Absolute space in its own nature, without relation to anything external, remains always similar and immovablewhich suggests that vectors at different times could be compared.

That Newton would have said this seems prescient for those are the very points that Einstein had to discard in the presence of new physical evidence. Had Newton considered alternatives to these principles? The Greeks dealt with a timeless geometry and I don’t know whether they would have imagined comparing vectors at different times.

Ernst Mach thought that comparing vectors at different times was a murky concept. In particular he thought that an absolute direction in space was as bogus a concept as Galileo had shown an absolute place or even an absolute velocity to be. Few physicists were sympathetic with Mach in his day.

About 1860 Bernhard Riemann developed a purely mathematical theory of curved space, which was an elegant extension of Gauss’ theory of curved surfaces. On the surface of the Earth there is no satisfactory concept of comparing distant surface vectors. Saying that the wind is blowing the same direction in New York and Boston seems a reasonable claim, but saying the wind is in the same direction at very distant cities is murky. Riemann showed how to compare two vectors when a path between them is specified. Riemann thought that perhaps real 3D space was curved but he quoted no evidence.

Einstein produced special relativity in 1905 but already knew that the theory contradicted itself. He struggled to make it consistent and succeeded only when he adopted the idea that space and time together formed a unified four dimensional whole, and that that whole was curved. This required the apparatus that Riemann had developed for the general curvature of n-dimensional space. It was noted that vectors at different times could not be compared for equality and the very meaning of “a gyroscope points in a constant direction” was brought into doubt. W. de Sitter predicted that a gyroscope which is carried in a path about a massive object would turn in the plane of the path.

Soon Lens and shortly thereafter Thirring noted that according to general relativity a spinning body would make a nearby gyroscope turn relative to a distant gyroscope, as Mach had suspected. It is due to the very shape of space near the spinning object. Einstein noted that Mach’s suspicion had been justified.

Stanford’s gravity probe B is an experiment designed to verify both of these effects. There are scheduled to present some preliminary results this April (2007).

As of the above report the de Sitter effect has been verified but the Lens Thirring effect, being much smaller, is still buried in unexpected polhode motions of gyroscopes. They plan to model these motions in order to retrieve the Lens Thirring effect.