Relativistic Illusion

There is science, and there are funny games, personal amusement. In science, we do not deal with manufacturing the tools and instruments: we just use them. Well, one might prefer certain brands to the others, but this does not make much difference. Afterwards, when the scientific product is ready, popular writers will explain what is really meant and how we can match it in our everyday life. Philosophers will justify the necessity of that very science, carefully concealing any pitfalls in the basic assumptions. The apparent contradictions are to be attributed to the weaknesses of the laymen's intuition, and a new kind if intuition is to provide a finer amusement, just to launch yet another coil of abstraction.

Special relativity theory has inspired numerous popular explanations. For a working physicist, there is nothing to explain, since, in science, a conceptual framework must be postulated before we start a meaningful discussion, and it is this common platform that makes science meaningful. For the rest of the humanity, formal manipulations are said to be justified enough as long as they conform to the so called principle of relativity, stating that the overall picture of a physical system's dynamics will be the same in all inertial frames of reference, moving with constant velocities in respect to each other. Unfortunately, most people are hardly aware of that this explanation only refers to the invariance of certain mathematical constructs under a class of coordinate transforms. Such relativity does not appeal to the heart of an average person, who would prefer palpable facts to any theoretical models, however perfect and beautiful. Theories come and go, while our acts remain basically the same. Here comes the other side and complement of the principle of relativity: the as famous correspondence principle demands that all the physical theories pertinent to the same physical domain agree with each other in its formal description.

Lack of understanding feeds all kinds of illusions. Which facts will survive a transition from one frame of reference to another? What is physical and what is not?

Take the simplest physical model ever, free mechanical motion. Slow down and think. Comparing the numerical estimates for lengths and times is a nontrivial activity requiring extensive procedural conventions. It seems like there is something more fundamental, a qualitative basis for quantitative comparison. When two observers perceive entirely different shapes, it's no use inquiring about the relative sizes of the details. If one observer detects an event, and the other doesn't, they can hardly compare the respective place and time. We have to admit, at least, that the two observers share some of their observations and are able to qualify them as referring to the same physical entity.

The next step is to pick out a particular common object and compare the overall character of motion in the same respect. This latter restriction may be important. Indeed, it is much easier to compare two trajectories than, say, a classical trajectory with a quantum system, or a statistical distribution. The spatial position (and shape) of the upper surface of a liquid column in a thermometer can be treated in a purely kinematic way, and we can trace its evolution with time; however, this is not exactly the same as temperature measurement.

To be sure, restricting ourselves to the same range of physical phenomena is not yet enough. With the same thing observed in different frames of reference, the observers are supposed to employ comparable observation techniques as well as interpret the results in a comparable manner. This may raise numerous questions about the validity of the corresponding reduction schemes. In the trivial mechanical model, we demand that the structure of all the frames of reference be the same. Three spatial coordinates and time will constitute the complete set of observables for every observer. The topology of the frame of reference is also universally fixed.

With all the precautions, we still cannot guarantee the overall agreement of the observed patterns of motion. As long as each observer entirely belongs to the corresponding frame of reference, there is no way to tell being at rest from steady (inertial) motion, and the inner scale of one frame does not need to comply with the scale of another. This may result in spurious forces interfering with the observed picture of dynamics, up to violating the very inertiality of the frame.

At school, we are told that a point steadily moving along a straight line in one frame of reference will also do that in any other inertial frame. But look at the picture below:

Let there be a stationary observer S, and a kind of solid rod placed in the XY-plane parallel to the axis X with the center positioned on the axis Y at the distance r from the frame's origin. There is another observer M moving relative to S along the stationary axis Y with some velocity V. It is traditionally assumed that M will see the rod steadily shifting towards the axis X, with a perfect accordance with the principle of relativity. The length of the rod will remain constant, and its ends will apparently draw straight lines in the frame of M, as it should be for the free motion of a material point.

It would have been so if the observer M saw the rod as belonging to the frame of S and moving entirely due to the relative motion of the two frames. However, there is no reason for M to associate the rod with the observer S and any frame of reference other than M's own. In M's place, everybody would observe a quite different behavior: some distant point gradually grows in an extended rod and then collapses back into a point. Observer S understands that the maximum size of the rod in M's eyes corresponds to the position just above the center of the rod. By why should M care for somebody else's impressions? For M, the seat of S looks like emitted from the same original point (big bang!), eventually being lost in the negative infinity. Isn't it like spatially expanding and collapsing Universe, with the position of S playing the role of time? Or entropy, if you wish. To enhance the resemblance, admit that M can see nothing but the expanding and collapsing rod (plus, probably the ticks of the clock represented by some signals from S). For a small portion of the whole story, with linearized dependences, we obtain the familiar cosmological picture.

By the way, the expansion and collapse of the rod in M's frame of reference is associated with certain acceleration, and hence forces. Are they spurious effects or true physical interactions? M has no evidence to tell.

A physicist would indicate that the calculated trajectories should exhibit certain numerical peculiarities that could be used to restore the overall picture of the two relatively moving frames of reference. The exact shapes of the dependences suggest a quite definite interpretation. This is how we deduce the global movements of the celestial bodies from our geocentric observations.

Is that convincing enough? Yes, for a physicist. A layman would wonder, why we should employ this calculation technique, and not that, and why the very idea of a frame common for all frames of reference should enter our heads. Isn't it against the principle of relativity? In a way, Einstein followed that very line of thought, to come to general relativity. On the other hand, to get a representative profile, we need to aggregate data from qualitatively different sources, which is quite an endeavor for a local observer (with a lifetime negligible on the cosmological scale), even in the trivial case, comparing the devices like a rule and a clock.

Numerical calculations are always questionable. They can be interpreted in many ways, and a slightest shift of perspective may result in a qualitatively different shape.

Indeed, just consider the popular example of the train passing a platform. When a passenger drops an apple, it will fall along a straight line, according to the notes of the passenger, while a person on the platform will see a curve. In a rotating frame, this curve would become a spiral. Are these shapes different? It depends. Thus, topology identifies all the shapes that can be continuously transformed into each other; still, an abrupt turn of a vehicle may cost somebody a life.

In other words, the validity of any calculus essentially depends on the context. To compare frames of reference, we need a common basis, a higher-level frame. To compare such upper frames we need yet another level of hierarchy. The overall picture of the world will vary from one such unfolding to another. The quality of the objects belonging to a specific level (and hence the range of their quantitative estimates) can only be defined in respect to a particular layered structure. If we guess it right, we call it a physical law. A wrong guess is mere illusion. Still, physics can never judge what is right or wrong; to do science, we need something beyond science. Within a wide class of activities, it may be important to know how it goes; in many other activities, we'd rather stick to how it feels. Restricting ourselves to a superficial impression, we lack depth; overestimating the power of science, we lack wisdom. However attractive in their apparent universality, our theories may be as illusory as a naive vision of a passer-by. Imposing the familiar modes of action regardless of the inner nature of the object is no better than sheer fantasy. Universal consent is not yet universal principle. To cultivate a bit of reason, let us never forget that our picture of the world (and the physical world in the first rank) isn't but relative, with all its frames and illusions being both a reflection of the current level of our local development and an indication of the movable nature of any horizon, as any relativity is relative. Including this one.