 |
 |
 |
 |
|
 |
 |
59
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part II: The General Theory of Relativity
Exact Formulation of the General Principle of Relativity
We are now in a position to replace the pro. visional formulation of the general principle of relativity
given in
Section 18 by an exact formulation. The form there used, "All bodies of reference K, K¹,
etc., are equivalent for the description of natural phenomena (formulation of the general laws of
nature), whatever may be their state of motion," cannot be maintained, because the use of rigid
reference?bodies, in the sense of the method followed in the special theory of relativity, is in
general not possible in space?time description. The Gauss co?ordinate system has to take the
place of the body of reference. The following statement corresponds to the fundamental idea of the
general principle of relativity: "All Gaussian co?ordinate systems are essentially equivalent for the
formulation of the general laws of nature."
We can state this general principle of relativity in still another form, which renders it yet more clearly
intelligible than it is when in the form of the natural extension of the special principle of relativity.
According to the special theory of relativity, the equations which express the general laws of nature
pass over into equations of the same form when, by making use of the Lorentz transformation, we
replace the space?time variables x, y, z, t, of a (Galileian) reference?body K by the space?time
variables x¹, y¹, z¹, t¹, of a new reference?body K¹. According to the general theory of relativity, on
the other hand, by application of arbitrary substitutions of the Gauss variables x1, x2, x3, x
4
,
the
equations must pass over into equations of the same form; for every transformation (not only the
Lorentz transformation) corresponds to the transition of one Gauss co?ordinate system into
another.
If we desire to adhere to our "old?time" three?dimensional view of things, then we can characterise
the development which is being undergone by the fundamental idea of the general theory of
relativity as follows : The special theory of relativity has reference to Galileian domains, i.e. to those
in which no gravitational field exists. In this connection a Galileian reference?body serves as body
of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the
uniform rectilinear motion of "isolated" material points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian domains to
non?Galileian reference?bodies also. A gravitational field of a special kind is then present with
respect to these bodies (cf. Sections 20 and 23).
In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the
fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks
is also influenced by gravitational fields, and in such a way that a physical definition of time which is
made directly with the aid of clocks has by no means the same degree of plausibility as in the
special theory of relativity.
For this reason non?rigid reference?bodies are used, which are as a whole not only moving in any
way whatsoever, but which also suffer alterations in form ad lib. during their motion. Clocks, for
which the law of motion is of any kind, however irregular, serve for the definition of time. We have
to imagine each of these clocks fixed at a point on the non?rigid reference?body. These clocks
satisfy only the one condition, that the "readings" which are observed simultaneously on adjacent
clocks (in space) differ from each other by an indefinitely small amount. This non?rigid