General Relativity

1915 · Relativity · The companion to Plate II

How far the Sun bends starlight, worked out step by step.

A Demonstration is a plate’s companion print: the reasoning behind the discovery, worked out on a sheet of its own.

Paper ground, a warm cream, on Hahnemühle Photo Rag, 308 gsm cotton

The plate it accompanies

Plate II, General Relativity: Light can circle a black hole. See it

What it shows

Plate II shows light passing a black hole. The closer a ray comes, the more it bends, until it circles the hole or falls in. This sheet works out why.

It starts from one equation for light near a round mass, which comes from Karl Schwarzschild’s solution of Einstein’s equations. From that equation, the sheet finds the one circle light can follow, at one and a half times the black hole’s radius. It also finds how close a ray can be aimed before it is captured. That limit is the edge of the black hole’s shadow.

Then it turns to the Sun. If light fell like matter under Newton’s law, starlight grazing the Sun would bend by 0.876 arcseconds. Einstein’s equation gives twice that: 1.751 arcseconds.

The eclipse of 1919 put the two predictions to the test. Its measurements sit on the sheet beside today’s radio measurements, which agree with Einstein’s 1.751 arcseconds to within a ten-thousandth of an arcsecond.

Close up, at about the size it prints at Medium

Step 4 of the sheet: a hill-shaped curve crossed by horizontal levels for rays of different aim, the highest clearing its top. — Step 4. Each level line stands for one ray. A ray aimed closer than 2.6 times the black hole’s radius passes over the top of the hill and falls in.

Step 6: the path of a ray past the mass, with a magnified circle at its far end. — Step 6. Each end of the ray swings away from a straight line by the same small angle that Newton’s law gives for the whole bend. With both ends counted, the ray bends twice as much as Newton’s law predicts. The magnified corner shows one end.

The answer line in vermilion: at the Sun’s edge starlight turns 4GM over c squared R, 1.751 arcseconds, twice the Newtonian 0.876. — For starlight grazing the Sun, we arrive at a bend of 1.751 arcseconds, twice the 0.876 arcseconds that Newton’s law predicts.

Hang them together

Plate II

Demonstration II

Demonstration II is the companion to Plate II, General Relativity, which shows light bending round a mass. Hung side by side in the same size and frame, the two cost 10% less bought together.

Plate II and Demonstration II, Medium, 42 × 59 cm each, in black frames.

The sheet

Light passing a mass, from Schwarzschild’s solution, with the eclipse of 1919 and today’s radio measurements

1A straight line, with no mass
2The one circle light can follow
3What stays fixed along a ray
4The aim inside which light is captured
5Light under Newton’s law
6The turn, twice Newton’s
7At the Sun’s edge, and the eclipse of 1919
†And a question to take further: step 6’s correction, found rather than checked

Sizes and prices

Size	Unframed	Framed	Print
Medium42 × 59 cm	£95$145	£225$295	420 × 594 mm
Large59 × 84 cm	£160$215	£390$515	594 × 841 mm
Extra large84 × 119 cm	£285$385	Unframed only	841 × 1189 mm

Paper: Hahnemühle Photo Rag, 308 gsm cotton, printed as a giclée by theprintspace.
Printed in: London for the UK, Brooklyn for the US. Unframed prints travel rolled in a tube.
Framed medium: A2 frame, glazed with 3 mm acrylic.
Framed large: A1 frame, glazed with 3 mm acrylic.
Extra large: unframed only. A0 is a standard frame size.
Frames: solid wood, finished in black, white or oak veneer, made in Germany. Each comes ready to hang.
Edition: open, with a certificate. File version v1.