# Characteristics of an inertial frame

"What does an empty inertial frame look like? Does it swim? Does it quack?"

TLDR: How is it possible to distinguish two completely empty classical inertial reference frames? What other characteristics (e.g. mass, charge) may be associated to them besides their relative velocity?

A reference frame (abbreviated as F) is a coordinate system with synchronized time everywhere.

A classical F is a theoretical concept: it doesn't need anything else to be present for its existence just like numbers don't either.

An inertial F is one with a relative velocity w.r.t. another F, $$F_{0}$$, constant in $$F_{0}$$'s time.

So out of thin air we have attributed a characteristic to a general empty F: its relative velocity. How is this possible? After all there is nothing to distinguish the two Fs -their coordinate systems aren't some measuring rods nor are their times being kept by tangible clocks.? In fact, how can one say there are even two empty inertial Fs?

In theory empty inertial Fs seem indistinguishable.

When considering the motion of point objects in $$F_{0}$$, the other inertial F is the objects rest frame. Here it makes sense to talk about two Fs. But note that this needs some object to be present which we might as well attribute to be a characteristic of F itself. Does this mean that any F must necessarily have something tangible to associate the relative velocity to? By tangible we mean detectable which requires F to contain either mass-energy,charge or color.

A frame is called inertial if it is orthonormal for the form $$-t^2+x^2+y^2+z^2$$. Two inertial frames differ by a linear transformation; those transformations are parameterized by a quantity called the relative velocity.
For example (in two dimensions) the velocity associated to $$\pmatrix{A&B\cr B&A\cr}$$ is $$-B/A$$.