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Special relativity

Lorentz transformations

In special relativity (SR), each standard inertial frame40 F is associated with a set {x0, x1, x2, x3} of four coordinates that covers the whole of Minkowski spacetime, M4. Here x0 = ct, where t is the laboratory time of clocks at rest in the frame and x1, x2, x3 are identified with standard spatial Cartesian coordinates commonly denoted by x, y, and я.41 SR deals with the transformation rules between different standard inertial frames. Note that all references to observations by F really mean the data collected by the observer chorus at rest in F, and so on.

There are two kinds of SR transformation. The more general are the Poincare transformations. These take the form

Here the coefficients A^ v form a set of numbers satisfying certain relationships, the x?“ are the ‘old’ coordinates used in F, the x'^ are the ‘new’ coordinates used in F', the a?“ represent a shift in the origin of spacetime coordinates, and there is an implied summation of the repeated index v. The less general transformations, which do not involve changes in the origin of coordinates, take the form

and are known as Lorentz transformations.

A useful simplification is to align the respective spatial axes so that there is no spatial rotation of any of them during the transformation. A final simplification is to take the velocity of the ‘new’ frame F' to be along the x-axis, with component v

  • 40 ‘Standard’ here means that each inertial frame chorus of observers has a copy of the latest version of the International Organization for Standards (ISO) handbook specifying the SI system of units and uses it to follow a universally agreed protocol for setting up their apparatus, system of units, and measurement protocols.
  • 41 Note that x2 does not mean x-squared but the second spatial coordinate, y.

Images of Time. First Edition. George Jaroszkiewicz.

© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.

in that direction, relative to the ‘old’ frame F. These simplifications reduce to the standard Lorentz transformation

where у = 1/^/1 - в2 is the Lorentz factor and в = v/c.

A crucial principle used by Einstein in his discussion of SR in 1905 [Einstein, 1905b] was that no inertial frame has a special status above any other inertial frame. Therefore, the inverse transformation taking us back to F from F' should take the same form, the only difference being a reversal of the relative velocity, that is, we replace в by -в. This gives the transformation

which is readily shown to be the inverse transformation of (17.3).

Implicit in this transformation is the assumption that the ‘one-way speed’ of light is the same in each direction, that is, the speed of light in the positive x- direction is the same as in the opposite direction. Although this is intuitively reasonable, it was noted by Tangherlini that the Michelson-Morley experiment did not prove that this one-way speed assumption is necessary: all that is needed is that the average speed around a closed path in a given frame is the speed of light [Tangherlini, 1958]. In Chapter 18 we discuss Generalized Transformations and Tangherlini’s version of SR.

 
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