reference frames - What does a Galilean transformation actually mean?

What does a Galilean transformation actually mean? I'm having trouble defining the equation for displacement shifts $x'=x-vt$. Does it mean that to any event $C$ the displacement in the primed coordinate system is the displacement to $C$ minus the velocity times time of the primed coordinate system? If so then don't both coordinate systems have to be at the same event at $t = 0$? I don't see this specified anywhere I look. Also, how does it directly show invariance of the euclidean distance between two points?

Answer

You have three question marks in your question. Addressing each of them separately, in chronological order;

1. Yes, provided the primed co-ordinate system is the moving one and it is moving along the positive direction of the x-axis.


2. Yes, they have to be; that is the initial condition of Galilean transformation.


3. Euclidean distance between $2$ points, ($x_1,y_1$) and ($x_2,y_2$) is given by: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Apply the Galilean transformation and check that the form of the expression written above remains the same.