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Lorentz transformations

In the years before Einstein’s 1905 paper, Lorentz attempted to resolve the problem by showing that Newtonian physics would indeed lead to length contractions as proposed by FitzGerald. By looking carefully at Maxwell’s equations, he came up with the so-called Lorentz transformations, which are the correct transformation rules between two standard inertial frames, assuming their coordinate axes are aligned and the origins of spatial and temporal coordinates coincide. Given two such frames, F and F', the Lorentz transformation from F to F' is

where у (v) is the Lorentz factor у (v) = 1/V1 - v[1] [2]/c[2]. We shall discuss this transformation further in the next chapter. Points to note are: every inertial frame say that objects moving relative to them have shrunk: Einstein’s SR is symmetrical in that respect.

2. The significance of the parameter v is seen as follows. Consider the origin of spatial coordinates (x',y', z')' = (0,0,0)' in F'. This corresponds to

or x = vt, the equation of the worldline of an object moving uniformly with respect to F with velocity v = (v, 0,0).

3. The form of the Lorentz factor tells us that we shall run into mathematical problems if the magnitude of the relative velocity between inertial frames is equal to c or greater. Then the Lorentz factor either diverges or is imaginary. Both situations are regarded as unphysical. This leads to the conclusion in SR that nothing can travel faster than the speed of light.

4. We may rewrite the Lorentz transformation in matrix terms, viz., where в = v/c.

5. The inverse transformation, from F' to F exists provided the determinant of the above 4 x 4 matrix is non-zero. A calculation gives this determinant to be unity, so the inverse transformation always exists. It is easy to obtain the inverse Lorentz transformation. We just recall that if frame F sees frame F' moving with velocity v, then frame F' will see F moving with velocity -v.

From this we deduce that the inverse transformation is

since у (-v) = y (v).

  • 6. If the frames F and F' are not in standard configuration (spatial axes aligned), then it is a straightforward matter to introduce purely spatial rotations which take this into account. Likewise, we can shift the origin of coordinates in space and time suitably.
  • 7. Lorentz transformations involve c in various places. In ordinary terms, that is, in Standard International (SI) units, c is an enormous number relative to unity. This gives us a way to understand the relationship between Galilean and Lorentz transformations. First, expand the Lorentz factor in powers of в:

Then the original Lorentz transformation (16.13) looks like

where O(fi) means ‘terms of the order в, в2, etc.’. For ordinary speeds, в is entirely negligible and so we recover the standard Galilean transformation (16.1) in the limit в ^ 0. Note that this limit could be reached in two ways: either v ^ 0 or c ^ю.

  • [1] Transformations in physics come in two varieties: active and passive.
  • [2] Mathematically, the above Lorentz transformation appears to be a mererelabelling of events in space-time, so it can be considered to be a passive transformation. On the other hand, FitzGerald, Larmor, and Lorentzthought that length contraction was a real phenomenon, so from their perspective, the transformation is very much an active one. In other words,they believed objects moving relative to the Aether actually shrink. However, from the perspective given by Einstein, discussed in the next chapter,objects do not shrink: it is the protocols of observation in different inertial frames that gives that impression. According to Einstein, observers in
  • [3] Mathematically, the above Lorentz transformation appears to be a mererelabelling of events in space-time, so it can be considered to be a passive transformation. On the other hand, FitzGerald, Larmor, and Lorentzthought that length contraction was a real phenomenon, so from their perspective, the transformation is very much an active one. In other words,they believed objects moving relative to the Aether actually shrink. However, from the perspective given by Einstein, discussed in the next chapter,objects do not shrink: it is the protocols of observation in different inertial frames that gives that impression. According to Einstein, observers in
 
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