Page:DeSitterGravitation.djvu/5

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

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

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

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,

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". 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

where