Natural Space-time in Mixed Coordinate System

Will Le

From Common space-time to Proper space-time

In physics, we're used to the common space-time that is shared between all objects and subjects. From Newton, in classic mechanics, it's convenient for us to have a common ruler to measure position of everything and a common clock to attach timestamp onto every event. Then in Relativity Theory, Einstein acknowledged that the space-time structure is different between reference frames of different velocities. So, beside the common space-time that we may all agree to take as the common reference, eg. UTC on Earth, each reference frame moving at velocity v relative to the common space-time does have another space-time which appears to us in the common space-time that "its clock runs slower" and "its ruler is shorter". However, the subjects in that different space-time does not feel any strangeness, the measurement of their objects using their "slower clock" and "shorter ruler" still results in the same physical laws as ours on Earth. That means their slowness and shortness that we see are just apparent effect, and their "slower clock" and "short ruler" are proper, are true to them, thus their space-time measured by those devices is their proper space-time.

And now the Natural space-time

Look straight to the nature of space and time, we can see that they are not only different but opposite and complement each other:
  • Space is an extrinsic feature reflecting the external state, i.e. the relation between objects.
  • Time is an intrinsic feature reflecting the internal state of each individual object.
Both space and time show relations, but while space shows relation with others, time shows relation with ourselves. So, taking a "middle way" between common space-time, the external extreme, and proper space-time, the internal extreme, we have the natural space-time:
The natural space-time comprises common space (s) and proper (private) time (τ).

Mixed coordinate system

Fusing both natural space-time and common space-time, we have the mixed coordinate system of 3 axes: common space (s), propertime (τ) and common time (t), with the light cone relates them all together.