Martin & Shaw, appendix A.

Recall from last lecture:


dot product of 4-vectors

example of energy available in fixed-target vs. colliding beam experiments

Consequences of the Lorentz transformation

Newton's laws are invariant under Galilean transformations:

However Maxwell's equations are not invariant under Galilean transformations. Maxwell's equations are invariant under Lorentz transformations, as originally noted by Lorentz. Special relativity arises from the conjecture that the Lorentz transformation is correct in all instances. The Galilean transformation is an approximation valid in the limit v«c, where most of our real world experiences lie (hence our prejudice for the Galilean transformation).

I'll discuss four of the important consequences of the Lorentz transformation.

  1. The relativity of simultaneity: If two events occur at the same time in frame S but at different locations, then they do not occur at the same time in frame S'. Specifically, for ta = tb and xa ≠ xb then t'a = t'b + γβ(xb-xa).

  2. Lorentz contraction: When viewed from a reference frame moving with respect to an object along the x-axis, the length of the object in x appears shortened. The y and z lengths are unchanged.

    Consider a stick of length L', oriented along the x axis. In its rest frame, if one end of the stick is at x'=0, the other end is at x'=L'. In a moving frame, we must record the positions of the ends of the stick at the same instant in time, say t=0. If we say that one end is at x=0 at this time, then the position of the other end is given by L'=γ(x-βt) = γx, or x = L'/γ. Since γ is always greater than one, the length of the stick appears shorter in the moving frame.

  3. Time dilation: Suppose in frame S' an interval of time runs from t'=0 to t'=T' always at x'=0. How long is this interval in S? It starts at t=γ(t' + βx') = 0, and it ends at t=γT'. Therefore the interval appears longer in the moving frame S, since γ is always greater than one.

    A clock is basically a device that counts the passage of intervals. As seen from S, a clock in S' appears to run slow. This fact is an important correction in the GPS system. Also, see how this applies in the muon decay homework problem.

  4. Velocity addition: Suppose an object is moving in the x direction with speed u' in S'. What is its speed u as measured in S?

    It travels a distance Δx = γ(Δx' + β Δt') in a time Δt = γ(Δt' + β Δx') so, dividing and recalling that Δx/Δt = u and Δx'/Δt' = u' yields:

    u = (u' + v)/(1 + u'v/c2).

    The numerator is the Galilean result, u = u' + v. The denominator represents the relativistic correction, and guarantees that upon addition, the resulting velocity is not greater than c.

Summation Notation

An alternative notation is often used for the transformation. This notation makes the equations a bit more compact and easier to manipulate. It comes about by realizing that we can organize the 4-vector as a 1-column by 4-row matrix. Then the Lorentz transformation is:

xμ = Λμν xν
and summation over the repeated index ν is implied.

Copyright © Robert Harr 2005