## X

Lorentz transformation on a space-time diagram. The transformation looks like a rotation of the axes except that the time and space axes rotate in opposite directions.

only one space coordinate, x, as well as the time coordinate. We can keep track of events in such a diagram by plotting the coordinates of the event. By convention, we have time running vertically. The effect of the Lorentz transformation is to rotate the axes through an angle whose tangent is v/c. The unusual feature is that the x-axis and t-axis rotate in opposite directions, so that the axes are no longer perpendicular to each other. Note that v = c puts both axes in the same place. It should not surprise us that something funny happens when v = c, because this is where the quantity y becomes infinite.

We know that in ordinary three-dimensional space, a rotation changes the coordinates of an object, but the lengths of things are unchanged. That is, if we have two objects, as shown in Fig. 7.9, whose separations are given in one coordinate system by (Ax, Ay, Az) and (Ax', Ay', Az') in another, then the distance between the two, which is the square root of the sum of the squares of the coordinate differences, doesn't change. That is

(Ax)2 + (Ay)2 + (Az)2 = (Ax')2 + (Ay')2 + (Az')2 (7.11)

We say that the length is invariant under rotation.

Since the Lorentz transformation has properties of a rotation, is there a corresponding concept

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