Euclidean and Non-Euclidean Continuum
Section Twenty-Four
Why do we have to give up Euclidean geometry in general relativity? Einstein tries to give an answer in this section. Recall the results of the last section; i.e. that clocks and rods are affected in a gravitational field. In order to determine the geometry of our physical space (and time), we've got to rely on these means of measurement.
Now, let us illustrate the point of Einstein's example of a marble slab.
Thus if we have some means to correct our measurements, detecting the influence of some factor on rods and clocks, we may be able to correct the measurements so that we can save Euclidean geometry. But in a gravitational field, rods and clocks are universally affected; hence we have to decide that if rods are identical in physical constitution and agrees exactly at one place, they exress the same unit of length even if they are transported to different places, with or without a gravitational field, and likewise for clocks. For, we have no means to correct the supposed deformation due to gravity. And on this supposition, there is no guarantee that our measurements agree with Euclideqan geometry; rather, we have good reasons for expecting that geometry is non-Euclidean, since both time and space warp (according to our clocks and rods).
Getting back to the marble slab, if our measurement tells that the width is 5 units in the upper part and less than 5 units in the central part, the width is different in these places; and if this makes the geometry non-Euclidean, the geometry is indeed non-Euclidean! Here are two examples of non-Euclidean continuum (2-dimensional curved surfaces, embedded in a 3-dimensional Euclidean space):
Don't say that light may help for correcting measurements; light is also affected in a gravitational field, as we have already seen in section 22. Einstein, at one time, used the change of the velocity of light as a measure for a gravitational field!
Later, Hans Reichenbach was to utilize a similar idea for defending his contention of conventionality in physical geometry. Reichenbach introduced the notion of "universal force", affecting everything in the same way. For this, and confusions and clarifications about this, see Gruenbaum on Reichenbach's "Universal Forces".
Last modified, May 21, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
