A.3 Galilean transformation

The Galilean transformation describes coordinate system transformations in nonrelativistic Newtonian physics. This note explains these transformation rules. Essentially the same analysis also applies to Lorentz transformations between observers using arbitrarily chosen coordinate systems. The small difference will be indicated.

Consider two observers A' and B' that are in inertial motion. In other words, they do not experience accelerating forces. The two observers move with a relative velocity of magnitude $V$ relative to each other. Observer A' determines the time of events using a suitable clock. This clock displays the time $t_{\rm {A'}}$ as a single number, say as the number of seconds since a suitably chosen reference event. To specify the position of events, observer A' uses a Cartesian coordinate system $(x_{\rm {A'}},y_{\rm {A'}},z_{\rm {A'}})$ that is at rest compared to him. The origin of the coordinate system is chosen at a suitable location, maybe the location of the reference event that is used as the zero of time.

Observer B' determines time using a clock that indicates a time $t_{\rm {B'}}$. This time might be zero at a different reference event than the time $t_{\rm {A'}}$. To specify the position of events, observer B' uses a Cartesian coordinate system $(x_{\rm {B'}},y_{\rm {B'}},z_{\rm {B'}})$ that is at rest compared to her. The origin of this coordinate system is different from the one used by observer A'. For one, the two origins are in motion compared to each other with a relative speed $V$.

The question is now, what is the relationship between the times and positions that these two observers attach to arbitrary events.

To answer this, it is convenient to introduce two additional observers A and B. Observer A is at rest compared to observer A'. However, she takes her zero of time and the origin of her coordinate system from observer B'. In particular, the location and time that A associates with her origin at time zero is also the origin at time zero for observer B':

\begin{displaymath}
(x_{\rm {A}},y_{\rm {A}},z_{\rm {A}},t_{\rm {A}}) = (0,0,0...
...\rm {B'}},y_{\rm {B'}},z_{\rm {B'}},t_{\rm {B'}}) = (0,0,0,0)
\end{displaymath}

The other additional observer, B, is at rest compared to B'. Like observer A, observer B uses the same origin and zero of time as observer B':

\begin{displaymath}
(x_{\rm {B}},y_{\rm {B}},z_{\rm {B}},t_{\rm {B}}) = (0,0,0...
...\rm {B'}},y_{\rm {B'}},z_{\rm {B'}},t_{\rm {B'}}) = (0,0,0,0)
\end{displaymath}

Observer B orients her coordinate system like A does.

That makes the relationship between A and B just like A and B as discussed for the Lorentz transform, figure 1.2. However, the classical Galilean transformation is much simpler than the Lorentz transformation. It is

\begin{displaymath}
\fbox{$\displaystyle
t_{\rm{B}} = t_{\rm{A}}
\qquad
...
...rm{B}} = y_{\rm{A}}
\qquad
z_{\rm{B}} = z_{\rm{A}}
$} %
\end{displaymath} (A.11)

Note however that these classical formulae are only an approximation. They can only be used if the relative velocity $V$ between the observers is much smaller than the speed of light. In fact, if you take the limit $c\to\infty$ of the Lorentz transformation (1.6), you get the Galilean transformation above.

The question still is how to relate the times and locations that observer A' attaches to events to those that observer B' does. To answer that, it is convenient to do it in stages. First relate the times and locations that A' attaches to events to the ones that A does. Then use the formulae above to relate the times and locations that A attaches to events to the ones that B does. Or, if you want the relativistic transformation, at this stage use the Lorentz transformation (1.6). Finally, relate the times and locations that B attaches to events to the ones that B' does.

Consider then now the relationship between the times and locations that A' attaches to events and the ones that A does. Since observer A and A' are at rest relative to each other, they agree about differences in time between events. However, A' uses a different zero for time. Therefore, the relation between the times used by the two observers is

\begin{displaymath}
\fbox{$\displaystyle
t_{\rm{A}} = t_{\rm{A'}} - \tau_{\rm{AA'}}
$} %
\end{displaymath}

Here $\tau_{\rm {AA'}}$ is the time that observer A' associates with the reference event that observer A uses as time zero. It is a constant, and equal to $-\tau_{\rm {A'A}}$. The latter can be seen by simply setting $t_{\rm {A'}}$ zero in the formula above.

To specify the location of events, both observers A' and A use Cartesian coordinate systems. Since the two observers are at rest compared to each other, they agree on distances between locations. However, their coordinate systems have different origins. And they are also oriented under different angles. That makes the unit vectors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ along the coordinate axes different. In vector form the relation between the coordinates is then:

\begin{displaymath}
\fbox{$\displaystyle
x_{\rm{A}} {\hat\imath}_{\rm{A}} +
...
... +
(z_{\rm{A'}}-\zeta_{\rm{AA'}}) {\hat k}_{\rm{A'}}
$} %
\end{displaymath} (A.12)

Here $\xi_{\rm {AA'}}$, $\eta_{\rm {AA'}}$, and $\theta_{\rm {AA'}}$ are the position coordinates that observer A' associates with the origin of the coordinate system of A. By putting $(x_{\rm {A'}},y_{\rm {A'}},z_{\rm {A'}})$ to zero in the expression above, you can relate this to the coordinates that A attaches to the origin of A'.

The above equations can be used to find the coordinates of A in terms of those of A. To do so, you will need to know the components of the unit vectors used by A' in terms of those used by A. In other words, you need to know the dot products in

\begin{eqnarray*}
{\hat\imath}_{\rm {A'}} & = &
({\hat\imath}_{\rm {A'}}\cdo...
...({\hat k}_{\rm {A'}}\cdot{\hat k}_{\rm {A}}) {\hat k}_{\rm {A}}
\end{eqnarray*}

Then these relations allow you to sum the ${\hat\imath}_{\rm {A}}$ components in the right hand side of (A.12) to give $x_{\rm {A}}$. Similarly the ${\hat\jmath}_{\rm {A}}$ components sum to $y_{\rm {A}}$ and the ${\hat k}_{\rm {A}}$ components to $z_{\rm {A}}$.

Note also that if you know these dot products, you also know the ones for the inverse transformation, from A to A'. For example,

\begin{displaymath}
({\hat\imath}_{\rm {A}}\cdot{\hat\imath}_{\rm {A'}}) =
(...
...rm {A'}}\cdot{\hat\imath}_{\rm {A}}) \qquad
\mbox{etcetera}
\end{displaymath}

(In terms of linear algebra, the dot products form a 3 $\times$ 3 matrix. This matrix is called “unitary,” or as a real matrix also more specifically “orthonormal,” since it preserves distances between locations. The matrix for the reverse transform is found by taking a transpose.)

The relationship between observers B and B' is a simplified version of the one between observers A and A'. It is simpler because B and B' use the same zero of time and the same origin. Therefore the formulae can be obtained from the ones given above by replacing A' and A by B and B' and dropping the terms related to time and origin shifts.