This paper presents Chapter VI (section 6) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.
The paragraphs of original material are accompanied by brief comments in color, based on the present understanding. Feedback on these comments is appreciated.
The heading below links to the original materials.
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Time Geometry
Einstein’s law of gravitation controls a geometrical quantity curvature in contrast to Newton’s law which controls a mechanical quantity force. To understand the origin of this geometrisation of the world in the relativity theory we must go back a little.
The curvature of Einstein refers to twisting of substance, which involves force at microscopic level. So curvature cannot be divorced from force.
The science which deals with the properties of space is called geometry. Hitherto geometry has not included time in its scope. But now space and time are so interlocked that there must be one science—a somewhat extended geometry—embracing them both. Three-dimensional space is only a section cut through four-dimensional space-time, and moreover sections cut in different directions form the spaces of different observers. We can scarcely maintain that the study of a section cut in one special direction is the proper subject-matter of geometry and that the study of slightly different sections belongs to an altogether different science. Hence the geometry of the world is now considered to include time as well as space. Let us follow up the geometry of time.
Space and time are not self-identified entities. They are characteristics of substance, namely, extension and duration, and that is how they are related.
You will remember that although space and time are mixed up there is an absolute distinction between a spatial and a temporal relation of two events. Three events will form a space-triangle if the three sides correspond to spatial relations—if the three events are absolutely elsewhere with respect to one another. (This would be an instantaneous space-triangle. An enduring triangle is a kind of four-dimensional prism.) Three events will form a time-triangle if the three sides correspond to temporal relations—if the three events are absolutely before or after one another. (It is possible also to have mixed triangles with two sides time-like and one space-like, or vice versa.) A well-known law of the space-triangle is that any two sides are together greater than the third side. There is an analogous, but significantly different, law for the time-triangle, viz. two of the sides (not any two sides) are together less than the third side. It is difficult to picture such a triangle but that is the actual fact.
Time is asserted by a sequence of changes. It is absolute from the universal viewpoint. It appears relative only from local viewpoints.
Let us be quite sure that we grasp the precise meaning of these geometrical propositions. Take first the space-triangle. The proposition refers to the lengths of the sides, and it is well to recall my imaginary discussion with two students as to how lengths are to be measured (p. 23). Happily there is no ambiguity now, because the triangle of three events determines a plane section of the world, and it is only for that mode of section that the triangle is purely spatial. The proposition then expresses that “If you measure with a scale from A to B and from B to C the sum of your readings will be greater than the reading obtained by measuring with a scale from A to C.”
For a time-triangle the measurements must be made with an instrument which can measure time, and the proposition then expresses that “If you measure with a clock from A to B and from B to C the sum of your readings will be less than the reading obtained by measuring with a clock from A to C.”
In order to measure from an event A to an event B with a clock you must make an adjustment of the clock analogous to orienting a scale along the line AB. What is this analogous adjustment? The purpose in either case is to bring both A and B into the immediate neighbourhood of the scale or clock. For the clock that means that after experiencing the event A it must travel with the appropriate velocity needed to reach the locality of B just at the moment that B happens. Thus the velocity of the clock is prescribed. One further point should be noticed. After measuring with a scale from A to B you can turn your scale round and measure from B to A, obtaining the same result. But you cannot turn a clock round, i.e. make it go backwards in time. That is important because it decides which two sides are less than the third side. If you choose the wrong pair the enunciation of the time proposition refers to an impossible kind of measurement and becomes meaningless.
Dependence of space and time on the motion of the observer is subjective only. Space and time can be seen as directly related to the quantization and inertia of substance. This view is objective and absolute.
You remember the traveller (p. 39) who went off to a distant star and returned absurdly young. He was a clock measuring two sides of a time-triangle. He recorded less time than the stay-at-home observer who was a clock measuring the third side. Need I defend my calling him a clock? We are all of us clocks whose faces tell the passing years. This comparison was simply an example of the geometrical proposition about time-triangles (which in turn is a particular case of Einstein’s law of longest track). The result is quite explicable in the ordinary mechanical way. All the particles in the traveller’s body increase in mass on account of his high velocity according to the law already discussed and verified by experiment. This renders them more sluggish, and the traveller lives more slowly according to terrestrial time-reckoning. However, the fact that the result is reasonable and explicable does not render it the less true as a proposition of time geometry.
The proposed time geometry has not been verified at all scales. A body’s inertia balances it natural motion. A material body can never travel at the speed of light because of its inertia.
Our extension of geometry to include time as well as space will not be a simple addition of an extra dimension to Euclidean geometry, because the time propositions, though analogous, are not identical with those which Euclid has given us for space alone. Actually the difference between time geometry and space geometry is not very profound, and the mathematician easily glides over it by a discrete use of the symbol √-1. We still call (rather loosely) the extended geometry Euclidean; or, if it is necessary to emphasise the distinction, we call it hyperbolic geometry. The term non-Euclidean geometry refers to a more profound change, viz. that involved in the curvature of space and time by which we now represent the phenomenon of gravitation. We start with Euclidean geometry of space, and modify it in a comparatively simple manner when the time-dimension is added; but that still leaves gravitation to be reckoned with, and wherever gravitational effects are observable it is an indication that the extended Euclidean geometry is not quite exact, and the true geometry is a non-Euclidean one—appropriate to a curved region as Euclidean geometry is to a flat region.
Euclidean geometry is valid only for the space and time corresponding to the inertia of material-substance. It is not valid for the space and time corresponding to the quantization levels of field-substance. That is represented by non-Euclidean geometry.
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