Space-Time Coordinates and Frames

Descriptions of classical Newtonian dynamics were traditionally based on the concept of a reference frame, which is usually introduced to students by asking them to imagine a rigid framework in some arbitrary state of motion, which may be rotational, translational, or both. A special class of reference frames, called inertial reference frames, is then said to consist of frames that are neither rotating nor accelerating translationally. It is then asserted that “Newton’s la ws of motion are valid in any inertial reference frame”. This approach has the merit of being intuitively acceptable to most beginning students, but unfortunately it entails several undefined and ambiguous elements, rendering it strictly meaningless.

An appeal to the intuitive (visceral) notion of a “rigid” physical object is not a logically meaningful definition, because the concept of rigidity itself pre-supposes some idea of a reference frame. No actual physical object or framework is perfectly rigid. In fact, many physical objects are very far from being rigid, so we obviously cannot simply define rigidity as a property of physical objects. It is only an approximate property of some physical objects, namely, the (approximately) rigid ones! But this just begs the question of how we define rigidity. Admitedly our intuitive notion of a fixed set of spatial relations is probably derived from our experience with approximately rigid objects, but the concept of perfect rigidity is logically distinct from the objects that suggested it, since we are able to say that no physical objects exactly conform to the concept. (Of course, special relativity shows that the concept of perfect rigidity itself is inherently untenable, but we set this aside for the moment, and restrict ourselves to the concepts of Newtonian dynamics.)

So what exactly do the introductory texts on classical dynamics mean when they refer to a “rigid framework”? One might think this could be answered by saying that each particle of the framework is in the same state of motion, but of course this requires us to identify and distinguish states of motion, which we can’t do until after the concept of a frame of reference has been established. To escape from this circularity, we might appeal to some a priori sense of absolute motion, but on this basis the proposition that all particles of a rigid frame are in the same state of motion is plainly false, as can easily be seen in the case of a rotating object. Particles on opposite edges of a spinning disk are not in the same state of motion, they are in opposite states of motion. So we must return to some concept of rigidity to characterize the motions of all the particles comprising the putative “rigid framework”.

Now, it’s easy enough to simply define a perfectly rigid object as an extended object whose parts maintain fixed spatial relations relative to each other, but this obviously requires a pre-existing basis for quantitative statements about spatial relations, which is precisely what the rigid frames are supposed to provide. Hence the definition of “reference frame” provided in introductory texts is nothing but an attempt to co-opt a student’s pre-existing vague intuitive notions of spatial relations, as if by attaching the name “reference frame” to those notions they are somehow transformed into a clear and precise scientific concept.

We should also note that, in order to describe motion, we need quantitative measures not only of space but also of time, and yet traditional presentations of reference frames almost always neglect this fundamental requirement, and instead merely assume a vague intuitive notion of absolute time and its measures. To show that these naïve notions are not logical necessities, it suffices to observe that they are actually false when relativistic effects are taken into account. Hence, the intuitive notions of time certainly cannot be presented as a priori or self-evident truths. Some empirically-based quantitative measure of time must be established, i.e., we must specify a coordination of time in order to describe motions. This coordination is not provided by the traditional concept of a frame, which is traditionally presented as an entity of purely spatial extension.

A further deficiency in the traditional definitions of inertial frames becomes apparent when the claim is made that “Newton’s laws of motion are valid in an inertial frame”. This assertion is, strictly speaking, nonsense, because it refers to being “in” an inertial frame. Since inertial frames (in Newtonian dynamics) are boundless, everything is “in” every possible inertial frame, so it makes no sense to talk about something being “in” one inertial frame and not “in” another. This isn’t just a quibble over a shorthand expression; the inability to explicitly articulate what is meant is clear evidence of a fundamental inadequacy in the traditional terms of discussion as presented in elementary texts. Notice that the word “in” is not being used to signify that an object is at rest relative to a certain frame, nor even that an object is in inertial motion relative to that frame, because Newton’s laws apply to objects in all states of motion. It might be argued that the intended meaning of “in” would be “relative to”, but this too is ambiguous and vague, because in order to quantify the motion of an object “relative to” another object, i.e., a frame, we (again) need a pre-existing quantification of motion external to the frame, whereas the frame is supposed to be providing us with our system of quantifying motion.

Thus the required meaning of “in” must be something like “in terms of”, which is to say, when motions of objects are described in terms of the specified reference frame. Unfortunately a frame does not come equipped with any explicit terms, so it isn’t possible to describe motions “in terms of” a frame. In order to give a quantitative explicit description of the motions of physical entities, we need a coordination not only of time (as noted above), but also of space. In other words, we must specify a system of coordinates, which assigns a unique set of four real numbers x,y,z,t to each distinct point (event) in time and space.

Needless to say, there are infinitely many mappings between coordinate sets and events that can serve this purpose. The first and strongest restriction we ordinarily place on such a mapping is the requirement that our coordinate system possess the same topological connectivity as the manifold of events. In other words, the mappings are collected into equivalence classes such that two mappings are said to be equivalent if they are related by a diffeomorphism, and we focus our attention on just one of these classes, namely, the one in terms of which all the instances of any given object constitute a continuous locus in the space of the coordinates. Thus our concept of the local topology of space-time ultimately derives from the notion of identifiable and individuated physical entities existing at unique spatial locations which vary continuously through time. For example, an individual particle of matter is considered to exist at precisely one spatial location x(t), y(t), z(t) for each value of the time coordinate t, and this position is a continuous function of t. More precisely, each physical particle is associated with a set of three single-valued continuous functions representing the locus of its spatial positions as a function of the time coordinate t.

Now, since all electrons (for example) are identical, it’s legitimate to ask how we know which instances of electrons are to be regarded as “the same object”. Basing our judgment on the continuity of instances for a single putative object would obviously be circular. Nevertheless, if we infer a mapping from one putative object and then find that this same mapping yields a coherent set of other objects – and no discontinuous objects at all – then we have a valid basis for this mapping (although this doesn’t rule out the possibility of multiple equally viable mappings.) Of course, in the context of quantum mechanics, there are no individuated electrons and no continuous classical trajectories, so it’s not possible to definitely identify multiple instances of “the same” electron. The wave functions of all electrons overlap, at least to some extent. Instead, a system of n particles is represented as a single wave function in a 3n-dimensional phase space, so the particles are not classically individuated, and do not possess definite single-valued positions as functions of time. In spite of this, the classical trajectories of macroscopic objects suffice to establish systems of space and time coordinates that turn out to be at least somewhat useful for coordinating quantum phenomena as well. Thus, as an organizing principle, our conceptual model of space and time has proven to be extremely successful - to such an extent that we may even find it difficult to imagine any other way of organizing our thoughts and experience - but it isn't logically inevitable.

When defining physically meaningful quantities such as velocity and acceleration we must be careful not to base those definitions on whatever intuitive pre-conceived notions we may have of an underlying description of events, prior to our initial coordination, because that would imply that it really isn’t our initial coordination, which means we need to shift our attention to the earlier coordination until we arrive at truly the initial coordination of space and time. Given the amount of arbitrariness in the class of diffeomorphically equivalent coordinations, it’s clear that no simple functions of the coordinates will be physically meaningful unless we restrict the class of coordinate system very severely. Moreover, we must do this based on the objects of our experience. We must identify systems of coordinates in terms of which (now we can use that phrase) the descriptions of physical phenomena are optimally simple.

We find, remarkably, that there exist systems of coordinates in terms of which each of the three space coordinates of every isolated particle is a linear function of the time coordinate. In addition, we find that in a subset of these coordinate systems the second derivatives of the space coordinates (with respect to the time coordinate) of any pair of identical particles, isolated from all other particles, are equal and opposite. The fact that physical phenomena admit the existence of such coordinate systems is not logically necessary, but it does appear to be a fact. We refer to these specialized coordinate systems as inertial. Notice that, notwithstanding the claims appearing in virtually every introductory text on dynamics, the satisfaction of Newton’s first law of motion is not sufficient to single out the class of coordinate systems in terms of which Newton’s three laws of motion are valid. This is because Newton’s first law does not fix the loci of simultaneity, which is necessary for the satisfaction of Newton’s third law. In effect, the law of action and re-action is an operational definition of simultaneity, although this was not recognized until the advent of special relativity (and is still not widely acknowledged even a century later).

After introducing systems of space and time coordinates, we can give a meaningful definition of a frame, which is simply an equivalence class of mutually stationary coordinate systems. The term mutually stationary means that any locus of constant spatial coordinates for one system is also a locus of constant spatial coordinates for the other system. We can then go on to define a inertial frame as a frame that includes an inertial coordinate system. However, not every coordinate system in an inertial frame (now we can use this phrase) is an inertial coordinate system. For example, a system of polar coordinates r,θ,ϕ,t may be mutually stationary with an inertial coordinate system x,y,z,t, and yet the polar coordinate system does not satisfy the conditions of an inertial coordinate: the polar spatial coordinates of an isolated particle are not linear functions of the time coordinate.

It might be argued that we could still call the polar coordinates an inertial coordinate system if we express the characteristic quality of the paths of isolated particles not explicitly in terms of the coordinates but implicitly as geodesics. The physically meaningful metrical relations of the manifold are established by inertial phenomena, and one those are established, we can define the inertially straight lines in terms of any system of coordinates. Of course, the equations of motion representing Newton’s laws will then involve more terms, because (for example) acceleration will no longer be simply the second derivative of the space coordinates with respect to the time coordinate, but nevertheless we can chose to adopt these definitions, and then we can declare that every coordinate system in an inertial frame is an inertial coordinate system. Thus, even though the spatial basis vectors change from place to place relative to an inertial coordinate system in the original sense, we can adjust the equations of motion to compensate for these variations.

Of course, we can do exactly the same thing with variations in the basis vectors from time to time. In other words, given any curvilinear coordinate system, regardless of whether it’s axes are accelerating or rotating relative to an inertial coordinate system in the original sense, we can adjust the Newtonian equations of motion so that they are satisfied in terms of these arbitrary (up to diffeomorphism) curvilinear coordinates. Thus we can say that every system of coordinates is inertial, as is every frame of reference. In effect, we are simply repeating in four dimensions what we already did in three dimensions. We defined a “frame” as an equivalence class of mutually stationary coordinate systems, meaning that a locus of fixed space coordinates in one maps to a locus of fixed space coordinates in the other. We decided, in effect, to disregard any difference in the spatial coordinates, and to focus instead on the underlying metrical structure. But our metrical structure extends over time as well as space. We can just as well decide to disregard any differences in any of the coordinate bases (spatial or temporal), and focus instead on the underlying metrical structure of space-time. Thus we enlarge our equivalence class of coordinate systems to be all those that are “mutually stationary” with respect to all four coordinates, which is to say, a locus of fixed space and time coordinates for one system corresponds to a locus of fixed space and time coordinates for the other system. This is nothing but the requirement for the mapping between systems to be one-to-one. In other words, each event in one system maps uniquely to a single event in the other. (This leads to the “eternal block” view of space-time.)

Symbolically, the laws of motion in terms of arbitrary space-time coordinates x⁰, x¹, x², x³ are expressed by the geodesic equations

where τ is a path parameter, f ^α is the force (per unit mass) and the coefficients of the second terms are the Christoffel symbols, which represent the necessary “corrections” to the second derivative to make the left hand side equal to “acceleration” in the geodesic sense.

Occasionally one or more of the Christoffel terms are brought over to the right side of the equation and regarded as fictitious forces. The choice of whether to treat any of the terms this way is, of course, arbitrary. None of the terms are more (or less) physically meaningful than another of the others. Despite this, there have been many long and heated arguments over the legitimacy of regarding one or another of these terms as fictitious forces. For example, when the equations of motion are written in terms of polar coordinates in an inertial frame, non-zero Christoffel symbols appear, and can be called a fictitious force (and if it has an “outward” sense, it can be called centrifugal force), but this enrages some individuals, because they’ve been taught that fictitious forces can appear only “in” non-inertial frames. The confusions implicit in that statement have been detailed above. We should say instead that fictitious forces appear when, and only when, the equations of motion are expressed in terms of non-inertial coordinates in the original sense of that term. As explained above, inertial frames contain non-inertial coordinate systems (such as polar coordinates) in the original sense. If, on the other hand, we choose to abstract away the differences in coordinates, and adopt the more intrinsic definition of inertial coordinates, then of course our polar coordinates are called inertial, but then so too are all coordinate systems. The only context in which it makes sense to say that terms arising in polar coordinates cannot be regarded as fictitious forces but terms arising in (for example) rotating coordinates can be regarded as fictitious forces is the half-way context where we have understood enough to abstract away the variations in basis vectors with spatial position but not enough to abstract away the variations in basis vectors with time. This half-way context has its origin in the confusions over the significance of reference frames. (Arguments along these lines invariably come down to debating the significance of the word “in”.)

Once the metrical properties of a region of space and time have been mapped, the speed of an object is defined as ds/dt where s is a measure of the spatial distance along the object’s path and t is the corresponding time. Obviously in order for this definition to be meaningful we must specify how spatial distances and time intervals are quantified. Letting x⁰ denote the time coordinate and x¹,x²,x³ denote the spatial coordinates, we treat the spatial part of space-time as a Riemannian manifold, so the incremental distance ds along an object’s path is given by (ds)² = g_μνdx^μdx^ν where the g_μν are the components of the metric tensor and summation from 1 to 3 is implied for the indices μ and ν. Thus the speed of the object with respect to these coordinates is

With respect to some other coordinate system (within the same diffeomorphism class) the object may have a different speed. Trivially this is true for changes in scale factors of the space coordinates relative to the time coordinate, but even if we fix the ratio of these scale factors, the speeds of objects can still be different with respect to different coordinate systems. For a simple example, consider an object moving with the speed v in one spatial dimension in terms of coordinates X,T. In this case we have v = dX/dT. Now consider another system of coordinates x,t related to the first according to the transformation x = X – uT and t = T, where u is an arbitrary constant. The speed of the object with respect to x,t is dx/dt = d(X – uT)/dT = v – u. Thus objects moving with speed u with respect to the X,T coordinates are stationary with respect to the x,t coordinates.

On the other hand, if we consider a coordinate system related to X,T everywhere by the transformation x = X – a and t = T – b for constants a,b then the speed of every object is the same with respect to both systems of coordinates. This represents a simple translation of the coordinates. For two or three spatial dimensions we can also consider pairs of coordinate systems related everywhere by a fixed rotation of the spatial coordinates about a specific point and axis. Since a rotation preserves the magnitude of each spatial interval, the speed of every object is unaltered by this transformation. Thus there is a potentially infinite set of coordinate systems that share the same speeds for all objects, so we can speak unambiguously about the speed of any object with respect to this entire set of coordinate systems. This furnishes another definition of a frame, as an equivalence class of coordinate systems in terms of which speeds are unambiguous.

As mentioned above, the motions of massive bodies are most easily and simply described in terms of a space and time coordinate system with respect to which inertia is homogeneous and isotropic, and such a system of coordinates is called inertial. Thus we speak of inertial coordinate systems and inertial frames. It’s worth emphasizing that these systems (and frames) are characterized by both the homogeneity and the isotropy of inertia. All too often we find the term “inertial frame” described as one with respect to which inertia is homogeneous, without the stipulation that inertia is also isotropic. Homogeneity by itself is not sufficient to single out a set of definite coordinate systems, because it merely implies that the spatial position of an object in inertial motion is a fixed linear function of the time coordinate, i.e., the object moves with constant speed in a straight line. This condition is satisfied by any system of coordinates whose axes are fixed linear functions of the paths of an independent set of inertially moving bodies. The key point is that homogeneity by itself does not constrain the synchronization of time at spatially separate locations, so it fails to fully specify a definite set of coordinate systems. Newton’s first law of motion is satisfied with respect to any inertially homogeneous coordinate system, but in general Newton’s second and third laws are not even approximately satisfied.

In order to fully specify a class of coordinate systems adequate for the expression of the basic laws of motion we must impose the requirement of inertial isotropy as well as homogeneity. These two conditions, together, are sufficient to identify the class of coordinate systems with respect to which the basic laws of motion (corresponding to Newton’s laws) are valid. These are the coordinate systems and reference frames that people have in mind when they talk about “inertial frames”, but unfortunately they often think these are characterized fully by homogeneity, forgetting Galileo’s illustrations of the need for isotropy as well. There are two related reasons for this common oversight. First, when people talk about an inertial object they mean an object moving inertially, so when they apply this notion to the term “inertial frame” they easily slip into thinking it signifies a frame that is moving inertially. They forget that a frame (or coordinate system) is not an instantaneous object, it covers an expanse of time as well as space – as it must if we are to define motion. Given that it covers both time and space, it’s necessary to specify the skew between those axes. This brings us to the second reason for the common tendency to overlook the requirement for isotropy, namely, the tacit assumption that there is a unique a priori synchronization of time between spatially separate locations. If this were so, then homogeneity would indeed be sufficient to specify the inertial coordinate systems, but only because we have tacitly assumed inertial isotropy. In essence, Galileo’s assumption of isotropic inertia amounts to an operational definition of synchronization, because identical particles repelling each other from a common state of motion can be used to synchronize separate clocks with respect to that state of motion.

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