, etc., are equally justifiable for the description of the general laws underlying the phenomena of physical nature, since it is, in general, not possible to make use of rigid bodies of reference for space-time descriptions of events in the manner of the special theory of relativity. Using Gaussian co-ordinates, i.e. labelling each point in space with four arbitrary numbers in the way specified above (three of these correspond to three space dimensions and one to time), the general principle of relativity may be enunciated thus:—
All Gaussian four-dimensional co-ordinate-systems are equally applicable for formulating the general laws of physics. This carries the principle of relativity, i.e. of equivalence of systems, to an extreme limit.
With regard to the relativity of rotations, it may be briefly mentioned that centrifugal forces can, according to the general theory of relativity, be due only to the presence of other bodies. This will be better understood by imagining an isolated body poised in space; there could be no meaning in saying that it rotated, for there would be nothing to which such a rotation could be referred: classical mechanics however, asserts that, in spite of the absence of other bodies, centrifugal forces would manifest themselves: this is denied by the general theory of relativity. No experimental test has hitherto been devised which could be carried out practically to give a decision in favour of either theory.
A favourable opportunity for detecting the slight curvature of light rays (which is predicted by the general theory) when passing in close vicinity to the sun occurred during the total eclipse of the 29th May, 1919. The results, which were made public at the meeting of the Royal Society on 6th November following, were reported as confirming the theory.
In addition to the slight motion of Mercury's perihelion, there is still a third test which is based upon a shift of the spectral line towards the infra-red, as a result of an application of Doppler's principle; this has not yet led to a conclusive experimental result.
I. NOTE ON NON-EUCLIDEAN GEOMETRY
In practical geometry we do not actually deal with straight lines, but only with distances, i.e. with finite parts of straight lines, yet we feel irresistibly impelled to form some conception of the parts of a straight line which vanish into inconceivably distant regions. We are accustomed to imagining that a straight line may be produced to an infinite distance in either direction, yet in our mathematical reasoning we find that in order to preserve consistency (in Euclid),[22] we may only allocate to this straight line one point at infinity: we say that two straight lines are parallel when they cut at a point at infinity, i.e. this point is at an .infinite distance from an arbitrary starting-point on either straight line, and is reached by moving forwards or backwards on either.
[22]According to the modern analytical interpretation of Euclid.
Many attempts have been made, without success, to deduce Euclid's "axiom of parallels," which asserts that only one straight line can be drawn parallel to another straight line through a point outside the latter, from the other axioms. It finally came to be recognized that this axiom of parallels was an unnecessary assumption, and that one could quite well build up other geometries by making other equally justified assumptions.