Space and Time (Prasad)

by Hermann Minkowski

For other English-language translations of this work, see Space and Time (Hermann Minkowski).

Space and Time.

Address delivered at Cologne on Sept. 21st, 1908, by the Late Prof. H. Minkowski of Göttingen.

[Translated by Ganesh Prasad.]

Gentlemen,

The views on space and time which I propose to expound to you have grown in the field of Experimental Physics. It is in this that their strength lies. Their tendency is a radical one. From this moment, space by itself and time by itself shall sink in the background, and only a certain union of these two shall retain substantiality.

I should like first to explain how, starting from the generally accepted Mechanics, one may, with the help of purely mathematical considerations, arrive at altered ideas about space and time. The equations of Newtonian Mechanics show a two-fold invariance. Their form remains unchanged first, if one subjects the underlying system of space co-ordinates to an arbitrary change of position; secondly, if one imposes any uniform translation on this system. Also the zero-point of time does not play any part. People are accustomed to look upon the axioms of Geometry as settled before they feel themselves ripe for the study of the axioms of Mechanics; for this reason, these two invariances are rarely mentioned in one breath. Each of them signifies a certain group of transformations in itself for the differential equations of Mechanics. The existence of the first group is looked upon as a fundamental attribute of space. On the other hand, people delight in punishing the second group with scorn, in order to thoughtlessly pass by it to the conclusion, that it is impossible to decide from the physical phenomena whether the space, which is supposed to be at rest, is not really in a state of uniform motion. Thus the two groups follow a separate career side by side. Their thoroughly dissimilar character may have frightened people from any attempt at combining them; but it is exactly the combination of these groups which gives us a good deal to meditate upon.

Let us try to represent the situation graphically. Let $x,y,z$ be the rectangular co-ordinates for space and let $t$ denote time. Space and time combined always form the subject of our perception. No one has noticed a place except at a time, and a time except at a place. However, I respect the dogma that space and time have each an independent significance. I will use the word world-point for a space-point corresponding to a time-point, i.e., for a system $x,y,z,t$. The totality of all conceivable systems $x,y,z,t$ may be called the world. I could boldly chalk out four world-axes on the board. Already one of the drawn axes consists of a number of vibrating molecules and, further, makes along with the Earth a voyage in All. Thus this axis already gives us enough to reflect upon; the somewhat greater reflection connected with the axis No. 4 does not do any harm to the mathematician. In order to leave nowhere a gaping void, we imagine to ourselves that something perceptible is existent at all places and at every moment. In order to avoid using the words matter or electricity, I will use the word substance for this "some thing." Let us direct our attention towards the substantial point, existent in the world-point $x,y,z,t$, and let us imagine to ourselves that we are in a position to recognize this substantial point at any other time. The changes $dz,dy,dz$ in the space co-ordinates of this substantial point may correspond to an element of time $dt$. Thus, as the picture – so to say — of the eternal life of the substantial point, we obtain a curve in the world, i.e., a world-line whose points admit of a one-to-one correspondence with the parameter $t$ from $-\infty$ to $\infty$. The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines.

In consequence of the notions, space and time, the $x,y,z$ totality $t=0$ and its two flanks $t>O$ and $t<0$ fall asunder. If, for the sake of simplicity, we keep the zero-point of space and time fixed, then the first group of Mechanics means that, corresponding to the homogeneous linear transformations of the expression

$x^{2}+y^{2}+z^{2}$

into itself, we may subject the $x,y,z$-axes in $t=O$ to an arbitrary rotation round the zero-point. But the second group means that, without having to alter the mechanical laws, we may also replace $x,y,z,t$ by $x-\alpha t,\,y-\beta t,\,z-\gamma t,t$, where $\alpha ,\beta ,\gamma$ are arbitrary constants. After this, the axis of time may be given a fully arbitrary direction towards the upper half-world $t>0$. Now, what has the demand of orthogonality in space to do with this complete freedom of the axis of time upwards?

To establish the connexion, let us take a positive parameter $c$ and consider the figure

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=1.$

Fig. 1

Like a hyperboloid of two sheets, it consists of two sheets separated by $t=0$. We consider the sheet in the region $t>0$, and we conceive now those homogeneous linear transformations of $x,y,z,t$ into four new variables $x'$, $y'$, $z'$, $t'$, by which the expression of these sheets in the new variables becomes similar. Evidently the rotations of space about the zero-point belong to these transformations. A complete understanding of the remainder of these transformations is acquired, if we fix our eyes upon such of them as leave $x$ and $z$ unaltered. Let us trace (Fig. 1) the section of these sheets with the plane of the $x$- and $t$-axes, viz., the upper branch of the hyperbole $c^{2}t^{2}-x^{2}=1$, together with its asymptotes. Further, let us mark an arbitrary radius vector $OA'$ of this hyperbolic branch from the zero-point $O$; lay down the tangent at $A'$ to the hyperbola up to $B'$, the point of intersection with the right-hand side asymptote; complete the parallelogram $OA'B'C'$; and, finally, produce $B'$ $C'$ to $D'$, its point of intersection with the $x$-axis. If we take now $OC'$ and $OA'$ as axes for the parallel co-ordinates $x'$, $t'$ with the scales $OC'=1$, $OA'={\tfrac {1}{c}}$, then the hyperbolic branch is again expressed by

$c^{2}t'^{2}-x'^{2}=1,\ t'>0$

and the transition from $x$, $y$, $z$, $t$ to $x'$, $y$, $z$, $t'$ is one of the transformations in question. We take up with these transformations the arbitrary displacements of the zero-points of space and time, and thus constitute a group of transformations which is evidently dependent on the parameter $c$, and which I denote by the symbol $G_{c}$.

Now let $c$ increase indefinitely, i.e., let ${\tfrac {1}{c}}$ converge to zero; then it is clear from the adjoined figure that the hyperbolic branch approaches closer and closer to the $x$-axis, the angle between the asymptotes becomes broader and broader, and the transformation $G_{c}$ changes in the limit in such a manner that the $t'$-axis can have an arbitrary direction upwards and $x'$ approaches closer and closer to $x$. Hence, it is clear that from $G_{c}$ in the limit when $c$ tends to $\infty$, i.e., as the group $G_{\infty }$, we have exactly the complete group belonging to the Newtonian Mechanics. In this state of affairs, and since $G_{c}$ is mathematically more understandable than $G_{\infty }$, a mathematician might very well, in the freedom of his imagination, come to the thought that really the phenomena of Nature possess an invariance not with the group $G_{\infty }$ with the group $G_{c}$, where $c$ is a definite finite number, which is only extremely large in the ordinary system of units. Such a presentiment would have been an extraordinary triumph of pure Mathematics. Now, if Mathematics shows here only an unnecessary witticism, still she has the satisfaction that, thanks to her favourable antecedents, she has the power, with her senses sharpened in the exercise of free foresight, to comprehend the deep-lying consequences of such a refashioning of our conception of Nature.

I will note immediately what value of $c$ is in question: in the place of $c$, the velocity of the propagation of light in empty space must make its appearance. In order not to speak of space or of emptiness, we may distinguish $c$ as the ratio of the electrostatic and the electromagnetic units of electricity.

The subsistence of the invariance of the laws of Nature for the group $G_{c}$ would be now expressed as follows :—

From the totality of natural phenomena, one may derive, by successive approximations, more and more exactly, a system of reference, $x,y,z$ and $t$ (i.e., space and time) by means of which these phenomena can be described according to definite laws. However, this system of reference is by no means uniquely determined by the phenomena. Corresponding to the transformations of the group $G_{c}$ , we may arbitrarily vary the system of reference, without the expression of the laws of Nature being changed thereby.

For example, in Fig. 1 we may call $t'$ time. But then we must necessarily define space by the totality of the three parameters $x',y,z$; and thus the physical laws would be as exactly expressed by means of $x',y,z,t'$ as by means of $x,y,z,t$. After this, we would have in the world no longer the space but an infinite number of spaces; just as in the three-dimensional space, there are an infinite number of planes. The three-dimensional Geometry becomes a chapter of the four-dimensional Physics. You now understand why I said at the outset that space and time shall sink in the background and only constitute a world with their union.

(I I)

Now arise the questions : What circumstances compel us to adopt the altered conception of space and time? Does this conception never really disagree with phenomena? Finally, does it offer advantages for the description of phenomena?

Before we enter into these matters, I will make an important remark. If we have somehow individualized space and time, then to a substantial point at rest corresponds as world-line a straight line parallel to the $t$-axis; to a uniformly moving substantial point, a straight line inclined to the $t$-axis; and to a non-uniformly moving substantial point a worldline curved in some manner. If we take up the world-line passing through an arbitrary world-point $x,y,z,t$, and find it to be there parallel to any radius vector $OA'$ of the hyperboloidal sheet mentioned above; then we may introduce $OA'$ as the new axis of time, and, with the new notions of space and time given thereby, the substance in the world-point in question will appear to be at rest. We will now introduce this fundamental axiom: The substance existent in any arbitrary world-point may be always regarded as at rest, if the space and time are suitably assigned.

This axiom means that in every world-point the expression

$c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$

turns out to be always positive, or, what comes to the same thing, every velocity $v$ turns out to be always smaller than $c$. Hence $c$ is the upper limit of the velocities of all substances, and herein lies the deep significance of the quantity $c$. In this other form the axiom gives at first the impression of being somewhat unsatisfactory. It should, however, he considered that now we have a modified Mechanics in which the square root appears with the above differential combination of the second degree, so that cases with velocity exceeding that of light will play a part somewhat like that which figures with imaginary co-ordinates play in Geometry.

The impulse and the true motive for the acceptance of the group $G_{c}$ was this, that the differential equation for the propagation of waves of light in empty space possesses that group $G_{c}$.^[1] On the other hand the notion of rigid bodies has a sense only in a Mechanics with the group $G_{\infty }$. Now, let there be an Optics with $G_{c}$ and, further, let there be rigid bodies. Then it is easy to see that with the help of the hyperboloidal sheets corresponding to $G_{c}$ and $G_{\infty }$ a $t$-direction would be marked out, and this would have the following consequence: we should be able to perceive a change of phenomena with the aid of suitable rigid optical instruments in the Laboratory, when these instruments are variously set against the direction of motion of the Earth. However, all exertions having this aim in view, in particular, a famous interference experiment of Michelson, have had a negative result. In order to obtain an explanation of this fact, Lorentz built up a hypothesis the success of which lies precisely in the invariance of Optics for the group $G_{c}$. According to Lorentz every moving body should experience a contraction in the direction of motion in the ratio $1:{\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}$, where $v$ is the velocity of the body. This hypothesis sounds extremely fantastic. For, the contraction is not to be thought of as anything like the consequence of resistances in æther but purely as a present from above — as a concomitant of the circumstances of motion.

I will now show with our figure that the hypothesis of Lorentz is fully equivalent to the new conception of space and time by means of which it becomes much more understandable. For the sake of simplicity let us not reflect upon $y$ and $z$, and let us imagine to ourselves a spacially one-dimensional world. Then a strip perpendicular like the $t$-axis and a strip inclined to the $t$-axis are respectively the images of the course of a body at rest and of that of a body in uniform motion; each of these bodies retaining a constant spacial extension. If $OA'$ is parallel to the second strip, we may introduce $t'$ as the co-ordinate of time and $x'$ as the co-ordinate of space, and then the second body appears to be at rest and the first to be in uniform motion. We suppose now that the first body conceived to be at rest has the length $l$; in other words, the section $PP$ of the first strip on the $x$-axis is equal to $l\cdot OC'$, where $OC$ represents the unit of measure on the $x$-axis. Again we suppose that the second body conceived to be at rest has the same length $l$; in other words, the cross-section of the second strip parallel to the $x'$-axis, i.e., $Q'Q'$, is equal to $l\cdot OC'$. We have now these two bodies as images of two equal Lorentzian electrons, one at rest and the other in uniform motion. However, if we retain the original co-ordinates $x,t$, then the extension of the second electron is to be considered to be $QQ$, the section of the corresponding strip parallel to the axis of $x$. Now, it is evident that $QQ=l\cdot OD'$ since $Q'Q'=l\cdot OC'$. An easy calculation shows that, if $v$ represent the ${\tfrac {dx}{dt}}$ for the second strip,

$OD'=OC\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}},$

and, consequently, also

$PP:QQ=1:{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.$

And this is the sense of Lorentz's hypothesis of the contraction of electrons on account of their motion. On the other hand, if we look upon the second electron as at rest and thus adopt the system of reference $x',t'$, then the length of the first electron will have to be denoted by the section $P'P'$ of the corresponding strip parallel to $OC'$. And we will find the first electron shortened in relation to the second one exactly in the ratio given above; for in the figure

$P'P':Q'Q'=OD:OC'=OD':OC=QQ:PP.$

Lorentz named the combination $t'$ of $x$ and $t$ the place-time of the uniformly moving electron and appropriated a physical construction of this notion for the better understanding of the hypothesis of contraction. Nevertheless, it is the merit of A. Einstein^[2] to have clearly recognized, that the time of one electron is as good as that of the other; in other words, that $t$ and $t'$ are to be treated alike. Therewith, first of all, time, as a notion uniquely determined by the phenomena, was dropped. Neither Lorentz nor Einstein shook the notion of space, perhaps for this reason that with the aforesaid special transformation, in which the $x',t'$-plane is congruent with the $x,t$-plane, an interpretation is possible as if the $x$-axis of space remained unchanged in its position. Also, to march over the notion of space (as in the case of the notion of time) is to be appraised as the rashness of mathematical refinement. After this further progress, which is, however, indispensable for the true understanding of the group $G_{c}$, the word postulate of relativity, for the demand of an invariance with the group $G_{c}$, appears to me very stale. As the sense of the postulate becomes, that only the four-dimensional world in space and time is determined by the phenomena while the projections in Space and in Time may be still taken in hand with a certain freedom, I should rather like to give this statement the name Postulate of the absolute world (or briefly world-postulate).

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1. A true application of this fact has been already given by W. Voigt in the Göttinger Nachrichten, 1887, p. 41.
2. Annalen der Physik, 1905, p. 891; Jahrbuch der Radioaktivität und Elektronik, 1907, p. 411.