Essays in Science by Albert Einstein Page A

to fulfil this last postulate of the field theory. The formal problem should be put as follows:—Is there a theory of the continuum in which a new structural element appears side by side with the metric such that it forms a single whole together with the metric? If so, what are the simplest field laws to which such a continuum can be made subject? And finally, are these field-laws well fitted to represent the properties of the gravitational field and the electromagnetic field? Then there is the further question whether the corpuscles (electrons and protons) can be regarded as positions of particularly dense fields, whose movements are determined by the field equations. At present there is only one way of answering the first three questions. The space structure on which it is based may be described as follows, and the description applies equally to a space of any number of dimensions.
    Space has a Riemannian metric. This means that the Euclidean geometry holds good in the infinitesimal neighborhood of every point P. Thus for the neighborhood of every point P there is a local Cartesian system of co-ordinates, in reference to which the metric is calculated according to the Pythagorean theorem. If we now imagine the length I cut off from the positive axes of these local systems, we get the orthogonal “local n-leg.” Such a local n-leg is to be found in every other point P’ of space also. Thus, if a linear element (PG or P’G’) starting from the points P or P’, is given, then the magnitude of this linear element can be calculated by the aid of the relevant local n-leg from its local co-ordinates by means of Pythagoras’s theorem. There is therefore a definite meaning in speaking of the numerical equality of the linear elements PG and P’G’.
    It is essential to observe now that the local orthogonal n-legs are not completely determined by the metric. For we can still select the orientation of the n-legs perfectly freely without causing any alteration in the result of calculating the size of the linear elements according to Pythagoras’s theorem. A corollary of this is that in a space whose structure consists exclusively of a Riemannian metric, two linear elements PG and P’G’, can be compared with regard to their magnitude but not their direction; in particular, there is no sort of point in saying that the two linear elements are parallel to one another. In this respect, therefore, the purely metrical (Riemannian) space is less rich in structure than the Euclidean.
    Since we are looking for a space which exceeds Riemannian space in wealth of structure, the obvious thing is to enrich Riemannian space by adding the relation of direction or parallelism. Therefore for every direction through P let there be a definite direction through P’, and let this mutual relation be a determinate one. We call the directions thus related to each other “parallel.” Let this parallel relation further fulfil the condition of angular uniformity: If PG and PK are two directions in P, P’G’ and P’K’ the corresponding parallel directions through P’, then the angles KPG and K’P’G’ (measurable on Euclidean lines in the local system) should be equal.
    The basic space-structure is thereby completely defined. It is most easily described mathematically as follows:—In the definite point P we suppose an orthogonal n-leg with definite, freely chosen orientation. In every other point P’ of space we so orient its local n-leg that its axes are parallel to the corresponding axes at the point P. Given the above structure of space and free choice in the orientation of the n-leg at one point P, all n-legs are thereby completely defined. In the space P let us now imagine any Gaussian system of co-ordinates and that in every point the axes of the n-leg there are projected on to it. This system of n2 components completely describes the structure of space.
    This spatial structure stands, in a sense, midway between the Riemannian and