Lorentz Transformation
Lorentz Derivation | Inverse Transform

Deriving the Lorentz Transformation

As shown in the illustratin below, the coordinates y and z are transverse to the direction of the relative motion between the spaceship and space station have the same values in both frames (where primes denote spaceship coordinates):

(eq. 1.0)

Then given

The invariance of the interval gives us a relation between t and t’,

from which

or

(eq. 2.0)

 For simplicity, and since this gives the ratio of observed clock rates, we will call and define a time stretch factor as

Substituting , equation 2.0 becomes

[when x’=0] (eq. 3.0)

Substituting again into equation above to find laboratory position in terms of spaceship measurements:

[when x’=0] (eq. 4.0)

 The following equations give the first answer to the question, "If we know the space and time coordinates of an event in one free-float frame, what are its space and time coordinates in some other overlapping free-float frame, what are its space and time coordinates in some other overlapping free-float frame?"

These equations are limited, however, since they apply only to a particular situation: one in which both events occur at the same place (x’=0) in the spaceship.

Since any event can be the reference event then the form of the Lorentz transformation will be a linear transformation. This means that the space station coordinates x and t are related to linear (first) power of spaceship coordinates x’ and t’ equations of the form

(eq. 5.0)

From here we need to find expressions fro the coefficients B, D, G, and H that do not depend on either the space station or the spaceship coordinates of a particular event, though the do depend on the relative speed .

Equations 3.0 and 4.0 provide coefficients D and H called for in equation 5.0;

(eq. 6.0)

We know nothing about the two constants B and G since events so far considered occurred at point x’ = 0 in the spaceship. Therefore, the two coefficients B and G could have any finite values without affecting the numerical results of the calculation. To determine B and G we must look at a more general events, one that occurs at a point with arbitrary spaceship coordinates x’ and t’. The we demand that the spacetime interval have the same numerical value in space station and spaceship frames for any event whatever:

Substitute expressions for t and x from equation .0:

On the left side, multiply out the squares. This leads to the result:

Group together coefficients of , coefficients of , and coefficients of the cross-term x’t’ to obtain

(eq. 7.0)

Now, t’ and x’ can take on any value since they represent the coordinates of an arbitrary event. Under these circumstances, it is impossible to satisfy equation 7.0 with a single choice of values of B and G unless these values make the coefficient of x’t’ on the left side of equation 7.0 vanish as it does on the right:

With the understanding that can never equal zero, the value of equals unity when and is great than this for any other values of . Hence the left side of this equation can only be zero only if

or (eq. 8.0)

Next, B and G must be such as to make the coefficient of x’2 equal on the left and right of equation 7.0; hence

(eq. 9.0)

Substitute B from equation 8.0 into equation 9.0:

or

Divide through by and take the square root of booth sides:

But the right side is just the definition of the time stretch factor , so that

Substitute this into equation 8.0 to find B:

These results plus equations 1.0 and 6.0 yield the Lorentz transormation equations:

(eq. 10.0)

or, substituting for the value of gamma, :

          (eq. 11.0)

and

In summary, the Lorentz transformation equations rest fundamentally on the required linearity of the transformation and on the invariance of the spacetime interval.

Lorentz Transformation
Lorentz Derivation | Inverse Transform