Spatial and temporal dimensions orthogonality

Question

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?

Answer 1

Relativistically, the right way to construe this notion of orthogonality is in terms of the inner product between four-vectors.

When two spacelike vectors are orthogonal, it means what we have in mind in Euclidean geometry.

When a timelike vector is orthogonal to a spacelike vector, it means that to an observer who is moving along the timelike vector, the spacelike vector is purely spatial, i.e., it connects events that are simultaneous.

A lightlike vector is orthogonal to itself.

Answer 2

1. What do you mean by orthogonal dimensions? It is not a well-defined notion. For starters, one doesn't have to draw coordinate axis orthogonal for a coordinate system to serve its purpose of coordinatizing a physical system. Even if one did draw them orthogonal, a Lorentz transformation could change that.

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.