Fields and Coordinates One of the original motivations for the scientific revolution the desire to do away with “occult” causes, and to base everything on (ostensibly) “clear and distinct” concepts such as the extension, motion, and impenetrability of substance. In this spirit, it was natural to insist that material entities can interact with each other only by the impulses resulting from direct contact. If two separate entities appeared to influence each other, it was to be assumed that there was an intervening series of contact interactions involving some other material entities. However, somewhat ironically, the first successful theory of physical interactions that emerged from the scientific revolution – Newton’s theory of universal gravitation – was originally interpreted as signifying that separate material entities exert a force on each other at a distance, with no explicit intermediate substance or mechanism. Not surprisingly, this was regarded as a return to “occult” causes by some philosophers of that time. Of course, there are other possible interpretations of Newtonian gravitation. For example, it can be formulated in terms of a continuous “potential field” permeating all of space. According to this view, each particle of matter produces (or is associated with) a spherically symmetrical field of gravitational “potential” centered on the respective particle, with an intensity that falls off inversely as the distance from that particle. The total potential field at any point in space is the sum of the fields contributed by all the particles in the universe. We then make the further assumption that each particle is subject to a force proportional to the gradient of the potential field at the location of the particle. Thus, instead of interacting with other particles at a distance, each particle “merely” responds to the gravitational field in its own vicinity. One might think that the field interpretation fails to eliminate distant action, because it still entails a distant interaction between each particle and its own potential field, which extends to infinite distances, and which must either “move” or change its value (instantaneously, according to Newton’s theory) whenever the source particle moves. However, this objection is mitigated somewhat by the fact that the potential field itself satisfies a simple partial differential equation (Laplace’s equation in empty space), according to which the field at any point is determined entirely by the field at the immediately surrounding points. Thus, if we grant to the gravitational field the status of a physical entity, we can (arguably) claim that gravity is indeed mediated by purely local contact interactions. In this case, though, it is not generally possible to consider only outward repulsive forces (associated with the concept of impenetrability), because the equation governing the potential field cannot be decomposed into impulses. Elementary attraction is also required. Thus, even by reifying a hypothetical potential field permeating all of space, we can restore (at best) only locality, not the complete doctrine of local contact impulses as the basis of all physical interactions. (Efforts were made to explain attractive forces by means of a shadow mechanism, as proposed by Fatio and Lesage, but this approach necessarily entails an infinite regress of supernatural substances.) One advantage often cited in favor of fields versus action at a distance is that many different configurations of material particles may result in the same field at a given location, so it is more convenient to characterize the response of a particle to the field at its location than to correlate the response to all the possible configurations of ambient particles that would produce the field. Essentially it is argued that the field at any given point has far fewer degrees of freedom than the set of configurations of the particles in the universe, all of which (in principle) contribute to the field at that point. This is somewhat ironic, considering that, in another sense, the introduction of an infinite and continuous field actually represents a huge increase in the overall degrees of freedom of the system, compared with the set of possible positions of a merely countably infinite number of discrete particles. There’s an interesting correspondence between the local-versus-distant action debate, and the issue of whether motion is relational or relativistic. (The concepts of relationism and relativism have often been conflated, and then jointly contrasted with absolutism, but the more interesting contrast is between relationism and relativism. Absolutism is actually just a form of relativism with an additional metaphysical commitment.) The issue can be explained in terms of how we conceive of and quantify motion. In accord with the concept of “action at a distance”, we could argue that the only physically meaningful spatial intervals are the distances between pairs of material particles. Letting s(t) denote the distance between two given particles at the time t (taking for granted a definite time foliation), we might then define motion as a derivative of the form ds/dt. However, this approach has several well-known shortcomings. For instance, if one particle is moving in a circle around another particle, then the distance between the particles is constant, and yet we know the particle is moving, so the quantity ds/dt doesn’t capture the full meaning of motion. We would need to supplement this with the derivatives of the distances to other particles, to somehow arrive at a single measure of “motion”. So, although the purely relational definition of motion in terms of the distances between physical entities seems plausible at first glance, we quickly realize that it is not very convenient, and it requires taking into account not just the distance to one reference particle, but to at least three or four reference particles, in order to give a full account of motion in three-dimensional space. In addition, it relies on an assumed unique temporal foliation. A different approach, which we may call the field approach to space, is to define a system of coordinates covering the entire space, and to conceive of the positions of entities in terms of these coordinates. Thus the motion of a particle is not represented by a quantity of the form ds/dt (where s is the distance to some reference particle), but rather by a quantity of the form dx/dt, where x is a certain kind of field quantity known as a space coordinate. We can regard x (and y and z) as analogous to potential field variables. Just as in the case of the potential field, the space coordinates can be seen as encoding the relevant information about the relations between particles. We find that the spatial positions of a particle have only three degrees of freedom, even though the number of possible combinations of distances to all the other particles in the universe is huge. The use of a space (and time) coordinate system evidently enables us to characterize the positions and motions of an individual particle, without explicit reference to any other particle. Of course, in order to establish the coordinate system, we must make use of certain stable configurations of particles, and repeatable processes involving those configurations, just as the potential field in a region is determined by the configurations of particles. The putative advantage of the coordinate/field approach is that it enables us to split the analysis of phenomena into two parts. First, the substantial entities are considered as “sources” which determine the field variables or coordinates in the region of interest. Due to the limited number of degrees of freedom of the field or coordinates, this can be done very economically by examining just a small number of entities. For example, to determine the field variables in a certain region, we might simply measure them by examining the response of some standard particles or charges placed at various locations in that region. This information, together with the differential equations that characterize the field, can then be used to determine the field variables. (In celestial mechanics the important sources of gravitational potential in an isolated system are often relatively small in number, given that we can treat astronomical bodies as point-like particles, and hence it is feasible to evaluate the potential contributions of the “sources” explicitly.) Similarly, to determine a suitable (i.e., inertial) system of space and time coordinates, it suffices to examine the spatial relations involving just a small number of entities. Then, once the field variables or coordinates in the region have been established, we can consider individual particles as passive entities, interacting with the fields and coordinates. One might challenge the idea that particles interact with (or respond to) inertial coordinates, since we are obviously free to choose arbitrary coordinates without affecting the phenomena. But this is quite analogous to the case of potential fields, which have a degree of arbitrariness. Recall that the effect of the potential field is to exert a force proportional to the gradient of the potential. We can obviously add an arbitrary constant to the values of potential without affecting the gradient of the field. (Likewise the electromagnetic potential field has a gauge freedom.) Thus the actual values of the potential are arbitrary – and yet the potential does encode information about the behavior of particles. In the same way, we have a considerable amount of arbitrariness in the choice of space and time coordinates, and yet the coordinates do encode constraints on the behavior of physical entities. This is seen most clearly by considering the coefficients of the spacetime (pseudo) metric. These metric coefficients serve as the potentials of the inertial (and, as it turns out, the gravitational) field. By choosing different coordinate systems (within the class of diffeomorphic equivalence) we obviously get different values of the metric coefficients, representing the gauge freedom of the field variables, but the absolute spacetime intervals are invariant, and these determine the geodesic paths of objects, accounting for the effects of both inertia and gravity. The metric coefficients corresponding to any choice of space and time coordinates satisfy Einstein’s field equation, just as the gravitational potential in Newton’s theory satisfies Poisson’s equation. (In fact, Einstein’s equation is just a generalization of Poisson’s equation.) So far we’ve omitted one very important and controversial aspect of field theories, namely, the question of whether fields are fully determined by the “sources”. One might think, based on the above discussion, that the answer must be yes, since the very concept of a field was presented as arising from an attempt to encode the relevant information about the influence of other objects. However, once we have taken the step of reifying the field, and we’ve begun to treat it as an entity in its own right, with behavior governed by it’s own equation, it becomes possible (and tempting) to imagine that the field is a primary entity and need not be regarded as just an auxiliary representation of interactions between material particles. In electrodynamics one finds debates about whether all electromagnetic fields are ultimately attributable to electric charges. Likewise there are on-going debates about whether mechanical inertia is ultimately attributable to interactions with distant masses, as suggested by Mach and others. The issue becomes more intriguing – and more important – when we recognize the time-dependence of the field variables, since this leads to the concept of waves propagating in the field. Such waves are conceived as possessing both energy and momentum, and, as Maxwell pointed out, one would expect that the energy must exist in some form between the emission and absorption, so we feel inclined to attribute ontological status to the field. On the other hand, this is to some extent an illusion, because all the phenomena usually associated with “radiation” are actually predicted by the “force at a distance” interpretation as well, provided we account for the temporal retardation of the potentials. Also, when seen in the context of Minkowski spacetime, the invariant intervals along which massless radiation propagates are seen to be null. Thus Maxwell’s argument is inconclusive at best. The preceding discussion should help to clarify the contrast between relationism and relativism, two concepts that are (as noted above) often conflated. When students first hear about special relativity, they often imagine that it must represent an attempt to express phenomena in terms of the relations between entities – a misconception that is unfortunately encouraged by some of the early literature on the subject. The difficulty often becomes acute when students struggle with the concept of the velocity of some object relative to some other object. They seem to have in mind a quantity such as ds/dt where s is the distance between the two objects. Of course, they are immediately told that such quantities are not actually speeds in relativity theory, although they are sometimes called “closing speeds” – an unfortunate and misleading name. As Voltaire observed, the Holy Roman Empire wasn’t holy, wasn’t Roman, and wasn’t an empire. We must be careful not to mistake names for descriptions. The quantity called a “closing speed” is not a speed in the strict sense of a relativistic theory (just as it is not a true speed in an absolutist theory). Classical relativistic theories are based firmly on the bifurcation between active and passive roles for physical entities, which is to say, they are based on the field/coordinate interpretation of physics. A velocity, in the context of special relativity, is a quantity of the form dx/dt, where x and t are space and time coordinates respectively. Unless otherwise specified, we stipulate that these constitute an inertial coordinate system, meaning that the “laws of mechanics (in their simplest form) hold good”. This basically signifies that x and t are defined in such a way that inertia is homogeneous and isotropic. It turns out (as Galileo noticed) that there are infinitely many such systems of coordinates, and that any material object in any state of motion is instantaneously at rest in terms of one such system of coordinates. This is the principle of relativity. Now, in the theory of relativity, when someone says (loosely) that object A has a certain velocity “relative to” object B, they are abusing the language, because objects don’t have velocities relative to other objects, they have velocities relative to (or, better to say, in terms of) a specified system of coordinates. We often want to specify the velocity of some object in terms of a coordinate system in terms of which some other object is stationary. Thus, for example, if particle A is moving in a circle around particle B, we can talk about the velocity A in terms of the inertial coordinate system in terms of which B is at rest. The linguistic gymnastics necessary to express the meaning of this quantity accurately are obviously discouraging, and it soon leads even the most conscientious person to begin using verbal short-cuts, such as calling this “the velocity of A relative to B”, even though the distance between A and B is not changing. The quantity is properly described as “the velocity of A in terms of the inertial rest coordinates of B”. The “inertial rest coordinates” of an object are defined as a system of space and time coordinates – essentially unique up to trivial translations and re-orientations – in terms of which the object is stationary. The content of special relativity is that relatively moving systems of inertial coordinates (in terms of which inertia is homogeneous and isotropic) have skewed space as well as time axes. The important point for understanding the distinction between relativistic and relational theories of motion is that relativistic theories are based on a metrical field, and all positions and velocities are defined in terms of spacetime coordinates, not in terms of distances between entities. In fact, in the context of special relativity, we cannot even uniquely define the distance between two entities, because distances depend on the choice of coordinate systems. The “distance” between two moving objects must be assessed based on the simultaneous positions of those objects, but the relativity of simultaneity implies that there is ambiguity in this mapping, which of course is why the spatial length of an object depends on the system of coordinates. Having chosen one particular system of inertial coordinates (which are, after all, along with rigid bodies, the foundation of our concepts of spatial and temporal intervals), we can express the velocity of each object. We may then consider the sum or difference between two of those velocities. This brings us to one of the more unfortunate features of the common parlance of relativity theory, going back to the earliest papers on the subject. The word “addition” is used to signify two very different things. The sum or difference of two velocities in special relativity is, of course, simply the sum or difference of those quantities. For example, if, in terms of a given system of inertial coordinates, a pulse of light is moving at the speed c directly toward an object, and if the object is moving toward the pulse at the speed v, then the sum of those two speeds is c + v. This is as true in special relativity as in Galilean relativity. Thus, if the pulse and the object at separated by a distance L at time t = 0, they will coincide at time t = L/(c+v). Unfortunately this can be confusing for beginning students, who leap to the conclusion that the quantity c+v represents the speed of the pulse “relative to the object”, and hence that this contradicts the purported invariance of the speed of light. They need to be reminded that velocities in relativity are not defined relative to objects, they are defined in terms of spacetime coordinate systems. When someone says (loosely) that the speed of light is always c “relative to any object”, this is just sloppy shorthand for saying that the speed of light is c in terms of the inertial rest coordinates of any object. Furthermore, given the velocities vA and vB of two objects in terms of one system of coordinates, the velocity of one of those objects in terms of the inertial rest coordinates of the other is not simply given by addition or subtraction of vA and vB. Given the values of vA and vB, the operation that yields the velocity of A in terms of the inertial rest coordinates of B would properly be called “composition” rather than “addition”. Unfortunately, beginning with Einstein, and continuing on to the present day, many authors have chosen to refer to the composition of velocities as the “addition of velocities”. This, then, unavoidably leads to confusion when students must deal with the actual addition of velocities, which of course is also a meaningful and useful concept. Many authors attempt to cope with this confusion by introducing the concept of “closing speeds”, as opposed to “speeds”, the idea being that the simple sum or difference of two speeds (defined on a consistent basis) yields a quantity of a different kind, referred to as a “closing speed”. This is a nominally workable scheme, but it does little to clarify the underlying concepts. It appears that some individuals never recover from the affront of being told in school or by some popular science book that the meaning of “addition” changed in 1905, and that 1+1 no longer equals 2. This is the kind of novelty that some popularizers enjoy imposing on the public, as if science can only be made interesting by making it sound as weird and inexplicable as possible. They apparently prefer not to say that the sum of vA and vB is, or course, vA + vB, and that although this is a meaningful and useful physical quantity, it does not represent the velocity of A in terms of the inertial rest coordinates of B. To find this latter quantity, we must perform the composition of the two given velocities. This operation is not the same as arithmetic addition, so it’s a misnomer to refer to it (as is done throughout the literature) as the “addition of velocities”. From one point of view, the co-opting of the word “addition” to signify composition may be understandable, because it is consistent with mathematical usage for things like the “addition laws” for trigonometric functions. It’s understood that this refers to “addition” in a different sense than simple arithmetic addition, which after all is used in the specification of the higher “addition laws”. Nevertheless, a great deal of confusion has been caused by inattention to the distinctions between addition and composition, as well confusion between relationism and relativism. Needless to say, the trouble cause by this confusion pales in comparison to that caused by confusion between relativity theory and the concept of moral relativism. Amusingly, the author of a recently published anti-relativity book explained to an interviewer that he had intentionally refrained from discussing in his book the close association between relativity theory and the abominable concept of moral relativism because he had been advised that this might lead readers to suspect that his objections to relativity theory were motivated by a belief that it was somehow associated with moral relativism. Perish the thought. Return to MathPages Main Menu