How can the Gallilean transformations form a group?

Question

In class my professor said the Galilean transformations form a group of order 10. $$ x'=x-vt\\ y'=y\\ z'=z\\ t'=t\\ $$ But how do these form a group? I don't see 10 things to interpret as elements. I also don't see what the group operation would be. What are the elements of the Galilean group and what is its operation?

Answer 1

It is slightly incorrect for your professor to state that the Galilean transformations form a group that is a Lie group of order 10. A group's order is the number of its elements. For example, a group of order 10 is a finite group. There are are two, and only two, possible groups with 10 elements. It is correct, however, to say that Galilean transformation form a group of dimension 10. The order of the Galilean group is uncountably infinite. This is because there are an uncountable infinte number of elements in the group.

I'm not sure how much continuous group theory (Lie group theory) you have done, but what Lie group of dimension 10 means roughly is that any transformation "near enough" to the identity can be specified by exactly 10 independent parameters. In this answer, these 10 independent parameters are represented by $\theta\,\gamma_x$, $\theta\,\gamma_y$, $\theta\,\gamma_z$, $v_x$, $v_y$, $v_z$, $ X$, $Y$, $Z$, and $\tau$.

The easiest way to concretely represent the Galilean group is with a set of $5\times 5$ matrices as the group elements, with matrix multiplication as the group operation. The group's elements act on $5\times 1$ column vectors called the homogeneous co-ordinates. The homogeneous co-ordinates are of the form:

$$\left(\begin{array}{c}x\\y\\z\\t\\1\end{array}\right)\tag{1}$$

where $x,\,y,\,z,\,t$ are the three spatial co-ordinates and one time co-ordinate of a point in one co-ordinate system. With the use of a Galilean transformation, this point is transformed to three spatial co-ordinates and one time co-ordinate in another inertial frame. The transform is done by one of the group elements.

We now sequentially consider three kinds of group elements. These group elements are Galilean translations, which we denote by $T$; Galilean boosts, which we denote by $B$; and Galilean rotations, which we denote by $R$.

The translations are transformations between two frames that are not in relative motion and not rotated with respect to each other. The translations are represented by $5\times 5$ matrices of the following form:

$$T(X,\,Y,\,Z,\,\tau) = \left(\begin{array}{ccccc}1&0&0&0&X\\0&1&0&0&Y\\0&0&1&0&Z\\0&0&0&1&\tau\\0&0&0&0&1\end{array}\right)\tag{2}.$$

Please note these group elements are translations in space as well as time. Try multiplying a column vector of the form (1) on the left by one of the matrices in (2). You will find that it represents a translation in space and time as the values in (1) transform by $x\mapsto x+X,\,y\mapsto y+Y,\,z\mapsto z+Z,\,t\mapsto t+\tau$.

The boosts are transformations between two frames that are in relative motion, but are not rotated with respect to each other. The boosts are represented by $5\times 5$ matrices of the following form:

$$B(v_x,\,v_y,\,v_z) = \left(\begin{array}{ccccc}1&0&0&v_x&0\\0&1&0&v_y&0\\0&0&1&v_z&0\\0&0&0&1&0\\0&0&0&0&1\end{array}\right)\tag{3}.$$

The rotations are transformations between two frames that are not in relative motion but that are rotated with respect to each other. The rotations are represented by $5\times 5$ matrices of the following form:

$$R(\theta\,\gamma_x,\,\theta\,\gamma_y,\,\theta\,\gamma_z) = \left(\begin{array}{c|c}\exp\left(\theta\left(\begin{array}{ccc}0&-\gamma_z&\gamma_y\\\gamma_z&0&-\gamma_x\\-\gamma_y&\gamma_x&0\end{array}\right)\right)&\mathbf{0}_{3\times2}\\\hline\mathbf{0}_{2\times3}&\mathrm{id}_2\end{array}\right)\tag{4}.$$

In (4), $\mathrm{id}_2$ is the $2\times 2$ identity matrix, $\mathbf{0}_{3\times2}$ three rows of two columns of noughts and $\mathbf{0}_{2\times3}$ two rows of three columns of noughts. Equation (4) represents a rotation of angle $\theta$ about an axis with direction cosines $\gamma_x,\,\gamma_y,\,\gamma_z$. Please note that though there are four symbols that are used for a rotation element, there are only three parameters. These parameters are $\theta\,\gamma_x$, $\theta\,\gamma_y$, and $\theta\,\gamma_z$.

Any Galilean transformation can be composed by a product of the form $T(X,\,Y,\,Z,\,\tau)\,B(v_x,\,v_y,\,v_z)\,R(\theta\,\gamma_x,\,\theta\,\gamma_y,\,\theta\,\gamma_z)$. Please note that there are precisely ten independent real parameters needed to specify the transformation: three parameters for Galilean rotations (i.e., $\theta\,\gamma_x$, $\theta\,\gamma_y$, and$\theta\,\gamma_z$); three parameters for Galilean boosts (i.e., $v_x$, $v_y$, $v_z$); and four parameters for Galilean translations (i.e., $ X$, $Y$, $Z$, $\tau$).