Space-Time Coordinates and Frames

 

Descriptions of spatio-temporal relations and motions are often expressed in terms of coordinate systems, which assign a unique set of four real numbers  x,y,z,t  to each point of an assumed four-dimensional manifold.  Needless to say, there are infinitely many mappings between coordinate sets and points that can serve this purpose.  The first and strongest restriction we ordinarily place on the mapping is the requirement that our coordinate system possess the same topological connectivity as the manifold.

 

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 spacetime 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. 

 

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 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 = gmndxmdxn where the gmn are the components of the metric tensor and summation from 1 to 3 is implied for the indices m and n.  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.  Such an equivalence class of coordinate systems is called a frame of reference.

 

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.  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.  We can use identical particles repelling each other from a mutual state of rest to synchronize separate clocks (with respect to that state of rest).

 

Inertial isotropy forms one of the bases for our intuitive concept of simultaneity.  Another is our visual sense, i.e., we tend to regard things we see as occurring now, at the same moment when we see them, even though they may be some distance away.  The speed of light is so great that in Galileo’s day it was still thought to possibly be infinite, so it could be used directly as a means of establishing simultaneity.  Naturally it agreed with the inertial concept of time (at least up to the precision that could be measured in those days.)  Even after it was realized that the speed of light is finite, it could still be used as the basis for synchronizing separate clocks, and it still appeared to be consistent with inertial synchronization.  This would be expected if we regard light as an inertial phenomenon, like small inertial particles, but it’s surprising if we regard light as a wave in a fixed medium – unless that medium somehow affects the inertia of material bodies as well.

 

This is the point at which classical ideas of a mechanistic luminiferous ether failed, because such an ether can’t account for the inertia of material bodies.  The classical ether (of the late 19th century) was seen as the basis for the propagation of forces such as electromagnetism, but classical forces cannot even account for the stability of the atom, let alone of sub-atomic particles.  According to classical electromagnetism an electron ought to radiate energy and quickly spiral into the nucleus.  There is no stable configuration of charges and electromagnetic fields, as shown by Earnshaw in a paper originally published in the 19th century in a paper about the luminiferous ether.  (Of course, we also know that Maxwell’s laws of electromagnetism lead to absurd predictions about cavity radiation - the ultra-violet catastrophe - and they are unable to account for the photo-electric effect and other quantum phenomena.)  Since classical force laws cannot account for stable massive objects, they obviously can’t account for how the shapes of such objects change with speed, let alone how their inertia changes with speed. 

 

Of course, it can easily be shown that equipotential surfaces around a charged point-like particle change shape in the required way, but equipotential surfaces do not constitute stable configurations of matter.  Likewise it can be shown that the so-called electro-magnetic mass (inertia) of a charged particle varies with speed in the required way, but it’s now known that only a very small fraction of the rest mass of stable particles is electromagnetic in origin, and moreover that the inertial rest mass of stable particles cannot be accounted for by any classical force.  The ultimate origin of inertia is still unknown, so we obviously cannot show it to vary with speed in any particular way.  This is why Lorentz found it necessary (in 1909 and 1915) to simply assume that the inertia of material objects transforms with velocity in the same way as electromagnetic forces.  In other words, he simply assumed mechanics is covariant with respect to Lorentz (not Galilean) transformations, and his justification for doing this was simply that this is a necessary assumption in order for the principle of relativity to be completely valid.  He did not in any sense derive the Lorentz covariance of mechanical inertia, he explicitly assumed it.  (See The End of My Latin.)

 

It’s often said that Einstein’s 1905 paper on special relativity abolished the 19th century ether, but actually it was his paper on light quanta that showed most clearly the inadequacy of Maxwell’s equations and the representation of light as a wave propagating in a medium.  Furthermore, Earnshaw’s theorem implied that classical forces could never account for stable configurations of charged particles, so it was clear that any theory encompassing the motions of massive objects could not be constructed on the basis of classical forces.

 

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