5.2.3 The Proper Time
We can clearly see from Figure 5.1 that space-time intervals can be obtained from:
, or
, because the modulus of hyperbolic or split-complex numbers
is calculated according to the formula:
.
Alternatively, instead of the classical space-time interval, we can use the time-time interval which is the modulus of the complex time:
, where
, or
for motion on x-axis only.
Thus it can be also readily seen from Figure 5.1 that the modulus of this complex time equals negative the proper time as defined in the current equations of Special Relativity:
(5.1)Hence:
(5.2)The reason why the modulus of complex time:
, or its imaginary part:
, is negative with relation to proper time:
, where
is the normal measured time, is because the (imaginary) flow of time in the outward level
is interrupting and delaying the real flow of time that is creating space and matter:
, so the actual (net value of) time is always smaller than the real time:
.