Deflection of Light by the Sun

A light ray that skims past the Sun does not travel in a straight line — it is bent toward the mass, and by a very specific amount. The classical corpuscular picture, in which light is a stream of fast particles falling in Newtonian gravity, together with the equivalence principle, predicts a deflection of $2GM/(c^2 b)$ . General relativity predicts exactly twice as much, $4GM/(c^2 b)$ , and that factor of two is the whole story: it is the cleanest classical fingerprint that gravity is spacetime curvature rather than a scalar force superimposed on flat space. The doubling has a transparent geometric origin. In the weak field the Schwarzschild metric warps both time and space by the same potential $\Phi$ : the time-time part $g_{00}\approx-(1+2\Phi/c^2)$ is the piece a slow planet feels, reproducing Newton, while the space-space part $g_{ij}\approx(1-2\Phi/c^2)\delta_{ij}$ is felt only by something moving fast. A photon travels at $c$ , so it samples the spatial curvature just as strongly as the temporal curvature. Each supplies $2GM/(c^2 b)$ ; together they make $4GM/(c^2 b)$ . For a ray grazing the solar limb this is $1.75''$ — under two arcseconds, small but measurable when the Sun's glare is removed during a total eclipse and background stars become visible. The 1919 expeditions to Sobral and Príncipe, led by Dyson and Eddington, photographed exactly this shift and found it consistent with Einstein, not with the half-value. A century later the same effect is a precision test: written in the PPN form $\tfrac12(1+\gamma)\,\theta_{\rm GR}$ , it pins $\gamma$ to unity at the $10^{-5}$ level.

Key takeaways

A light ray grazing a mass is deflected by $\Delta\phi = 4GM/(c^2 b)$ , where $b$ is the impact parameter — exactly twice the value a Newtonian corpuscle plus the equivalence principle predicts.
For a ray grazing the solar limb ( $b=R_\odot$ ) this is $1.75''$ ; the Newtonian-only value is $0.875''$ . The 1919 eclipse measured the full value, confirming general relativity.
The factor of two arises because both the time-time and space-space parts of the metric bend the path; a photon moving at $c$ feels the spatial curvature that a slow planet does not.
Equivalently the weak field acts as an optical medium of refractive index $n=1+2GM/(c^2r)$ — the "2" inside is the doubling, and it makes gravitational lensing the macroscopic consequence.
In the PPN formalism the deflection is $\tfrac12(1+\gamma)\cdot 4GM/(c^2b)$ ; VLBI and the Cassini Shapiro-delay give $\gamma-1=(2.1\pm2.3)\times10^{-5}$ , and Gaia now models the bending for every star on the sky.
Key idea: light bending is twice the Newtonian value because a photon moving at $c$ samples the spatial curvature of spacetime as strongly as the time curvature; each contributes $2GM/(c^2b)$ to the total deflection $4GM/(c^2b)$ .

Where this lives today

The eclipse method is now a museum piece; the effect itself is a workhorse of precision gravity. Very Long Baseline Interferometry watches compact radio quasars pass near the Sun and — because radio waves pierce daylight and the corona, needing no eclipse — measures their deflection to microarcseconds, confirming the $4GM/c^2b$ law and, with the Cassini Shapiro-delay run of 2003, fixing $\gamma-1=(2.1\pm2.3)\times10^{-5}$ . ESA's Gaia mission takes it to the whole sky: since even a star $90^\circ$ from the Sun is deflected by about $4\,\mathrm{mas}$ , the GR bending must be modelled for every one of its two billion targets, turning routine astrometry into an all-sky test of $\gamma$ . On larger scales the same physics is gravitational lensing — galaxies and clusters split background sources into multiple images, arcs, and Einstein rings whose radius $\theta_E=\sqrt{4GM D_{LS}/(c^2 D_L D_S)}$ is a direct descendant of the solar formula, and weak lensing statistically distorts galaxy shapes to weigh dark matter. At the extreme, the Event Horizon Telescope's images of M87\ and Sgr A\ show the bright "photon ring" produced as the deflection diverges near $b_c$ — the strong-field entry in the limiting-cases table, made visible.

The physical picture

Picture starlight skimming past the Sun on its way to Earth. The Sun curves spacetime, and the photon — forced to follow the straightest available path, a null geodesic — veers toward the mass. The closer it grazes, the steeper the local curvature and the larger the bend, so the deflection scales as $1/b$ . To the observer on Earth the ray arrives tilted, and tracing that tilted ray backward puts the star's apparent position slightly farther from the Sun than its true position. The first figure fixes this geometry and carries the headline number.

Figure

Starlight grazing the solar limb at impact parameter b = R⊙ is bent toward the Sun by Δφ = 4GM/c²R⊙ = 1.75 arcseconds, so the star appears shifted away from the Sun; the observer sees the apparent position, and the bend angle is hugely exaggerated here

Why exactly double Newton? Split the metric into a "time warp" and a "space warp," each governed by the same potential $\Phi=-GM/r$ . A slowly moving planet lingers in the time warp — clocks run slow deeper in the well, and that gradient is gravity à la Newton. But a photon races through space at $c$ and therefore registers the space warp just as strongly. The two equal contributions of $2GM/(c^2b)$ simply add. The 1919 test was designed to tell these apart: not "is there any bending?" (Newton already says yes) but "is it the full amount or half?" The second figure makes that split quantitative.

Figure

The deflection as a sum of two equal curvatures: the Newtonian corpuscle feels only the time-time metric g₀₀ and gets 0.875 arcsec, while general relativity adds an equal 0.875 arcsec from the space-space metric gᵢⱼ, so the total is 4GM/c²b = 1.75 arcsec, exactly twice Newton

An equivalent and very useful picture makes the same point optically: in the weak field a photon behaves as if it moves through a medium of refractive index $n(r)=1-2\Phi/c^2=1+2GM/(c^2r)$ . Light bends toward higher index, exactly as in a glass lens — and the "2" in $n-1$ is the very same doubling. Gravitational lensing, from Einstein rings to cluster arcs, is the macroscopic descendant of this one number.

The core equations

The headline result is a single formula: the total angle through which a ray is bent depends only on the deflecting mass and the impact parameter. It is the leading term in an expansion in the small quantity $GM/(c^2b)$ , which for the Sun is of order $10^{-6}$ .

\Delta\phi = \frac{4GM}{c^2 b} = \frac{2R_s}{b} \qquad (1)

where: $\Delta\phi$ — total deflection angle of the ray; $G$ — Newton's constant; $M$ — mass of the deflector; $c$ — speed of light; $b$ — impact parameter (the perpendicular distance from the mass to the incoming asymptote); $R_s=2GM/c^2$ — Schwarzschild radius. Reading: double the Schwarzschild radius, divide by the impact parameter, and you have the bend in radians — larger for a closer graze, falling off as $1/b$ . Everything else is the machinery that produces equation (1) from the geometry. The starting point is the Schwarzschild metric, the unique static spherically symmetric vacuum solution of Einstein's equations, which sets the arena in which the photon moves.

\mathrm{d}s^2 = -\left(1-\frac{R_s}{r}\right)c^2\mathrm{d}t^2 + \left(1-\frac{R_s}{r}\right)^{-1}\mathrm{d}r^2 + r^2\mathrm{d}\Omega^2 \qquad (2)

where: $r,t$ — Schwarzschild radial and time coordinates; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\varphi^2$ — the metric on the unit sphere. Reading: far from the mass ( $r\gg R_s$ ) it reduces to flat spacetime; the two factors $(1-R_s/r)$ in front of the time and radial parts are the "time warp" and "space warp" that will each contribute to the bend. The metric has two symmetries — it does not depend on $t$ or on the azimuth $\varphi$ — and each gives a conserved quantity along a geodesic. Their ratio fixes the impact parameter of the orbit.

E = \left(1-\frac{R_s}{r}\right)c^2\frac{\mathrm{d}t}{\mathrm{d}\lambda}, \qquad L = r^2\frac{\mathrm{d}\varphi}{\mathrm{d}\lambda}, \qquad b \equiv \frac{cL}{E} \qquad (3)

where: $E$ — conserved energy (time-translation symmetry); $L$ — conserved angular momentum (axial symmetry); $\lambda$ — affine parameter along the null geodesic; $b$ — impact parameter, which equals the geometric perpendicular distance of the incoming straight-line ray. Reading: for a photon only the ratio $b=cL/E$ matters — the affine parametrization and overall scale drop out, so a single number sets the whole trajectory. Feeding the conserved quantities into the null condition $\mathrm{d}s^2=0$ yields a first-order "orbit equation" for the inverse radius $u=1/r$ as a function of azimuth. Differentiating it once gives the cleaner second-order form used in the derivation below.

\left(\frac{\mathrm{d}u}{\mathrm{d}\varphi}\right)^2 = \frac{1}{b^2} - u^2\left(1 - R_s u\right), \qquad \frac{\mathrm{d}^2 u}{\mathrm{d}\varphi^2} + u = \frac{3}{2}R_s u^2 \qquad (4)

where: $u\equiv 1/r$ ; $\varphi$ — azimuthal angle in the orbital plane; the term $\tfrac{3}{2}R_s u^2$ is the purely relativistic source that drives the bending — set it to zero and the ray is a perfect straight line. Reading: without gravity the right side vanishes and $u''+u=0$ gives straight lines; the small cubic-in- $u$ correction is the entire effect, and it is what we treat perturbatively. The optical restatement is often the fastest route to the answer and makes the doubling manifest. Reading the coordinate speed of light off the weak-field metric defines an effective refractive index.

n(r) = 1 + \frac{2GM}{c^2 r} = 1 - \frac{2\Phi}{c^2} \qquad (5)

where: $n(r)$ — effective index of refraction for light in the weak field; $\Phi=-GM/r$ — the Newtonian potential. Reading: both metric potentials contribute equally, so the coefficient is $2GM/c^2$ rather than the $GM/c^2$ a Newtonian scalar theory would give — the geometric origin of the factor of two, spelled out in Derivation 3. Finally, to compare with experiment one generalizes to the parametrized post-Newtonian (PPN) form, in which a single parameter $\gamma$ measures how much spatial curvature a metric theory of gravity actually produces.

\Delta\phi = \frac{1+\gamma}{2}\cdot\frac{4GM}{c^2 b}, \qquad \gamma_{\rm GR}=1 \qquad (6)

where: $\gamma$ — the PPN parameter multiplying the space-space curvature; $\gamma=1$ in general relativity, $\gamma=0$ in a theory with no spatial curvature. Reading: measuring the deflection is a direct measurement of $\gamma$ ; the full GR value ( $\gamma=1$ ) doubles the deflection relative to the pure-time-curvature ( $\gamma=0$ ) case.

Derivation ABending angle from the null geodesic equation

The goal is to obtain the headline result $\Delta\phi=4GM/(c^2b)$ directly from the Schwarzschild orbit equation, treating the relativistic term as a small perturbation on straight-line motion. We start from $u''+u=\tfrac32 R_s u^2$ , solve the unperturbed straight ray, feed it back as a source for a first-order correction, and then ask where the corrected ray reaches infinity. The angular gap between the two asymptotes, minus the $\pi$ of a straight line, is the deflection. Every step is elementary once the perturbative bookkeeping is set up, and the answer is exact to first order in the tiny number $R_s/b$ .

Step 1. Solve the unperturbed equation $u_0''+u_0=0$ with $u_0\to 0$ (i.e. $r\to\infty$ ) at $\varphi=0$ and $\varphi=\pi$ : the solution is $u_0(\varphi)=\dfrac{\sin\varphi}{b}$ .

u_0(\varphi) = \frac{\sin\varphi}{b}

Why: this is the polar equation of a straight line whose perpendicular distance from the origin is $b$ , confirming that in flat space the impact parameter is exactly $b$ and the ray sweeps from $\varphi=0$ to $\varphi=\pi$ — a bend of zero.

Step 2. Insert $u_0$ into the relativistic source term to get the equation for the first-order correction $u_1$ , using $\sin^2\varphi$ as the driving term.

u_1'' + u_1 = \frac{3}{2}R_s\,u_0^2 = \frac{3R_s}{2b^2}\sin^2\varphi

Why: because $R_s/b$ is of order $10^{-6}$ for the Sun, the correction $u_1$ is genuinely small and it is legitimate to evaluate the source using the zeroth-order ray — this linearization is the whole approximation.

Step 3. Solve for $u_1$ . Writing $\sin^2\varphi=\tfrac12(1-\cos2\varphi)$ and solving term by term gives a particular solution that can be repackaged neatly.

u_1(\varphi) = \frac{R_s}{2b^2}\left(1 + \cos^2\varphi\right)

Why: direct substitution confirms it — $u_1''+u_1=\tfrac{R_s}{2b^2}(-2\cos2\varphi + 1 + \cos^2\varphi)$ collapses, using $\cos2\varphi=2\cos^2\varphi-1$ , to $\tfrac{3R_s}{2b^2}\sin^2\varphi$ , matching the source exactly.

Step 4. Add the two orders to get the full trajectory to first order in $R_s$ .

u(\varphi) = \frac{\sin\varphi}{b} + \frac{R_s}{2b^2}\left(1 + \cos^2\varphi\right)

Why: the ray is now the straight line plus a small inward pull; the correction is largest near closest approach ( $\varphi=\pi/2$ ) and pushes the asymptotes off the values $0$ and $\pi$ .

Step 5. Find the asymptotes by setting $u=0$ (the ray at infinity). This gives a small-angle condition for the two directions in which the photon escapes.

u=0 \;\Rightarrow\; \sin\varphi \approx -\frac{R_s}{2b}\left(1+\cos^2\varphi\right)

Why: the ray starts and ends at $r=\infty$ , so it hits $u=0$ twice; without the correction these were at $\varphi=0,\pi$ , and the small right-hand side shifts each by a tiny angle $\epsilon$ .

Step 6. Solve for the shift near $\varphi=0$ , where $\cos^2\varphi\approx1$ and $\sin\varphi\approx\varphi$ .

-\epsilon \approx -\frac{R_s}{2b}(1+1) \;\Rightarrow\; \epsilon = \frac{R_s}{b}

Why: the ray enters not at $\varphi=0$ but at $\varphi=-\epsilon$ , and by the symmetry of the trajectory it exits at $\varphi=\pi+\epsilon$ ; each end tilts by the same $\epsilon$ .

Step 7. The total angle swept is $\pi+2\epsilon$ ; the excess over the straight-line $\pi$ is the deflection.

\Delta\phi = 2\epsilon = \frac{2R_s}{b} = \frac{4GM}{c^2 b}

Why: the ray bends by $\epsilon$ on the way in and $\epsilon$ on the way out, and the two add — this is the physical meaning of the " $2$ " in $2R_s/b$ , and it lands on the famous $1.75''$ at $b=R_\odot$ .

Go deeper · Derivation B · 1 minThe factor of two from the optical analogy

Derivation BThe factor of two from the optical analogy

The goal is to reproduce $\Delta\phi=4GM/(c^2b)$ with no geodesic machinery at all, by treating the weak gravitational field as a medium with a position-dependent refractive index and integrating the transverse deflection a ray accumulates as it flies past. This route isolates where the factor of two comes from: it sits entirely in the coefficient of the index, $n-1=2GM/(c^2r)$ , and the Newtonian $n-1=GM/(c^2r)$ would give exactly half. The calculation is a single standard integral once the geometry is set up along the nearly straight path.

Step 1. Take the effective index from equation (5) and parametrize the almost-straight ray by a coordinate $z$ along its undeflected direction, with fixed transverse offset $b$ , so that $r=\sqrt{b^2+z^2}$ .

n(r) = 1 + \frac{2GM}{c^2 r}, \qquad r=\sqrt{b^2+z^2}

Why: for a ray that barely bends, evaluating the index along the straight path is the same first-order approximation used in the geodesic derivation — the error is higher order in $R_s/b$ .

Step 2. A ray in a slowly varying index turns toward higher $n$ at a rate given by the transverse gradient of $\ln n\approx n-1$ ; the total bend is the integral of that gradient along the path.

\Delta\phi = \int_{-\infty}^{\infty} \frac{\partial}{\partial b}\!\left(\frac{2GM}{c^2 r}\right)\mathrm{d}z = \int_{-\infty}^{\infty}\frac{2GM}{c^2}\,\frac{b}{(b^2+z^2)^{3/2}}\,\mathrm{d}z

Why: this is Fermat's principle in the small-angle limit — each slab of medium deflects the ray by the transverse gradient of its optical thickness, and the contributions accumulate additively along the trajectory.

Step 3. Evaluate the elementary integral using the antiderivative $z/(b\sqrt{b^2+z^2})$ .

\int_{-\infty}^{\infty}\frac{b\,\mathrm{d}z}{(b^2+z^2)^{3/2}} = \left[\frac{z}{b\sqrt{b^2+z^2}}\right]_{-\infty}^{\infty} = \frac{1}{b}-\left(-\frac{1}{b}\right)=\frac{2}{b}

Why: the two limits each contribute $1/b$ with the same sign, so the symmetric fly-by doubles the single-sided value — a geometric doubling on top of, and independent from, the metric doubling.

Step 4. Combine the prefactor with the integral to recover the geodesic result.

\Delta\phi = \frac{2GM}{c^2}\cdot\frac{2}{b} = \frac{4GM}{c^2 b}

Why: the answer is identical to the null-geodesic calculation, and the manifest presence of $2GM/c^2$ rather than $GM/c^2$ shows the factor of two lives in the space-plus-time index — a Newtonian corpuscle would land on $2GM/(c^2b)$ .

Go deeper · Derivation C · 1 minThe effective refractive index of the weak field

Derivation CThe effective refractive index of the weak field

The goal is to justify equation (5) — that the weak field acts optically like a medium of index $n=1+2GM/(c^2r)$ — by reading the coordinate speed of light directly off the metric, and thereby to expose the physical difference between a slow planet and a fast photon. The key is that the metric warps time and space by the same potential $\Phi$ , and while a slow body responds essentially only to the time warp, light at speed $c$ weights the two equally. This is the cleanest statement of why the deflection is exactly twice Newton, and it takes only a first-order expansion of two square roots.

Step 1. Write the isotropic weak-field metric and impose the null condition $\mathrm{d}s^2=0$ for light, with $\Phi=-GM/r$ and $\mathrm{d}\ell$ the spatial line element.

0 = -\left(1+\frac{2\Phi}{c^2}\right)c^2\mathrm{d}t^2 + \left(1-\frac{2\Phi}{c^2}\right)\mathrm{d}\ell^2

Why: light follows null paths, so setting the interval to zero directly relates elapsed coordinate time to distance travelled — this is exactly what a "speed of light in a medium" means.

Step 2. Solve for the coordinate speed $\mathrm{d}\ell/\mathrm{d}t$ and expand both square roots to first order in the small quantity $\Phi/c^2$ .

\frac{\mathrm{d}\ell}{\mathrm{d}t} = c\,\frac{\sqrt{1+2\Phi/c^2}}{\sqrt{1-2\Phi/c^2}} \approx c\left(1+\frac{2\Phi}{c^2}\right)

Why: the time factor contributes $+\Phi/c^2$ and the space factor contributes another $+\Phi/c^2$ ; the two equal pieces are the "time warp" and "space warp," and their sum is what doubles the effect.

Step 3. Define the index as $n=c/v$ with $v=\mathrm{d}\ell/\mathrm{d}t$ and substitute $\Phi=-GM/r$ .

n = \frac{c}{v} \approx 1 - \frac{2\Phi}{c^2} = 1 + \frac{2GM}{c^2 r}

Why: had only the time warp been present — the Newtonian case — the coefficient would be $GM/c^2$ and the deflection would come out at half; the second, spatial contribution is the entire relativistic surplus, and it is precisely what a photon at $c$ is fast enough to feel.

Limiting cases

The single formula $\Delta\phi=4GM/(c^2b)$ is the weak-field leading term; the table shows how it changes as one dials the theory or the field strength.

Case	Result	Meaning
Newtonian / equivalence-principle only	$\Delta\phi_{\rm N}=\dfrac{2GM}{c^2 b}$	Light as a corpuscle accelerated by $g_{00}$ alone — half the GR value; $0.875''$ at the solar limb. This is what the 1919 test ruled out.
Weak-field GR (Solar System)	$\Delta\phi=\dfrac{4GM}{c^2 b}=\dfrac{2R_s}{b}$	Leading order, valid because $R_s/R_\odot\sim4\times10^{-6}$ ; gives $1.75''$ at the limb.
PPN generalization	$\Delta\phi=\dfrac{1+\gamma}{2}\,\dfrac{4GM}{c^2 b}$	GR has $\gamma=1$ ; a pure scalar (no spatial curvature) has $\gamma=0$ and halves the deflection.
Strong field / near photon sphere	$\Delta\phi\to\infty$ as $b\to b_c=\dfrac{3\sqrt3}{2}R_s\approx2.60\,R_s$	As the impact parameter approaches the critical value $b_c=3\sqrt3\,GM/c^2$ the ray winds many times around the photon sphere; the linear formula fails completely and one gets logarithmically divergent looping.

Common trapNewton predicts no bending because light has no mass

This is the single most common misconception, and it inverts the history. Newtonian gravity, applied to light either as a stream of corpuscles or via the equivalence principle, predicts a perfectly definite, nonzero deflection of $2GM/(c^2b)=0.875''$ at the solar limb — Soldner computed essentially this value in 1801, and Einstein's own 1911 paper (before he had the full field equations) reproduced it. The 1919 eclipse was therefore never a test of "bending versus no bending." It was a test of the amount: half ( $0.875''$ , the pure time-warp value) versus full ( $1.75''$ , time plus space warp). General relativity's real prediction is the doubling, not the existence of the effect. A separate trap: the quoted $1.75''$ is specifically for a ray grazing the limb ( $b=R_\odot$ ); rays at larger impact parameter are bent less, as $1/b$ , and $b$ must not be conflated with the coordinate radius $R_\odot$ or the radius of closest approach $r_0$ (they agree only to leading order).

Worked examples

1 · The solar limb deflection (fully worked)

A ray of starlight grazes the Sun at the limb, $b=R_\odot$ . Compute the deflection in arcseconds. Use $M_\odot=1.989\times10^{30}\,\mathrm{kg}$ , $R_\odot=6.96\times10^{8}\,\mathrm{m}$ , $G=6.674\times10^{-11}$ , $c=2.998\times10^{8}\,\mathrm{m/s}$ .

Step 1. Form the Sun's Schwarzschild radius, which packages the mass into a length: $R_s=2GM_\odot/c^2=2(6.674\times10^{-11})(1.989\times10^{30})/(2.998\times10^{8})^2=\mathbf{2.95\,km}$ .

Step 2. The deflection is $\Delta\phi=2R_s/R_\odot$ in radians: $\Delta\phi=2(2.95\times10^{3})/(6.96\times10^{8})=\mathbf{8.49\times10^{-6}\,rad}$ . This is the dimensionless bend — note how tiny $R_s/R_\odot\approx4\times10^{-6}$ is, which is why the weak-field formula is essentially exact here.

Step 3. Convert to arcseconds with $1\,\mathrm{rad}=206265''$ : $\Delta\phi=8.49\times10^{-6}\times206265''=\mathbf{1.75''}$ . The Newtonian-only value is half this, $0.875''$ . The GR prediction is 1.75 arcseconds; a ray at $b=10R_\odot$ would be bent only $0.175''$ , since $\Delta\phi\propto1/b$ .

2 · Deflection by Jupiter's limb (fully worked)

Radio interferometry can measure the bending of a background quasar by planets. Compute the deflection for a ray grazing Jupiter's limb. Use $M_{\rm J}=1.898\times10^{27}\,\mathrm{kg}$ , $R_{\rm J}=7.15\times10^{7}\,\mathrm{m}$ .

Step 1. Jupiter's Schwarzschild radius: $R_s=2GM_{\rm J}/c^2=2(6.674\times10^{-11})(1.898\times10^{27})/(2.998\times10^{8})^2=\mathbf{2.82\,m}$ . Jupiter is far less compact than the Sun, so its $R_s$ is metres, not kilometres.

Step 2. Deflection in radians: $\Delta\phi=2R_s/R_{\rm J}=2(2.82)/(7.15\times10^{7})=\mathbf{7.88\times10^{-8}\,rad}$ .

Step 3. In arcseconds: $7.88\times10^{-8}\times206265''=0.0163''=\mathbf{16.3\,mas}$ (milliarcseconds). This ~16 mas grazing deflection was actually measured by Fomalont and Kopeikin in 2002 using VLBI as Jupiter passed near a quasar — a modern, non-eclipse confirmation of the same $4GM/c^2b$ law.

3 · Turning a deflection measurement into a bound on γ

Suppose an experiment measures the solar-limb deflection to a fractional precision of $1\%$ . What does that imply for the PPN parameter $\gamma$ ?

Step 1. From equation (6), $\Delta\phi/\Delta\phi_{\rm GR}=\tfrac12(1+\gamma)$ , so $\gamma=2(\Delta\phi/\Delta\phi_{\rm GR})-1$ . A measurement of the ratio to $\pm1\%$ propagates to $\Delta\gamma=2\times0.01=\mathbf{0.02}$ .

Step 2. So a $1\%$ deflection measurement constrains $\gamma$ to $\pm0.02$ — already enough to exclude many alternative theories. The Cassini Shapiro-delay experiment did enormously better, reaching $\gamma-1=(2.1\pm2.3)\times10^{-5}$ , i.e. a deflection consistent with GR to about one part in $10^{5}$ , because timing a radio signal's delay is far more precise than imaging a star's shift.

Physics of the portrait

Light bending is the effect that made Einstein a household name, and its story runs straight through his own two predictions — the Newtonian half in 1911 and the relativistic full value in 1915 — to the eclipse that decided between them.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · the 1911 and 1915 predictions of starlight deflection

Einstein's road to $1.75''$ was not a single stroke but a two-act drama that nearly ended in embarrassment. In 1911, working only from the equivalence principle and the gravitational redshift — before he had a theory of curved space — he predicted that starlight grazing the Sun would be deflected by $0.83''$ . This is the Newtonian value, the same $2GM/(c^2b)$ that Soldner had found in 1801, and Einstein even urged astronomers to test it at the next eclipse. Had the 1912 or 1914 eclipse expeditions succeeded (they were rained out or interrupted by the outbreak of war), they would have confirmed a number he would soon disown. Between 1911 and 1915 Einstein built general relativity, and in November 1915 the completed field equations gave both the time and the space curvature — doubling the deflection to $1.75''$ . When Dyson and Eddington announced in November 1919 that the eclipse data favoured the full relativistic value, the result crossed from physics into legend: newspapers ran it on the front page, and Einstein became the first celebrity scientist of the modern age. The near-miss of the war-cancelled expeditions is one of physics' great pieces of luck — the theory was allowed to make its real prediction before it was tested.

History

1801

— Soldner computes a Newtonian starlight deflection of about 0.84 arcsec

1911

— Einstein predicts about 0.83 arcsec from the equivalence principle alone

1915

— the completed field equations double the prediction to 1.75 arcsec

1919

— Dyson and Eddington measure the eclipse deflection, favouring Einstein

2003

— Cassini pins the PPN parameter to gamma minus one near two parts in a hundred thousand

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §40.3 (Light Deflection) · Wald, General Relativity, §6.3 · Carroll, Spacetime and Geometry, §7.5 · Hartle, Gravity, §9.4 · Schutz, A First Course in General Relativity, §11.2 · Will, Theory and Experiment in Gravitational Physics (PPN tests and the living review). The particular solution $u_1=\tfrac{R_s}{2b^2}(1+\cos^2\varphi)$ can be confirmed by direct substitution into the orbit equation. The solar limb value $1.75''$ follows from $R_s=2GM_\odot/c^2\approx2.95\,\mathrm{km}$ , $R_\odot=6.96\times10^{8}\,\mathrm{m}$ , and $1\,\mathrm{rad}=206265''$ ; the Jupiter limb deflection $\approx16\,\mathrm{mas}$ follows from $R_{s,{\rm J}}\approx2.82\,\mathrm{m}$ . The Cassini bound is $\gamma-1=(2.1\pm2.3)\times10^{-5}$ (Bertotti, Iess & Tortora 2003); the strong-field critical impact parameter is $b_c=3\sqrt3\,GM/c^2=\tfrac{3\sqrt3}{2}R_s\approx2.60\,R_s$ , and the optical integral evaluates to $\int b\,\mathrm{d}z/(b^2+z^2)^{3/2}=2/b$ .

Prerequisites. Schwarzschild Geodesics and the Effective Potential · Weak-Field Limit and Linearized Gravity · Newtonian Limit of General Relativity. Leads to. Shapiro Time Delay · Parametrized Post-Newtonian (PPN) Formalism · PPN Parameters Beta and Gamma.

Ask the tutor: Ask me to walk through why a photon feels exactly twice the Newtonian deflection, or to derive the $1.75''$ from the null geodesic step by step.

Why is the deflection exactly twice the Newtonian value, not some other factor?

Because the Schwarzschild metric warps time and space by the same potential $\Phi$ , and light at speed $c$ weights the two contributions equally. A slow planet spends almost all its "motion" in the time direction, so it feels essentially only the time-time part $g_{00}$ — that is Newtonian gravity, worth $2GM/(c^2b)$ . A photon, moving as fast through space as through time, feels the space-space part $g_{ij}$ just as strongly, and that adds an equal $2GM/(c^2b)$ . Two equal halves make the full $4GM/(c^2b)$ . In the optical language, the refractive index is $1+2GM/(c^2r)$ with a "2," not a "1"; the extra unit is the spatial curvature.

Did Newton's theory really predict light bending?

Yes — a nonzero, specific amount. If light is a stream of corpuscles falling in a gravitational field, or if one simply applies the equivalence principle, Newtonian gravity gives $2GM/(c^2b)=0.875''$ at the solar limb. Soldner published essentially this figure in 1801, and Einstein's own 1911 paper reproduced it before he had general relativity. So the famous eclipse was not testing whether light bends, but whether it bends by the Newtonian half or the relativistic full value. The popular story that "Newton says light goes straight" is simply wrong; light has energy, energy gravitates, and even the pre-relativistic theory knew that.

What exactly is the impact parameter b, and is it the same as the closest approach?

The impact parameter $b$ is the perpendicular distance from the mass to the incoming straight-line asymptote — where the ray would have passed if there were no gravity. Formally it is the ratio of conserved angular momentum to energy, $b=cL/E$ . The radius of closest approach $r_0$ is the smallest $r$ the real, bent ray actually reaches. Because gravity pulls the ray inward, $r_0$ is slightly smaller than $b$ , but the difference is of order $R_s$ and vanishes to leading order — for the Sun they agree to about a part in $10^{6}$ . In the $1.75''$ formula one uses $b=R_\odot$ , and the distinction only matters at second order.

How can a two-arcsecond shift possibly have been measured in 1919?

With great difficulty and honest controversy. Two arcseconds is comparable to the atmospheric "seeing" that blurs stars, so the expeditions photographed faint stars around the Sun during the few minutes of totality and compared their positions to night-time reference plates of the same field. The Sobral and Príncipe teams had to control for temperature-induced changes in their telescopes, plate scale, and refraction. The results scattered, and one Sobral instrument gave a poor value that Eddington set aside — a decision later critics questioned. Modern re-analysis of the original plates confirms the teams' conclusion, and the effect has since been measured thousands of times more precisely, so the 1919 result, though marginal, pointed the right way.

Why does radio interferometry beat the eclipse method today?

Three reasons. First, radio waves from compact quasars pass through the daytime sky and the solar corona, so no eclipse is needed and observations can be made whenever a source lies near the Sun's path. Second, Very Long Baseline Interferometry achieves angular resolution of microarcseconds — thousands of times finer than optical astrometry through the atmosphere — by combining telescopes across continents. Third, one can observe a source repeatedly as its line of sight sweeps past the Sun, mapping the full $1/b$ dependence rather than a single grazing value. The upshot is that VLBI, and the closely related Cassini timing experiment, constrain the deflection to parts in $10^{5}$ rather than the tens of percent of 1919.

Is gravitational lensing the same physics as the solar deflection?

Fundamentally, yes — it is the identical $4GM/(c^2b)$ law applied to more mass and larger distances. When the deflector is a galaxy or a cluster instead of the Sun, the deflection is large enough to produce multiple images, giant arcs, and Einstein rings; the ring radius $\theta_E=\sqrt{4GM D_{LS}/(c^2 D_L D_S)}$ is built directly from the deflection formula with the source, lens, and observer distances folded in. Weak lensing is the same effect statistically, gently distorting distant galaxy shapes. So mapping dark matter with lensing and weighing $\gamma$ with the Sun are two applications of one equation.