---
title: Light Deflection and Gravitational Lensing
module: Tests of General Relativity
moduleNumber: 7
lessonNumber: 2
order: 702
summary: >
  A light ray grazing the Sun bends by 4GM/(c²b), exactly twice the value a
  Newtonian corpuscle would give; the extra factor is the curvature of space. The
  1919 eclipse confirmed it. The same bending focuses light from distant sources
  into Einstein rings, multiple images, and microlensing brightenings, making
  lensing a direct probe of mass, including mass that emits no light.
topics: [Tests of General Relativity]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 10 — Solar-System Tests (deflection of light); Ch. 11 — Gravitational Lensing"
  - book: Tipler
    ref: "Modern Physics, 5th ed., Ch. 2 §2-5 — General Relativity (deflection of light, the Eddington eclipse expedition)"
---

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](/relativity/the-schwarzschild-solution/light-bending-and-null-geodesics);
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

$$
\delta = \frac{4GM}{c^2 b},
$$

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 $\Delta v_\perp = 2GM/(c b)$, and the
deflection is $\Delta v_\perp / c$:

$$
\delta_{\text{Newt}} = \frac{2GM}{c^2 b} = \tfrac12\,\delta_{\text{GR}}.
$$

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

$$
\d s^2 = -\left(1 + \frac{2\Phi}{c^2}\right)c^2\,\d t^2
        + \left(1 - \frac{2\Phi}{c^2}\right)\bigl(\d x^2 + \d y^2 + \d z^2\bigr),
\qquad \Phi = -\frac{GM}{r}.
$$

- 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.

> **Result.** The deflection of a light ray with impact parameter $b$ by a mass
> $M$ is $\delta = 4GM/(c^2 b)$, twice the value from Newtonian gravity plus the
> equivalence principle. The extra factor of two is the contribution of spatial
> curvature.

For a ray grazing the Sun, $b = R_\odot = 6.96\times10^8\ \text{m}$ and
$GM_\odot/c^2 = 1.477\times10^3\ \text{m}$:

$$
\delta = \frac{4(1.477\times10^3)}{6.96\times10^8}
       = 8.49\times10^{-6}\ \text{rad}
       = 1.75'' .
$$

$$
% caption: A ray grazing the Sun is deflected by the angle delta; the observer on
% Earth sees the star displaced away from the Sun, along the line back to the
% straightened ray, by the same angle. The true and apparent directions differ by
% delta.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % Sun
  \draw (2.6,0) circle (0.34);
  \node at (2.6,0) {Sun};
  % Earth observer
  \fill[black!75] (0,0) circle (1.8pt);
  \node[black!70, left] at (-0.05,0) {Earth};
  % true star direction (straight, grazing top of Sun)
  \draw[black, dashed] (0,0) -- (6.4,1.05);
  \node[black, right] at (6.4,1.05) {true position};
  % bent ray: from star, grazes Sun, arrives at Earth horizontally-ish
  \draw[acc, very thick] (6.4,1.72) .. controls (3.6,1.15) and (2.9,0.55) .. (0,0);
  \node[right] at (6.4,1.72) {star};
  % apparent direction (tangent at Earth)
  \draw[black, dashed] (0,0) -- (5.0,0.42);
  \node[black, right] at (5.0,0.42) {apparent position};
  % deflection angle marker near Earth
  \draw[->, black] (1.6,0.135) arc (5:14:1.6);
  \node[black] at (1.85,0.5) {bend angle};
\end{tikzpicture}
$$

$$
% caption: Along the same grazing path the Newtonian corpuscle model bends the
% ray by 0.87 arcsec, half the general-relativistic 1.75 arcsec; the two
% predictions differ by exactly a factor of two, and the eclipse data fell on
% the upper value.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (0,3.6) node[above] {bending (arcsec)};
  \draw[black] (-0.1,0) -- (5.2,0);
  % scale: 1.75 -> 3.0 ; 0.87 -> 1.49
  \draw[black] (0.7,0) rectangle (1.7,1.49);
  \node[black!70, above] at (1.2,1.55) {half};
  \node[black!70, below] at (1.2,-0.05) {time only};
  \fill[acc, draw=black] (3.0,0) rectangle (4.0,3.0);
  \node[acc, above] at (3.5,3.2) {double};
  \node[black!70, below] at (3.5,-0.05) {space $+$ time};
  % data marker on the GR bar
  \draw[black] (3.5,2.82) -- (3.5,3.02);
  \draw[black] (3.4,2.82) -- (3.6,2.82);
  \draw[black] (3.4,3.02) -- (3.6,3.02);
  \node[black, right] at (4.05,2.92) {1919 data};
\end{tikzpicture}
$$

## The 1919 eclipse

A star whose light grazes the Sun is displaced outward, away from the solar limb,
by up to $1.75''$. 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 $1.75''$ and excluded the
Newtonian $0.87''$. 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

$$
\theta_E = \sqrt{\frac{4GM}{c^2}\,\frac{D_{LS}}{D_L D_S}} .
$$

> **Definition (Einstein radius).** The Einstein radius $\theta_E$ is the angular
> radius of the ring image formed when source, lens, and observer are collinear.
> It sets the characteristic angular scale of all lensing by the mass $M$: images
> form at separations of order $\theta_E$, and the enclosed mass follows from a
> measured ring, $M = \theta_E^2\,c^2 D_L D_S / (4G D_{LS})$.

$$
% caption: With source, lens, and observer collinear, rays bent symmetrically on
% every side of the lens converge to the observer as a ring of angular radius
% theta_E rather than a single point.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % observer
  \fill[black!75] (0,0) circle (1.8pt);
  \node[black!70, left] at (-0.05,0) {observer};
  % lens
  \draw (3.0,0) circle (0.22);
  \node[below] at (3.0,-0.28) {lens};
  % source behind lens
  \fill[black!70] (6.2,0) circle (1.8pt);
  \node[black!70, right] at (6.3,0) {source};
  \draw[black, dashed] (0,0) -- (6.2,0);
  % upper and lower bent rays
  \draw[acc, very thick] (6.2,0) .. controls (4.2,0.62) and (3.4,0.9) .. (0,0);
  \draw[acc, very thick] (6.2,0) .. controls (4.2,-0.62) and (3.4,-0.9) .. (0,0);
  % apparent ring directions from observer
  \draw[black, dashed] (0,0) -- (2.4,0.72);
  \draw[black, dashed] (0,0) -- (2.4,-0.72);
  \node[acc, above] at (1.5,0.62) {ring image};
  \node[acc, below] at (1.5,-0.62) {ring image};
  \draw[->, black] (1.35,0.0) arc (0:17:1.35);
  \node[black, right] at (1.42,0.26) {ring radius};
\end{tikzpicture}
$$

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.

$$
% caption: A slight offset between source and lens breaks the ring into two
% images on opposite sides of the lens, one just inside and one just outside the
% Einstein radius; a galaxy-scale lens can split a quasar into four.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black!75] (0,0) circle (1.8pt);
  \node[black!70, left] at (-0.05,0) {observer};
  \draw (3.0,0) circle (0.2);
  \node[below] at (3.0,-0.26) {lens};
  % true source, offset upward
  \fill[black!70] (6.2,0.7) circle (1.8pt);
  \node[black!70, right] at (6.3,0.7) {true source};
  \draw[black, dashed] (0,0) -- (6.2,0.7);
  % ray bent over the top (outer image) and under (inner image)
  \draw[acc, very thick] (6.2,0.7) .. controls (4.0,0.95) and (3.4,1.05) .. (0,0);
  \draw[acc, very thick] (6.2,0.7) .. controls (4.0,-0.45) and (3.4,-0.7) .. (0,0);
  \draw[black, dashed] (0,0) -- (2.6,0.95);
  \draw[black, dashed] (0,0) -- (2.6,-0.55);
  \node[acc, above] at (1.6,0.85) {outer image};
  \node[acc, below] at (1.7,-0.5) {inner image};
\end{tikzpicture}
$$

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.[^hartle][^tipler]

[^hartle]: Hartle, _Gravity: An Introduction to Einstein's General Relativity_,
§10.3 (the deflection of light and its measurement) and Ch. 11 (gravitational
lensing, the Einstein radius, and image formation).
[^tipler]: Tipler and Llewellyn, _Modern Physics_, 5th ed., §2-5 (general
relativity: the deflection of starlight and the 1919 eclipse expeditions).