Are Relativistic Effects Kinematic?

Kinematics is the study of the motions of particles and rigid bodies, disregarding the forces associated with those motions. It is purely mathematical in nature and does not involve any physical laws such as Newton’s laws.

                                                                      Greenwood’s “Principles of Dynamics”

In a tradition dating back to Einstein’s 1905 paper, the basic effects of special relativity, such as length contraction and time dilation, are often said to be “kinematic”, but is this a correct use of that term? To answer this question we first need to agree on what the word “kinematic” means. Strictly speaking, it refers to purely descriptive propositions, which are tautological, not involving any physical laws. Classical definitions of kinematics often referred explicitly to the motions of rigid bodies, but of course, one of the lessons of special relativity is that the concept of a perfectly rigid body actually contradicts the laws of physics. Therefore, the portion of “kinematics” that deals with rigid bodies is obviously not applicable to special relativity. Even in a non-relativistic context, the naive notion of a “rigid body” should really be replaced by the concept of a solid body, meaning an equilibrium configuration of particles, which of course already involved physical laws. But if we agree to set this aside for the moment, we can certainly consider the motions of particles, and ask whether the effects of special relativity can properly be described as kinematic. Needless to say, kinematics can be applied to the motions of particles, in the following sense. Given any well-defined and complete labeling of the locations and times in a given region, the positions of a set of particles in that region at various times can be expressed in terms of those labels. Moreover, if we use continuously varying numerical labels, called coordinates, we can compute things such as the derivatives of the space coordinates with respect to the time coordinate for contiguous instances of a particle (classically assumed to have a unique and persistent identity). We may refer to these coordinate derivatives as a particle’s velocity, acceleration, and so on. But this does not enable us to predict the motions of the particles or how they interact or respond to external influences. Kinematic propositions are purely descriptive.

The labeling of events contains a high degree of arbitrariness, and we can define a different system of coordinates (i.e., set of labels) in the same region and times, by specifying the mapping from one system of coordinates to the other. Given the description of a set of particles in terms of the first coordinate system, and the mapping between the first coordinate system and the second, we can describe the particles in terms of the second coordinate system. We can also now compute the coordinate derivatives (i.e., the velocities, accelerations, and so on) in terms of these new coordinates. Again, this is purely descriptive and tautological. It tells us nothing about the laws of physics that govern the motions of the particles. In particular, it doesn’t tell us if the laws of physics are Lorentz invariant.

Now consider a solid rod of length L at rest in a system S of inertial coordinates x,t, defined as coordinates in terms of which all three of Newton’s equations of motion are quasi-statically valid. When we say the rod (oriented parallel to the x axis) has length L, we mean that if x₁(t) and x₂(t) denote the values of the x coordinates of the two ends of the rod at any given coordinate time t, then x₁(t) – x₂(t) = L. Next we ask for the length of the rod in terms of a coordinate system that is moving uniformly in the positive x direction with speed v. One such coordinate system, which we will denote as Sʹ, is given by applying a Galilean transformation to the original coordinates, so we have xʹ = x – vt, tʹ = t. This is a perfectly well-defined numerical labeling of events. In terms of these coordinates we have xʹ₁(tʹ) – xʹ₂(tʹ) = L, so there is no length contraction. However, another such coordinate system, which we will denote by Sʺ, is given by applying a Lorentz transformation to the original coordinates, so we have xʺ = γ(x – vt), tʺ = (t – vx)γ, where γ = (1−v²)^−1/2, and in terms of these coordinates we have xʺ₁(tʺ) – xʺ₂(tʺ) = L/γ, so there is length contraction. So how are we to decide if length contraction actually exists (and what do we mean by “actually exists”)?

We could go on to describe the same rod in terms of many different coordinate systems, and the “lengths” of the rod in terms of those coordinates will have various values, some greater than and some less than the length in terms of the original coordinates. These are purely kinematic statements, involving nothing but alternative descriptions of the rod in a given condition in terms of specified systems of coordinates, given only the description of the rod in terms of one system, and the mappings from that system to the other systems. But this tells us nothing about the physics. These are purely mathematical and tautological statements.

What, then, do we mean when we say that the length of the rod is “really” contracted in terms of the relatively moving frame of reference? The answer is that, of all the alternate coordinate systems moving uniformly at the speed v relative to the original inertial coordinate system S, only one of them is an inertial coordinate system as defined above. In our example, Sʺ is an inertial coordinate system whereas Sʹ is not. Inertial coordinate systems are related by Lorentz transformations, not by Galilean or any of the other possible transformations. This doesn’t mean that those other transformations are impermissible or ill-defined, it simply means that when applied to an inertial coordinate system they do not yield another inertial coordinate system.

For many people the previous paragraph may seem baffling, because we’ve considered a class of coordinate systems, such as Sʹ and Sʺ, each of which is stipulated to be moving uniformly relative to a given inertial coordinate system S, and yet we are claiming that only one of these is an inertial coordinate system. This would be self-contradictory if (as is all-too commonly said) any coordinate system moving uniformly relative to an inertial coordinate system is an inertial coordinate system. The problem is that there are two different definitions of “inertial coordinate system” in common usage, one that simply requires Newton’s first law to be satisfied, and one that also requires Newton’s second and third laws to be satisfied (at least quasi-statically). Worse, many authors tacitly conflate these two definitions.

The underlying reason for the confusion is that, prior to the advent of special relativity, a coordinate system was commonly regarded as consisting of only space coordinates. Time was not treated formally as a coordinate. (We can see evidence of this even at the beginning of Einstein’s June 1905 paper on special relativity.) This was possible, not because Newtonian physics lacked an operational definition of time (it actually had a perfectly good operational definition of time, with distant clocks synchronized so as to satisfy the third law of equal action and reaction from a state of rest), but because it was believed that the operational definition of time would yield the same time coordination for every uniformly moving system of coordinates. Hence by stipulating uniformly moving (and rectilinear) space coordinates, together with the tacit assumption of a unique temporal foliation applicable to all inertial coordinate systems, the two definitions of inertial coordinate systems were essentially equivalent. But this was based on the failure to realize that all forms of energy – including kinetic energy – possess inertia, and therefore the time coordinations of relatively moving systems of inertial coordinates in the full sense (i.e., such that all three of Newton’s laws are at least quasi-statically valid) are necessarily skewed, and hence the two definitions of inertial coordinate systems are not equivalent. Today this confusion is shown by the fact that almost all texts define an inertial coordinate system as one in terms of which Newton’s first law is formally satisfied, but then they immediately state (incorrectly) that all three of Newton’s laws formally hold good in terms of any such system. Typically all subsequent references to inertial coordinate systems in those texts actually refer to coordinate systems in term of which all three of Newton’s laws are satisfied, so they tacitly adopt the full definition, even though they explicitly give only the partial definition. In our discussion we will consistently define “inertial coordinate system” to mean a system of space and time coordinates in terms of which all three of Newton’s laws, expressed in their isotropic and homogeneous form, hold good quasi-statically. (We could remove the quasi-static condition by replacing Newton’s laws with the equations of mechanics in isotropic and homogeneous form.)

We note in passing that the beginning of the “kinematic” part of Einstein’s 1905 paper says “Let us take a system of coordinates in which the equations of Newtonian mechanics hold good (to the first approximation)”. Premising a discussion of “kinematics” on the equations of Newtonian mechanics is obviously problematic, since by definition kinematics are independent of the laws of mechanics. And yet the invocation of mechanics is crucial for all the statements that follow in the paper, despite the fact that they are labeled “kinematic”. This issue is somewhat obscured in the paper due to the fact that, at this stage, Einstein was still thinking in terms of purely spatial coordinate system, and hadn’t yet adopted the four-dimensional approach later emphasized by Minkowski. He also overlooked (or at least neglected to mention) the operational definition of simultaneity already implicit in Newton’s third law.

Incidentally, there exist some so-called proofs that inertial simultaneity (i.e., the simultaneity of inertial coordinate systems) is the only possible simultaneity consistent with causality, but this amounts to nothing other than the observation that coordinate systems related by (for example) Galilean transformations are not generally inertial coordinates, and so the equations of mechanics expressed in terms of such coordinates must contain fictitious forces, analogous to those in accelerating coordinates, which are typically regarded as acausal (barring some kind of Machian account of inertial forces).

With these clarifications, we return to the question of whether the effects of special relativity can correctly be termed “kinematic”.  As noted above, the fact that we can re-label events by applying a Lorentz transformation to a system of inertial coordinates does not actually tell us anything about whether the laws of physics are Lorentz invariant, since we could carry out such a re-labeling even if the laws of physics were not Lorentz invariant. A rod with coordinate length L at rest in the inertial coordinate system S will have various coordinate lengths in terms of other systems of coordinates, and this is indeed purely kinematic, but for that reason it has no physical significance. The coordinate length of the rod in terms of the Sʺ system, related to S by a Lorentz transformation, is regarded as a physically meaningful length only because Sʺ is (like S) an inertial coordinate system. If the solid rod initially at rest in S is slowly accelerated (slowly enough that it’s limit of elastic deformation is not exceeded) until it is at rest in Sʺ, and allowed to reach equilibrium, it will have coordinate length L in terms of Sʺ, and of course, by reciprocity of the Lorentz transformation, it will then have length L/γ in terms of S. (For a more detailed discussion of the sources of confusion regarding length contraction, see the note on Length Contraction, Passive and Active.)

The most glaring deficiency of Newtonian mechanics was its failure to treat energy as a palpable thing, as shown by the failure to correctly predict that the center of energy of an isolated system moves uniformly in a straight line. Inertial coordinate systems have skewed loci of simultaneity because all forms of energy, including kinetic energy, have inertia. This single fact about the laws of nature accounts for all relativistic effects. In particular, note that two identical particles exerting mutual repulsive force on each other, beginning from an initial tandem condition of motion in S, will respond to the mutual forces in such a way that the particle recoiling in the direction of their initial motion will recoil from the center of mass-energy with a lesser speed difference than the particle that recoils in the opposite direction (in terms of S), because the faster particle has more kinetic energy and hence more inertia. Thus an equal force results in less acceleration for one particle than for the other, when expressed in terms of the inertial coordinates S in which the center of mass-energy of the particles is moving. This is why inertial coordinate systems are related by Lorentz transformations rather than (for example) Galilean transformations. In order to claim that relativistic effects are kinematic we would need to claim that the relativistic inertia of energy is kinematic – but this is a contradiction in terms, because the response of particles to applied forces is the very definition of dynamics, and is explicitly excluded from the definition of kinematics.

Analogous reasoning applies to the ordinary classical geometry of rigid bodies. (Looking forward to special relativity, we could avoid the problematic concept of rigidity by talking about equilibrium configurations, but in the context of classical geometry we will use the concept of rigid bodies as a shorthand for equilibrium configurations.) This geometry can be characterized as the group of isometries under translations and rotations. However, we can consider two distinct kinds of translations and rotations. One kind is purely descriptive, consisting of a change of the coordinate system (labeling), and the other kind involves actually changing the positions and/or orientations of some rigid bodies by the application of forces.

Suppose the coordinates of the constituent parts of a rigid body are {x}_S in terms of a Cartesian coordinate system S, and let {x}_Sʹ denote the coordinates of those constituent parts in terms of a different Cartesian coordinate system S’ with the same orientation but with its origin shifted to the left (i.e., the negative x direction in terms of S) by a distance D. Now suppose we apply gentle forces to the body, moving it to a new position and bringing it to rest again at a distance D to the right of it’s original position, and let {xʹ}_S denote the coordinates of the constituent parts of the body in its new position in terms of the original system S. We expect to find that {xʹ}_S = {x}_Sʹ. In other words, the coordinate descriptions of a rigid body are the same, whether the coordinate labels are all moved to the left, or the rigid body is moved to the right by the same amount. The reason for this identity is easy to see: When we “moved” the labels of S to create Sʹ, we tacitly moved them in a rigid pattern. It is not enough to define S’ as a coordinate system with origin a distance D to the left, because there are infinitely many such coordinate systems, and in most of those the identity in question does not hold good. In order to assign the S’ labels so that they constitute a (shifted) Cartesian system equivalent to S, such that {xʹ}_S = {x}_Sʹ in general, we must move all the labels as if they were part of a rigid body. The situation is perfectly reciprocal because our suitable coordinate systems, and the quality of being “Cartesian”, are defined to conform to the behaviors of rigid bodies under changes in position and orientation. This is not a new insight. Newton, Gauss, and others recognized that geometry (as it pertains to physics) is part of mechanics, not apriori like arithmetic. We regard the Euclidean transformation between two Cartesian coordinate systems as the most physically meaningful one precisely because the equilibrium states of solid bodies transform in accordance with the Euclidean transformation.

When we consider the descriptions of objects in motion we must include time labels, and the concept of a “Cartesian coordinate system” is expanded to “inertial coordinate system”, so that not only are the space coordinates mutually orthogonal but also the time coordinate is mutually “orthogonal” to yield equal action and re-action. Again, just as with the purely spatial geometry, the reciprocity between passive and active transformations (i.e., setting the coordinate system in motion or setting a body in the opposite motion) is not merely coincidental. Just as the Euclidean transformation between Cartesian coordinate systems represents (and is defined in terms of) the physical behavior of objects, the Lorentz transformation between inertial coordinate systems represents (and is defined in terms of) the behavior of physical objects and processes. The inertia of energy, which is equivalent to the conservation of mass-energy, is the essence of special relativity, from which all the well-known consequences follow, and this is obviously a physical law involving the response of bodies to applied forces, etc., and hence cannot rightly be called “kinematic”.

Why, then, is the conventional view so strongly insistent that the effects of relativity are purely kinematic? This is partly due to historical reasons, not least being the fact that Einstein entitled the first section of his 1905 paper as the “Kinematical Part”. But he chose that title in reaction to the then prevailing ideas related to the 19^th century notion of a mechanical luminiferous ether, according to which the effects of special relativity have a mechanistic explanation in terms of interactions with this putative mechanical ether. Einstein described the fundamental propositions of special relativity as “kinematic” in order to clearly disassociate them from the context of a mechanistic ether. (This is reminiscent of how Newton described his concept of space and time as absolute, not to dispute their relativistic nature, but to disassociate them from any attachment to palpable bodies, since it may be that no body is perfectly inertial.) He formulated special relativity as a “theory of principle”, and apparently thought this implies that the propositions are purely kinematic, since they didn’t involve interactions with a mechanical medium. Granted, if we accept Einstein’s two principles, then Lorentz invariance follows tautologically, and hence the effects could be described as “kinematic” given those premises. This is presumably what Einstein had in mind when he decided to use the word kinematic. But it does not follow that the effects can be properly described as kinematic, because the premises themselves are explicitly dynamical, referring to systems of coordinates in which the equations of Newtonian mechanics and inertia hold good (to the first approximation).

Prior to special relativity, anything that didn’t involve mechanical interactions could rightly have been called kinematic, but one of the lessons of special relativity is that dynamical facts such as the inertia of energy do not necessarily involve mechanical interactions. Therefore, although Einstein’s use of the word is understandable, we can see in retrospect that the inertia of energy (and all its implications) cannot properly be called kinematic, if only because “kinematics” by definition excludes any consideration of inertia. (It’s worth noting that Einstein did not even explicitly identify the inertia of energy as a key feature of special relativity until a follow-up paper in September of 1905.) Of course, we are free to re-define words if we choose, but according to the current standard definition, we cannot say the effects of special relativity are kinematic.

Notice that both of the “principles” in Einstein’s 1905 paper are assertions about physical laws and the behavior of physical processes: (1) The equations describing physical laws are the same for any system of inertial coordinates, and (2) The speed of light has the same value in terms of any system of inertial coordinates. It’s true that these principles don’t involve any interaction with a mechanical ether, but they cannot be regarded as purely kinematic, since they refer to inertial coordinate systems and physical phenomena. Kinematics does not favor any system or class of coordinate systems above any others. From a purely kinematic standpoint, there is no reason to regard the Lorentz transformation as more correct or physically significant than the Galilean (or any other) transformations, just as there is no purely kinematical reason to think inertial coordinate systems are distinguished for the expression of physical laws. Kinematics is tautological, but Lorentz invariance is not. For example, the familiar “twins paradox” would really be paradoxical from a purely kinematic standpoint, because kinematics makes no distinction between inertial coordinate systems and any others.

Another source of confusion is the dispute about absolute time. Many individuals who promote a neo-Lorentzian view of special relativity, and who (rightly) deny the intelligibility of the claim that the effects of special relativity are purely kinematic, also argue that a dynamical interpretation implies absolute time. This is clearly not true, since each inertial coordinate system has it’s own time coordinate, as shown by the empirical fact that inertial coordinate systems are related by Lorentz transformations. People are free to advocate a metaphysical concept of absolute time, or to argue that some cosmologically preferred foliation should be regarded as the “true” time, etc., but this doesn’t conflict with the empirical fact that the time coordinates of relatively moving local inertial coordinate systems are mutually skewed. This is all that special relativity asserts.

Some authors argue that the word “kinematic” applied to special relativity signifies that the effects all have a common cause, rather than being a set of independent coincidences as they are in the neo-Lorentzian interpretation. They claim that this common cause is the Minkowskian metric of spacetime. However, the Minkowski metric dτ² = dt² – dx² is defined in terms of inertial coordinates x,t. The metric has no physical significance prior to the identification of a system of inertial coordinates. Once we have established such a system, based on the characteristic behavior of physical objects and processes, we find that the quantity dτ between any two given events is invariant for any such system. We can identify this with the phase of the quantum wave function of any physical system passing between those two events, and from this we could infer the conservation of mass-energy (i.e., the inertia of energy). In a sense, the invariance of the Minkowski interval is equivalent to the conservation of mass-energy. Either of these can be taken as the underlying principle on which special relativity is based, but neither of them can properly be called a “kinematic” principle, because they both explicitly involve inertia, i.e., the response of bodies to forces, which the definition of “kinematic” explicitly excludes.

In summary, the effects of special relativity cannot correctly be attributed to “kinematics” – at least not without changing the definition of that word. Kinematics deals strictly with descriptive tautologies, with no regard for inertia, whereas Lorentz invariance inherently involves inertia.

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