Arthur Eddington

“ We ought to construct our geometry in such a way as to show that there are intermediate steps, and that the comparison of the interval with the ultimate standard is not a kind of action at a distance.
To compare intervals in different directions at a point in space and time does not require this comparison with a distant standard. The physicist's method of describing phenomena near a point is to lay down for comparison (1) a mesh-system, (2) a unit of length (some kind of material standard) , which can also be used for measuring time, the velocity of light being unity. ”

“ We ought now to seek the law of the electromagnetic field on the same lines as we sought for the law of gravitation, laying down the condition that it must be independent of mesh-system and gauge-system since it seeks to limit the possible kinds of world which can exist in nature. ”

“ If he is in flat space there will be no difficulty; he will have to proceed by steps, a kind of triangulation, but the route chosen is of no importance. We know definitely that there is just one direction at parallel to the original direction at ; and it is in ordinary geometry supposed that the length is equally determinate. But if space is not flat the case is different. Imagine a two-dimensional observer confined to the curved surface of the earth trying to perform this task. As he does not appreciate the third dimension he will not immediately perceive the impossibility ”

“ An angle is commonly represented as a pure number, but it has not really this character; an angle can only be measured in terms of a unit of angle, just as a length is measured in terms of a unit of length. ”

“ Anything which behaves exactly like electricity must manifest itself to us as electricity. Distinction of form is the only distinction that physics can recognise; and distinction of individuality, if it has any meaning at all, has no bearing on physical manifestations. ”

“ The gravitational field is the expression of some state of the world, which also manifests itself in the natural geometry determined with measuring appliances; the electric field must also express some state of the world, but we have not as yet connected it with natural geometry. May there not still be unnecessary assumptions to be removed, so that a yet more comprehensive geometry can be found, in which gravitational and electrical fields both have their place? ”

“ The connection between the form of the law of gravitation and the total amount of matter in the world now appears less mysterious. The curvature of space indirectly provides the gauge which we use for measuring the amount of matter in the world. ”

“ We can only explore the world with apparatus, which is itself part of the world. Our idealised apparatus is reduced to a few simple types—a neutral particle, a charged particle, a rigid scale, etc. The absolute constituents of the world are related in various ways, which we have studied, to the indications of these test-bodies. The main features of the absolute world are so simple that there is a redundancy of apparatus at our disposal ”

“ The relativity theory deduces from geometrical principles the existence of gravitation and the laws of mechanics of matter. Mechanics is derived from geometry, not by adding arbitrary hypotheses, but by removing unnecessary assumptions, so that a geometer like Riemann might almost have foreseen the more important features of the actual world. ”

“ How, for instance, can the change in the length of a rod taken round a circuit in space and time be responsible for the sensations of an electric shock? The geometrical potentials () obey the recognised laws of electromagnetic potentials, and each entity in the physical theory—charge, electric force, magnetic element, light, etc.—has its exact analogue in the geometrical theory; but is this formal correspondence a sufficient ground for identification? The doubt which arises in our minds is due to a failure to recognise the formalism of all physical knowledge. ”

“ But the three-dimensional space in which a curved two-dimensional space is conceived to exist is quite arbitrary. A two-dimensional observer cannot ascertain by any observation whether he is on a plane or a cylinder, a sphere or any other convex surface of the same total curvature. ”

“ Presumably, we could obtain a unified mechanical conception by taking the moving uncharged particle as standard indicator; or a unified electrical conception by taking the charged particle. For particular purposes one test-body is generally better adapted than others. The gravitational field is more sensitively explored with a moving particle than a scale. Although the electrical field can theoretically be explored by the change of length of a scale taken round a circuit, a far more sensitive way is to use a little bit of the scale—an electron. ”

“ Actual rods as they are moved about change their lengths compared with this absolute unit according to the route taken, and the differences correspond to the electromagnetic field. Einstein's curved space appears in a perfectly natural manner in this theory; no part of space-time is flat, even in the absence of ordinary matter, for that would mean infinite radius of curvature, and there would be no natural gauge to determine, for example, the dimensions of an electron—the electron could not know how large it ought to be, unless it had something to measure itself against. ”

“ Actually we prefer to look at the world as revealed by exploration with scales and clocks—the former for measuring so-called imaginary intervals, and the latter for real intervals; this gives us a unified geometrical conception of the world. ”

“ Experiment has isolated what are believed to be units of action, which at least in many phenomena behave as indivisible atoms called quanta; but the theory, as at present developed, does not permit us to represent the quantum of action by the number 1. The quantum is a very minute fraction of the absolute unit. ”

“ The 's refer to the arbitrary gauge-system of the observer; but he cannot by altering his gauge-system alter their values entirely at will. Intrinsic properties of the world are contained in their values, unaffected by any change of gauge-system. Further we may expect that they will have to satisfy some law corresponding to the law of gravitation, so that not all arbitrary values of the 's are such as can occur in nature. ”

“ And in general for practical efficiency, we do not use any simple type of apparatus, but a complicated construction built up with a view to a particular experiment. The reason for emphasising the theoretical interchangeability of test-bodies is that it brings out the unity and simplicity of the world ”