Thin gravitational lens equation ($\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }$)

A thin lens maps apparent image angle $\mathbf{\theta }$ to unlensed source angle $\mathbf{\beta }$. In the usual outward reduced-deflection convention, $\mathbf{\alpha }={\nabla }_{\theta }\psi$ for a projected lensing potential. For an axisymmetric lens its magnitude is $4G{M}_{\mathrm{projected}}(<D_d\theta )D_{ds}/(c^2{D}_d{D}_s\theta )$. The mass here is projected inside a cylinder, not the three-dimensional mass inside a sphere.

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 $\hat{\alpha }=4GM/(c^{2}b)$ 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.^[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 $\mathbf{\beta }$ is the direction the source would have without this lens; the observed image coordinate is $\mathbf{\theta }$. Both are measured relative to the observer-lens optical axis.^[ref]

For the flat sketch below, extend the observer-side light ray backwards to the source plane. Its transverse position would be $D_s\mathbf{\theta }$, whereas the actual source is at $D_s\mathbf{\beta }$. Their separation is $D_{ls}\hat{\mathbf{\alpha }}$. Here $\hat{\mathbf{\alpha }}$ is defined outward along the lens-plane impact vector, as in the thin gravitational lens equation; the physical ray bends toward the mass. Substituting $\mathbf{b}=D_l\mathbf{\theta }$ givesThe 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.

Choose a signed angular coordinate along the source-lens line. The point-mass gravitational lens becomesFor $\beta \ne 0$, multiplication by $\theta$ gives the quadratic equation ${\theta }^2-\beta \theta -{\theta }_E^2=0$. Its positive discriminant yields the two displayed roots. Their product is ${\theta }_{+}{\theta }_{-}=-{\theta }_E^2$, so they lie on opposite sides of the lens. With $\beta >0$, ${\theta }_{+}>{\theta }_E$ and $-{\theta }_E<{\theta }_{-}<0$: the outer image lies on the source side. For $\beta <0$, the outer image is instead the minus root. These are image directions seen by the observer, not two emitting objects.^[ref]

For a dimensionless example, set $u=\beta /{\theta }_E=0.6$ and $x=\theta /{\theta }_E$. Solving $u=x-1/x$ gives $x_{+}\simeq 1.344$ and $x_{-}\simeq -0.744$, with separation $\sqrt{u^2+4}\simeq 2.088$. The lensing magnification follows from image parity in gravitational lensing: ${\mu }_{+}\simeq 1.442$, ${\mu }_{-}\simeq -0.442$, and total amplification $\simeq 1.884$. Changing $M$ or the distances rescales the angles through the Einstein angular radius, while these dimensionless values remain the same.

For $u\gg 1$, the quadratic equation gives $x_{+}\simeq u+1/u$ and $x_{-}\simeq -1/u$. Using $\mu =1/(1-x^{-4})$, the outer image tends to unit lensing magnification and the inner image has $|{\mu }_{-}|\simeq u^{-4}$. Two mathematical images can therefore look like one at finite sensitivity or angular resolution. Furthermore, the inner impact parameter eventually becomes too small for the stated weak-field approximation or passes through the opaque lens: the ideal formula must then be replaced by an appropriate physical model.

Deflection changes one light ray's direction; focusing concerns relative directions of neighboring light rays and their bundle's cross section. To see the difference for a point-mass gravitational lens, send parallel incident light rays past the lens. A ray at impact parameter $b$ meets the optical axis approximately $f(b)=b/\hat{\alpha }=c^2{b}^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common focal length for all rays, although it can produce lensing magnification and an Einstein ring for a specified source-observer geometry.

The point-mass gravitational lens also illustrates why focusing must not be identified just with positive lensing convergence. Away from the central mass, differentiate the reduced deflection $\mathbf{\alpha }={\theta }_E^2\mathbf{\theta }/r^2$: its two-dimensional divergence is zero, so $\kappa =0$ there. Nevertheless its radial and tangential derivatives differ, giving lensing shear of magnitude ${\theta }_E^2/r^2$ and $\det A=1-{\theta }_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent geometrical optics model, surface brightness is preserved; the larger apparent solid angle supplies the increased received light.

This circular lensing potential has reduced deflection $\alpha (r)={\theta }_{E}r/\sqrt{r^2+{\theta }_c^2}$, mean lensing convergence $\bar{\kappa }={\theta }_E/\sqrt{r^2+{\theta }_c^2}$ and local convergence $\kappa ={\theta }_E({\theta }_c^2+r^2/2)/(r^2+{\theta }_c^2{)}^{3/2}$. Its central convergence is finite for ${\theta }_c>0$. As the core shrinks to zero, the deflection tends to that of a singular isothermal sphere lens. This softening of the projected lensing potential must not be confused with replacing a three-dimensional density by a different cored profile.

For $0<{\theta }_c<{\theta }_E$, the signed circular lens mapping $\beta =\theta [1-{\theta }_E/\sqrt{{\theta }^2+{\theta }_c^2}]$ has radial stationary points at ${\theta }_r^2=({\theta }_E{\theta }_c^2{)}^{2/3}-{\theta }_c^2$. Substituting them gives the displayed caustic radius. An offset point-like astronomical source has three images when $0<|\beta |<{\beta }_c$, a merging pair plus a separate outer image on the fold, and one image outside. At alignment, the two off-centre images form a ring while a central image remains. For ${\theta }_c\ge {\theta }_E$ the mapping is monotone and there is no finite multiple-image region. The zero-core singular limit loses the central image and has a different image-count exception.