= Point-mass gravitational lens
{title2=$\boldsymbol\beta=\boldsymbol\theta-\theta_E^2\boldsymbol\theta/|\boldsymbol\theta|^2$}
A <point-mass gravitational lens> is the ideal <gravitational lens> obtained by concentrating a positive <mass> $M$ at one location. Use <geometrical optics>, the <weak-field approximation>, small angles in <radians>, and a stationary lens whose deflecting region is short compared with the observer-lens and lens-source distances. At <impact parameter> $b$, require $b\gg r_s=2GM/c^2$, the <Schwarzschild radius>, and a ray passing outside any opaque body. The physical bend has magnitude $\widehat\alpha=4GM/(c^2b)$ from <Schwarzschild light deflection>; $G$ is the <gravitational constant> and $c$ the <speed of light>. Multiple images can occur within this <weak-field approximation>: <strong gravitational lensing> does not require a ray to approach a <black hole> closely.<Gravitational Lensing by Matthias Bartelmann>[Section 1.2, printed p. 2: geometrical optics, weak fields and slowly moving lenses; compact-object near-field propagation excluded]{ref}
Let $D_l$, $D_s$ and $D_{ls}$ be <angular diameter distances> from observer to lens, observer to source, and lens to source, respectively. They refer to the background geometry used in the <thin gravitational lens equation>. In nearby <Euclidean geometry>, $D_{ls}=D_s-D_l$; in <cosmology>, do not make that subtraction. The angular source coordinate $\boldsymbol\beta$ is the direction the source would have without this lens; the observed image coordinate is $\boldsymbol\theta$. Both are measured relative to the observer-lens <optical axis>.<Lectures on Gravitational Lensing by Ramesh Narayan and Matthias Bartelmann>[Section 2.1.3, pp. 8-9, equations (13)-(15), including the warning that lens-source angular diameter distance is not the difference of the observer distances]{ref}
For the flat sketch below, extend the observer-side <light ray> backwards to the source plane. Its transverse position would be $D_s\boldsymbol\theta$, whereas the actual source is at $D_s\boldsymbol\beta$. Their separation is $D_{ls}\widehat{\boldsymbol\alpha}$. Here $\widehat{\boldsymbol\alpha}$ is defined outward along the lens-plane impact vector, as in the <thin gravitational lens equation>; the physical ray bends toward the <mass>. Substituting $\boldsymbol b=D_l\boldsymbol\theta$ gives
$$
\boldsymbol\beta=\boldsymbol\theta-\frac{D_{ls}}{D_s}\widehat{\boldsymbol\alpha},\qquad
\widehat{\boldsymbol\alpha}=\frac{4GM}{c^2D_l}\frac{\boldsymbol\theta}{|\boldsymbol\theta|^2},\qquad
\theta_E^2=\frac{4GM}{c^2}\frac{D_{ls}}{D_lD_s}.
$$
The same <thin gravitational lens equation> uses <angular diameter distances> in the cosmological setting. The factor $D_{ls}/D_s$ converts the physical bend into the reduced angular deflection; these are different angles.
\Image[/point-mass-gravitational-lens-ray-geometry.png]
{title=Two light paths and their apparent directions for a point-mass gravitational lens}
{description=Theoretical nearby Euclidean sketch, with transverse distances exaggerated. Solid paths connect the same source to the observer; dashed lines extend the observer-side rays to their apparent image positions. The dotted line marks the unlensed source direction.}
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