This Quantum World/Appendix/Relativity/Lorentz transformations

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Lorentz transformations (general form)[edit]

We want to express the coordinates and of an inertial frame  in terms of the coordinates and of another inertial frame  We will assume that the two frames meet the following conditions:


  1. their spacetime coordinate origins coincide ( mark the same spacetime location as ),
  2. their space axes are parallel, and
  3.  moves with a constant velocity  relative to 


What we know at this point is that whatever moves with a constant velocity in  will do so in  It follows that the transformation maps straight lines in  onto straight lines in  Coordinate lines of  in particular, will be mapped onto straight lines in  This tells us that the dashed coordinates are linear combinations of the undashed ones,



We also know that the transformation from to  can only depend on  so and  are functions of  Our task is to find these functions. The real-valued functions and  actually can depend only on so and A vector function depending only on must be parallel (or antiparallel) to  and its magnitude must be a function of  We can therefore write and (It will become clear in a moment why the factor is included in the definition of ) So,



Let's set equal to This implies that As we are looking at the trajectory of an object at rest in  must be constant. Hence,

Let's write down the inverse transformation. Since moves with velocity relative to it is



To make life easier for us, we now chose the space axes so that Then the above two (mutually inverse) transformations simplify to



Plugging the first transformation into the second, we obtain


     


The first of these equations tells us that

 and 

The second tells us that

 and 

Combining with (and taking into account that ), we obtain

Using to eliminate  we obtain and

Since the first of the last two equations implies that we gather from the second that

 tells us that  must, in fact, be equal to 1, since we have assumed that the space axes of the two frames a parallel (rather than antiparallel).

With and yields Upon solving for  we are left with expressions for and depending solely on :

Quite an improvement!

To find the remaining function we consider a third inertial frame  which moves with velocity relative to  Combining the transformation from to 



with the transformation from to



we obtain the transformation from to :


     
     


The direct transformation from to must have the same form as the transformations from to and from to , namely



where is the speed of relative to  Comparison of the coefficients marked with stars yields two expressions for  which of course must be equal:



It follows that and this tells us that



is a universal constant. Solving the first equality for we obtain

This allows us to cast the transformation

into the form



Trumpets, please! We have managed to reduce five unknown functions to a single constant.