Translation:Graphical representation of the four-dimensional space-time universe

by Paul Gruner

Gruner, P. (Berne). – A) Graphical representation of the space-time universe in four dimensions.

The author develops the ideas he had presented last year at the Physical Society (Arch. Sc. Phys. Nat. (5) 3, 295, 1921). The motion of a point can be given by the following four equations:

$\Phi (xy)=0,\quad {\mathsf {X}}(yz)=0,\quad \Psi (zx)=0$ et $x=f(t)$.

which represent the projections of a four-dimensional curve on four coordinate planes in the space-time universe. By folding these projections in the same plane, it becomes easy to represent the phenomena of the universe in four dimensions by simple methods of descriptive geometry.

Thus the uniform rectilinear motion of a point will be represented in terms of $XOY$ by a straight line in the plane of $XOT$ (called "subspace"), corresponding to the straight line $x=v\cdot t$ (the world line of the motion). To develop the phenomena of the theory of relativity, it is useful to measure time by the path traveled by light $u=c\cdot t$, $c$ being the speed of light equal to unity. Also it is best to choose a non-orthogonal system of coordinates $XOR$ as "subspace"; with this choice it is possible to relate the two systems $XOY$ and $X'O'Y'$ that move parallel to axis $OX$ with a relative velocity $\alpha$, to two "subspaces" whose axes $OX\perp O'U'$ and $OU\perp O'X'$ are mutually orthogonal and for which the angle $XOX'=UOU'=\varphi$ determines the relative velocity $\alpha =\sin \varphi$.

Now we project the world line $x=v\cdot t$ constructed for subspace $XOR$ into subspace $X'O'U'$ and from there into space $X'O'Y'$, we obtain the point's motion as it appears in the system $X'O'Y'$. The figures provide immediately the Lorentz-Einstein transformation formulas, the speed of the point, the addition theorem, aberration, etc..

The same construction can be applied to the wave propagation phenomena, either planar or spherical. In those somewhat complicated constructions, one shall never confuse phenomena that are synchronous in a coordinate system with those synchronous in the other system: in these subspaces, synchronous phenomena should always be on a world-line parallel to the $OX$ axis, respectively $O'X'$. Taking into account these remarks it is easy to directly construct wavelengths and frequencies of undulatory motion in both systems; one exactly obtains the expressions given by Einstein, whose deductions thereby receive a new geometric confirmation.

b) Graphical representation of universal time in the theory of relativity.

In the constructions given in the preceding article, the bisectors of angle $\varphi$ play a special role. They form an orthogonal coordinate system for length $x$ and time $u$ which is symmetrical with respect to two systems $XOU$ and $X'O'U'$ of the subspace. It is therefore very natural to relate the subspace phenomena to this unique and orthogonal system, and introduce for both systems a common coordinate of time, the "universal time" $t$, as well as for length, the "universal length" $r$. A simple geometric consideration provides a glimpse that the coordinates $xx'$ $uu'$ can be projected in a proper manner on the axis of $r$, and $t$ respectively, by changing the value of units.

One then finds between $x$, $x'$ and $t$ the old formula of relativity of Galileo-Newton: $x=x'+2\tan {\tfrac {\varphi }{2}}\cdot t$. It is the merit of Ed. Guillaume to have found and developed these results already several years ago.

But from this construction it follows in a striking manner what was already noted by Mirimanoff (Arch. Soc. Phys. nat. (5) 3, 46, 1921). The indications $u$ and $u'$ of the clocks of each system may naturally be adjusted so that they provide the universal time $t$; but this correction depends on $\varphi$, that is to say on the relative velocity $\alpha$ of the two compared systems. Thus the corrections given to the clock regulated to local time, depend on the system with which it is intended to be compared; the concept of universal time then becomes illusory.

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