1. Introduction

It is well known that two inertial frames in special relativity are related via the LT. What appears to be unknown is how to derive the LT without making any additional assumptions about the form of the transformation. In most cases, authors presume that the transformation is linear, i.e., that it has the form

(1)

where are constants to be determined. As will be elaborated upon in the following section, some authors will, at best, give a “hand-waving” argument as to why the transformation is linear. Failure to provide a rigorous proof of why the two postulates imply linearity leaves open the possibility that there may be a nonlinear transformation that is consistent with the two postulates. Our ultimate goal is to eliminate nonlinearity as a possibility.

In the following section we will discuss several example derivations of the LT that appear in the literature. Subsequently, we will provide a rigorous proof that the LT is a consequence of the two postulates without making any additional assumptions.

2. Example Derivations

Let us first describe the familiar setting that is common to all of the examples that we intend to discuss. Consider two inertial frames, and . We’ll assume that the frame is moving at a speed in the direction of increasing values of . Moreover, our derivation will be based on the conventional that whenever .

Since there is no motion in either the or direction it is taken that and . So all that remains to determine the interrelatedness between and .

2.1. Spacetime Physics

Arguably, Spacetime Physics by Taylor and Wheeler, [1], is among one of the most popular texts on special relativity. On page 100, it is asserted that the LT has the form (1).

Let us now discuss the argument given in [1] of why the two reference frames are related via a linear transformation. An observer at rest in frame S observes three events that occur in the same spatial location at three equally spaced times, say t=1, t=2, and t=3. The time dilation formula, was previously established with rigor. Hence, an observer at rest in S’ will also see these same three events occur at equally spaced times, and .

Subsequently, the transformation given by where is a constant, is considered. Under this particular transformation, the authors demonstrate that an observer in does not see the three events occurring at three equally spaced times.

The fact that equally spaced times in one frame yields equally spaced times in the other may indeed only be possible if the frames are related via a linear transformation. But to provide a rigorous proof, the task of ruling out an arbitrary nonlinear transformation must be completed.

2.2. Hyperbolic Trigonometric Functions

The invariance of the speed of light leads to the equation

(2)

where are the coordinates of an arbitrary inertial reference frame. This motivates the likes of [2] and [3] to consider a transformation of the form

(3)

What motivates (3) is the fact that follows from both (2) and (3). However, once again, we have a situation whereby is tacitly assumed to be independent of both and . So not only is (3) a linear transformation, it’s a linear transformation of a particular form. It’s worth re-emphasizing that we are not arguing that the (3) is incorrect. We will ultimately prove that it is correct. But if is simply assumed to be independent of and , then a transformation of the form (3) in which depends on and/or remains possible.

It is worth pursuing the argument presented in [2] a bit further. In letting it follows from (3) that

But is the velocity of relative to the . Thus,

(4)

From this it is easily shown that

(5)

where is known as the Lorentz factor.

Substituting (5) into (3) yields the familiar LT. The drawback of this approach is that there’s still no guarantee that is independent of x and t. For example, in choosing the above argument can be implemented to draw the same conclusion.

2.3. Special Relativity & Classical Field Theory

Next, we discuss [4] which is one of four books of Susskind and Friendman’s “The Theoretical Minimum” series. They begin by pointing out that if and only if . This is certainly true in the way we described the frames and . Then they claim that

(6)

where is a function to be determined. Clearly, (6) certainly implies that if and only if . The converse, however, is not true. As a random example,

also ensures that if and only if . Yet again we have a situation in which prior knowledge of the LT is used to derive the LT.

2.4. On the Electrodynamics of Moving Bodies

On page 6 of [5], it is written “In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.” There are two points to be made about this assertion. The first is that Einstein is now going beyond his two postulates to support his claim.

Secondly, it isn’t immediately clear how homogeneity guarantees that two inertial reference frames are related linearly. A common definition of the term homogeneous is that there is no special place in the universe. More precisely, at any given point in time, the universe is everywhere the same assuming that we consider sufficiently large spatial regions. In my best effort to put this in mathematical terms, let denote a physical quantity of some kind at a spatial location . Homogeneity requires the function to be invariant under spatial translations, . How this implies that two inertial reference frames are related by a linear transformation is unclear.

3. A Rigorous Derivation of the Lorentz Transformation

In this section we will derive the Lorentz Transformation using only the two postulates of special relativity. We will do so without making the presupposition that the transformation has any particular form.

For easy reference, let’s state the two postulates.

1. The Principle of Relativity: The laws of physics are the same in all inertial reference frames.

2. Invariance of the Speed of Light: The speed of light in a vacuum is constant and independent of the motion of the light source or the observer.

The length contraction and time dilation formulas are easily and rigorously derived in essentially any source on the topic. Thus, we will simply state these two formulas.

The time between two distinct events that occur at the same spatial location in the frame is related to the time in the frame by the equation

(7)

Furthermore, suppose that a rod that is at rest in the frame, and is of length as measured by an observer who is at rest in that frame. An observer in the frame will determine the length of the rod to be

(8)

We will now proceed to determine how can be written in terms of and . As asserted in Section 2, travels in the direction of increasing values of at a speed . That is, given an arbitrary constant the plane in the frame travels in the direction of increasing values of at speed . So a stationary observer in will describe the motion of such a traveling plane via the kinematic equation where is the distance between the traveling plane and the plane at according to the observer in .

Let us consider a specific example in which .

Figure 1. The plane at time

Figure 1 depicts a snapshot of the plane at the moment in which the planes and align. While a stationary observer in at will see the plane as stationary and 17 units away, a stationary observer in at will see the plane at a contracted length of units away at that moment. Thus, we substitute for in the kinematic equation. Doing so yields

which can be rewritten as

Of course there is nothing special about the number 17. An arbitrary but fixed value of refers to a stationary plane in and a plane traveling at speed in . Thus, the very same argument can be applied to that was applied to 17 to obtain

(9)

Our next task will be to obtain in terms of and . In proving (9) we viewed planes of the form as traveling in the direction of increasing values of at speed . It is equally legitimate to consider as the moving frame and as the stationary frame. That is, we can repeat the same method that we used to obtain (9) to deduce

(10)

It follows from (10) that or equivalently

By (9), we can substitute for in the above equation. Doing this and a little algebra yields

(11)

as desired.

Proving (9) and (11) was our primary objective for this section. It’s worth noting that (9) and (11) can be combined to conclude that

which holds whenever .

4. Conclusions

We have presented several textbook examples of how the LT is derived. In every case, the author took advantage of their prior knowledge regarding the LT’s form. This raises the question of whether the LT must be of the stated form. In particular, on the topic of whether the transformation must be linear, a hand-waving argument is presented at best. We also discussed the shortcomings of Einstein’s reasoning of why the LT is linear as presented in [5].

Next, we provided a rigorous derivation of the Lorentz transform without presupposing that it has any particular form from the start.