“DECEMBER 28, 1916] 
NATURE 329 
Until recently the application of the principle of 
relativity was limited to one particular transforma- 
tion, namely, a uniform translation of the axes. 
In this case there is a wide range of experimental 
evidence in support of the principle. In 1915 
Prof. A. Einstein* finally succeeded in developing 
the complete theory by which the postulate of 
relativity can be satisfied for all transformations 
of the co-ordinates. Gravitation plays a part of 
great importance in the new theory, and therein 
lies much of the practical interest of Einstein’s 
work. 
No attempt is made to explain the cause of 
gravitation—as a kink in space or anything of 
that nature. But the extended law of gravitation 
is determined, to which Newton’s law is an ap- 
proximation under ordinary conditions. It has 
long been suspected that there must be some 
modification of the law when the bodies concerned 
are in rapid relative motion; moreover, the 
“mass” of a moving body no longer has a unique 
meaning, so that a further definition, if not exten- 
sion, of Newton’s law is clearly needed. Now, 
although we do not seek a cause of gravitation in 
the properties of space, it may well happen that 
the law of gravitation is determined by these pro- 
perties. The inverse-square law represents the 
natural weakening of an effect through spreading 
out in three dimensions; we may say that it is 
determined by the properties of Euclidean space. 
There is, therefore, nothing unreasonable in pro- 
ceeding, as Einstein does, to examine whether a 
more extended law is suggested by the properties 
- of generalised space—that is, by geometry. 
The way in which gravitation enters into the 
discussion may be seen from the following 
example. Suppose an observer is in a closed lift; 
let the supports break and the lift fall freely. To 
the observer everything in the lift will now appear 
to be without weight; gravity has been suddenly 
annihilated. The acceleration of his frame of 
reference (the lift) is equivalent to an alteration 
of the gravitational field. Now an acceleration of 
the axes is one of the transformations contem- 
plated by the general principle of relativity, and it 
is therefore necessary to allow that the gravitational 
field depends on the choice of co-ordinates. There 
-is a “local” gravity, just as there is a “local” 
time or magnetic field depending on the co- 
ordinates selected. 
We can now take a brief survey of Einstein’s 
procedure. Suppose that space and time are 
measured by a system of co-ordinates x, Xo, %3, 
x4; X4 is the time, but there is no need to dis- 
criminate between it and the others. If there is a 
gravitational field at any point, it can be abolished 
(as in the pores of the att) by choosing new co- 
ordinates x,!, x5/, x/, x4! accelerated with respect 
to the old. The necessary transformation could 
be specified in various ways; the way chosen in- 
volves the element of length ds, measured by co- 
incidences with a standard scale, and therefore 
1 Einstein, “Die Grundlage der allgemeinen Relativitdtstheorie.” 
(Leipzig: J. A. Barth, 1016.) A detailed account appears in Monthly 
Notices, 1916, No. 9, by Prof. W. de Sitter, giving the astronomical applica- 
tions. See also an article by de Sitter in the Odserwatory, October, 1916. 
NO. 2461, VOL. 98] 
independent of the choice of co-ordinates. Since in 
the x/ co-ordinates there is no gravitation to com- 
plicate matters, we can safely use the usual 
formula, 
as" = dy? + avy? + 
and this when transformed to the old co-ordinates 
takes the most general form, 
AS = AVY? + Lov + 
The ten g’s depend on the transformation, and can 
be used to specify it. But they do more than 
that; they define the original gravitational field, 
since they specify how it can be got rid of. They 
usually vary from point to point, because a 
different transformation is needed to eliminate 
gravity at different points. In the new theory the 
g’s are regarded as ten gravitational potentials 
specifying the field; and, in fact, one of them, 
£44, iS approximately the same as the Newtonian 
potential ¢, except for a factor. 
The inverse-square law can be expressed by the 
well-known different'al equation y2¢=0.  Evi- 
dently the new law of gravitation must be a’ 
generalisation equation, or set of 
equations, involving the ten potentials instead of 
one. Also, to conform to relativity, the 
equations must be unaltered by a change of co- 
ordinates. If new co-ordinates are used the g’s 
will be different, but the relations between the 
new g’s and new co-ordinates must be the same 
as those between the old g’s and old co-ordinates. 
The possible sets of relations which satisfy this are 
very limited in number. This subject, known as 
the theory of tensors, has been worked out very 
fully by Riemann, Christoffel, and others, and the 
possible sets of equations can be classified and 
enumerated. From this limited choice we have 
further to pick out a set of equations which will 
reduce to V7g44=0 as a first approximation, and 
that is found to leave only one possibility. There 
is just one set of ten differential equations (of 
which, however, only six are independent) which 
satisfy both conditions. Einstein takes these as 
expressing his generalised law of gravitation. It 
is important to notice exactly how much of this 
is geometry. Geometry shows that if the equa- 
tions hold for a particular set of co-ordinates, 
they hold for every set. We abolish the “if,” and 
so assert a new jaw of Nature. 
It is further necessary to consider what must 
be the generalised equivalent of Poisson’s equa- 
tion y2¢= — 47p, which supplants Laplace’s equa- 
tion when matter is present. The extension is not 
difficult, since it is found that the ten equations 
above mentioned are the expression of a generalised 
principle of least action as applied to gravitational 
energy. Now mass is considered to be simply 
electromagnetic energy, and since there is no 
reason to believe that electromagnetic energy will 
behave differently from gravitational energy in 
regard to least action, we have only to include 
both forms of energy together in the equations, 
treating them as equivalent. It is not possible to 
write down here these final equations, since a very 
elaborate notation is needed for their expression. 
+ 29,904 AX y+ 28,01 AX, eres 
new - 
