[ Home - Physics - Philosophy - Archive - Reference - Authors]

Lorentz transformation

The Lorentz Transformations are mathematical equations that allow us to transform from one coordinate system to another. Why would we want to do this? Because special relativity deals with frames of reference. When you analyze properties from one frame to another, it is necessary to first transform from one coordinate system to another. Thus, we can utilize the Lorentz Transforms to convert length and time from one frame of reference to another. For example, if you are flying in an airplane and I am standing still on the ground, you could apply the transformations to transform my frame of reference into your frame of reference and I could do the same for you in my frame of reference. The previous statements imply that lengths and times are not the same for objects that are in motion with respect to each other. As unbelievable as this may seem, it is a result of SR. Einstein utilized the transformations because they provide a method of translating the properties from one frame of reference to another when the speed of light is held constant in both.

Let the reference frame S be one in which a light source is at rest. Positions and times measured by an observer in this frame will be denoted by the unprimed symbols x, y, z, t. If a light source is at the origin of the frame S and a spherical wave front is emitted at t = 0, the front will be at a distance ct from the origin at time t. The equation of the spherical wave front is

Let S' be the moving reference frame. Positions and times measured by an observer in this frame are denoted by the primed symbols x', y', z', t'. For convenience we suppose that the zero of t' coincides with the zero of t, and that two origins coincide this zero of time when the wave front under consideration leaves. Then to an observer in S' the equation of the spherical wave front must be

These two equations contain the spirit of both of Einstein's hypotheses. Firstly, the speed of light is taken to be c in both frames. Also, there is no preferred frame. We have distinguished the frames by priming one and not the other, but this was arbitrary and we could work out the transformation with things interchanged. Neither frame is "preferred".

There is really just one wave front. We are trying to describe it in two different inertial reference frames. Stated another way, we want a transformation from primed to unprimed coordinates so that there is just one wave front.

For example, let's say we still wished the G.T worked. Assume S' is moving in the +x direction relative to S at velocity V. This is a sign convention -- we could let V be negative, in which case S' would move in the -x direction. The G.T. connects the primed and unprimed coordinates according to:

• x' = x - Vt
• y' = y
• z' = z
• t' = t

If we substitute these into the equation for the spherical wave front in the primed coordinate system, we get

This is not the equation for a spherical wave front centered at the origin of the unprimed coordinate system! No surprises -- we did not expect the G.T. to work, given Einstein's postulates.

We suspect that the new transformation must be trivial for y’ and z’. We need a transformation which is linear in x and t, because we want to get a sphere which expands at a uniform rate. It is clear from the result using the G.T. that we cannot leave the t' = t transformation unchanged if we want to cancel out the undesired terms containing t. Let us try next a transformation of the form

• x' = x - Vt
• y' = y
• z' = z
• t' = t + fx

"f" is a constant to be determined -- it may depend on V and c, but the coordinates themselves. The motivation for this, once again, is to mix the x and t transformations in such a way that the undesirable terms cancel in the equation of the spherical wave front in the unprimed coordinate system. If we substitute these into the S' equation

we get

Notice that the terms in "xt" on the two sides cancel if we set

With this value of f, the equation in the unprimed coordinate system becomes

This is closer to the desired result, but there remains an unwanted scale factor multiplying x^2 and t^2 terms. We can dispose of the scale factor by taking the transformation to be

This is the Lorentz transformation. It is

• linear in x and t
• reduces to the Galilean transformation for V << c, and
• when substituted in

  

  it gives

  

  as required.

Thus, we say that

is invariant under a Lorentz transformation. The form of the equation describing the wave front is the same in all frames moving with uniform relative velocity. The Lorentz transformation is the unique solution to all our difficulties.

With the now standard definitions

the Lorentz transformation adopts the more visually appealing form

The inverse transformation is easily obtained by changing signs in front of beta and changing primed variables into non-primed, and vice versa:

Now,the main features of the Lorentz transformation equations are these:

1. Lengths perpendicular to the relative motion are measured to be the same in both frames;
2. the time interval indicated on a clock is measured to be longer by an observer for whom the clock is moving than by one at rest with respect to the clock;
3. lengths parallel to the relative motion are measured to be contracted compared to the rest lengths

[ Home - Physics - Philosophy - Archive - Reference - Authors]