Light Deflection and Gravitational Lensing

Light has no rest mass, so in Newtonian gravity it should feel no force at all. Treating a light pulse as a stream of corpuscles moving at speed $c$ and applying the equivalence of gravitational and inertial mass gives a nonzero but small deflection; general relativity gives exactly twice as much. The doubling is not a numerical accident. Half the bending comes from the Newtonian gravitational potential acting on the energy of the light, and the other half comes from the curvature of space itself, a purely relativistic contribution with no Newtonian counterpart. Measuring the deflection therefore distinguishes general relativity from every theory that keeps flat space, and the 1919 eclipse measurement did so.

The bending of a single ray is worked out from the null geodesics of the Schwarzschild geometry; this lesson quotes the deflection angle, interprets the factor of two, and builds the lensing phenomenology on top of it. The signature is $(-,+,+,+)$.

The deflection angle and its factor of two

A light ray that passes a mass $M$ with impact parameter $b$ (the perpendicular distance from the mass to the undeflected straight-line path) is bent by a total angle

valid when $b$ greatly exceeds the gravitational radius $GM/c^2$, so the bending is small. The impact parameter for a grazing ray is the radius of the deflecting body.

The Newtonian estimate treats the photon as a particle of speed $c$ falling freely past $M$. Integrating the transverse acceleration along a straight-line trajectory gives a transverse velocity kick $\Delta v_{\perp }=2GM/(cb)$, and the deflection is $\Delta v_{\perp }/c$:

The relativistic result is twice this. The two halves have distinct origins, visible in the weak-field metric

• The time part $g_{tt}$ carries the Newtonian potential. A ray follows the gradient of the effective refractive index set by $g_{tt}$; this alone reproduces ${\delta }_{\text{Newt}}=2GM/(c^{2}b)$.
• The space part $g_{ij}$, the factor $(1-2\Phi /c^2)$ multiplying the spatial distances, is the curvature of space. It contributes an equal $2GM/(c^{2}b)$.

Their sum is $4GM/(c^{2}b)$. A theory with curved time but flat space predicts half the general-relativistic value, so the measured deflection is a direct test of spatial curvature.

For a ray grazing the Sun, $b=R_{\odot }=6.96\times 10^8\text{m}$ and $G{M}_{\odot }/c^2=1.477\times 10^3\text{m}$:

The 1919 eclipse

A star whose light grazes the Sun is displaced outward, away from the solar limb, by up to $1.75^{\prime \prime }$. The displacement is unobservable in daylight because the Sun's glare drowns the star field, so a total solar eclipse is required: with the solar disk covered, stars near the limb become visible and their positions can be compared against the same field photographed at night months earlier, when the Sun is elsewhere.

Two expeditions organized by Eddington observed the total eclipse of 29 May 1919, at Sobral in Brazil and on the island of Príncipe. The measured limb-star displacements clustered near the general-relativistic $1.75^{\prime \prime }$ and excluded the Newtonian $0.87^{\prime \prime }$. The result, announced that November, was the first confirmation of general relativity's prediction for light and the one that carried it into public view. Modern radio interferometry (VLBI) tracks the deflection of quasar signals as the Sun passes near them and confirms the coefficient $4GM/c^2$ to better than $0.02\%$.

Lensing: rings, images, and magnification

The same deflection that displaces a single star focuses light from a distant source when a massive body lies close to the line of sight. The deflecting body is a gravitational lens. Because rays passing on opposite sides of the lens are both bent inward, they can converge toward the observer, producing multiple images, arcs, and rings of a single background source.

Consider a source S, a lens L of mass $M$, and an observer O nearly collinear, at angular-diameter distances $D_L$ (observer to lens), $D_S$ (observer to source), and $D_{LS}$ (lens to source). A ray from the source deflected by $\delta =4GM/(c^{2}b)$ reaches the observer when the geometry closes. With the source exactly behind the lens, symmetry sends the deflected rays into a full ring, the Einstein ring, at angular radius

When source and lens are not perfectly aligned the ring breaks into a small number of images. A point-mass lens produces two images straddling the lens, one inside and one outside the Einstein radius; an extended lens such as a galaxy can produce four bright images of a background quasar in a cross or fold configuration, with arcs when the source is resolved.

Lensing is graded by the scale of the images relative to the resolution:

• Strong lensing — image separations of arcseconds, resolvable, giving rings, arcs, and multiple quasar images from galaxies and clusters.
• Microlensing — a foreground star lenses a background star with an Einstein radius far below telescopic resolution. The images are not separated, but their combined brightness rises and falls as the lens drifts across the line of sight, a symmetric, achromatic light curve. This detects unseen masses (faint stars, compact objects, exoplanets around the lens star) by their gravity alone.
• Weak lensing — distant background galaxies are slightly sheared by the mass along the line of sight; averaging the distortion over many galaxies maps the total mass, most of it dark.

Because the deflection depends only on mass and geometry, not on whether the mass shines, lensing weighs the total gravitating mass. It was among the first tools to show that clusters of galaxies contain far more mass than their visible stars and gas, and it now maps dark matter directly. The magnification also acts as a natural telescope: a well-aligned cluster brightens and resolves background galaxies otherwise too faint to study.¹²