Minkowski Space

A geometric view of the Lorentz transformation. The moving frame uses the blue coordinates. The lab frame uses the gray coordinates.

Coordinate transformations are ubiquitous in physics. Every freshman physics student quickly learns that it is easier to solve a block on an incline plane if the x and y axis are chosen to lie parallel and perpendicular to the incline. A rotation of spatial coordinates doesn't change the physics but it makes the math necessary to do the problem simpler. Much of the physics of the free electron laser is easiest to calculate in the rest frame of the electron. Unfortunately, the electron in traveling close to the speed of light and so the Lorentz transformation derived from the special theory of relativity is needed. We will observe that this transformation produces very odd effects on space and time. Other Java applets will show show how the Lorentz transformation effects the electric field of an electron thereby producing both magnetic fields and synchrotron radiation.

Rotation

Use the Minkowski applet to first study the rotation of the Cartesian coordinate gird. Assume that the x axes of the two systems coincide, i.e., they overlap. Then the two coordinate systems, call them the lab and incline systems, have the same x values but different y and z values. The transformation from the lab, (x,y,z), to the incline,(x',y',z') coordinate frames can be written in convenient matrix notation as:

Since the x values do not change, a compact 2x2 matrix that transforms only the y and z values is all that is really needed.

Select rotation as the transformation type and observe the coordinate grid as you change the angle. The grid remains undistorted and the distance between any two points is the same in both coordinate systems. In fact, the distance between points in the x,y plane does not depend on the angle of rotation. Such quantities are said to be invariant.


The Lorentz Transformation

The Lorentz transformation was used by Albert Einstein in his 1905 paper on special relativity. It is also a coordinate transformation although the coordinates space has four dimensions. Every physical event-- such as a collision between two cars or the birth of a child--can be defined by three spatial and one temporal coordinate, usually written as (x,y,z,t). The Lorentz transformation allows us to predict the spatial and temporal measurement obtained by another observer moving with velocity V with respect to the first observer. We will call the values obtained by the second observer (x',y',z',t'). We are, of course, free to choose an origin in each coordinate systems and the direction for the relative motion. In order to make the math as simple as possible, we set x=x'=0, y=y'=0, and z=z'=0 at the time t=t'=0. We also choose the z and z' axes to lie along the direction of relative motion. Two of the spatial coordinates are now the same for both observers, x=x' and y=y', and their transformation is uninteresting. The two remaining coordinates, z and t, can be plotted on a two dimensional grid. There transformation is again expressed most easily in matrix notation as:


where tanh(q)= v/c.

The Minkowski Java applet implements this transformation. Space and time units have been chosen to make the speed of light, c, equal to unity. To see how the applet works enter the points (z'=3, t'=2) and (z'=4, t'=2) in the moving frame. Notice that these points lie on the intersection of the blue coordinate lines in the applet. Notice also that these points do not have the same time (i.e., vertical) component in the lab frame. Events that are simultaneous in the moving frame are not simultaneous in the lab frame. Notice also that the z separation between the two points is different in the two coordinate systems.

Use the slider to vary the relative speed between the two reference frames and notice that change in the blue coordinate grid. The coordinate grid is squeezed and stretched. (Only a few of the coordinate lines are actually drawn on the screen.) The distance between any two points on the diagram is not the same when measured in the other frame. There is, however, an invarient that is not changed by the Lorentz transformation. This invarient is (c t)2 - z2.

Play with the applet. What are the coordinates for a light beam that starts at x=0 at t=0? (i.e., Where is the light beam at t=1,2,3,4....) Enter the coordinate values in the lab frame and observe the path of the light beam through space-time. Now enter the coordinate values for a moving observer who performs the same experiment. Remember that the speed of light is the same, i.e., the speed is one, in both frames.

Question: Visualize what the Galilean transformation would do to the (z,t) coordinates using the same assumptions about the origin as were made for the Lorentz transformation?