Sylvester and Clifford on Curved Space

Einstein realized that gravity is due to the curvature of spacetime, but let’s go back earlier:

On the 18th of August 1869, the eminent mathematician Sylvester gave a speech arguing that geometry is not separate from physics. He later published this speech in the journal Nature, and added a footnote raising the possibility that space is curved:

the laws of motion accepted as fact, suffice to prove in a general way that the space we live in is a flat or level space […], our existence therein being assimilable to the life of the bookworm in a flat page; but what if the page should be undergoing a process of gradual bending into a curved form?

Then, even more dramatically, he announced that the mathematician Clifford had been studying this!

Mr. W. K. Clifford has indulged in more remarkable speculations as the possibility of our being able to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space of three dimensions being in the act of undergoing in space of four dimensions (space as inconceivable to us as our space to the supposititious bookworm) a distortion analogous to the rumpling of the page.

This started a flame war in letters to Nature which the editor eventually shut off, saying “this correspondence must now cease”. Clifford later wrote about his theories in a famous short paper:

• William Clifford, On the space-theory of matter, Proceedings of the Cambridge Philosophical Society 2 (1856), 157–158.

It’s so short I can show you it in its entirety:

Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.

I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact

(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.

(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.

(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

I am endeavouring in a general way to explain the laws of double refraction on this hypothesis, but have not yet arrived at any results sufficiently decisive to be communicated.

To my surprise, the following paper argues that Clifford did experiments to test his ideas by measuring the polarization of the skylight during a solar eclipse in Sicily on December 22, 1870:

• S. Galindo and Jorge L. Cervantes-Cota, Clifford’s attempt to test his gravitation hypothesis.

Clifford did indeed go on such an expedition, and did indeed try to measure the polarization of skylight as the Moon passed the Sun. I don’t know of any record of him saying why he did it.

I’ll skip everything the above paper says about why the polarization of skylight was interesting and mysterious in the 1800s, and quote just a small bit:

The English Eclipse Expedition set off earlier in December 1870, on the steamship H.M.S. Psyche scheduled for a stopover at Naples before continuing to Syracuse in Sicily. Unfortunately before arriving to her final call, the ship struck rocks and was wrecked off Catania. Fortunately all instruments and members of the party were saved without injury.

Originally it was the intention of the expedition to establish in Syracuse their head-quarters, but in view of the wreckage the group set up their base camp at Catania. There the expedition split up into three groups. The group that included Clifford put up an observatory in Augusta near Catania. The leader of this group was William Grylls Adams, professor of Natural Philosophy at King’s College, London.

In a report written by Prof. Adams, describing the expedition, we learn that the day of the eclipse, just before the time of totality, “… a dense cloud came over the Moon and shut out the whole, so that it was doubtful whether the Moon or the clouds first eclipsed the Sun […] Mr. Clifford observed light polarized on the cloud to the right and left and over the Moon, in a horizontal plane through the Moon’s centre [….] It will be seen from Mr. Clifford’s observations that the plane of polarization by the cloud…was nearly at right angles to the motion of the Sun”.

As was to be expected, Clifford’s eclipse observations on polarization did not produce any result. His prime intention, of detecting angular changes of the polarization plane due to the curving of space by the Moon in its transit across the Sun´s disk, was not fulfilled. At most he confirmed the already known information, i.e. the skylight polarization plane moves at right angles to the Sun anti-Sun direction.

This is a remarkable prefiguring of Eddington’s later voyage to the West African island of Principe to measure the bending of starlight during an eclipse of the Sun in 1919. Just one of many stories in the amazing prehistory of general relativity!