(3) _ds²_ = σ_{στ}_g__{στ}_dx__{σ}_dx__{τ}
where _g__{στ} will be functions of _x__{σ}, but will no longer depend
upon the orientation and motion of the ‘local’ co-ordinates; for _ds²_
is a definite magnitude belonging to two point-events infinitely near in
space and time and can be got by measurements with rods and clocks. The
_g__{τσ}’s are here to be so chosen, that _g__{τσ} = _g__{στ}; the
summation is to be extended over all values of σ and τ, so that the sum
is to be extended, over 4 × 4 terms, of which 12 are equal in pairs.
From the method adopted here, the case of the usual relativity theory
comes out when owing to the special behaviour of _g__{στ} in a _finite_
region it is possible to choose the system of co-ordinates in such a way
that _g__{στ} assumes constant values—
{ -1, 0, 0, 0
(4) { 0 -1 0 0
{ 0 0 -1 0
{ 0 0 0 +1
We would afterwards see that the choice of such a system of co-ordinates
for a finite region is in general not possible.
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.