On Inertial Rotation
Starting from Galileo, we are strongly convinced: while no outer forces act upon a physical body, it will steadily move along a straight line. Consequently, two observers moving one relative to another with a constant velocity will deal with the same structure of forces in any physical system.
Later on, a couple of scientific revolutions has attached forces to fields and connectivities on manifolds, for uniform linearity to give way to geodesic displacements. This, however, does not really change anything, since the whole theory stands on the same notion of a locally plain space (regardless of how we formally marry it with time).
Well, but what if space is not indeed that plain? Even locally. Or, maybe, it is plain, but not in an entirely traditional sense?
A modern physicist, who has long since got accustomed to mathematical indiscretion, would hardly ever object to the very formulation of the question: all right, let it be non-plain; then, what is it like? In the toy theories of everything, there are tons of topological perversions; should anybody seek for more? Well, in the quantitative sense, that's enough; but there is always way to dig a little bit deeper.
I admit that an outlook of a nontrivial philosophy is far from making the scientists of today entirely happy. Still, the problem is essentially philosophical, as we have to decide on primality. Which, of course, is relative, but never to the degree of utter disappearance in any particular case. Now, our question in the philosophical language looks like that: what can be considered as primary to inertial motion in the traditional sense? It seems like there is nothing else to lift from a completely free thing, which is simply allowed to do as it likes, flying on, just following its nose... At this point, a politically mature person will yield to vague suspicions, since there is no freedom on itself, we can only be free of something and for something. It is in the tall tales of the bourgeois propaganda that the ideas of absolute and unrestricted freedom get spread like butter on bread, regardless of epoch and range; their "universal" values eventually turn out to be the values of a bourgeois, and their "common" liberty becomes the "natural" right of a capitalist to plunder all around and, with all kinds of arms, cut short any attempt to take a bite of his sacrosanct property. Similarly, the talks of free motion in physics might well disguise the same bourgeois absolutes; in this context, the delivery of Galilean mechanics at the very dawn of European capitalism is in no way a coincidence. Never mind, this was a lyrical digression for the most inquisitive; from the physical consideration, we only spare ourselves the right of doubting the very presence of independent and non-interacting bodies, and hence the workability of the classical inertial motion.
Indeed, how should we learn about an absolutely free body? It does not know anybody, and nobody can see it. It's of no use for anything. In other worlds, it does not exist at all. Or, even if we grant it a kind of existence, this would put it outside the physical realm, as yet another immaterial abstraction. Keeping off religious savageries, such (ideal) formations are nothing but the features of things and their relations, rather than things as such; it is in this sense that we speak of their secondary (or derivative) character. Surely, any contrast is relative, and we always need a definite context, to distinguish the levels of hierarchy. This, however, does not remove the presence of hierarchical order in any specific situation.
One can only speak about motion (for instance, mechanical) in respect to something taken (in the current context) for immobile. What stands behind such a relatedness? In classical mechanics, the observer almighty arbitrarily connects and reconnects things, to produce a common frame of reference. This is an instance of the same abstract totality, since philosophy adopts the universal connectability of all things as a most general definition of the conscious subject. Now, if we (as appropriate for natural sciences) exclude the observer, all what is left is interaction, the influence of one body onto another. The outer distinction of the bodies must affect their inner state. In certain cases, and in certain respects, the state of one of the interacting bodies would vary less than the state of the other bodies, so that this relatively stable physical system could objectively play the role of a frame of reference. The traces of interaction with many other bodies mutually interfere within such a (material) reference system to allow the common presence of several bodies in one. Just rescind this amalgamation, and there will no longer be any interconnection, with the physical system splitting into a number of independent systems.
Classical education still suggests a one-way bias: there is a frame of reference, and there are physical bodies moving in that frame. Even admitting a shift to another observation point, we yet fancy the same overall construction in a different observer. When a physicist speaks of a transition to the frame of reference associated with one of the moving bodies, this is a metaphor, a sort of useful trickery; logically, a frame of reference can never be attached to any of the bodies represented within it, since this is a higher-level construct, a stable system of bodies. Nevertheless, since the very representedness in a frame of reference means interaction, one can never entirely rule out the influence of the frame on the motion of the bodies it embraces, which also assumes the indirect interaction of the physical bodies through the common frame of reference. Sometimes, this will count to a negligible correction; the bulk of classical physics piles up in this way. In other cases, conversely, a frame of reference would actively shape the physical system, imposing a specific mode of motion that does not pertain to the bodies "as such", that is, the same bodies taken in a different respect. This holds for quantum correlations; similarly, the society makes the human brain serve a culturally determined activity, though, on itself, this piece of flesh is devoid of any sublime inspiration, it does not assume any consciousness at all.
Well, suspending the primacy of uniform rectilinear motion, where should we look for a new conceptual basis? Let's approach it from a different direction. For instance, I (as a frame of reference) sit in the same place and do not move; but I can disturb the outer world in many ways (say, emitting some probing bodies, outgoing signals) to register its reaction (incoming signals). Admitting that my signals travel with the same speed in any direction (and what else could I assume in my stationary world?), I evaluate the positions and velocities of any physical bodies. Note that the detection of an incoming signal does mean anything on itself; any unauthorized intrusion is normally treated as environmental or instrumental noise, which many generations of physicists tried hard to eliminate, all through the history of science. Only those signals are considered as informative that come in response to our exploratory actions (though this parentage can sometimes be extremely distant, highly indirect, up to the illusion of "pure", unconditioned contemplation). That is, our notions of physical motion are based on the "inquiry–response" scheme; in philosophy, we call it a "reflexive act" (often, simply "reflection"), with a thing as if going out of itself and coming back. Reflection is in no way a god's gift; it can be found in a smallest portion of the Universe, and equally, in the Universe as a whole. What goes around comes around. Just turn around to turn up. All the parts of the world participate in universal reflection on different levels; with any particular part, it looks like regular recurrence, self-reproduction of a (relatively) durable integrity.
All we have to do is to express the same in the language of a physical theory. No good news for the adepts of static knowledge, for those engaged in erecting the only true depiction of the world, valid for all times. The same can differently manifest itself in many possible circumstances, and no formal model can grasp this versatility in full. There are no theories of everything; such constructs are beyond science. However, each level of research will coin abstractions representing the universal ability of reflection in a "natural" manner. Periodic motion in general (and harmonic oscillator in particular) are, probably, the most common examples. As soon as we pass to additional spatial dimensions, we can picture it as rotation.
In everyday life, we find that any motion at all is somehow related to rotation. The relocation of business freights and laboratory objects gets detected by the changing angles of observation; all that is pinned to a spinning planet involved in periodic movements of many a scale, including, at least, orbiting the Sun and following in its rotation around the center of the Galaxy. In fact, the very existence of compact cosmic objects assumes some limits for the relative motion of its components, which means oscillatory change of their positions and velocities within a final volume. Even considering collective states typical, say, of the inside of a neutron star, we do not avoid the issue of the oscillatory modes of motion that determine the possibility and spectrum of the observable radiation. Angular momentum conservation law is no worse than momentum conservation, while rotation exchange is as common in space as redistribution of translational velocities.
A logical conjecture: it is rotation that should be taken as the most general and fundamental form of motion, while infinitely straight lines are nothing but illusion arising from our usual experience limited to almost zero angles and relatively long radii, with the corresponding arc paths (linear displacements) significantly surpass the characteristic size of the observer. The exploration of the outer space has demanded a revision of our notions and freed us of many illusions; however, the idea of the primacy of rectilinear motion has proved to be stronger than the idea of the privileged role of the Earth (or the Sun) in the Universe: the primordial anthropocentrism is more deep-rooted than geocentrism. Our body is our first scale, the level of hierarchy to start any unfolding. We need much more time to inwardly accept the possibility of other scales.
In respect to rotation, the principle of inertia takes a slightly different form: in the absence of any outer influences, any rotation will proceed at the same rate for all times. This does not entirely match the classical conservation laws derived from the fundamental role of spatial translation. When a mass goes round on a circle, its angular momentum is preserved; however, classical inertial motion along a straight line will also conserve angular momentum, with the rotation speed slowing down at farther distances from the origin. In the world of inertial rotation, this trick won't work, as we demand the constancy of rotation rate regardless of the radius. This new vision of inertia implies that no "free" body can infinitely go along a straight line, as its trajectory will necessarily "curl". Linearity is still kept as a local approximation valid for short tracks far away from the rotation center, the origin of the corresponding reference frame.
At this stage, the formal applicability of traditional theories is in no way restricted. All we need is to admit that there is no plain space-time, or, in other words, that gravity is the fundamental mechanism of reference frame formation. Nobody’s going to contest that. Moreover, we don't necessarily need any point masses, which are not just physically real. On the other hand, we have long since grown accustomed to all kinds of "inner" motion in quantum systems. For a banal example of a self-supporting rotation, take the spin of microscopic particles usually pictured as rotation in some special space, where we do not know any physical forces that could act inside such spaces and make them anisotropic; it is not utterly impossible that some interesting discoveries wait for us in this line of thought.
Technically, mathematical physics makes use of two complementary concepts: power expansions and trigonometric series. Both are practically useful. The same function can be represented either by a power series or by Fourier integral. Still, the theoretical load of these formally equivalent representations is not the same. Power expansions lie in the basis of dynamic description, while the Fourier transform is mostly considered as an auxiliary, a matter of convenience. What if we reverse the roles? Why not picture the Universe as a hierarchy of all possible rotations (oscillations), considering a change in the state of rotation fundamentally important, primary to mere displacement?
It is not evident, whether such a viewpoint could be of any practical use. Power expansions are built in our everyday life; they incorporate the idea of stage-by-stage advancement, transition from one goal to another. Reflexivity is thus represented in its most folded form: we mark every return to the origin, but we do not care for anything in between, in the "dead" time. After all, any positional numeration system is all power series; yes, we know that there are other cultures differently structuring quantity, but we treat such notation as something rude and primitive, as temporary aids that are only valuable until we come to the "true" science of numeration. Quite probably, numeration systems based on Fourier expansions are as interesting, but they alien to our habits. Except, possibly, a few very special areas (for example, music). From philosophical considerations, however, one would deny any absolute "primacy". The inhabitants of some worlds will find rotation more natural; some other cultures will produce a counterpart of our "perturbation series" approach. One is also certain to encounter situations where neither of the two properly reflects the structure of activity, and we'll need deeply original inventions (though one might meet the old acquaintances somewhere behind the curtain).
In the era of the triumph of the Standard Model and intricate cosmologies, the appellations to the ages-old common intuition may seem utterly improper, if not obscene. A physical theoretician would possess an intuition of a higher rank capable of operating with abstract combinations of abstractions. And this is right, as it helps the humanity to tramp on to the bright future filled with advanced science and technologies. The practical importance is above all.
The souls of less concrete and quarks may sometimes grow vaguely suspicious: isn't the world too prompt in yielding to our prescriptions of the truth? Once we have postulated the constancy of the speed of light, we get tons of positive evidence at every corner. We have built an impressive theory of all the fundamental interactions, and voilà, all the particles we could predict. Just draw a couple of universal inequalities, and stage an immediate experimental justification. Black holes, gravitational waves... Anything you like. Just wish and get.
No doubt, our discoveries refer to some real portions of nature, and our smart theories can perfectly cope with that splendor. Still, inebriation with success is no better than any other drag addiction. Nature can never be reduced to what we have already found and mastered. Who can tell in which part our achievements are due to the merits of the method, and what is rather a (practically important) artefact? The Universe does not offer us more than we are able to ask for. See above, about the frames of reference. So, we roll on and on by the circles of cultural inertia, and we cannot drop out without an outer force. Maybe. But its inevitable intervention may yet be not so far away.