Now I want to show you how we can arrive at the changed concepts about time and space from mechanics, as accepted now-a-days, from purely mathematical considerations. The equations of Newtonian mechanics show a twofold invariance, (_i_) their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, (_ii_) when we change the system in its nature of motion, _i. e._, when we impress upon it any uniform motion of translation, the null-point of time plays no part. We are accustomed to look upon the axioms of geometry as settled once for all, while we seldom have the same amount of conviction regarding the axioms of mechanics, and therefore the two invariants are seldom mentioned in the same breath. Each one of these denotes a certain group of transformations for the differential equations of mechanics. We look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off the second group to itself, and with a light heart conclude that we can never decide from physical considerations whether the space, which is supposed to be at rest, may not finally be in uniform motion. So these two groups lead quite separate existences besides each other. Their totally heterogeneous character may scare us away from the attempt to compound them. Yet it is the whole compounded group which as a whole gives us occasion for thought.
We wish to picture to ourselves the whole relation graphically. Let (_x_, _y_, _z_) be the rectangular coordinates of space, and _t_ denote the time. Subjects of our perception are always connected with place and time. _No one has observed a place except at a particular time, or has observed a time except at a particular place._ Yet I respect the dogma that time and space have independent existences. I will call a space-point plus a time-point, _i.e._, a system of values _x_, _y_, _z_, _t_, as a _world-point_. The manifoldness of all possible values of _x_, _y_, _z_, _t_, will be the _world_. I can draw four world-axes with the chalk. Now any axis drawn consists of quickly vibrating molecules, and besides, takes part in all the journeys of the earth ; and therefore gives us occasion for reflection. The greater abstraction required for the four-axes does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to specify either matter or electricity, we shall simply style these as substances. We direct our attention to the _world-point_ _x_, _y_, _z_, _t_, and suppose that we are in a position to recognise this substantial point at any subsequent time. Let _dt_ be the time element corresponding to the changes of space coordinates of this point [_dx_, _dy_, _dz_]. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point),—a curve in the _world_—the _world-line_, the points on which unambiguously correspond to the parameter _t_ from +∞ to -∞. The whole world appears to be resolved in such _world-lines_, and I may just deviate from my point if I say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines.
By this conception of time and space, the (_x_, _y_, _z_) manifoldness _t_ = 0 and its two sides _t_ < 0 and _t_ > 0 falls asunder. If for the sake of simplicity, we keep the null-point of time and space fixed, then the first named group of mechanics signifies that at _t_ = 0 we can give the _x_, _y_, and _z_-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression
_x²_ + _y²_ + _z²_.
The second group denotes that without changing the expression for the mechanical laws, we can substitute (_x_ - α_t_, _y_ - β_t_, _z_ - γ_t_ for (_x_, _y_, _z_) where (α, β, γ) are any constants. According to this we can give the time-axis any possible direction in the upper half of the world _t_ > 0. Now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half?
To establish this connection, let us take a positive parameter c, and let us consider the figure
_c²__t²_ - _x²_ - _y²_ - _z²_ = 1
According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by _t_ = 0. Let us consider the sheet, in the region of _t_ > 0, and let us now conceive the transformation of _x_, _y_, _z_, _t_ in the new system of variables; (_x’_, _y’_, _z’_, _t’_) by means of which the form of the expression will remain unaltered. Clearly the rotation of space round the null-point belongs to this group of transformations. Now we can have a full idea of the transformations which we picture to ourselves from a particular transformation in which (_y_, _z_) remain unaltered. Let us draw the cross section of the upper sheets with the plane of the _x_- and _t_-axes, _i.e._, the upper half of the hyperbola _c²__t²_ - x² = 1, with its asymptotes (_vide_ fig. 1).
Then let us draw the radius rector OA′, the tangent A′ B′ at A′, and let us complete the parallelogram OA′ B′ C′; also produce B′ C′ to meet the x-axis at D′. Let us now take Ox′, OA′ as new axes with the unit measuring rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again expressed in the form _c²__t′²_ - x′² = 1, t′ > 0 and the transition from (_x_, _y_, _z_, _t_) to (_x′_ _y′_ _z′_ _t_) is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points; then we get a group of transformation depending only on _c_, which we may denote by G_{_c_}.
Now let us increase _c_ to infinity. Thus (1/c) becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the _x_-axis, the asymptotic angle becomes a straight one, and every special transformation in the limit changes in such a manner that the _t_-axis can have any possible direction upwards, and _x′_ more and more approximates to _x_. Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group G_{_c_}, with the value of _c_ = ∞. In this state of affairs, and since G_{_c_} is mathematically more intelligible than G_{∞}, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G_{∞}, but in fact also for a group G_{_c_}, where _c_ is finite, but yet exceedingly large compared to the usual measuring units. Such a preconception would be an extraordinary triumph for pure mathematics.
At the same time I shall remark for which value of _c_, this invariance can be conclusively held to be true. _For c, we shall substitute the velocity of light c in free space._ In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity.
We can form an idea of the invariant character of the expression for natural laws for the group-transformation G_{_c_} in the following manner.
Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (_x_, _y_, _z_, _t_); by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely determined by the phenomena. _We can change the system of reference in any possible manner corresponding to the above-mentioned group transformation G_{c}, but the expressions for natural laws will not be changed thereby._
For example, corresponding to the above described figure, we can call _t′_ the time, but then necessarily the space connected with it must be expressed by the manifoldness (_x′_ _y_ _z_). The physical laws are now expressed by means of _x′_, _y_, _z_, _t′_,—and the expressions are just the same as in the case of _x_, _y_, _z_, _t_. According to this, we shall have in the world, not one space, but many spaces,—quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning that time and space shall reduce to mere shadows and we shall have a world complete in itself.
