Change of inertial frames: what transforms are possible?

1 Introduction

Inertial frames of reference (or more concisely frames thereafter) and the transforms of space and time coordinates of an event in one frame into its coordinates in another frame (or more concisely transforms thereafter) are one of the pillar of physics . The purpose of this article is to discover all the viable transforms under a minimalistic set of postulates.

As it is well known, the transforms which have been in use for more than a century are the Lorentz transforms. Their historical derivation by Einstein (1905) in his seminal paper relies on the postulate that the speed of light is the same in every direction and in every frame. But a much stronger theorem relates Lorentz transforms and electromagnetism: Maxwell equations are invariant under them, as shown by Einstein in the same paper. In the course of the 20th century, two other fundamental interactions were discovered, the weak interaction and the strong interaction, whereas electromagnetism got quantified. Eventually, two theories emerged: Quantum Chromodynamics, for the strong interaction, and the Electroweak Theory, for the unification of the weak and electromagnetic interactions. Each of these theories is invariant under Lorentz transforms. Therefore their historical derivation is a biased one as it unduly emphasises a special connection to electromagnetism.

The only other transforms to have found a widespread use in physics are the Galilean transforms, which had been unchallenged from the beginning of modern physics in the 16th century to the middle of the 19th century when tensions started to appear with electromagnetism. At the time, Galilean transforms were motivated by the postulates of absolute time and absolute length, which were stronly motivated experimentally. Nowadays we know that Lorentz transforms degenerate into Galilean transforms when the relative speed \(v\) between the frames is small compared to the speed of light \(c\), and that this is why they had been successful before interferometry made possible experiments sensitive enough to probe effects proportional to \(\left(\frac{v}{c}\right)^2\) which appear as Lorentz transform deviates from Galilean transforms (Michelson-Morley experiments for example). Thus, Galilean transforms seem to derived either from obsolete concepts or from a special role played by the speed of light.

However, it has been realised just a couple of years after the publication of Einstein’s seminal paper that very general postulates which do not involve the properties of any particular physical theory, and especially which do not rely on any assumption regarding the speed of light, can be used to demonstrate that the only possible transforms are either Galilean transforms or Lorentz transforms. One of the earliest complete analysis is that of Rothe and Frank (1911), based on an even earlier work by Wladimir Ignatowsky. The most recent papers¹ I have studied are by Lee and Kalotas (1975) and Lévy-Leblond (1976) and Lévy-Leblond and Provost (1979). The purpose of this article is to give another demonstration of this result using a different approach: Lie Groups.

The main advantage is that I will be able to reason with three dimensions of space throughout, contrary to Lévy-Leblond who first derived transforms for one dimension of time and one dimension of space, then extrapolated the results to three dimensions of space by using an argument of isotropy. As for Lee and Kalotas, they started with the assumption that the transforms for the two dimensions of space perpendicular to the speed between the frames was the identity, only referring to “well-known arguments” to justify it. I will not need such preparatory arguments. Moreover Lee and Kalotas make heavy use of the definition of the speed between the two frames, further assuming that the speed of \(R'\) with respect to \(R\) is the opposite of the speed of \(R\) with respect to \(R'\). Those hypotheses are very well motivated, of course, but our demonstration will make this relative speed “magically” appear as we proceed, which is another nice touch.

The drawback is that those simplifications and extra physical postulates allowed the first two cited papers (Lee and Kalotas, 1975) and (Lévy-Leblond, 1976) to rely only on elementary mathematics. It should be noted that last paper (Lévy-Leblond and Provost, 1979) used a powerful theorem about matrix group whose elements are parametrised by a single scalar (the speed between the two frames), allowing the authors to greatly simplify the discussion. This theorem is actually a special case of the theory of Lie groups and my demonstration is therefore conceptually closer to the paper of Lévy-Leblond and Provost (1979).

Lie groups form a large and complex field of mathematics but fortunately we will only need a handful of their most simple properties. Nevertheless my demonstration does require more mathematical background than those of Lee and Kalotas (1975), and Lévy-Leblond (1976). I greatly encourage readers to read them.

2 Postulates

We will first list our postulates and then we will briefly motivate them.

The mapping of any time \(t\) and any triplet of spatial coordinates \(x\) in one frame onto their equivalent \(t'\) and \(x'\) in another frame has a matrix \(A\) and we choose the following convention:

The problem is therefore transformed into the search of a set of viable \(4 \times 4\) matrices. We will denote this set by \(G\). We will write the matrices by block as

where \(a\) is a scalar, \(u\) and \(v\) are 3-vectors (in column), and \(M\) is a \(3 \times 3\) matrix – we will denote by \(\reals^3\) the set of all such vectors. This block decomposition means that

Spatial isometries are the matrices

where \(R\) is orthogonal, i.e. \(R R^T = I\) where \(I\) is the identity matrix in dimension 3. This includes the subgroup of rotations, as well as the space inversion, which physicists call parity,

A change of scale has a diagonal matrix with positive coefficients, i.e.

where \(x_i\) and \(x'_i\) are the spatial coordinates, components of the vectors \(x\) and \(x'\) used so far, and \((a_0, a_1, a_2, a_3)\) are positive constants.

3 Motivation of the postulates

Postulate 1. The classic argument for linearity is the homogeneity of space-time but a better, more modern argument is simply to state that we search the transforms in the tangent space of “curved spacetime” (i.e. spacetime manifold in rigorous mathematical terms). This does not mean assuming General Relativity, not even the existence of a spacetime metric.

Postulate 2. Isometries are practically important because they change the orientation and the handedness of frame axes, an operation which is a very common occurence in the practice of physics. They are also theoretically important because verifying whether a physical model is invariant under rotations or space inversion is mundane. It should be noted that we do not make any preconceived statement about whether the model has such an invariance. For example, the Electroweak Model mentioned in the introduction is not invariant under parity.

Postulate 3. We assume we have fixed units of time and lengths in all frames once and for all, and that from that point on, we are interested only in all the other transforms.

Postulate 4. The group structure directly emerges from the definition of changes of frame:

1. the identity matrix correspond to no change of frame;
2. given the change of frame \((t,x) \mapsto (t',x')\), we shall also have the change of frame \((t',x') \mapsto (t,x)\), and if \(A\) is the matrix of the former, then \(A^{-1}\) is the matrix of the latter;
3. given the change of frame \((t,x) \mapsto (t',x')\) followed by the change of frame \((t',x') \mapsto (t'',x'')\), we shall also have the change of frame \((t,x) \mapsto (t'',x'')\), and if \(A\) and \(A'\) are the matrix of the first and of the second respectively, then \(AA'\) is the matrix of the last.

The postulated Lie structure is as mundane, since every single non-discrete matrix group of practical use in physics is a Lie group. Instead of postulating the Lie structure, we could have postulated that \(G\) is topologically closed and then invoked Cartan’s theorem to conclude that \(G\) is a Lie subgroup of the Lie group of \(4 \times 4\) matrices. Whether requiring that any convergent sequence of matrices in \(G\) converges in \(G\), for example, is more intuitive than requiring a Lie structure in the first place is a moot point for a theoretical physicist in the 21st century, in my humble opinion!

4 Primer on matrix Lie groups

We will succintly give here the essential properties we will use in our demonstration.

Commutator: for any matrix \(A\) and \(B\), it is defined as \([A,B] = AB - BA\). Mathematicians name it “Lie bracket” instead.

Matrix Lie algebra: it is a linear subspace \(H\) of the linear space of all matrices with the extra property that, for any \(A\) and \(B\) in \(H\), \([A,B]\) is also in \(H\).

Matrix exponential: for any matrix \(A\), it is defined as the convergent series

Matrix Lie group: it is a group \(G\) of matrices such that there exists a Lie algebra \(H\) with the property that \(\exp H\) is a subgroup of \(G\). We will denote \(H\) by \(\Lie{G}\).

There is an important property of the Lie algebra of a Lie group which will play an important role in our demonstrations:

Proof. First, \((A^{-1}BA)^n = A^{-1}B^nA\) for any integer \(n\), and therefore the exponential series satisfies

which proves the result.

Proof. Since \(\exp\) is continuous and it maps the zero matrix onto the identity matrix, the result is obvious.

Lie group of matrices may seem rather abstract but it is actually just a different, powerful, way to look at all the classic matrix groups. Let us study an example, which will be useful later: the group \(O_3\) of the \(3\times 3\) orthogonal matrices, i.e. the matrices of the isometry of 3-dimensional space, and its subgroup \(SO_3\) consisting of matrix rotations (those notations are standard but usually one specifies the field the matrix elements belong to, which is the field \(\reals\) of real numbers here). It is important since it is the spatial part of the group \(O\) introduced in postulate 2.

We will start by discussing the structure of \(O_3\) and \(SO_3\). Given a rotation \(R\) of angle \(\theta\) about some vector \(u\), there is a continuous path from \(R\) to the identity matrix, by continuously varying \(\theta\) toward 0. By definition, this means that \(SO_3\) is the connected component of the identity in \(O_3\). Thus to prove that \(O_3\) is a Lie group, we only need to prove that the matrix exponential maps some Lie algebra to be determined onto \(SO_3\). As a side note, \(O_3\) not only contains rotations but also the product of rotations by the spatial inversion. The latter form the other connected component of \(O_3\), one which does obviously not contain the identity.

It is actually easy to find what this Lie algebra should be if it exists. Indeed, keeping only terms linear in \(\theta\) when \(\theta \to 0\),

Thus

and then

Thus if \(R\) is orthogonal, then \(RR^T = I\), and therefore all terms of order \(\theta\) or higher must be zero in series of \(RR^T\), which implies that \(A\) is antisymmetric.

Thus we shall consider the set \(A_3\) of all \(3\times 3\) antisymmetric matrix. It is a subspace of the algebra of \(3\times 3\) matrices as any linear combination of antisymmetric matrices is antisymmetric. Then given two antisymmetric matrices \(A\) and \(B\), \([A,B]\) is clearly antisymmetric by the very definition of the Lie bracket. Thus \(A_3\) is a Lie algebra and we just need to prove that \(\exp A_3 = SO_3\) to conclude.

For any antisymmetric matrix \(A\), the exponential series implies that

But then \(-A\) commutes with \(A\), and therefore, for any integer \(n\), the coefficient of \(A^n\) in the series of \(\exp(-A)\exp A\) is the same as the coefficient of \(x^n\) in the series \(\exp(-x)\exp x\) for a real number \(x\): they are all equal to zero, except for the order 0. Thus \(\exp(-A)\exp A = I\) and therefore the equation above shows that \(\exp A\) is orthogonal. Finally, any antisymmetric matrix \(A\) has a zero eigenvalue. Indeed, the generic form of \(A\) is

for some vector \(u\), and clearly \(Au = 0\). Then \((\exp A)u = u\) and therefore \(\exp A\) shall be a rotation, as if it belonged to the other connected component, consisting of rotations times the inversion, its only real eigenvalue would be -1.

It should be noted that we could actually have written a constructive proof instead, based on Rodrigues’ rotation formula in matrix form. But the demonstration would have been rather similar to the one we will encounter when Lorentz boosts will emerge later in this article, and we therefore preferred the demonstration above which involves only very simple calculations. In any case, we have demonstrated

5 Derivation of the group \(G\) of viable transforms

Our strategy will be to characterise \(\Lie{G}\) and then use the matrix exponential to find \(G\). We start by demonstrating a simple lemma we will reuse a couple of time.

Proof. With

we can write

On the right-hand side, the first term belongs to \(\Lie{O}\), as \(M^-\) is antisymmetric, and therefore the second term \(B\) belongs to \(\Lie{G}\) too. Since \(M^+\) is symmetric, there is an orthogonal matrix \(R\) and a diagonal matrix \(\Delta\) such that \(R^{-1}M^+R = \Delta\). But then with

we have

With property 1, \(B'\) is in \(\Lie{G}\), and therefore

is in \(G\). But this is a change of scale, and therefore postulate 3 requires that \(a=0\) and \(\Delta=0\). This completes the proof.

Then we move to the keystone of our demonstration, as the subsequent work will simply be the computation of matrix exponentials.

Proof. Let us consider a generic element \(A \in \Lie{G}\),

Since \(P \in G\), \(PAP^{-1}\) is also in \(\Lie{G}\), and therefore \(B = \frac{1}{2}(A + PAP^{-1})\) too. But

and lemma 1 implies that \(a=0\) and \(M\) is antisymmetric. Thus

and the last term is in \(\Lie{G}\), which proves that

Thus we have shown that any element of \(\Lie{G}\) is the sum of an element of \(\Lie{O}\) and of an element of the vector space

and we are left with characterising \(\KLG\). The case \(\KLG = \{ 0 \}\) is ruled out by our second postulate, as this would leave only isometries. Thus we have three cases to study.

Case 1: \(\KLG\) has an element \(A=\mat{0}{0}{v}{0}\) with \(v \ne 0\).

Then for any rotation

and any real \(\lambda\), \(A'=\lambda QAQ^{-1} \in \Lie{G}\) but

and \(\lambda R v\) spans \(\reals^3\) when \(\lambda\) spans \(\reals\) and \(R\) spans all the rotations. Thus

Then let

be an element of \(\KLG\). For any

we have therefore \([A,B] \in \Lie{G}\) but

lemma 1 requires that \(p^Tv=0\). Since \(v\) can be any 3-vector, this implies that \(p=0\). We have therefore demonstrated that

This completes the proof for this case.

Case 2: \(\KLG\) has an element \(A=\mat{0}{u}{0}{0}\) with \(u \ne 0\).

The analysis is completely similar to the previous case, replacing \(K_G\) with \(K_S\).

Case 3: \(\KLG\) has an element \(A=\mat{0}{u}{v}{0}\) with \(u \ne 0\) and \(v \ne 0\).

Let \(R_0\) be the rotation of angle \(\pi\) around the bissector of \((u,v)\). Then it exists \(\kappa > 0\) such that \(R_0 v = \kappa u\) and \(R_0 u = \frac{1}{\kappa} v\). Then, with

we have \(B=Q_0AQ_0^{-1} \in G\) and therefore \([A,B] \in G\). But

Thus

Then lemma 1 requires that

Multiplying both sides of the second equation with \(v\) on the right shows that \(v\) is colinear to \(u\), and the first equation then shows that \(v=\epsilon\kappa u\) where \(\epsilon=\pm 1\). Introducing \(c=\sqrt{\kappa}\) and \(w=cu\), we have \(v=\epsilon c w\) and therefore

The scalar \(c\) has the dimension of a speed and we see that it appeared solely because rotations are a subgroup of the group of transforms we seek, and because we have a Lie group (since we used the fact that \([A,B]\) is in the Lie algebra).

Then for any rotation

and any real \(\lambda\), \(A'=\lambda QAQ^{-1} \in \Lie{G}\) but

and since \(\lambda Rw\) spans \(\reals^3\) when \(\lambda\) spans \(\reals\) and \(R\) spans all \(3 \times 3\) rotations, we have proved that

Let

be an element of \(\KLG\). For any \(A \in K_L^\epsilon\),

\([A,B] \in \Lie{G}\) but

and lemma 1 then requires that

Multiplying the second equation by \(w\) on the right gives \(\left(\epsilon c p^T w - \frac{1}{c}q^T w \right)w = \left(\frac{1}{c}q - \epsilon c p \right)w^2\)

and therefore, using the first equation, and \(w \ne 0\), we get \(q=\epsilon c^2 p\), i.e.

is therefore in \(K_L^\epsilon\). We have therefore demonstrated that

With the previous opposite inclusion (\ref{KepsSubsetKLG}), this completes the proof of this case and therefore of the proof of lemma 2 too.

We can now prove the following theorem.

Proof. Those four cases obviously correspond to the four cases of lemma 2. Thus let us review them in turn.

Case I and II: it is trivial to verify that any \(K \in K_G\) and any \(K \in K_S\) is such that \(K^2 = 0\) and therefore \(K^n = 0\) for any integer \(n \ge 2\). As a result, the exponential series is trivial: \(\exp K = I + K\). Thus in case I, this gives

and the transforms

These transformations are synchrony transformations. This ties in the well-known presentation of Special Relativity in which a clock sits at each location \(x\), motionless in that system of coordinates, clock which defines the time \(t\) at that location. Because those transformations leave the position invariant while shifting the time in a location-dependent manner, we are therefore just changing how those clocks are synchronised with each others. Hence the name synchrony group for \(K_S\). It should be noted that Lévy-Leblond calls them the Carroll transforms for reasons he developped in (Lévy-Leblond, 1965), an article which expounds in great details this group of transforms, including its quantum representations.

In case II, this gives

and the transforms are the Galilean transforms

Case III^±^: the computation of the exponential is only slightly more involved. We will compute both cases at the same time by keeping \(\epsilon\) unspecified. First, let us write \(u = \varphi \hat{u}\) where \(\hat{u}\) is a unit vector. Then \(\newcommand{Keps}{\hat{K}_\epsilon} K_\epsilon = \varphi\Keps\) where

Then \(\newcommand{\PPu}{\mat{1}{0}{0}{P_u}} \Keps^2 = \epsilon\mathcal{P}_u\) where

and \(P_u = \hat{u}\hat{u}^T\) is the projection onto \(u\). Therefore, on one hand, for any \(n \ge 1\), \(\Keps^{2n} = \epsilon^n \mathcal{P}_u\) since \(P_u^n = P_u\) because \(P_u\) is a projector. On the other hand, \(\Keps^{2n+1} = \epsilon^n \mathcal{P}_u \Keps = \epsilon^n\Keps\). Thus the exponential series reads, by separating even and odds terms,

Thus we have either case (III\(^-\)),

or case (III\(^+\)),

where \(\projperp{u} = I - P_u\) is the projection onto the plane \(u^\perp\) perpendicular to \(u\). Thus the transformations are either, in case (III\(^-\)),

or, in case (III\(^+\)),

where \(x_\parallel\) is the component of \(x\) parallel to \(u\) whereas \(x_\perp\) is the component perpendicular to \(u\).

In case (III\(^+\)), we recognise a Lorentz boost, parametrised by the rapidity \(\varphi\) and the direction of the boost \(\hat{u}\). The case (III\(^-\)) is not mainstream but time and space are mixed as two spatial coordinates would be by a rotation of angle \(\varphi\), hence the name “rotation in spacetime”.

Thus it would seem we have failed, since we not only recovered Galilean and Lorentz transforms but two other groups. The rotations of spacetime are definitively “exotic” in the sense that they have never played any role in physics: it is rather puzzling they appear and it is very desirable we find a criteria to eliminate them. The synchrony group is important on the contrary as a way to formalise the conventionality of synchronicity which plays an essential role when considering experimental aspects, either real experiments or gedanken ones. Actually, there is a deeper point to be made here.

Indeed, it would make sense to combine the synchrony group with either of the Galilean group, the Lorentz group, or the spacetime rotations. Indeed, the latter embodies the physical content of the theory whereas the former embodies a conventional aspect. But then the Lie algebra in those three cases will turn out to be the set of matrices \(\mat{0}{u}{v}{0}\) where \(u\) and \(v\) span \(\reals^3\). But then the commutator of two of them, \(A_1=\mat{0}{u_1}{v_1}{0}\) and \(A_2=\mat{0}{u_2}{v_2}{0}\) is

As a result, our group now features changes of scale. We have therefore demonstrated that our postulate 3 banning any change of scale does eventually prevent us for including synchrony transformations in our group of transforms. But this is only because postulates 1–4 lead only to the Galilean group, the Lorentz group and the group of spacetime rotations, beside the synchrony transformations.

The question we would like to address now is whether we can find an extra postulate so as to eliminate the synchrony transformations and the spacetime rotations. For the former, we could require on a physical ground that we cannot have only synchrony transformations. But it turns out we can appeal to a more general idea.

6 Causality

It would be highly convenient if any two observers could agree which of two events happens first. This amount to require the existence of a synchronisation procedure resulting in that property. It is therefore largely conventional but, as we have just seen, we have already banned general synchronies. Therefore it makes sense to “bite the bullet”. Hence the following requirement

Because of linearity, this is equivalent to state that for any \(t > 0\), there exists spatial coordinates \(x\) such that for every \(M \in G\), \(\vec{t'}{x'} = M \vec{t}{x}\) is such that \(t'>0\). Then we wish to prove

For the synchrony group, \(t' = t\left(1 - u^T \frac{x}{t}\right)\). But for any \(x \in \reals^3\) and any \(t > 0\), one can find big enough a vector \(u\) such that the factor of \(t\) is negative. This was intuitively obvious: causality as we defined it cannot hold with an arbitrary synchronisation of clocks but we see that we need \(u\) to span the whole of \(\reals^3\) for this argument to work mathematically, a property which, in turns, comes from requiring a group in the first place as we saw in the previous section.

For the rotations in spacetime, \(t' = t\cos\varphi\left(1 - \frac{1}{c}\hat{u}^T \frac{x}{t}\tan\varphi\right)\). Since \(\lim_{\varphi \to \frac{\pi}{2}} \tan\varphi = +\infty\), for any \(x \in \reals^3\) and any \(t > 0\), one can choose \(\varphi\) such that the factor of \(t\cos\varphi\) is negative. This concludes the proof.

This is trivial for the former. For the latter, we have \(t'=t\cosh\varphi\left(1-\frac{1}{c}\hat{u}^T \frac{x}{t}\tanh\varphi\right)\). Since for any \(\varphi \in \reals\), \(\tanh\varphi\) is between -1 and 1, for any \((t,x)\) such that \((ct)^2 - x^2 > 0\) and \(t > 0\), we have \(t' > 0\). Thus the set of event \(E\) is the interior of the so-called light cone.

We can now conclude with

References

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Lévy-Leblond, J.-M. (1976). One more derivation of the Lorentz transformation. American Journal of Physics, 44, 271.

Lévy-Leblond, J.-M., & Provost, J.-P. (1979). Additivity, rapidity, relativity. American Journal of Physics, 47(12), 1045.

Lévy-Leblond, J.-M. (1965). Une nouvelle limite non-relativiste du groupe de Poincaré. Annales De l’I.H.P. Physique Théorique, 3(1), 1–12.

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