Illustrated Physics
physics, drawn
Concept

Deflection angle — where it appears

The total turning a trajectory suffers in passing a centre of force, taken between its two straight asymptotes far away. It depends on the impact parameter and the speed, and the way it does so is what a scattering cross-section is computed from.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit.

The bend Newton got half right

A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry.

astrophysics · Light deflection Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator.

The ray that bends without a surface

Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

optics · Fermat

Named alongside it

The objects these essays reach for when they reach for this one.

Fermat's principleRefractive indexConservation lawsEquivalence principleFocal lengthGeodesicGradientImpact parameterOptical path lengthScale heightSnell's lawSpacetime curvature

All concepts