Essays in Science by Albert Einstein

method of the special theory of relativity is characterized by the following principle:—Only those equations are admissible as an expression of natural laws which do not change their form when the co-ordinates are changed by means of a Lorentz transformation (co-variance of equations in relation to Lorentz transformations).
    This method led to the discovery of the necessary connection between impulse and energy, the strength of an electric and a magnetic field, electrostatic and electro-dynamic forces, inert mass and energy; and the number of independent concepts and fundamental equations was thereby reduced.
    This method pointed beyond itself. Is it true that the equations which express natural laws are co-variant in relation to Lorentz transformations only and not in relation to other transformations? Well, formulated in that way the question really means nothing, since every system of equations can be expressed in general co-ordinates. We must ask, Are not the laws of nature so constituted that they receive no real simplification through the choice of any one particular set of co-ordinates?
    We will only mention in passing that our empirical principle of the equality of inert and heavy masses prompts us to answer this question in the affirmative. If we elevate the equivalence of all co-ordinate systems for the formulation of natural laws into a principle, we arrive at the general theory of relativity, provided we stick to the law of the constant velocity of light or to the hypothesis of the objective significance of the Euclidean metric at least for infinitely small portions of four-dimensional space.
    This means that for finite regions of space the existence (significant for physics) of a general Riemannian metric is presupposed according to the formula

     
    whereby the summation is to be extended to all index combinations from 11 to 44.
    The structure of such a space differs absolutely radically in one respect from that of a Euclidean space. The coefficients gμν are for the time being any functions whatever of the co-ordinates x 1 to x 4 , and the structure of the space is not really determined until these functions gμν are really known. It is only determined more closely by specifying laws which the metrical field of the gμν satisfy. On physical grounds this gave rise to the conviction that the metrical field was at the same time the gravitational field.
    Since the gravitational field is determined by the configuration of masses and changes with it, the geometric structure of this space is also dependent on physical factors. Thus according to this theory space is—exactly as Riemann guessed—no longer absolute; its structure depends on physical influences. Physical geometry is no longer an isolated self-contained science like the geometry of Euclid.
    The problem of gravitation was thus reduced to a mathematical problem: it was required to find the simplest fundamental equations which are co-variant in relation to any transformation of co-ordinates whatever.
    I will not speak here of the way this theory has been confirmed by experience, but explain at once why Theory could not rest permanently satisfied with this success. Gravitation had indeed been traced to the structure of space, but besides the gravitational field there is also the electro-magnetic field. This had, to begin with, to be introduced into the theory as an entity independent of gravitation. Additional terms which took account of the existence of the electromagnetic field had to be included in the fundamental equations for the field. But the idea that there were two structures of space independent of each other, the metric-gravitational and the electro-magnetic, was intolerable to the theoretical spirit. We are forced to the belief that both sorts of field must correspond to verified structure of space.
    The “unitary field-theory,” which represents itself as a mathematically independent extension of the general theory of relativity, attempts