392
Prof. de Sitter, On the bearing of the Principle
LXXI. 5,
The invariants of the transformation are all of the form—,
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(4)
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where
, or
may be replaced by any set of quantities, which are transformed by the same formulæ, such as
etc.
The equation
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is thus not altered by the transformation. If now we define a new variable
by the equations
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(5)
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this variable is the same function of
as of
and is consequently independent of the system of reference. We have, of course,
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The variable
is called by Minkowski the "Eigenzeit" of the point whose coordinates are
which may be translated by "proper-time".[1] In many problems it is more convenient as an independent variable than
.
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 (
) is
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
defining the position of a particle of mass
in the system of reference (
), may be called an "event." Two events are called simultaneous if their values of
are the same. Two events which are simultaneous in one system (
) are in general not simultaneous in another system (
). 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
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where
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- ↑ It should be remarked that this is not the same as "local" time, as originally defined by Lorentz.