expressions of time delay between relativistic images in strong field limits for asymptotic flat spacetime without a cosmological horizon has been presented in [22].…”