Definition: Frame of Reference

Let \(\mathcal R=(\mathbb R,V_{\mathbb R},v)\) be the \(3\)-dimensional affine space with the associated normed vector space \((V_{\mathbb R},||~||)\) over the field of real numbers \(\mathbb R\). Any non-empty affine bounded subset \(\mathcal I\subset\mathcal R\) is also called a frame of reference.

Practical Consequences from this Definition

Please note that any frame of reference \(\mathcal I\) has a norm \(||~||\) induced by the associated vector space, i.e. we can measure the distance between any two points \(P,Q\in\mathcal I\) by measuring the norm of the corresponding vector \[||\overrightarrow{PQ}||\ge 0\in\mathbb R.\]

Please also note that the boundedness of \(\mathcal I\) is chosen in the definition for technical reasons to avoid the necessity to deal with infinite (unbounded) frames of reference to model the physics of real world.

Examples

Frames of reference can be:

\(0\) dimensional (e.g. a particle)
\(1\) dimensional (e.g. being a gravitational model of a binary star system)
\(2\) dimensional (e.g. a (bounded) surface of the Earth small enough to ignore the Earths curvature)
\(3\) dimensional (e.g. a vehicle, a spaceship, a light clock, etc.)