4.2  Inertial and Gravitational Separations

And I am dumb to tell a weather’s wind

How time has ticked a heaven round the stars.

                                                    Dylan Thomas, 1934

The special theory of relativity is formulated as a local theory, so its natural focus is on the worldlines of individual particles. In addition, special relativity presupposes a preferred class of worldlines, those representing inertial motion. The idea of a worldline is inherently “absolute” in the sense that it is nominally defined with reference only to a system of space and time coordinates, not to any other objects. This is in contrast to a truly relational theory, which would take the "dual" approach, and regard the separations between particles as the most natural objects of study. In fact, as mentioned in Section 4.1, we could go to the relationist extreme of regarding separations as the primary ontological entities, and considering particles to be merely abstract concepts that we use to psychologically organize and coordinate the separations. The relationist view arguably has the advantage of not presupposing a fixed background or even a definite dimensionality of space, since each “separation” could be considered to represent an independent degree of freedom. Of course, this freedom doesn’t seem to exist in the real world, since we cannot arrange five particles all mutually equidistant from each other. Indeed it appears that the n(n-1)/2 separations between n particles can be fully encoded as just 3n real numbers, and moreover that those real number vary continuously as the individual particles “move”. This is the justification for the idea of particles moving in a coherent three-dimensional space.

Nevertheless, it’s interesting to examine the spatial separations that exist between material particles (as opposed to the space and time coordinates of individual particles), to see if their behavior can be characterized in a simple way. From this point of view, the idea of "motion" is secondary; we simply regard separations as abstract entities having certain properties that may vary with time. In this context, rather than discussing inertial motion of an individual particle, we consider the spatial separation (as a function of time) between two inertial particles. However, since we don’t presuppose a background of absolute inertial motion, we will refer to the particles as being “co-inertial”, meaning simply that the spatial separation between them behaves like the separation between two particles in absolute inertial motion, regardless of whether the two particles are actually in absolute inertial motion.

Is it possible to characterize in a simple way the spatial separations that exist between co-inertial particles? Consider, for example, the spatial separation s(t) as a function of time between a stationary particle and a particle moving uniformly in a straight line through space, as depicted in the figure below for the condition when the direction of motion of the moving particle B is perpendicular to the displacement from the stationary particle A.

Obviously the separation between objects A and B in this configuration is stationary at this instant, i.e., we have ds/dt = 0, and yet we know from experience that this physical situation is distinct from one in which the two objects are actually stationary with respect to each other’s inertial rest frames. For example, the Moon and Earth are separated by roughly a constant distance, and yet we understand that the Moon is in constant motion perpendicular to its separation from the Earth. It is this transverse motion that counter-acts the effect of gravity and keeps the Moon in its orbit. This is another reason that we ordinarily find it necessary to describe motion not in purely relational terms, but in terms of absolutely non-rotating systems of inertial coordinates. Of course, as Mach observed, the apparent existence of “absolute rotation” doesn’t necessarily refute relationism as a viable basis for coordinating events. It could also mean that we must take more relations into account. (For example, the Moon’s motion is always tangential to the Earth, but it is not always tangential to other bodies, so it’s orbital motion does show up in the totality of binary separations.) Whether or not a workable physics could be developed on a purely relational basis is unclear, but it’s still interesting to examine the class of co-inertial separations as functions of time. It turns out that co-inertial separations are characterized by a condition that is nearly identical to the condition for linear gravitational free-fall, as well as for certain other natural kinds of motion.

The three orthogonal components Dx, Dy, and Dz of the separation between two particles in unaccelerated motion relative to a common reference frame must be linear functions of time, i.e.,

where the coefficients a_i and b_i are constants.  Therefore the magnitude of any "co-inertial separation" is of the form

where

Letting the subscript n denote nth derivative with respect to time, the first two derivatives of s(t) are

The right hand equation shows that s₂ s₀³ = k, and we can differentiate this again and divide the result by s₀² to show that the separation s(t) between any two particles in relatively unaccelerated (i.e., co-inertial) motion in Galilean spacetime must satisfy the equation

Now we consider the separation that characterizes an isolated non-rotating two-body system in gravitational free-fall. Assume the two bodies are identical particles, each  of mass m. According to Newtonian theory the inertial and gravitational constraints are coupled together by the auxiliary quantity called "force" by the following equations

where G is a universal constant.  (Note that each particle's "absolute" acceleration is half of the second derivative of their mutual separation with respect to time.)  Equating these two forces gives s₂ s₀² = -2Gm. Differentiating this again and dividing through by s₀, we can characterize non-rotating gravitational free-fall by the purely kinematic equation

The formal similarity between equations (1) and (2) is remarkable, considering that the former describes strictly inertial separations and the latter describes gravitational separations. We can show how the two are related by considering general free motion in a gravitational field. The Newtonian equations of motion are

where r is the magnitude of the distance from the center of the field and w is the angular velocity of the particle. If we solve the left hand equation for w and differentiate to give dw/dt, we can substitute these expressions into the right hand equation and re-arrange the terms to give

which applies (in the Newtonian limit) to arbitrary free paths of test particles in a gravitational field. Obviously if m = 0 this reduces to equation (1), representing free inertial separations, whereas for purely radial motion we have d²r/dt²  = -m/r², and so this reduces to equation (2), representing radial gravitational separation.

Other classes of physical separations also satisfy a differential equation similar to (1) and (2). For example, consider a particle of mass m attached to a rod in such a way that it can slide freely along the rod.  If we rotate the rod about some point P then the particle in general will tend to slide outward along the rod away from the center of rotation in accord with the basic equation of motion

where s is the distance from the center of rotation to the sliding particle, and w is the angular velocity of the rod. Differentiating and multiplying through by s0 gives

Then since s₂ = w²s₀, we see that s(t) satisfies the equation

                                             (3)

So, we have found that arbitrary co-inertial separations, non-rotating gravitational separations, and rotating radial separations are all characterized by a differential equation of the form

                                                    (4)

for some constant N. (Among the other solutions of this equation (with N = -1) are the elementary transcendental functions e^t, sin(t), and cos(t).) Solving for N, to isolate the arbitrary constant, we have

Differentiating this gives the basic equation

If none of s₀, s₁, s₂, and s₃ is zero, we can divide each term by all of these to give the interesting form

This could be seen as a (admittedly very simplistic) “unification” of a variety of physically meaningful spatial separation functions under a single equation. The “symmetry breaking” that leads to the different behavior in different physical situations arises from the choice of N, which appears as a constant of integration.

Incidentally, even though the above has been based on the Galilean spatial separations between objects as a function of Galilean time, the same conditions can be shown to apply to the absolute spacetime intervals between inertial particles as a function of their proper times.  Relative to any point on the worldline of one particle, the four components Dt, Dx, Dy, and Dz  of the absolute interval to any other inertially moving particle are all linear functions of the proper time t along the latter particle's worldline.  Therefore, the components can be written in the form

where the coefficients a_i and b_i are constants.  It follows that the absolute magnitude of any "co-inertial separation" is of the form

where

Thus we have the same formal dependence as before, except now the parameter s represents the absolute spacetime separation. This shows that the absolute separation between any fixed point on one inertial worldline and a point advancing along any other inertial worldline satisfies equation (1), where subscripts denote derivatives with respect to proper time of the advancing point. Naturally the reciprocal relation also holds, as well as the absolute separation between two points, each advancing along arbitrary inertial worldlines, correlated according to their respective proper times.

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