# Spatial and temporal dimensions orthogonality

It seems that the spatial dimensions are orthogonal: a particle can move along one axis without changing its position in relation to other two axes.

It seems that the temporal dimension is somewhat orthogonal:

• a particle can move along the time axis but not along the space axes by standing still
• if a particle is to move along space axes, it necessarily also moves along time axis

Is there something deep and meaningful in this seemingly incomplete orthogonality, or is it just a consequence of some theory (perhaps general relativity)?

If we take into account quantum mechanic, the above statements are not entirely correct. A particle cannot really be standing completely still (due to uncertainty principle). But does it matter?

Do we need to treat massive and massless particles separately in this deliberation?

2. The lesson to be learned is that we need an objective/physical definition of orthogonality, that doesn't depend on how we draw it on paper. This is provided by the metric tensor or inner product $${\bf u} ~\perp~ {\bf v}\quad \Leftrightarrow \quad g({\bf u} , {\bf v})~=~0.\tag{1}$$ This definition (1) is Lorentz covariant.