The Principle of Relativity · Albert Einstein
V
From the considerations in § 2 and § 3 it is clear, that from the physical stand-point the quantities _g__{στ} are to be looked upon as magnitudes which describe the gravitation-field with reference to the chosen system of axes. We assume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for some particular choice of co-ordinates. The _g__{στ}’s then have the values given in (4). A free material point moves with reference to such a system uniformly in a straight-line. If we now introduce, by any substitution, the space-time co-ordinates _x₁_..._x₄_ then in the new system _g__{μν}’s are no longer constants, but functions of space and time. At the same time, the motion of a free point-mass in the new co-ordinates, will appear as curvilinear, and not uniform, in which the law of motion, will be _independent of the nature of the moving mass-points_. We can thus signify this motion as one under the influence of a gravitation field. We see that the appearance of a gravitation-field is connected with space-time variability of _g__{στ}’s. In the general case, we can not by any suitable choice of axes, make special relativity theory valid throughout any finite region. We thus deduce the conception that _g__{στ}’s describe the gravitational field. According to the general relativity theory, gravitation thus plays an exceptional rôle as distinguished from the others, specially the electromagnetic forces, in as much as the 10 functions _g__{στ} representing gravitation, define immediately the metrical properties of the four-dimensional region.
B Mathematical Auxiliaries for Establishing the General Covariant Equations. We have seen before that the general relativity-postulate leads to the condition that the system of equations for Physics, must be co-variants for any possible substitution of co-ordinates _x₁_, ... _x₄_; we have now to see how such general co-variant equations can be obtained. We shall now turn our attention to these purely mathematical propositions. It will be shown that in the solution, the invariant _ds_, given in equation (3) plays a fundamental rôle, which we, following Gauss’s Theory of Surfaces, style as the line-element. The fundamental idea of the general co-variant theory is this:—With reference to any co-ordinate system, let certain things (tensors) be defined by a number of functions of co-ordinates which are called the components of the tensor. There are now certain rules according to which the components can be calculated in a new system of co-ordinates, when these are known for the original system, and when the transformation connecting the two systems is known. The things herefrom designated as “Tensors” have further the property that the transformation equation of their components are linear and homogeneous; so that all the components in the new system vanish if they are all zero in the original system. Thus a law of Nature can be formulated by putting all the components of a tensor equal to zero so that it is a general co-variant equation; thus while we seek the laws of formation of the tensors, we also reach the means of establishing general co-variant laws.
This can be true, for any choice of B^{στ′} only when the term within the bracket vanishes. From which by referring to (11), the theorem at once follows. This law correspondingly holds for tensors of any rank and character. The proof is quite similar. The law can also be put in the following form. If B^{μ} and C^{ν} are any two vectors, and if for every choice of them the inner product A_{μν} B^{μ} C^{ν} is a scalar, then A_{μν} is a co-variant tensor. The last law holds even when there is the more special formulation, that with any arbitrary choice of the four-vector B^{μ} alone the scalar product A_{μν} B^{μ} B^{ν} is a scalar, in which case we have the additional condition that A_{μν} satisfies the symmetry condition. According to the method given above, we prove the tensor character of (A_{μν} + A_{νμ}), from which on account of symmetry follows the tensor-character of A_{μν}. This law can easily be generalized in the case of co-variant and contravariant tensors of any rank. Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind of tensor: If the quantities A_{μν} B^{ν} form a tensor of the first rank, when B^{ν} is any arbitrarily chosen four-vector, then A_{μν} is a tensor of the second rank. If for example, C^{μ} is any four-vector, then owing to the tensor character of A_{μν} B^{ν}, the inner product A_{μν} C^{μ} B^{ν} is a scalar, both the four-vectors C^{μ} and B^{ν} being arbitrarily chosen. Hence the proposition follows at once.
Ideas discussed in this chapter
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