Alternative Interpretations of Special Relativity

Steelmanning Poincaré and Lorentz

Henri Poincarè is one of the greatest scientists who ever lived. He understood the principle of relativity and worked out the math of special relativity before Einstein. In fact, Einstein’s annus mirabilis was largely a product of him discussing Poincaré’s book Science and Hypothesis at his reading group Akademia Olympia.

And yet, it’s Einstein who typically gets full credit for Special Relativity.¹ The common modern view is that Poincarè was simply too stubborn to draw the necessary conclusions.

Einstein recognized that the Michelson-Morley experiment and underlying math simply tell us that space and time themselves transform precisely in such a way to keep the speed of light constant in all inertial frames of reference. For Einstein, relativity is a conspiracy of space and time.

Poincaré, on the other hand, firmly believed that the same results and equations are dynamical phenomena. For him, Lorentz contractions are a real physical deformation of matter (including meter sticks). Poincaré saw relativity as a conspiracy of physical effects (ruler contraction, clock deformation).

Four years after Einstein published his results, Poincaré was still writing papers that argued for an alternative interpretation.

He was “clinging onto an idea that is a mistake from the modern point of view”. Poincaré knew too much whereas Einstein was able to look at the facts with fresh eyes.

This is certainly a neat story. But it also has the typical shape of a story told to students to make them shut up when they start asking questions that challenge the standard narrative.

Poincarè thought much longer and deeper about space and time than virtually anyone else at the time or ever since.² What are the odds that he was completely wrong here?

A Simple Toy Model

Consider a one-dimensional mattress: a row of point masses connected by springs.

Let

\(q_n(t)\)

be the position of the n-th mass. The microscopic equation of motion is

\(m\ddot q_n = k(q_{n+1}-2q_n+q_{n-1}).\)

Here m is the mass of each point particle, k is the spring stiffness, and a is the equilibrium spacing.

The force on mass n from the right spring is

\(F_{n}^{\mathrm{right}} = k\big[(q_{n+1}-q_n)-a\big],\)

and the force from the left spring is

\(F_{n}^{\mathrm{left}} = -k\big[(q_n-q_{n-1})-a\big].\)

Thus Newton’s equation gives us

\(m\frac{d^2 q_n}{dt^2} = k\big[(q_{n+1}-q_n)-a\big] -k\big[(q_n-q_{n-1})-a\big] \\ = k(q_{n+1}-2q_n+q_{n-1}).\)

Now write the absolute position as equilibrium position plus displacement:

\(q_n(t)=na+u_n(t). \)

The acceleration reads

\(\frac{d^2 q_n}{dt^2}=\frac{d^2 u_n}{dt^2}, \)

So the exact displacement equation is

\(m\frac{d^2 u_n}{dt^2}=k(u_{n+1}-2u_n+u_{n-1}). \)

In the continuum limit this gives us

\(\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}, \qquad c^2=\frac{ka^2}{m}.\)

This is the continuum wave equation.

To see what kinds of waves are allowed, try a plane-wave pattern

\( u(x,t)=A\cos(\kappa x-\omega t). \)

Here k encodes the spatial rate of phase change, while w is the temporal rate of phase change.

Substituting the plane wave into the wave equation gives

\( \partial_t^2 u = -\omega^2 u, \qquad \partial_x^2 u = -\kappa^2 u .\)

Therefore

\( -\omega^2 u + c^2\kappa^2 u = 0, \)

so an allowed wave must satisfy

\( \omega^2 = c^2\kappa^2 . \)

This is the dispersion relation. It is the rule that connects the spatial pattern of the wave to its temporal rhythm.

The equation admits two kinds of traveling solutions. A right-moving wave is

\(u_R(x,t)=A\cos\big(\kappa x-\omega t\big), \)

A crest satisfies

\( \kappa x-\omega t=\text{constant}, \)

so

\( x=\frac{\omega}{\kappa}t+\text{constant}=ct+\text{constant}. \)

Hence, this is a wave that moves to the right with speed +c.

Analogously:

\(u_L(x,t)=A\cos\big(\kappa x+\omega t\big).\)

This is a wave that moves to the left with speed -c.

The important point is the mattress does not allow arbitrary phase patterns. It only allows phase patterns satisfying

\( \omega^2-c^2\kappa^2=0 . \)

Galilei Transformation

Now let’s consider an observer that moves at velocity V along the chain:

\(x_G=x-Vt, \qquad t_G=t.\)

The right-moving solution now reads:

\(u_R =A\cos\big(\kappa(x_G+Vt_G)-\omega t_G\big)\\ =A\cos\big(\kappa x_G-(\omega-\kappa V)t_G\big)\\ =A\cos\big(\kappa x_G-k(c-V)t_G\big).\)

This is now a wave moving with velocity c-V. Analogously, the left-moving solution now has velocity c+V.

Intuitively, this simply means that when you move to the right relative to the mattress you are chasing right-moving waves which therefore now move slower. Left-moving waves, on the other hand, move faster.

Also note that the wave equation under this transformation becomes

\(\frac{\partial^2 u}{\partial t'^2} - 2v \frac{\partial^2 u}{\partial t' \partial x'} + v^2 \frac{\partial^2 u}{\partial x'^2} = c^2 \frac{\partial^2 u}{\partial x'^2} \)

The wave equation is not invariant under Galilei transformations.

Lorentz Transformation

However, there is a unique coordinate choice that ensures the velocity of right-moving and left-moving waves is always equal and leaves the wave equation unchanged:

\(x' = \gamma (x - vt), \qquad t' = \gamma \left(t - \frac{v}{c^2} x \right), \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \)

Under these Lorentz transformations, the wave equation retains its form:

\(\frac{\partial^2 u}{\partial t'^2} = c^2 \frac{\partial^2 u}{\partial x'^2} \)

The left-moving and right-moving transformations also retain their simple form:

\(u_R(x',t')=A\cos\big(\kappa x'-\omega t'\big),\)

\(u_L(x,t)=A\cos\big(\kappa x'+\omega t'\big).\)

This means, after applying a Lorentz transformation, left-moving and right-moving waves are still moving with velocity c.

It’s interesting that the Lorentz formulas show up here. This is a system that lives in absolute Euclidean space with absolute Newtonian time.

And yet the Lorentz transformation is there, hiding inside the wave equation, as the coordinate choice in which the wave equation looks exactly the same in all inertial frames of reference.

The key question then is, of course, what does this mean?

What Symmetries are Real?

At its core, this boils down to a confusion that has plagued physics for centuries: the difference between real symmetries and mere redundancies.

A passive transformation is a change of description. You relabel your coordinates. The system hasn’t changed, only the way you write things down. You can always make equations invariant under a set of coordinate transformation if you introduce the appropriate bookkeepers. Invariance under passive transformation is a redundancy.

An active transformation, on the other hand, is a change of the physical state. You take the system and do something to it: rotate it, boost it, move it. If the result is physically indistinguishable from the original, then you have a genuine symmetry. This is the kind of invariance that has physical content.

So is the Lorentz transformation here a real symmetry or a mere redundancy aka an artefact of our description?

Consider an experimenter inside a ship looking at our mattress system. Will he be able to detect any difference if the ship moves with constant velocity V?

The two states of the entire ship are related by a Galilean boost:

\(q_n(t) \mapsto q_n'(t)=q_n(t)+Vt. \)

Then

\(\ddot q_n'(t)=\ddot q_n(t), \)

and

\(q_{n+1}'-2q_n'+q_{n-1}' = (q_{n+1}+Vt)-2(q_n+Vt)+(q_{n-1}+Vt) = q_{n+1}-2q_n+q_{n-1}.\)

Therefore the microscopic equations are invariant:

\(m\ddot q_n'=k(q_{n+1}'-2q_n'+q_{n-1}'). \)

Internal distances and relative velocities are also unchanged. Thus spring lengths, spring tensions, internal wave behavior, clocks, rulers, and all other internal observables are unchanged.

In summary, a Galilean boost of the whole ship is internally undetectable. This is a real physical symmetry of the full Newtonian mattress system.

---

What about two states of the ship connected by a Lorentz transformation?

Write the position of each mass as equilibrium position plus displacement:

\(q_n(t)=na+u_n(t). \)

Take state A to be a relaxed mattress in the ship:

\(q_n=na, \qquad \dot q_n=0.\)

Then neighboring masses are separated by

\(q_{n+1}-q_n=a, \)

So every spring is unstretched:

\(F_n=k(q_{n+1}-q_n-a)=0. \)

Now compare this with a state B obtained by actively Lorentz-transforming the microscopic configuration. We find

\(a' = \frac{a}{\gamma}. \)

Thus neighboring masses are separated by

\(q_{n+1}'-q_n'=\frac{a}{\gamma}. \)

But the springs’ natural rest length is still a. Therefore each spring is compressed by

\(\Delta a = a'-a = a\left(\frac{1}{\gamma}-1\right). \)

The spring force is then

\(F_n'=k(a'-a) =ka\left(\frac{1}{\gamma}-1\right).\)

This is nonzero unless v=0.

Thus an experimenter inside the ship can detect the Lorentz-transformed microscopic state simply by measuring local spring tension.

So, our mattress system does not have a real Lorentz symmetry.³

What is going on here? If Lorentz transformations are not a real symmetry here, why do they show up in the first place? And what does this have to do with Special Relativity?

The Lorentz-Poincaré Perspective

Let’s consider what happens for an observer who has no way to measure spring tension or anything else related to the microscopic structure of the mattress.

He also has no clocks or rulers measuring absolute Newtonian time intervals and distances. All he has access to is the wave-pattern that are subject to the wave equation:

\(\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}.\)

To build a ruler, he might use a standing wave:

\(u_0(x,t)=2A\cos(\kappa x)\cos(\omega t), \qquad \omega=c\kappa.\)

If its endpoints are separated by N half-wavelength intervals, then its length is

\(L_0=N\frac{\pi}{\kappa}.\)

A standing-wave clock at rest has period

\(T_0=\frac{2\pi}{\omega}, \qquad \omega=c\kappa.\)

Now put this wave-made laboratory into uniform motion through the mattress with velocity v.

Naively, one might try the Galilean-shifted pattern

\(u_G(x,t)=2A\cos(\kappa(x-vt))\cos(\omega t).\)

But this is not a solution of the wave equation unless v=0.

To see this note that this pattern has phase pieces

\(\kappa(x-vt)\pm \omega t = \kappa x-(\kappa v\mp \omega)t.\)

So its two traveling components have frequencies

\(\omega_+=\omega+\kappa v, \qquad \omega_-=\omega-\kappa v.\)

But allowed waves must satisfy

\(\omega_\pm=c\kappa.\)

Instead, as already mentioned, the correct transformation laws that preserves the dispersion relation are the Lorentz transformation laws

\(u_v(x,t) = 2A\cos\!\left(\kappa\gamma(x-vt)\right) \cos\!\left(\omega\gamma\left(t-\frac{vx}{c^2}\right)\right).\)

This can be decomposed into two traveling waves:

\(u_v = A\cos(\kappa_R x-\omega_R t) + A\cos(\kappa_L x+\omega_L t),\)

where

\(\omega_R=c\kappa_R, \qquad \omega_L=c\kappa_L.\)

So both components are allowed waves.

The nodes of the Lorentz transformed wave satisfy

\(\kappa\gamma(x-vt)=\text{constant}.\)

So at fixed Newtonian time t, the distance between corresponding nodes is

\(L_v=\frac{L_0}{\gamma}.\)

In words, this means that the ruler constructed from our standing wave is contracted if we have access to Newtonian time t.

However, our observer here only has access to his wave-based clock.

The center of the clock is moving along

\(x=vt.\)

Along that trajectory, the clock phase is

\(\omega\gamma\left(t-\frac{v(vt)}{c^2}\right) = \omega\frac{t}{\gamma}.\)

So its period in Newtonian time is

\(T_v=\gamma T_0.\)

Now, what if we put these two puzzle pieces together to get the observer’s actual description that is independent of Newtonian time?

Consider a specific experiment: the observer sends a pulse from one end of the moving wave-ruler to the other and back.

At rest, the standing-wave ruler has length and period

\(L_0=N\frac{\pi}{\kappa}, \qquad T_0=\frac{2\pi}{\omega}.\)

Since the pulse speed is c, the round-trip time is

\(t_{\rm round}^{(0)}=\frac{2L_0}{c}.\)

The number of ticks counted is

\(M_0 = \frac{t_{\rm round}^{(0)}}{T_0} = \frac{2L_0/c}{2\pi/\omega}.\)

Using

\(L_0=N\frac{\pi}{\kappa}, \qquad \omega=c\kappa,\)

we get

\(M_0 = \frac{2(N\pi/\kappa)/c}{2\pi/(c\kappa)} = N.\)

So, at rest, the pulse returns after N ticks.

Now put the same wave-made lab into uniform motion with velocity v.

The outgoing pulse catches the far end, so

\(t_{\rm out}=\frac{L_v}{c-v}.\)

The returning pulse meets the near end, so

\(t_{\rm back}=\frac{L_v}{c+v}.\)

Therefore

\(t_{\rm round} = \frac{L_0}{\gamma} \left( \frac{1}{c-v} + \frac{1}{c+v} \right) = \frac{L_0}{\gamma} \frac{2c}{c^2-v^2} = \frac{2\gamma L_0}{c}.\)

Again, this is the description in Newtonian absolute time that is inaccessible to our observer.

The only clock our observer has access to is also transformed

\(T_v=\gamma T_0.\)

Hence the actual number of ticks counted by the moving observer during the round trip is

\(M_v = \frac{t_{\rm round}}{T_v} = \frac{2\gamma L_0/c}{\gamma T_0} = \frac{2L_0/c}{T_0}.\)

Using

\(L_0=N\frac{\pi}{\kappa}, \qquad T_0=\frac{2\pi}{c\kappa},\)

we find

\(M_v=N.\)

So for our observer using wave-based instruments, the pulse returns after the same number of ticks as when the wave-lab was at rest.

So to recap, “at rest” his measured round-trip time is

\(\Delta \tau_0 = M_0 T_0 = N T_0.\)

The measured round-trip distance is

\(2L_0.\)

So the measured two-way speed is

\(c_{\text{meas}}^{(0)} = \frac{2L_0}{\Delta \tau_0} = \frac{2N\pi/\kappa}{N(2\pi/\omega)} = \frac{\omega}{\kappa} = c.\)

Now when moving with constant velocity v relative to this initial measurement setup, his ruler is still defined by the same node count N. He calls its length

\(L_{\text{own}}=N\frac{\pi}{\kappa}.\)

His clock still defines one tick as one standing-wave period

\(T_{\text{own}}=\frac{2\pi}{\omega}.\)

Therefore his measured two-way speed is

\(c_{\text{meas}}^{(v)} = \frac{2L_{\text{own}}}{M_vT_{\text{own}}} = \frac{2N\pi/\kappa}{N(2\pi/\omega)} = \frac{\omega}{\kappa} = c.\)

So the conclusion is:

\( c_{\text{meas}}^{(v)}=c_{\text{meas}}^{(0)}=c. \)

To anyone with access to Newtonian absolute clocks and rulers or equivalently the microscopic mattress, this is an astonishingly puzzling result. Waves are moving with the exact same speed no matter how you move? How is that possible?

Clearly, space and time conspire to transform in just such a way to keep the speed of waves unchanged. There is no deeper explanation for this conspiracy. It is a deep fact encoded in the very fabric of space and time itself.

But what we’ve just seen is that there is an alternative explanation once you drop the fantasy of having access to Newtonian absolute clocks and rulers.

If your rulers and clocks are made from the same stuff governed by the wave equation, this astonishing result is almost a tautology.

The wave speed comes out invariant because the very procedures used to measure it are built from the same wave dynamics.

The internal observer never sees “transformed Newtonian length” or “transformed Newtonian time.” He only counts the number of node intervals crossed and number of phase ticks elapsed. And these counts come out the same.

From the inside, this is simply the consistency condition of a world whose measuring devices and signals are all made from the same wave-physics.

Lorentzian symmetry is not a feature of spacetime but the invariant structure of the observable event-relations. Lorentz geometry is what a wave-world sees when it measures itself.

Or, to put it differently, we don’t need to interpret Lorentz invariance as a mysterious kinematic fact about spacetime itself. We can understand it as a consistency condition arising when signals, rods, and clocks are all governed by the same Lorentz-covariant dynamics.

This is a toy version of the central Poincaré-Lorentz idea: the Lorentzian structure can be understood operationally through the dynamics of rods, clocks, and signals rather than postulated as primitive geometry.⁴

Unsplitting the World

Einstein, of course, was aware of this. He noted:⁵

One is struck [by the fact] that the theory [of special relativity] . . . introduces two kinds of physical things, i.e., (1) measuring rods and clocks, (2) all other things, e.g., the electromagnetic field, the material point, etc. This, in a certain sense, is inconsistent; strictly speaking measuring rods and clocks would have to be represented as solutions of the basic equations (objects consisting of moving atomic configurations), not, as it were, as theoretically self-sufficient entities.

The standard formulation of special relativity treats measuring instruments as primitives that stand outside the physics they probe.

The world is out there; we access it using rods and clocks that are somehow exempt from the field equations governing everything else. This is a manifestation of the Cartesian subject-object split⁶, and it is a wonderfully simplifying assumption.

But Einstein himself recognized that it is, strictly speaking, inconsistent. Rods and clocks are made from “material” that deep down is governed by field equations. The material consists of solutions of the same equations they are being used to test.

To be clear, there is no proof that Einstein’s interpretation is wrong. Interpretations, by their very nature, cannot be proven wrong.

However, the Lorentz-Poincaré interpretation has not been falsified either. It is, as far as I know, equally consistent with every experiment ever performed.

So Poincaré was not clinging to a mistake. Instead, he was simply looking at the same facts through a different lens. And having more than one lens is immensely useful, as Sir William Lawrence Bragg famously put it:

“The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them.”

1. Heinzmann gives four reasons why Poincaré has not attracted biographers in the way Einstein has. First, his career followed French academic norms without major obstacles, and his family life was orderly and free of scandal, so a biography lacks the ruptures, outsider status, and dramatic turns on which modern critical biography typically depends. Second, Poincaré was wary of exaggeration and never cultivated iconic poses or programmatic self-interpretations, leaving none of the material that turned Einstein into a myth after 1920. Third, his standing as a press icon during his lifetime did not persist after his death. Fourth, his cousin Raymond Poincaré’s tenure as President of the Republic during the First World War turned the family chronicle into a public matter, overlaying the mathematician with political prominence and paradoxically obstructing any independent biographical treatment. See G. Heinzmann, ‘Poincaré (A)’, Encyclopédie philosophique (2017), https://encyclo-philo.fr/poincare-a.

2. Poincaré 1904 book The Value of Science contains the deepest discussion of the problems with naive conceptions of time I’ve encountered so far.

3. The Lorentz symmetry belongs to the effective wave equation, not to the full microscopic mattress system. The full system has Galilean symmetry. If we actively Lorentz-transform the lattice itself, the springs become compressed and the state is physically detectable. So Lorentz invariance here is not a symmetry of the substrate but a symmetry of the observable wave dynamics.

4. As far as I know, one of the few physicists who took the Poincaré-Lorentz interpretation somewhat serious is John Bell. See the chapter “How to teach special relativity” in his book “Speakable and Unspeakable in Quantum Mechanics”.

A standard criticism of Lorentz–Poincaré-flavored interpretations of special relativity is the “conspiracy” objection: if Lorentz contraction and time dilation are dynamical effects of matter, why do all rods, clocks, atoms, forces, bound states, and measuring devices transform according to exactly the same Lorentz rules? In a crude mechanical ether theory, this looks like a large collection of independent miracles. But from a modern “There are no particles, there are only fields” perspective the objection seems to lose much of its force. Rods, clocks, atoms, and observers are not independent mechanical add-ons; they are stable configurations of underlying fields. If those fields obey Lorentz-invariant equations, then the coordinated behavior of all composite systems is no longer surprising. It can arise naturally in wave-based systems, even when the underlying substrate has a completely different structure.

The mattress model still contains a hidden Newtonian substrate, so it should not be read as a literal proposal about our world. It is only used here to show that Lorentzian structure can arise from the invariant behavior of physical systems rather than being put in by hand as fundamental spacetime geometry. The next step is to drop the mattress altogether. If there is no accessible substrate, and perhaps no substrate at all, then spacetime is not the thing behind rods, clocks, and signals but the invariant structure encoded in their possible relations.

I have not fully thought this through, but it seems to point naturally toward a relational view: spacetime as the stable structure of possible physical coincidences, not as an independently given container.

5. Einstein, A. (1969), “Autobiographical Notes”, in P.A. Schilpp, (ed.), Albert Einstein: Philosopher-Scientist, Vol 1. Illinois: Open Court, pp. 1–94.

6. See https://forums.fqxi.org/d/4037-how-to-unsplit-the-world-quantum-mechanics-cognitive-science-and-the-subject