General Relativity: Light Bending & Curved Spacetime

Equivalent forms

einstein ring radius

\theta_E = \sqrt{\frac{4GM}{c^{2}}\,\frac{d_{LS}}{d_L d_S}}

newtonian half value

\theta_{\text{Newton}} = \frac{2GM}{c^{2}b}

gw strain scale

h \sim \frac{2G\ddot{Q}}{c^{4}d}

One factor of 2 — space curvature added to time curvature — separated Newton from Einstein, and a 1919 eclipse decided it.

Symbol	Name	SI unit	Dimension	Range
$\theta$	deflection angleoutput	rad	1	—
$G$	gravitational constant	6.674\times10^{-11} N·m²/kg²	M^{-1} L^{3} T^{-2}	—
$M$	mass of the deflecting body	kg	M	—
$c$	speed of light	2.998\times10^{8} m/s	L T^{-1}	—
$b$	impact parameter (closest approach of the ray)	m	L	—

Discovery

Albert Einstein · Arthur Eddington · 1919

Einstein predicted in 1915 that the Sun bends grazing starlight by 1.75″ — double the naive Newtonian value, because space itself curves, not just trajectories in time. On 29 May 1919 Eddington photographed stars near the eclipsed Sun from the island of Príncipe. The measured shift matched Einstein, headlines declared Newton overthrown, and Einstein became the most famous scientist alive overnight. The same physics now finds exoplanets by microlensing and weighs dark matter by how it lenses galaxies.

Real-world applications

Gravitational lensing: galaxy clusters image background galaxies as arcs and Einstein rings — our best scale for weighing dark matter.
Microlensing surveys (OGLE, Roman telescope) detect exoplanets when a star + planet briefly magnify a background star.
GPS would drift $\sim 10\,\mathrm{km/day}$ without general-relativistic clock corrections (gravitational time dilation, the same curvature of time).
LIGO/Virgo detect gravitational waves — curvature ripples that stretch their 4 km arms by 1/10,000 the width of a proton.
Event Horizon Telescope photographed the photon ring around M87* and Sgr A*, light orbiting $at \sim 1.5$ Schwarzschild radii.

Common misconceptions

“Gravity pulls on the photon's mass.” — Photons are massless; they travel on geodesics of curved spacetime. That's why Newton's value is half the observed one.
“Gravitational waves are light.” — They are oscillations of geometry itself, propagating at c but stretching space transversely as they pass.
“Black holes suck.” — Far away, a black hole's gravity is identical to a star of equal mass; only inside $\sim 1.5\,\mathrm{Rs}$ does light begin to be trapped.

Experimental verification

1919 Príncipe/Sobral eclipse

(1.75''

confirmed); VLBI radio measurements now verify the deflection to 0.01%; Hipparcos/Gaia must correct every stellar position for it; LIGO's GW150914 (2015) matched the predicted binary-black-hole chirp waveform; EHT imaged the predicted photon ring (2019, 2022).

Derivation

In the Schwarzschild metric, a light ray with impact parameter b obeys $d^{2}u/d\varphi ^{2} + u = 3GM u^{2}/c^{2}$ with $u = 1/r$ .
Zeroth order (no mass): a straight line $u = \sin (\varphi )/b$ .
Treat the $3GMu^{2}/c^{2}$ term as a perturbation and integrate over the trajectory.
Each half of the path picks up $2GM/(c^{2}b)$ ; the total bend $is \theta = 4GM/(c^{2}b)$ .
Newtonian ballistics for a particle at speed c gives only $2GM/(c^{2}b)$ — time curvature alone. Space curvature doubles it: the famous factor of 2.

Limiting cases

b ≫ R_s⟶

\theta = 4GM/c^{2}b

≪ 1Weak-field lensing — the regime of all astronomical lensing surveys.

b \to (3\sqrt{3}/2) R_s

⟶

\theta \to \infty

Photon capture: light winds around the photon sphere — the edge of a black hole's shadow.

What if…

What if light passed a black hole at

b \approx 2.6\,\mathrm{GM/c}^{2} \times 2

?

Below the critical impact parameter $b_c = 3\sqrt{3} GM/c^{2} \approx 2.6\,\mathrm{R}_s$ , the photon spirals into the photon sphere and is captured — that's the dark shadow the Event Horizon Telescope photographed.

What if the Sun collapsed to a black hole?

Earth's orbit wouldn't change at all — same mass, same external curvature. Only within a few kilometers of the new horizon would GR effects become extreme.

1

Starlight grazing the Sun

Given ·

M = 1.989\times 10^{30} kg

,

b = R_sun = 6.96\times 10^{8} m

Find · Deflection angle

\theta in

arcseconds

Steps

$\theta = 4GM/(c^{2}b) = 4 \times 6.674\times 10^{-11} \times 1.989\times 10^{30} / ((2.998\times 10^{8})^{2} \times 6.96\times 10^{8})$
$\theta = 5.31\times 10^{20} / 6.256\times 10^{25} = 8.49\times 10^{-6} rad$
Convert: $8.49\times 10^{-6} \times 206265$ arcsec/ $rad \approx 1.75''$ .

Result ·

\theta \approx 1.75

arcseconds — the 1919 eclipse number that made Einstein famous.

2

Einstein ring of a lensing galaxy

Given · Lens mass

10^{12}

solar masses, lens at

d_L = 1\,\mathrm{Gpc}

, source at

d_S = 2\,\mathrm{Gpc}

(so

d_LS \approx 1\,\mathrm{Gpc})

Find · Einstein angular radius

\theta _E

Steps

$\theta _E = \surd (4GM/c^{2} \times d_LS/(d_L d_S))$
$4GM/c^{2} = 4 \times 6.674\times 10^{-11} \times 1.989\times 10^{42} / 8.99\times 10^{16} \approx 5.9\times 10^{15} m$
$d_LS/(d_L d_S) = 3.086\times 10^{25} / (3.086\times 10^{25} \times 6.17\times 10^{25}) = 3.24\times 10^{-26} m^{-1}$
$\theta _E = \sqrt{5.9\times 10^{15} \times 3.24\times 10^{-26}} = \sqrt{1.9\times 10^{-10}} \approx 1.4\times 10^{-5} rad \approx 2.9''$ .

Result ·

\theta _E \approx 3

arcseconds — the size of the luminous arcs seen around massive galaxy clusters.

Schwarzschild Radius→Gravitational Time Dilation→Gravitational Redshift→Hawking Temperature→Spacetime Interval→