Consider a truck (just for a change) moving with velocity v1 in the x-direction with respect to the ground. Inside the truck a ball is thrown with velocity v2 with respect to the truck, also in the x- direction. Call the frame of the truck F1 and the frame of the ground F2. The question is this: what is the velocity of the ball with respect to the ground? Under the Galilean transformations the answer is intuitive and obvious: the ball is moving with velocity v = v1 + v2 with respect to the ground. Things are quite different in relativity. We know that v, the velocity of the ball with respect to the ground is given by v = , where the subscripts refer to the frame F2. Since F1 is moving with respect to F2, we can use the lorentz transformations to write:
|Δx2 = //Δt2 =|
|v = =|
|v = =|
A Minkowski diagram or spacetime diagram is a convenient way of graphically representing the lorentz transformations between frames as a transformation of coordinates. They are especially useful for gaining a qualitative understanding of relativistic problems. We make a spacetime diagram by representing frame F as the coordinate axes x (horizontal) and ct (vertical). We are ignoring the y and z directions, since they are uninteresting. The plot of an object's x- position versus time on the Minkowski diagram is called its worldline. Notice that light, traveling one unit of ct for every unit of x will follow the line x = ct, inclined at a 45o angle.
What do the axes of F', moving with velocity v along the x-axis of F look like? Take the point (x', ct') = (0, 1). From the lorentz transformations we can find that this point transforms to (x, ct) = (γv/c, γ). As shown in the angle between the ct' and ct axes is given by: tanθ1 = x/ct = v/c. Actually, the ct' axis is just the worldline of the origin of F'. The point (x, ct) = (γv/c, γ) is a distance = γ from the origin, so the ratio of units on the ct' axis to those on the ct axis is just this value, namely:
The advantage of a Minkowski diagram is that the same worldline applies to both sets of coordinate axes (that is, to x and ct, as well as to x' and ct'). The Lorentz transformation is made by changing the coordinate system underneath the worldline rather than the worldline itself. In many situations this allows us to visualize the perspectives of the different observers more easily. If we had a very detailed and accurate Minkowski diagram we could use it to read off the values for Δx, Δct, Δx', and Δct'. To find the spacetime coordinates of an event in F, one can read the value off the x and ct axes; to find the coordinates in a moving frame the x' and ct' axes corresponding to the appropriate velocity can be constructed (using the angle formulas explained above), and the value read off using the units derived for x' and ct', above.