Schwarzschild black hole lensing

TL;DR

The paper investigates strong-field gravitational lensing by a Schwarzschild black hole, predicting an infinite sequence of relativistic images formed by light bending near the photon sphere. It derives a lens equation capable of handling large deflections and applies it to the Galactic center, demonstrating that the outer relativistic images appear at about tens of microarcseconds with extreme demagnification. Although observing these images is unlikely with current technology, their detection would provide a stringent test of general relativity in strong fields and constrain the compactness of the lens (r0/M ≈ 3.21). The work also outlines how such observations would bolster the black-hole interpretation of galactic centers and refine our understanding of relativistic lensing phenomena.

Abstract

We study strong gravitational lensing due to a Schwarzschild black hole. Apart from the primary and the secondary images we find a sequence of images on both sides of the optic axis; we call them {\em relativistic images}. These images are formed due to large bending of light near r = 3M (the closest distance of approach r_o is greater than 3M). The sources of the entire universe are mapped in the vicinity of the black hole by these images. For the case of the Galactic supermassive ``black hole'' they are formed at about 17 microarcseconds from the optic axis. The relativistic images are not resolved among themselves, but they are resolved from the primary and secondary images. However the relativistic images are very much demagnified unless the observer, lens and source are very highly aligned. Due to this and some other difficulties the observation of these images does not seem to be feasible in near future. However, it would be a great success of the general theory of relativity in a strong gravitational field if they ever were observed and it would also give an upper bound, r_o = 3.21 M, to the compactness of the lens, which would support the black hole interpretation of the lensing object.

Figures (3)

• Figure 1: The lens diagram: $O, L$ and $S$ are respectively the positions of the observer, deflector (lens) and source. $OL$ is the reference (optic) axis. $\angle LOS$ and $\angle LOI$ are the angular separations of the source and the image from the optic axis. $SQ$ and $OI$ are respectively tangents to the null geodesic at the source and observer positions; $LN$ and $LT$, the perpendiculars to these tangents from $L$, are the impact parameter $J$. $\angle OCQ$, is the Einstein bending angle. $D_{s}$ represents the observer-source distance, $D_{ds}$ the lens-source distance and $D_{d}$ the observer-lens distance.
• Figure 2: This gives relativistic image positions for a given source position. $\alpha$ and $\tan\theta-\tan\beta$ are plotted against the angular position $\theta$ of the image; these are represented by the continuous and the dashed curves, respectively. For a given position of the source, the points of intersections of the continuous curves (the two outermost ones on each side being shown) with the dashed curves give the angular positions of relativistic images. The Galactic "black hole" (mass $M= 2.8 \times 10^6 M_{\odot}$ and the distance $D_d = 8.5 kpc$ so that $M/D_d \approx 1.57 \times 10^{-11}$) serves as the lens, $D_{ds}/D_s = 1/2,$ and $\beta = \mp 0.075$ radian ($\approx \mp 4.29718\mathop{\rm {{}^\circ}}$). $\theta$ is expressed in microarcseconds. The angular position of a relativistic image changes very slowly with respect to a change in the source position.
• Figure 3: The tangential magnification$\mu_t$ denoted by dotted curves and the total magnification$\mu$ denoted by continuous curves are plotted against the image position (expressed in microarcseconds) near relativistic tangential critical curves. The figures on the right side give the magnification for the outermost relativistic image, whereas those on left side are for a relativistic image adjacent to the previous one. The lens is the Galactic "black hole" ($M/D_d \approx 1.57 \times 10^{-11}$) and $D_{ds}/D_s = 1/2$. The singularities in magnifications show the angular positions of the relativistic tangential critical curves. The origin of the $\theta$-axis for the figures on the left side is $16.87715$ and for each on right side is $16.89825$.