A Lecture delivered before the Naturforscher Versammlung (Congress of
Natural Philosophers) at Cologne—(21st September, 1908).
Gentlemen,
The conceptions about time and space, which I hope to develop before you
to-day, has grown on experimental physical grounds. Herein lies its
strength. The tendency is radical. Henceforth, the old conception of
space for itself, and time for itself shall reduce to a mere shadow, and
some sort of union of the two will be found consistent with facts.
I
Now I want to show you how we can arrive at the changed concepts about
time and space from mechanics, as accepted now-a-days, from purely
mathematical considerations. The equations of Newtonian mechanics show a
twofold invariance, (_i_) their form remains unaltered when we subject
the fundamental space-coordinate system to any possible change of
position, (_ii_) when we change the system in its nature of motion, _i.
e._, when we impress upon it any uniform motion of translation, the
null-point of time plays no part. We are accustomed to look upon the
axioms of geometry as settled once for all, while we seldom have the
same amount of conviction regarding the axioms of mechanics, and
therefore the two invariants are seldom mentioned in the same breath.
Each one of these denotes a certain group of transformations for the
differential equations of mechanics. We look upon the existence of the
first group as a fundamental characteristics of space. We always prefer
to leave off the second group to itself, and with a light heart conclude
that we can never decide from physical considerations whether the space,
which is supposed to be at rest, may not finally be in uniform motion.
So these two groups lead quite separate existences besides each other.
Their totally heterogeneous character may scare us away from the attempt
to compound them. Yet it is the whole compounded group which as a whole
gives us occasion for thought.
We wish to picture to ourselves the whole relation graphically. Let
(_x_, _y_, _z_) be the rectangular coordinates of space, and _t_ denote
the time. Subjects of our perception are always connected with place and
time. _No one has observed a place except at a particular time, or has
observed a time except at a particular place._ Yet I respect the dogma
that time and space have independent existences. I will call a
space-point plus a time-point, _i.e._, a system of values _x_, _y_, _z_,
_t_, as a _world-point_. The manifoldness of all possible values of _x_,
_y_, _z_, _t_, will be the _world_. I can draw four world-axes with the
chalk. Now any axis drawn consists of quickly vibrating molecules, and
besides, takes part in all the journeys of the earth ; and therefore
gives us occasion for reflection. The greater abstraction required for
the four-axes does not cause the mathematician any trouble. In order not
to allow any yawning gap to exist, we shall suppose that at every place
and time, something perceptible exists. In order not to specify either
matter or electricity, we shall simply style these as substances. We
direct our attention to the _world-point_ _x_, _y_, _z_, _t_, and
suppose that we are in a position to recognise this substantial point at
any subsequent time. Let _dt_ be the time element corresponding to the
changes of space coordinates of this point [_dx_, _dy_, _dz_]. Then we
obtain (as a picture, so to speak, of the perennial life-career of the
substantial point),—a curve in the _world_—the _world-line_, the points
on which unambiguously correspond to the parameter _t_ from +∞ to -∞.
The whole world appears to be resolved in such _world-lines_, and I may
just deviate from my point if I say that according to my opinion the
physical laws would find their fullest expression as mutual relations
among these lines.
By this conception of time and space, the (_x_, _y_, _z_) manifoldness
_t_ = 0 and its two sides _t_ < 0 and _t_ > 0 falls asunder. If for the
sake of simplicity, we keep the null-point of time and space fixed, then
the first named group of mechanics signifies that at _t_ = 0 we can give
the _x_, _y_, and _z_-axes any possible rotation about the null-point
corresponding to the homogeneous linear transformation of the expression
_x²_ + _y²_ + _z²_.
The second group denotes that without changing the expression for the
mechanical laws, we can substitute (_x_ - α_t_, _y_ - β_t_, _z_ - γ_t_
for (_x_, _y_, _z_) where (α, β, γ) are any constants. According to this
we can give the time-axis any possible direction in the upper half of
the world _t_ > 0. Now what have the demands of orthogonality in space
to do with this perfect freedom of the time-axis towards the upper half?
To establish this connection, let us take a positive parameter c, and
let us consider the figure
_c²__t²_ - _x²_ - _y²_ - _z²_ = 1
According to the analogy of the hyperboloid of two sheets, this consists
of two sheets separated by _t_ = 0. Let us consider the sheet, in the
region of _t_ > 0, and let us now conceive the transformation of _x_,
_y_, _z_, _t_ in the new system of variables; (_x’_, _y’_, _z’_, _t’_)
by means of which the form of the expression will remain unaltered.
Clearly the rotation of space round the null-point belongs to this group
of transformations. Now we can have a full idea of the transformations
which we picture to ourselves from a particular transformation in which
(_y_, _z_) remain unaltered. Let us draw the cross section of the upper
sheets with the plane of the _x_- and _t_-axes, _i.e._, the upper half
of the hyperbola _c²__t²_ - x² = 1, with its asymptotes (_vide_ fig. 1).
Then let us draw the radius rector OA′, the tangent A′ B′ at A′, and let
us complete the parallelogram OA′ B′ C′; also produce B′ C′ to meet the
x-axis at D′. Let us now take Ox′, OA′ as new axes with the unit
measuring rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again
expressed in the form _c²__t′²_ - x′² = 1, t′ > 0 and the transition
from (_x_, _y_, _z_, _t_) to (_x′_ _y′_ _z′_ _t_) is one of the
transitions in question. Let us add to this characteristic
transformation any possible displacement of the space and time
null-points; then we get a group of transformation depending only on
_c_, which we may denote by G_{_c_}.
Now let us increase _c_ to infinity. Thus (1/c) becomes zero and it
appears from the figure that the hyperbola is gradually shrunk into the
_x_-axis, the asymptotic angle becomes a straight one, and every special
transformation in the limit changes in such a manner that the _t_-axis
can have any possible direction upwards, and _x′_ more and more
approximates to _x_. Remembering this point it is clear that the full
group belonging to Newtonian Mechanics is simply the group G_{_c_}, with
the value of _c_ = ∞. In this state of affairs, and since G_{_c_} is
mathematically more intelligible than G_{∞}, a mathematician may, by a
free play of imagination, hit upon the thought that natural phenomena
possess an invariance not only for the group G_{∞}, but in fact also for
a group G_{_c_}, where _c_ is finite, but yet exceedingly large compared
to the usual measuring units. Such a preconception would be an
extraordinary triumph for pure mathematics.
At the same time I shall remark for which value of _c_, this invariance
can be conclusively held to be true. _For c, we shall substitute the
velocity of light c in free space._ In order to avoid speaking either of
space or of vacuum, we may take this quantity as the ratio between the
electrostatic and electro-magnetic units of electricity.
We can form an idea of the invariant character of the expression for
natural laws for the group-transformation G_{_c_} in the following
manner.
Out of the totality of natural phenomena, we can, by successive higher
approximations, deduce a coordinate system (_x_, _y_, _z_, _t_); by
means of this coordinate system, we can represent the phenomena
according to definite laws. This system of reference is by no means
uniquely determined by the phenomena. _We can change the system of
reference in any possible manner corresponding to the above-mentioned
group transformation G_{c}, but the expressions for natural laws will
not be changed thereby._
For example, corresponding to the above described figure, we can call
_t′_ the time, but then necessarily the space connected with it must be
expressed by the manifoldness (_x′_ _y_ _z_). The physical laws are now
expressed by means of _x′_, _y_, _z_, _t′_,—and the expressions are just
the same as in the case of _x_, _y_, _z_, _t_. According to this, we
shall have in the world, not one space, but many spaces,—quite analogous
to the case that the three-dimensional space consists of an infinite
number of planes. The three-dimensional geometry will be a chapter of
four-dimensional physics. Now you perceive, why I said in the beginning
that time and space shall reduce to mere shadows and we shall have a
world complete in itself.