# This Quantum World/Appendix/Relativity/Lorentz transformations

### 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:

- their spacetime coordinate origins coincide ( mark the same spacetime location as ),
- their space axes are parallel, and
- 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.