Relativistic images of Schwarzschild black hole lensing

We model massive dark objects at centers of many galaxies as Schwarzschild black hole lenses and study gravitational lensing by them in detail. We show that the ratio of mass of a Schwarzschild lens to the differential time delay between outermost two relativistic images (both of them either on the primary or on the secondary image side) is extremely insensitive to changes in the angular source position as well as the lens-source and lens-observer distances. Therefore, this ratio can be used to obtain very accurate values for masses of black holes at centers of galaxies. Similarly, angular separations between any two relativistic images are also extremely insensitive to changes in the angular source position and the lens-source distance. Therefore, with the known value of mass of a black hole, angular separation between two relativistic images would give a very accurate result for the distance of the black hole. Accuracies in determination of masses and distances of black holes would however depend on accuracies in measurements of differential time delays and angular separations between images. Deflection angles of primary and secondary images as well as effective deflection angles of relativistic images on the secondary image side are always positive. However, the effective deflection angles of relativistic images on the primary image side may be positive, zero, or negative depending on the value of angular source position and the ratio of mass of the lens to its distance. We show that effective deflection angles of relativistic images play significant role in analyzing and understanding strong gravitational field lensing.

Figure 1

Left: The deflection angle $\widehat{\alpha }$ is plotted against the dimensionless scaled impact parameter $J/M$. The arrow attached to the curve indicates that $\widehat{\alpha }$ decreases with increase in the value of $J/M$. Right: $S$, $L$, and $O$ represent positions of the source, the lens, and the observer, respectively. $S{P}_1$ and $S{P}_2$ are tangents on 2 null geodesics at the source position, whereas $P_{1}O$ and $P_{2}O$ are tangents on, respectively, the same pair of null geodesics at the observer position. ${\widehat{\alpha }}_1$ and ${\widehat{\alpha }}_2$ are light bending angles, whereas ${\gamma }_1$ and ${\gamma }_2$ are their respective supplementary angles. $\beta$ and $\theta p$ stand, respectively, for the angular positions of the source and the primary image. Angles in this schematic diagram are greatly exaggerated. ${\gamma }_2>{\gamma }_1\Rightarrow {\widehat{\alpha }}_2<{\widehat{\alpha }}_1$. According to the $\widehat{\alpha }$ vs $J/M$ plot (on left side), ${\widehat{a}}_2<{\widehat{\alpha }}_1$ is possible only if $S{P}_2$ and $P_{2}O$ are, respectively, tangents at the source and the observer positions on a null geodesic which loops around the lens at least once giving rise to a relativistic image.

Figure 2

Schematic diagram showing the variation in effective deflection angles of relativistic images with respect to increase in the value of angular source position $\beta$. $S$, $L$, and $O$ stand for the position of the source, the lens, and the observer, respectively. $S{C}_1$ and $S{C}_2$ are tangents on 2 null geodesics at $S$, and $C_{1}O$ and $C_{2}O$ are tangents on corresponding null geodesics at $O$. ${\widehat{\alpha }}_{1p}^e$ and ${\widehat{\alpha }}_{1s}^e$ stand for effective deflection angles of relativistic images of order 1, respectively, on the primary and secondary image sides; ${\theta }_{1p}$ and ${\theta }_{1s}$ are their respective angular image positions. Angles are greatly exaggerated.

Figure 3

Top left and middle: The angular positions of primary images ${\theta }_p$, secondary images $|{\theta }_s|$, and their separations ${\theta }_p-{\theta }_s$ are plotted against the angular source position $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005. Top right: The angular positions of relativistic images (on the same side as the primary image) of the first order ${\theta }_{1p}$ and of the second order ${\theta }_{2p}$ are plotted against $\beta$ for the same values of $\mathcal{D}$ as in the figures on left. The curves for ${\theta }_{1p}$ (for different values of $\mathcal{D}$) intersect for ${\beta }_{1c}\approx 24.3028\mu \mathrm{as}$, whereas those for ${\theta }_{2p}$ intersect for ${\beta }_{2c}\approx 24.2724\mu \mathrm{as}$. Below: The angular separations among relativistic images versus the angular source position $\beta$ are plotted for $\mathcal{D}=0.5$, 0.05 and 0.005. ${\theta }_{np}$ and ${\theta }_{ns}$ ($n=1$, 2) stand for angular positions of relativistic images on the primary and the secondary image sides, respectively. The Galactic MDO is modeled as the Schwarzschild lens, which has mass $M=3.61\times {10}^6{M}_{\odot }$ and is situated at the distance $D_d=7.62\mathrm{kpc}$ so that $M/D_d\approx 2.26\times {10}^{-11}$.

Figure 4

Top left: The deflection angles of primary images ${\widehat{\alpha }}_p$ and secondary images ${\widehat{\alpha }}_s$ are plotted against the angular source position $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005. Top middle and right: The effective deflection angles of the first order relativistic images on the same side as the primary image ${\widehat{\alpha }}_{1p}^e$ and on the same side as the secondary image ${\widehat{\alpha }}_{1s}^e$ are plotted against $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005. Below left and right: The effective deflection angles of the second order ${\widehat{\alpha }}_{2p}^e$ (left) and of the first order ${\widehat{\alpha }}_{1p}^e$ (right), both on the same side as the primary image, are plotted against $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005 in the vicinity of zero effective deflection angle. ${\beta }_{2c}\approx 24.2724\mu \mathrm{as}$ and ${\beta }_{1c}\approx 24.3028\mu \mathrm{as}$, where ${\beta }_{2c}$ and ${\beta }_{1c}$ are, respectively, critical angular source positions for the second and first order relativistic images. The gravitational lens is the same as for the Fig. 3.

Figure 5

Top left and middle: The magnifications of primary images ${\mu }_p$, the absolute magnifications of secondary images $|{\mu }_s|$, and their ratios $|{\mu }_p/{\mu }_s|$ are plotted against the angular source position $\beta$ for different values of $\mathcal{D}$. Top right and below left: The magnifications of relativistic images (on the same side as the primary image) of the first order ${\mu }_{1p}$ and the second order ${\mu }_{2p}$ are plotted against the angular source position $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005. Below middle and right: The magnifications ratios $|{\mu }_{1p}/{\mu }_{1s}|$ (where ${\mu }_{1s}$ stands for magnification of the first order relativistic image on the same side as the secondary image) and ${\mu }_{1p}/{\mu }_{2p}$ vs $\beta$ are plotted for the same values of $\mathcal{D}$ as in the figure on below left. The lens is the same as for the Fig. 3.

Figure 6

Top: The time delays of primary images ${\tau }_p$ and secondary images ${\tau }_s$, and the differential time delays of the secondary images with respect to their respective primary images (i.e., ${\tau }_s-{\tau }_p$) are plotted against the angular source position $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005. Below: The time delays of relativistic images (on the same side as the primary image) of the first order ${\tau }_{1p}$, second order ${\tau }_{2p}$, and the differential time delays (i.e., ${\tau }_{2p}-{\tau }_{1p}$) are plotted against $\beta$ for the same values $\mathcal{D}$ as in figures on top. The lens is the same as for the Figs. 3 through 5.

Figure 7

The ratios of mass $M$ of the lens to differential time delays of images are plotted against the angular source position $\beta$ for $\mathcal{D}=0.5$, 0.05 and 0.005. ${\tau }_p$ and ${\tau }_s$ stand for time delays of primary and secondary images, respectively. ${\tau }_{1p}$ and ${\tau }_{2p}$, respectively, represent time delays of relativistic images (on the same side as the primary image) of orders 1 and 2. The gravitational lens is the same as for the Figs. 3 through 6.

Figure 8

Top left and middle: The angular positions of primary images ${\theta }_p$ (represented by continuous curves), secondary images $|{\theta }_s|$ (represented by dotted curves), and their separations ${\theta }_p-{\theta }_s$ are plotted against $M/D_d$ of MDOs at centers of many galaxies for $\mathcal{D}=0.5$, 0.05 and 0.005. Top right: The angular positions of the relativistic images (on the same side as the primary image) of the first order ${\theta }_{1p}$ are plotted against $M/D_d$ for the same values of $\mathcal{D}$ as in figures on left. The curves for ${\theta }_{1p}$ for different values of $\mathcal{D}$ intersect for $(M/D_d{)}_{1c}\approx 9.31854\times {10}^{-13}$. Below middle and right: The angular separations ${\theta }_{1p}-{\theta }_{2p}$ and ${\theta }_{1p}-{\theta }_{1s}$ among relativistic images versus $M/D_d$ are plotted for the same values of $\mathcal{D}$ as in figures on top. ${\theta }_{2p}$ and ${\theta }_{1s}$ stand for angular positions of relativistic images of second order on the primary image side and of first order on the secondary image side, respectively. The angular source position $\beta =1\mu \mathrm{as}$ for all figures.

Figure 9

Top left: The deflection angles of primary images ${\widehat{\alpha }}_p$ (represented by continuous curves) and of secondary images ${\widehat{\alpha }}_s$ (represented by dotted curves) are plotted against $M/D_d$ of MDOs at centers of many galaxies for $\mathcal{D}=0.5$, 0.05 and 0.005. Top middle and right: The effective deflection angles of relativistic images of the first order on the same side as the primary image ${\widehat{\alpha }}_{1p}^e$ and on the secondary image side ${\widehat{\alpha }}_{1s}^e$ are plotted against $M/D_d$ of MDOs for the same values of $\mathcal{D}$ as in the left figure. Below: The effective deflection angles of relativistic images on the primary image side of the second order ${\widehat{\alpha }}_{2p}^e$ and of the first order ${\widehat{\alpha }}_{1p}^e$ versus $M/D_d$ are plotted in the vicinity of zero effective deflection angle for the same values of $\mathcal{D}$. The curves for different values of $\mathcal{D}$ on left and right figures intersect, respectively, for $(M/D_d{)}_{2c}\approx 9.33022\times {10}^{-13}$ and $(M/D_d{)}_{1c}\approx 9.31854\times {10}^{-13}$. The angular source position $\beta =1\mu \mathrm{as}$ for all figures.

Figure 10

Top left and top middle: The magnifications of primary images ${\mu }_p$ (represented by continuous curves) and the absolute magnifications of secondary images $|{\mu }_s|$ (represented by dotted curves), and their ratios $|{\mu }_p/{\mu }_s|$ vs $M/D_d$ of MDOs at centers of many galaxies are plotted for $\mathcal{D}=0.5$, 0.05 and 0.005. Top right and below left: The magnifications of relativistic images (on the same side as the primary image) of the first order ${\mu }_{1p}$ and of the second order ${\mu }_{2p}$ are plotted against $M/D_d$ of MDOs for same values of $\mathcal{D}$. Below middle: The ratio of absolute magnifications of relativistic images of first order on the primary image side ${\mu }_{1p}$ to that on the secondary image side $|{\mu }_{1s}|$ are plotted against $M/D_d$ for $\mathcal{D}=0.5$, 0.05 and 0.005. Below right: The ratio of absolute magnifications of relativistic images (on the primary image side) of the first order ${\mu }_{1p}$ to the second order ${\mu }_{2p}$ are plotted against $M/D_d$ for same values of $\mathcal{D}$ as in other plots. The curves ${\mu }_{1p}/{\mu }_{2p}$ vs $M/D_d$ for different values of $\mathcal{D}$ intersect for $M/D_d\approx 7.07130\times {10}^{-13}$. For all figures, the angular source position $\beta =1\mu \mathrm{as}$.

Figure 11

Top left, top right, and below left: The ratios of mass $M$ of the lens to the differential time delays of images vs $M/D_d$ (the ratio of the mass of the lens to its distance) are plotted for $\mathcal{D}=0.5$, 0.05 and 0.005. ${\tau }_p$ and ${\tau }_s$ stand, respectively, for time delays of primary and secondary images, whereas ${\tau }_{1p}$ and ${\tau }_{2p}$, respectively, represent time delays of relativistic images (on the same side as the primary image) of orders 1 and 2. The curves on below left plot intersect for $M/D_d\approx 9.32438\times {10}^{-13}$. For a given value of $\mathcal{D}$, ratios of mass to differential time delay are strictly increasing function of $M/D_d$. Below right: The ratios of mass $M$ of the lens to the differential time delay $({\tau }_{2p}-{\tau }_{1p})$ of images are plotted against the distance $D_d$ of the lens for same values of $\mathcal{D}$. The ratio $M/({\tau }_{2p}-{\tau }_{1p})$ is not a function of $D_d$. The angular source position $\beta =1\mu \mathrm{as}$ for all figures.

Figure 12

Left: The percentage difference in values for differential time delays $({\tau }_2-{\tau }_1)$ obtained by Bozza and Mancini [18] (using analytical method) and by us (using numerical method) is plotted against $M/D_d$ of MDOs at centers of a few galaxies. ${\tau }_1$ and ${\tau }_2$, respectively, stand for time delays of relativistic images of orders 1 and 2. Right: The ratios of mass $M$ of the lens to the differential time delay $({\tau }_2-{\tau }_1)$ vs $M/D_d$ are plotted for both cases. Our numerical results show that the ratio $M/({\tau }_2-{\tau }_1)$ is strictly increasing function of $M/D_d$. Masses and distances of MDOs are taken from 18 and are given in Table . $\mathcal{D}=0.5$ and the angular source position $\beta =0$ in both figures.