Intuitions:

While the special relativity of Einstein, Lorentz, and Minkowski is typically considered less intuitive than Galilean relativity, I daresay that most engineers haven’t bothered to consider some of its implausible implications. That is, in the classical approximation, the force acting on particles is given by the potential energy function which is a function of the coordinates of the interacting particles. So it is implicitly assumed that signal propagation is instantaneous.

This necessarily leads us to the counter-intuitive and problematic situation that the velocity of signal propagation among interacting particles must be infinite. In this context, the objective of this article is to present special relativity as an empirically-valid remedy to this rather unphysical assumption as well as a coherent explanation for the Michelson-Morley experiments in 1887.

In what follows, I present the principles of Galilean relativity and special relativity as competing approximate descriptions of physical processes, emphasising that they rest upon very different ontologies. Furthermore, I show that the special theory of relativity leads to fundamentally new insights.

The Galilean group, or the isometries of Galilean frames:

Galilean relativity is essentially the combination of two fundamental principles. First, in order to describe any physical process in nature we need a reference frame. Second, it is possible to obtain arbitrarily many such frames that describe identical dynamics provided that these frames move at constant velocity relative to each other.

Such frames are known as inertial frames, and given that classical dynamics depends entirely upon relative positions of interacting particles, it is possible to describe the set of inertial frames succinctly by analysing the isometries of Galilean reference frames.

The reader may verify that given a particular reference frame, arbitrarily many inertial frames may be generated by Galilean transformations:

  1. Uniform motion with velocity \(\vec{v}\):

    \begin{equation} \forall \vec{v} \in \mathbb{R}^3, (\vec{x},t) \mapsto (\vec{x} + t\vec{v},t) \end{equation}

  2. Given \(\vec{y}\) and \(s \in \mathbb{R}\), we may define spatiotemporal translations via:

    \begin{equation} (\vec{x},t) \mapsto (\vec{x} + \vec{y},t+s) \end{equation}

  3. Rotations, \(G: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) are given by:

    \begin{equation} (\vec{x},t) \mapsto (G\vec{x},t) \end{equation}

In addition, the reader may check that the composition of Galilean transformations is another Galilean transformation and that the composition operation is not commutative.

Newton, Lagrange, and Hamilton successfully built upon the Galilean description of physical systems, and it is the approximation of reality most engineers are familiar with. However, the Michelson-Morley experiment demonstrated that Galilean relativity does not hold for physical systems that move close to the speed of light.

So the Galilean model not only contained unphysical assumptions but became experimentally untenable. It had to be replaced.

Lorentz transformations and the wave nature of light:

Some authors say that the Michelson-Morley experiment was a test of Einstein’s theory of special relativity. However, this is both historically and scientifically inaccurate. The Michelson and Morley experiment which required immense work and spanned several years was completed in 1887 whereas physicists struggled to account for the Michelson-Morley observations until Einstein published his theory of special relativity in 1905. In fact, it would be more accurate to say that the Michelson-Morley experiments were a test of Galilean relativity.

The aether hypothesis:

Robust wave-theories of light date back to Huygens(1678), and this perspective grew in importance when Maxwell developed his theory of electromagnetic waves. However, given what was known about wave-like signals(such as sound), most physicists suspected that light travelled through a medium. This was known as the aether hypothesis.

Meanwhile, Lorentz(a proponent of the aether hypothesis) discovered that Maxwell’s equations possessed different invariants that were unlike Galilean transformations.

Lorentz transformations:

Without loss of generality, let’s suppose we have two reference frames that are parallel to the x-axis. One fame \(K\) is stationary and measures the velocity of light emitted by \(K'\) which moves at velocity \(u\) in the direction opposite the velocity of light. The theory of Galilean relativity suggests that we should measure:

\begin{equation} \dot{x}’ = \dot{x}-u= c-u \end{equation}

However, this is not what experimentalists observed. Instead, Lorentz discovered the following invariant of Maxwell’s equations:

\begin{equation} x’ = \frac{x-ut}{\sqrt{1-\frac{u^2}{c^2}}} \end{equation}

\begin{equation} y’=y \end{equation}

\begin{equation} z’=z \end{equation}

\begin{equation} t’ = \frac{t-ux/c^2}{\sqrt{1-\frac{u^2}{c^2}}} \end{equation}

It’s worth noting that these transformations are general. If \(K\) moves relative to \(K'\) with constant velocity \(V\) we may choose a coordinate system such that \(K\) moves with respect to \(K'\) along a single dimension(\(x\),\(y\), or \(z\)).

The Michelson-Morley experiment, a test of Galilean relativity:

figure 1: A sketch of the Michelson-Morley interferter

Given that light travels at a ridiculous speed, physicists would have to wait a little before Michelson and Morley completed their clever experiment in 1887. Using the principle of Galilean relativity, we may analyse the experiment as follows.

If the apparatus moves at velocity \(u\), and the time taken for light to go from \(B\) to \(E'\) is \(t_1\):

\begin{equation} c \cdot t_1 = L + u \cdot t_1 \implies t_1 = \frac{L}{c-u} \end{equation}

If the time taken for light to return from \(E'\) to \(B'\) is given by \(t_2\):

\begin{equation} (c+u) \cdot t_2 = L \implies t_2 = \frac{L}{c+u} \end{equation}

and the total time is given by:

\begin{equation} t_1 + t_2 = \frac{2Lc}{c^2-u^2}= \frac{2L/c}{1-u^2/c^2} \end{equation}

Now, we may also calculate the time \(t_3\) for light to go from \(B\) to \(C'\):

\begin{equation} (c \cdot t_3)^2 = L^2 + (u \cdot t_3)^2 \implies L^2 = t_3^2 \cdot (c^2-u^2) \end{equation}

so we have:

\begin{equation} t_3 = \frac{L}{\sqrt{c^2-u^2}} = \frac{L/c}{\sqrt{1-u^2/c^2}} \end{equation}

Due to symmetry, the total time is given by:

\begin{equation} 2 \cdot t_3 = \frac{2L/c}{\sqrt{1-u^2/c^2}} \end{equation}

and if we compare the times \(t_1 + t_2\) with \(2 \cdot t_3\) we should find a discrepancy:

\begin{equation} \frac{t_1+t_2}{2 \cdot t_3} = \frac{1}{\sqrt{1-u^2/c^2}} > 1 \implies t_1 + t_2 > 2 \cdot t_3 \end{equation}

At least, this is what we should expect from a Galilean perspective. However, this is not what Michelson and Morley observed although their interferometer was sufficiently sensitive to detect such a subtle difference.

Relativistic analysis of the Michelson-Morley experiment:

Given that the movement of the interferometer is parallel to the velocity of the light wave from A, we may consider that we have two reference frames on the x-axis.

At \(E'\), the rod of length \(L\) is perceived to move along the x-axis with a speed \(u>0\) whereas at \(B\), a rod of length \(L\) is perceived to be at rest because \(B\) moves with this rod at the same velocity.

Since \(E'\) is observed to approach \(B\), the relation between \(B\) and \(E'\) is given by:

\begin{equation} x = \frac{x’-ut}{\sqrt{1-\frac{u}{c}^2}} \end{equation}

\begin{equation} \implies \Delta x = \frac{\Delta x’}{\sqrt{1-\frac{u}{c}^2}} \end{equation}

and if we set \(\Delta x := L\), the proper length of the rod:

\begin{equation} L’ = L \cdot \sqrt{1-\frac{u}{c}^2} \end{equation}

Substituting (18) into (11) would give us:

\begin{equation} t_1 + t_2 = 2 \cdot t_3 \end{equation}

so the relativistic analysis effectively gives us the correct answer.

The principle of special relativity and the hyperbolic geometry of Lorentz transforms:

The principle of special relativity:

Looking back, it appears that Lorentz, Michelson, Poincaré and the other scientists that preceded Einstein’s 1905 publication, possessed the mathematical devices that were sufficient to develop the special theory of relativity. However, Michelson and Morley’s 1887 discovery that there was no aether meant that a fundamentally different ontology was required.

The simplest possible explanation, the one adopted by Einstein is that all spacetime descriptions of physical processes are invariant to Lorentz transformations, and not Galilean transforms. This new principle of relativity, what we call special relativity, has the particular implication that the velocity of signal propagation of gravitational and electromagnetic interactions is the same in all inertial systems. From this it follows that the velocity of signal propagation is a universal constant, the speed of light in empty space.

The hyperbolic geometry of the Lorentz transformation:

If we ignore the stationary terms in the Lorentz transform (5-8), we find:

\begin{equation} x’ = \frac{x-ut}{\sqrt{1-u^2/c^2}} \end{equation}

\begin{equation} t’ = \frac{t-ux/c^2}{\sqrt{1-u^2/c^2}} \end{equation}

Now, if we consider that all velocities only make sense relative to the speed of light, we ought to use the term \(u/c\). It’s also worth noting that time only makes sense with respect to signal propagation \(ct\) we may transform the above equation as follows:

\begin{equation} x’ = \frac{x-\frac{u}{c}(ct)}{\sqrt{1-u^2/c^2}} \end{equation}

\begin{equation} ct’ = \frac{ct-x \cdot \frac{u}{c}}{\sqrt{1-u^2/c^2}} \end{equation}

If we employ the substitution \(\tanh \phi = \frac{u}{c}\), we find:

\begin{equation} \frac{1}{\sqrt{1-\tanh^2 \phi}} = \cosh \phi \end{equation}

so we have:

\begin{equation} x’ = x \cdot \cosh \phi - ct \cdot \sinh \phi \end{equation}

\begin{equation} ct’ = ct \cdot \cosh \phi - x \cdot \sinh \phi \end{equation}

and the reader may check that:

\begin{equation} (x’)^2 - (ct’)^2 = x^2 - c^2 \cdot t^2 \end{equation}

which means that the interval between these events are preserved by the Lorentz transforms.

In fact, we may discover the hyperbolic geometry of the Lorentz transforms by analysing this interval using the variable \(s\):

\begin{equation} s^2 = x^2 - c^2 \cdot t^2 \implies \frac{x^2}{s^2}- \frac{c^2 t^2}{s^2} = 1 \end{equation}

which is the equation of a hyperbola.

Galilean relativity as an approximation of Special relativity:

Given equations (16) and (17), it is worth noting that if we compute the limit as \(c \to \infty\) we find that we recover Galilean transformations:

\begin{equation} x’ = \lim\limits_{c \to \infty} \frac{x-ut}{\sqrt{1-u^2/c^2}}= x - ut \end{equation}

\begin{equation} t’ = \lim\limits_{c \to \infty} \frac{t-ux/c^2}{\sqrt{1-u^2/c^2}} = t \end{equation}

Furthermore, we note that when \(\frac{u}{c} \approx 0\) (16) and (17) simplify to:

\begin{equation} x’ \approx x - ut \end{equation}

\begin{equation} t’ \approx t \end{equation}

so Galilean relativity may be understood as an approximation of special relativity.

Further reflections:

To some extent, the principles of Galilean relativity may be understood as reasonable approximations of Einstein’s theory of special relativity. However, these two paradigms rest upon fundamentally different ontologies and the theory of special relativity leads to fundamentally new insights.

For concreteness, the theory of special relativity correctly predicts the much greater apparent lifetime of muons. And the theory also predicts an unexpected mass-energy relationship which became central to the development of nuclear physics. In fact, the mass-energy relationship probably deserves its own article.

References:

  1. Michelson, Albert A.; Morley, Edward W. . On the Relative Motion of the Earth and the Luminiferous Ether. 1887.
  2. A. EINSTEIN. The electrodynamics of moving bodies. 1905.
  3. R. Feynman. Feynman Lectures: The Special Theory of Relativity. Basic Books. 1963.
  4. L. Landau & E. Lifschitz. The Classical theory of fields. 1959.
  5. A. Einstein & L. Infeld. The Evolution of ideas in physics. Cambridge University Press. 1938.