Illustrated Physics
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Series

Light deflection — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · astrophysics
  2. A radar echo past the Sun, delayed by 233 microseconds. The extra time a round-trip radar signal takes when its path passes close to the Sun, against how close, for a reflector 0.723 AU away. Grazing the Sun's limb the delay is 233 microseconds — about 70 kilometres of light travel, on a path of hundreds of millions — and it falls only as the logarithm of the impact parameter, so the effect is still tens of microseconds ten solar radii out. That slow falloff is what makes the measurement possible: the delay can be watched building and fading as the geometry changes, rather than having to be caught at one instant. The dashed curves are the delay the solar corona's plasma adds at three radio frequencies. It is the competing effect, it is larger than the gravitational one close in, and it falls as the square of the frequency while the gravitational delay does not depend on frequency at all — which is how the two are separated, and why the sharpest measurement of this was made with a spacecraft carrying three radio links instead of one.

    The delay that is not a bend

    The same metric that bends a ray also slows it, and the two are different tests. A radar echo from Venus arrives 233 microseconds late when its path grazes the Sun — and half of that delay is accumulated more than twenty solar radii away, in a field thousands of times weaker.

    part 2 · astrophysics
  3. A lens with no focal length. Where a ray crosses the axis, against how far off the axis it passed the deflecting body, for 1 solar mass of radius 1 solar radius. A glass lens deflects a ray by an angle proportional to its distance off axis, which is precisely the condition for every ray to arrive at one point — the flat dashed line. Gravity deflects by 4GM/c²b, which grows smaller further out, so the crossing distance goes as b² and each ray has its own focus. The grazing ray crosses at 548 astronomical units and a ray passing at 12 radii crosses at 78857; the square law is verified on the drawn curve to 1.5e-16. So there is no image plane at all, only a half-line of foci beginning at the first of those and running outward for ever. Anything placed on that line sees not an image but a ring, and moving along it does not refocus anything — it selects which rays are being seen.

    The lens with no focal length

    A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.

    part 3 · astrophysics

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