# Defining what it means for a reference frame to move with a velocity $\mathbf{u}$ with respect to another

In describing a Galilean transformation, for example, one might say that if a reference frame $$S'$$ is moving at a velocity $$\mathbf{u}$$ with respect to $$S$$, then an object traveling at a velocity $$\mathbf{v}$$ in $$S$$ would be traveling at a velocity $$\mathbf{v}'$$ in $$S'$$, where $$\mathbf{v}' = \mathbf{v} - \mathbf{u}.$$ What, though, would it precisely mean for $$S'$$ to be moving at a velocity $$\mathbf{u}$$ with respect to $$S$$?

It is easiest to simply consider $$\mathbf{u}$$ to be the velocity of the origin of $$S’$$ relative to the origin of $$S$$. Let $$\mathbf{R}$$ be the vector from the origin of $$S$$ to the origin of $$S’$$. Then $$\mathbf{u}=d\mathbf{R}/dt$$.

• But that doesn't express the implicit assumption that the reference frames aren't rotating with respect to each other. – PiKindOfGuy Apr 3 at 6:40
• @PiKindOfGuy No. In my frame I can see your origin moving by me at some speed even if I take your frame to also be rotating about its origin. – Aaron Stevens Apr 3 at 10:00

A frame of reference is a way of being able to location a point in space at a given time.
In the Cartesian coordinate system three perpendicular axes are drawn and the axes labelled $$x, y$$ and $$z$$.
Any point in space can therefore be represented by the coordinates $$(x,y,z)$$.
One point in space is chosen to be the origin with coordinates $$(0,0,0)$$.
There are an infinite number of such reference frames.

Let reference frame $$S$$ have an origin with coordinates $$(0,0,0)$$.

Suppose that another point in space is chosen as the origin with coordinates $$(0',0',0')$$ and any other point would be represented by coordinates $$(x',y',z')$$.
This defines reference frame $$S'$$.

For ease of drawing I switch to two dimensions.

It so happens that the point with coordinates $$(0',0')$$ using reference frame $$S'$$ is moving at a constant velocity $$\vec u = u\,\hat u$$ relative to the point with coordinates $$(0,0)$$ using reference frame $$S$$.

After a certain time let the point with coordinates $$(0',0')$$ using reference frame $$S'$$ undergo a displacement of $$a \,\hat u$$ as measured using reference frame $$S$$.

During that same time interval points on the axes with coordinates $$(3,0), (2,0) \,.\,.(0,2) \,.\,.$$ etc using reference frame $$S'$$ all undergo the same displacement, $$a\, \hat u$$, as measured using reference frame $$S$$.

We can say that reference frame $$S'$$ (the axes $$x',\, y'$$ and $$z'$$) is moving at a velocity $$\vec u$$ relative to reference frame $$S$$ (the axes $$x,\, y$$ and $$z$$).
Think of two sheets of graph paper, labelled $$S$$ and $$S'$$, with axes drawn on them moving at a constant velocity relative to one another.

A Galilean transformation is a mapping of spacetime co-ordinates $$(x,y,z,t)$$ from one system of co-ordinates (a.k.a. "reference frame") to another. It describes how the spacetime co-ordinates for a given point in spacetime (a.k.a. an "event") change when we move from one reference frame to another.

If an event has co-ordinates $$(x,y,z,t)$$ in reference frame $$S$$ and co-ordinates $$(x',y',z',t')$$ in reference frame $$S'$$ which is related to $$S$$ by a Galilean transformation then

$$(x',y',z',t') = (x-u_xt, \space y-u_yt, \space z-u_zt, \space t)$$

where the fixed parameters $$(u_x,u_y,u_z)$$ are the co-ordinates of a 3-vector $$\mathbf{u}$$ which is called the "velocity" of $$S'$$ with respect to $$S$$.

A physical interpretation of this is that the origin of the co-ordinates in $$S'$$ is moving at a fixed velocity $$\mathbf{u}$$ relative to the origin in $$S$$.