Space and Time (Saha)

by Hermann Minkowski

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

SPACE AND TIME

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

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

The second group denotes that without changing the expression for the mechanical laws, we can substitute $x-\alpha t,\,y-\beta t,\,z-\gamma t,t$ 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

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

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 $c^{2}t^{2}-x^{2}=1$, 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'= ${\tfrac {1}{c}}$; then the hyperbola is again expressed in the form $c^{2}t'^{2}-x'^{2}=1,t'>0$ and the transition from (x, y, z, t) to (x' y' z' t) is one of the 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 ${\frac {1}{c}}$ 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 abovementioned 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.

II

Now the question may be asked, — what circumstances lead us to these changed views about time and space, are they not in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena?

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualised time and space in any manner; then a world-line parallel to the t-axis will correspond to a stationary point; a world-line inclined to the t-axis will correspond to a point moving uniformly; and a world-curve will correspond to a point moving in any manner. Let us now picture to our mind the world-line passing through any world point x, y, z, t; now if we find the world-line parallel to the radius vector OA' of the hyperboloidal sheet, then we can introduce OA' as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom: —

The substance existing at any world point can always be conceived to he at rest, if we establish our time and space suitably. The axiom denotes that in a world-point the expression

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

shall always be positive or what is equivalent to the same thing, every velocity V should be smaller than c. c shall therefore be the upper limit for all substantial velocities and herein lies a deep significance for the quantity c. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.

The impulse and real cause of inducement for the assumption of the group-transformation G_c is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation G_c. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G_∞. Now if we have an optics with G_c, and on the other hand if there are rigid bodies, it is easy to see that a t-direction can be defined by the two hyperboloidal shells common to the groups G_∞, and G_c, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group G_c. According to Lorentz every substance shall suffer a contraction $1:\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)$ in length, in the direction of its motion ${\frac {l}{l'}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\qquad l'=l\left(1-{\frac {v^{2}}{c^{2}}}\right).$. This hypothesis sounds rather phantastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of condition always accompanying a state of motion.

I shall show in our figure that Lorentz's hypothesis is fully equivalent to the new conceptions about time and space. Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (y, z) and fix our attention on a two dimensional world, in which let upright strips parallel to the t-axis represent a state of rest and another parallel strip inclined to the t-axis represent a state of uniform motion for a body, which has a constant spatial extension (see fig. 1). If OA' is parallel to the second strip, we can take t' as the t-axis and x' as the x-axis, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length l, i.e., the cross section PP of the first strip upon the x-axis = l · OC, where OC is the unit measuring rod upon the x-axis — and the second body also, when supposed to be at rest, has the same length l, this means that, the cross section Q'Q' of the second strip has a cross-section l·OC, when measured parallel to the x'-axis. In these two bodies, we have now images of two Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the strip belonging to it measured parallel to the x-axis. Now it is clear since Q'Q' = l·OC', that QQ = l·OD'. If ${\frac {dx}{dt}}=v$, an easy calculation gives that

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

This is the sense of Lorentz's hypothesis about the contraction of electrons in ease of motion. On the other hand, if we conceive the second electron to be at rest, and therefore adopt the system (x', t') then the cross-section P'P' of the strip of the electron parallel to OC' is to be regarded as its length and we shall find the first electron shortened with reference to the second in the same proportion, for it is,

${\frac {P'P'}{Q'Q'}}={\frac {OD}{OC'}}={\frac {OD'}{OC}}={\frac {QQ}{PP}}$

Lorentz called the combination t' of (t and x) as the local time (Ortszeit) of the uniformly moving electron, and used a physical construction of this idea for a better comprehension of the contraction-hypothesis. But to perceive clearly that the time of an electron is as good as the time of any other electron, i.e. t, i' are to be regarded as equivalent, has been the service of A. Einstein [Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis... 4-4-1 1—1907] There the concept of time was shown to be completely and unambiguously established by natural phenomena. But the concept of space was not arrived at, either by Einstein or Lorentz, probably because in the case of the above-mentioned spatial transformations, where the (x', y') plane coincides with the x-t plane, the significance is possible that the x-axis of space some-how remains conserved in its position.

We can approach the idea of space in a corresponding manner, though some may regard the attempt as rather fantastical.

According to these ideas, the word "Relativity-Postulate" which has been coined for the demands of invariance in the group G, seems to be rather inexpressive for a true understanding of the group G_c, and for further progress. Because the sense of the postulate is that the four-dimensional world is given in space and time by phenomena only, but the projection in time and space can be handled with a certain freedom, and therefore I would rather like to give to this assertion the name "The Postulate of the Absolute world" [World-Postulate].

III

By the world-postulate a similar treatment of the four determining quantities x, y, z, t, of a world-point is possible. Thereby the forms under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more striking and clear.

I shall again use the geometrical method of expression. Let us call any world-point O as a "Space-time-null-point." The cone

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=0$

consists of two parts with O as apex, one part having t < 0', the other having t > 0. The first, which we may call the fore-cone consists of all those points which send light towards O, the second, which we may call the aft-cone, consists of all those points which receive their light from O. The region bounded by the fore-cone may be called the fore-side of O, and the region bounded by the aft-cone may be called the aft-side of O. (Vide fig. 2).

On the aft-side of O we have the already considered hyperboloidal shell $F=c^{2}t^{2}-x^{2}-y^{2}-z^{2}=l,\ t>0$.

The region inside the two cones will be occupied by the hyperboloid of one sheet

$-F=x^{2}+y^{2}+z^{2}-c^{2}t^{2}=k^{2}$,

where k² can have all possible positive values. The hyperbolas which lie upon this figure with O as centre, are important for us. For the sake of clearness the individual branches of this hyperbola will be called the "Interhyperbola with centre O." Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for t = — ∞ and t = ∞ asymptotically approaches the velocity of light c.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness x, y, z, t a vector, then we have to distinguish between a time-vector directed from O towards the sheet +F = 1, t > O and a space-vector directed from O towards the sheet -F = 1. The time-axis can be parallel to any vector of the first kind. Any world-point between the fore and aft cones of O, may by means of the system of reference be regarded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily always earlier, every point on the aft side of O, later than O. The limit c = ∞ corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness t = 0. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards (x, y, z, t) into its components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces ±F = 1, and of a tangent RS at the point R of the surface, then the vectors shall be called normal to each other. Accordingly

$c^{2}tt_{1}-xx_{1}-yy_{1}-zz_{1}=0,$

which is the condition that the vectors with the components (x, y, z, t) and $\left(x_{1}\ y_{1}\ z_{1}\ t_{1}\right)$ are normal to each other.

For the measurement of vectors in different directions, the unit measuring rod is to be fixed in the following manner; — a space-like vector from to -F = 1 is always to have the measure unity, and a time-like vector from O to +F = 1 , t >0 is always to have the measure ${\tfrac {1}{c}}$.

Let us now fix our attention upon the world-line of a substantive point running through the world-point (x, y, z, t); then as we follow the progress of the line, the quantity

$d\tau ={\frac {1}{c}}{\sqrt {c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}}},$

corresponds to the time-like vector-element (dx, dy, dz, dt).

The integral $\tau =\int d\tau$, taken over the world-line from any fixed initial point P₀ to any variable final point P, may be called the "Proper-time" of the substantial point at P₀ upon the world-line. We may regard (x, y, z, t), i.e., the components of the vector OP, as functions of the "proper-time" τ; let $\left({\dot {x}},\ {\dot {y}},\ {\dot {z}},\ {\dot {t}}\right)$ denote the first differential-quotients, and $\left({\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}\right)$ the second differential quotients of (x, y, z, t) with regard to τ, then these may respectively be called the Velocity-vector, and the Acceleration-vector of the substantial point at P. Now we have

$\left.{\begin{array}{l}c^{2}{\dot {t}}^{2}-{\dot {x}}^{2}-{\dot {y}}^{2}-{\dot {z}}^{2}=c^{2}\\c^{2}{\dot {t}}{\ddot {t}}-{\dot {x}}{\ddot {x}}-{\dot {y}}{\ddot {y}}-{\dot {z}}{\ddot {z}}=0\end{array}}\right\}$

i.e., the ’Velocity-vector’ is the time-like vector of unit measure in the direction of the world-line at P, the ’Acceleration-vector’ at P is normal to the velocity-vector at P, and is in any case, a space-like vector.

Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a 'fore-cone' and an 'aft-cone.' This hyperbola may be called the "hyperbola of curvature" at P (vide fig. 3). If M be the centre of this hyperbola, then we have to deal here with an 'Inter-hyperbola' with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is a vector of magnitude ${\frac {c^{2}}{\rho }}$ in the direction of MP.

If ${\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}$ are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.

IV

In order to demonstrate that the assumption of the group G_c for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of "Thermodynamics and Radiation,"^[1] for "Electromagnetic phenomena",^[2] and finally for "Mechanics with the maintenance of the idea of mass."

For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (x, y, z, t), where the velocity-vector is $\left({\dot {x}},\ {\dot {y}},\ {\dot {z}},\ {\dot {t}}\right)$, then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group G_c is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components

${\dot {t}}X,\ {\dot {t}}Y,\ {\dot {t}}Z,\ {\dot {t}}T,$

where

$T={\frac {1}{c^{2}}}\ \left({\frac {\dot {x}}{\dot {t}}}X+{\frac {\dot {y}}{\dot {t}}}Y+{\frac {\dot {z}}{\dot {t}}}Z\right)={\frac {1}{c^{2}}}$

(the rate of which work is done at the world-point), remains unaltered. This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a moving force-vector at P.

Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the impulse-vector, and m-times the acceleration-vector at P as the force-vector of motion, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector^[3]:

The force-vector of motion is equal to the moving force-vector.

This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the "Energy-law." Accordingly c²-times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass. The expression for this is

$mc^{2}{\frac {dt}{d\tau }}=mc^{2}/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

i.e., if we deduct from this the additive constant mc², we obtain the expression ½mv² of Newtonian-mechanics up to magnitudes of the order of ${\frac {1}{c^{2}}}$. Hence it appears that the energy depends upon the system of reference. But since the t-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting: ease c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.^[4]

From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write ${\sqrt {-1}}t=l$, in the place of t, then the differential expression

$d\tau ^{2}=-\left(dx^{2}+dy^{2}+dz^{2}+dl^{2}\right),$

becomes symmetrical in (x, y, z, l); this symmetry then enters into each law, which does not contradict the world-postulate. We can clothe the essential nature of this postulate in the mystical, but mathematically significant formula

3·10⁵ km = ${\sqrt {-1}}$ Sec.

V

The advantages arising from the formulation of the world-potulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge (e), and let us introduce upon it the "Proper-time" τ reckoned from any possible initial point. In order to obtain the field caused by the electron at any world-point P₁ let us construct the fore-cone belonging to P₁ (vide fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P₁ the normal to this tangent. Let r be the measure of P₁Q. According to the definition of a fore-cone, r/c is to be reckoned as the measure of PQ. Now at the world-point P₁, the vector-potential of the field excited by e is represented by the vector in direction PQ, having the magnitude ${\frac {e}{cr}}$, in its three space components along the x-, y-, z-axes; the scalar-potential is represented by the component along the t-axis. This is the elementary law found out by A. Lienard, and E. Wiechert.^[5]

If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed; the two forces considered together bears some analogy to the force-screw in mechanics; the analogy is, however, imperfect.

I shall now describe the ponderomotive force which is exerted by one moving electron upon another moving electron. Let us suppose that the world-line of a second point-electron passes through the world-point P₁. Let us determine P, Q, r as before, construct the middle-point M of the hyperbola of curvature at P, and finally the normal MN upon a line through P which is parallel to QP₁. With P as the initial point, we shall establish a system of reference in the following way: the t-axis will be laid along PQ, the x-axis in the direction of QP₁. The y-axis in the direction of MN, then the z-axis is automatically determined, as it is normal to the x-, y-, z-axes. Let ${\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}$ be the acceleration-vector at $P,\ {\dot {x}}_{1},\ {\dot {y}}_{1},\ {\dot {z}}_{1},\ {\dot {t}}_{1}$ be the velocity-vector at P₁. Then the force-vector exerted by the first electron e, (moving in any possible manner) upon the second electron e, (likewise moving in any possible manner) at P₁ is represented by

$-ee_{1}\left({\dot {t}}_{1}-{\frac {{\dot {x}}_{1}}{c}}\right)\mathrm {F}$

For the components $\mathrm {F} _{x},\ \mathrm {F} _{y},\ \mathrm {F} _{z},\ F_{t}$ of the vector $F$ the following three relations hold: —

$c\mathrm {F} _{t}-\mathrm {F} _{x}={\frac {1}{r^{2}}},\ \mathrm {F} _{y}={\frac {\ddot {y}}{c^{2}r}},\ \mathrm {F} _{z}=0$

and fourthly this vector F is normal to the velocity-vector P₁, and through this circumstance alone, its dependence on this last velocity-vector arises.

I£ we compare with this expression the previous formulæ^[6] giving the elementary law about the ponderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions; but in three dimensions, they have very complicated projections.

In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics, and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses m and m₁ describe their world-lines; a moving force-vector is exercised by m upon m₁, and the expression is just the same as in the case of the electron, only we have to write +mm₁ instead of -ee₁. We shall consider only the special case in which the acceleration-vector of m is always zero; then t may be introduced in such a manner that m may be regarded as fixed, the motion of m is now subjected to the moving-force vector of m alone. If we now modify this given vector by writing ${\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ instead of ${\dot {t}}$ (${\dot {t}}=1$ up to magnitudes of the order ${\tfrac {1}{c^{2}}}$), then it appears that Kepler's laws hold good for the position (x₁, y₁, z₁), of m₁ at any time, only in place of the time t₁, we have to write the proper time τ₁ of m₁. On the basis of this simple remark, it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the Newtonian law of attraction in combination with Newtonian mechanics.

Also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate. I shall also show on a later occasion that the deduction of these equations, as taught by Lorentz, are by no means to be given up.

The fact that the world-postulate holds without exception is, I believe, the true essence of an electromagnetic picture of the world; the idea first occurred to Lorentz, its essence was first picked out by Einstein, and is now gradually fully manifest. In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of the postulate; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts, will be reconciled to the new ideas of time and space, — in the prospect that they will lead to pre-established harmony between pure mathematics and physics.

---

1. Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p. 1.
2. H. Minkowski; the passage refers to paper (2) of the present edition.
3. Minkowski — Mechanics, appendix, page 65 of paper (2).
   Planck — Verh. d. D. P. G. Vol. 4, 1906, p. 136.
4. Schütz, Gött. Nachr. 1897, 110.
5. Lienard, L'Eclairage électrique T. 16, 1896, p. 53,
   Wiechert, Ann. d. Physik, Vol. 4.
6. K. Schwarzschild. Gött-Nachr. 1903.
   H. A. Lorentz, Enzyklopädie der Math. Wissenschaften V. Art 14, p. 199.