Page:DeSitterGravitation.djvu/5

The invariants of the transformation are all of the form—,

$x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}-ct_{1}ct_{2},$

(4)

where $x_{1},\ y_{1},\ z_{1},\ ct_{1}$, or $x_{2},\ y_{2},\ z_{2},\ ct_{2},$ may be replaced by any set of quantities, which are transformed by the same formulæ, such as $(\xi ),(\eta ),(\zeta ),(\kappa ),$ etc.

The equation

$(\kappa )^{2}-(\phi )^{2}=1$

is thus not altered by the transformation. If now we define a new variable $c\tau$ by the equations

${\frac {d\tau }{dt}}={\sqrt {1-\phi ^{2}}}={\frac {1}{(\kappa )}},\ {\text{or}}\ {\frac {dt}{d\tau }}=(\kappa )={\sqrt {1-(\phi )^{2}}},$

(5)

this variable is the same function of $x',y',z',t'$ as of $x,y,z,t,$ and is consequently independent of the system of reference. We have, of course,

$(\xi )={\frac {dx}{cd\tau }},\quad (\eta )={\frac {dy}{cd\tau }},\quad (\zeta )={\frac {dz}{cd\tau }},\quad (\kappa )={\frac {dct}{cd\tau }}.$

The variable $\tau$ is called by Minkowski the "Eigenzeit" of the point whose coordinates are $x,y,z,$ which may be translated by "proper-time".^[1] In many problems it is more convenient as an independent variable than $t$.

Every point has thus its own proper-time, which is independent of the system of reference, but depends on the state of motion of the point and on its previous history. The proper-time of a point rigidly connected with the axes of the system of reference ($x,y,z,t$) is $t$ itself. As a convenient abbreviation, we may speak of "heliocentric time," "geocentric time," etc., meaning the proper-time of the Sun, the Earth, etc.

4. A set of values of $m,x,y,z,t,$ defining the position of a particle of mass $m$ in the system of reference ($x,y,z,t$), may be called an "event." Two events are called simultaneous if their values of $t$ are the same. Two events which are simultaneous in one system ($x,y,z,t$) are in general not simultaneous in another system ($x',y',z',t'$). And, within certain restrictions, which are of no importance for our purpose, a system can always be found in which two arbitrarily given events are simultaneous.

We have

$c\left(t'_{1}-t'_{2}\right)={\frac {c\left(t_{1}-t_{2}\right)-q\Delta _{q}}{\sqrt {1-q^{2}}}},$

where

$\Delta _{q}=\alpha \left(x_{1}-x_{2}\right)+\beta \left(y_{1}-y_{2}\right)+\gamma \left(z_{1}-z_{2}\right).$

1. It should be remarked that this is not the same as "local" time, as originally defined by Lorentz.