Relativity in Ten Minutes
Imagine slowly passing a simple, solid, three-dimensional object through the surface of a pool of perfectly still water. Keep in mind that the surface is the boundary between the air and the water, and so does not include the water of the pool itself (nor the air above the water). An oblong, American-style football serves this purpose well because it is a simple, (effectively) solid, three-dimensional object. As the football were to pass through the plane of the surface of the pool, it would intersect this two-dimensional surface, first as a point, then as the changing contour of a football, and finally, as a point again, as it finished crossing through this surface, leaving the surface unrippled, flat, and still, just as it was before the football began entering.
Let us imagine further, that the surface of the pool represents a two-dimensional perspective of observation. The changing outline of the object, the football, as it passes through the surface, is how the three-dimensional object is perceived, from a two-dimensional 'perspective'. From a two-dimensional perspective, it is impossible to 'perceive' a three-dimensional object, in its entirety, as a three-dimensional perspective – which ours happens to be – can. The two-dimensional perspective allows only two-dimensional cross-sections, of the three-dimensional solid object (the football). So, no matter how many dimensions an object might have, from a two-dimensional perspective, no more than two dimensions can ever be perceived, in any given moment, directly.
Naturally, one can only ask, what does a shape (the football) passing through the surface of a pool of water have to do with explaining a more correct and fundamental way to look at motion? How can this image lead us to a simpler way to explain the relativity of space and time measures? We can answer this question by imagining passing, instead of the simple shape of a football, a much more complex shape through this same surface, a single object that would intersect the surface at many points simultaneously, thus creating many different, individual cross-sections at the same time, instead of just a single one as the football would. A classic, outdoor, television antenna, like those before parabolic antennas, it being a complicated lattice of rods, would serve this purpose well. It is a complex, three-dimensional shape that would intersect the surface at many individual and distinct places, all of them small (in area), like ‘points’ are small (in area).
We can imagine passing the complex shape of a classic outdoor television antenna, with its interconnected lattice of rods, through the surface, along a single, unchanging direction, straight down, perpendicular to the surface, and equally important, at a steady, uniform rate of motion (speed). We will orient the antenna in such a way that no rod will be perpendicular to this direction (the direction in which it is introduced to the surface). Passing it in such an orientation prevents any rod from being parallel to the intersecting surface, and thus excludes any rod from intersecting this perceiving surface in more than a single "point" (meaning discrete region). From the point of view of this perceiving surface, the 'points' of intersection would change their position, as the object passed through, either approaching or receding, from one another (ignoring any parallel rods). The sharper the angle of the intersecting rod, with respect to the surface of the pool, the faster its intersections would change position.
The changing positions of these points, intersecting the surface, could easily be mistaken for something other than the changing contours of an antenna, passing through the surface. They could easily be mistaken for something else because they would look exactly like something else. Instead of realizing that all that they were really perceiving was a classic, three-dimensional television antenna, being pushed through their world, any two-dimensional creatures inhabiting the two-dimensional surface of the pool that we are imagining, could be easily convinced, and quite reasonably so, that what they were perceiving couldn't 'really' be a 'higher dimensional' shape (the antenna), intersecting with their lesser dimensional surface, as it passed through their world, but instead, the 'motion' of discrete and individual points, or rather, the motion of discrete and individual 'particle points', moving across their two-dimensional surface (this, their two-dimensional surface, being their version of 'space').
According to our perspective, imagining the changing intersections of the antenna, we imagine the antenna dropping, as a whole, with the points of intersection changing. We imagine the rods as rods, in their entirety. But, remembering that the number of dimensions to one’s perspective is the limit on the number of dimensions that one can perceive (that is, within the context of a two-dimensional perspective, a rod of the antenna, being part of a three-dimensional shape, cannot be perceived in its three-dimensional entirety in just two dimensions –one fewer than the number of dimensions that it requires), then, by virtue of this limitation, the moving intersections of the antenna on the surface would be perceived as 'bounded' points in two dimensions only, NOT in the three dimensions that the antenna would really have. They would not seem like intersecting three-dimensional ‘rods’ at all, but rather, like two-dimensional particles in motion, across their seemingly flat (i.e. Euclidian), two-dimensional universe!
We can use this picture of an antenna, creating motion upon the surface of a pool of water, by it intersecting with the surface of the pool as it passes through this surface, for explaining relativity in a very simple and very understandable geometric way. And so we shall. We shall follow, through the initial history of their modern physics, the achievements of these two-dimensional creatures inhabiting the surface of our pool, as we explore exactly how in these first fifteen minutes, they discover Special Relativity, which works no differently for them than it does for us, except with one dimension fewer; and next, how they use Special Relativity’s simple explanation of tilting spaces for deriving a more inclusive form of relativity, General Relativity, which accurately describes the effects of gravity. We can consider relativity in this fewer number of dimensions because, provided there is more than one dimension, the number of dimensions there may be is an irrelevant consideration to either theory. By exploring how, like us, these creatures finally arrive at their physical theory, in the simplicity of fewer (two) dimensions, we can see, in easily imaginable, and clearly understandable terms, just how this theory "works".
When we imagine passing the television antenna through the surface of the pool, we might assume that it is the antenna that moves, while the water and its surface remain fixed. However, from the perspective of the television antenna, which, of course, can consider itself to be perfectly at rest (just as we can, while we’re reading, yet still be moving with the motion of our planet, solar system, and galaxy through space), it is not itself (the television antenna) that moves, but rather, the level of the pool that rises. Let us assume such a state of affairs, so that instead of dropping the antenna into the water, we simply hold it in place, letting the water, and its surface, do the rising, again, at a steady and unchanging rate.
Realizing that any clock always ran at the very same pace, anywhere on their surface, early two-dimensional physicists among the two-dimensional creatures inhabiting the surface of the pond would quickly realize that this meant that they could specify the time as being the same everywhere, since such a property would seem to be preserved as their surface rose. They would therefore logically conclude that their entire surface (in which the time is specified as being the same anywhere upon it) was, just as it is in our three-dimensional 'surface' for us, always rising, anywhere and everywhere, at the same steady rate (speed), regardless of one's 'position', on it. They would utilize this phenomenon to eventually map motion, with respect to the steady rate with which their surface was always rising.
They would specify 'speed', according to the relationship between the rate with which the surface was rising, and surface distances. Consequently, they would treat rising as a dimension. They would treat it as such because they would, in their mathematical models, associate the line defining the dimension of time as being parallel to that of an observer at rest on a rising surface. Considered in such a way, time would seem perfectly perpendicular to the two-dimensional 'flat' surface of their pool (meaning perpendicular to any two perpendicular lines on it) Two-dimensional physicists would utilize time in this way, as a convenience, just as our early three-dimensional physicists did.
Just like these first three-dimensional physicists, their two-dimensional equivalents would initially believe time was something completely independent of, and clearly distinguishable from, their 'space', which would be their surface. Such a thing would be clearly obvious, because they would believe that no one could intuitively 'reason' motion itself being an actual physical dimension, 'inseparable' from space, any more than one could ‘reasonably’ consider time being a physically real part of the geometry of their universe. They’d have reasoned most incorrectly; time (according to relativity's impeccable description of nature) is identical to space. Unquestioned faith in their conventional, intuitive reasoning would lead to many a misunderstanding about nature. Still, misunderstanding would stubbornly persist, because they, like so many of us, would incorrectly reason it simply far too difficult or altogether pointless, to even try to understand.
Because their surface would be, just like our three-dimensional space is, always rising everywhere at the same rate, everywhere, regardless of one's position on it, two-dimensional physicists would have no basis for believing that this would not also be true regardless of one's motion upon it, as well. They would so presume, because motion in their realm would be confined to such slow speeds that they would never observe any thing marking time moving fast enough to suggest otherwise. Under closer examination however, they would realize that this could not possibly be true (that a surface could not possibly rise at the same rate for something 'moving'), because they would notice repeatedly that their speed of light was, just as it is in our universe, a 'constant' (and finite) speed, everywhere, even on (i.e. with respect to)things inmotion. It never changed, as a consequence of moving anything emitting or observing light – NEVER – which is not necessarily what they might imagine happening, since the speed of such motion should, intuitively, be expected to compound (add to or subtract from) the speed of the emitted light, along the direction of motion. Yet, it would not, just as it doesn’t in our universe.
Intuitively, we know, just like the two-dimensional creatures on our rising surface would surely know as well, that by pushing something, it goes faster. So naturally, if light travels at a particular speed, and whatever it is that is creating the light is pushed, to make it move faster, then, intuitively, the light should go faster, in that direction, by being pushed in that direction by the motion, thereby increasing its speed. Likewise, in the direction opposite to the motion, light 'should', correspondingly, slow down, its speed reduced, or 'canceled', by the motion. Conversely, moving the observer, with respect to the light, ‘should’ have the same effect too, since that is what happens when we push anything else, anything except light. For light, it doesn't; in both directions of the motion as well as in any other direction, its speed always measures the same (only wavelength does not, as its measure can change with motion [e.g. the "red shifting" of rapidly receding stars and the "blue sifting" of rapidly approaching stars]).
You can neither speed up light – nor slow it down – using motion, no matter how much (how fast) you push it or pull it (or anyone observing it). In other words, this speed would, anywhere and everywhere at any time, never change, which is to say that this speed would always be "constant." Correspondingly, so would the laws of nature (i.e. the laws of physics) never change, either. Early two-dimensional physicists would be most perplexed by this fact, and, they would lack a logical and demonstrable explanation of how this could be (which is to say that they would lack an explanation maintaining the same laws of nature for things moving as for things that were not – so as to maintain the common convention of a universal, flat rising surface for all 'physical existence'), unless there happened to appear among them a very astute and insightful physicist, so astute and insightful that he or she recognized just what the speed of light being constant really meant (as the reader ultimately will too), and so, was able to explain how this speed could NEVER be changed by "moving" the light (or moving anyone observing it either) – as well as how.
Once revealed, this explanation would be immediately obvious, its logic remarkably simple and straightforward, its implications absolutely amazing, its validity irrefutably undeniable, and its truth inescapably compelling. This physicist would have imagined what probably none had ever imagined before: time and space together, as one. It would be the greatest conceptual leap in two-dimensional history short of harnessing fire or inventing language, standing in stark defiance to convention and tradition, and providing a description of nature’s space, time, and motion as had never existed before.
By explaining 'why' (the laws of nature never changing) and 'how' (tilting surfaces) the speed of light never changed, this exceptionally astute and insightful physicist recognizing what an unchanging (again, constant) speed of light meant would recognize many things, all of them as a result of realizing just one thing, which is that his or her surface was NOT the sole and only surface, but rather, JUST ONE, AMONG MANY, many OTHER SURFACES in his or her universe, all 'tilted' uniquely with respect to each and every other surface, each corresponding to a unique motion in this universe. Such a physicist would have discovered an extremely significant quality of their world – of their universe. That quality, to be precise, would be the inseparability (i.e. relativity) of time and space, with respect to motion. That physicist would have recognized EXACTLY what the great, three-dimensional physicist of our own universe, Albert Einstein, recognized over a century ago. He or she would have discovered the 'first half' of a model for describing the geometry oftheir realm.
This remarkable two-dimensional physicist would have discovered the Theory of 'Special' Relativity. With Special Relativity, this physicist (or any) could describe, among many other things, just how another surface 'tilted', changing when and where events occurred, as a consequence of uniform motion on their or any other observer's surface – motion that does not change speed or direction. To be precise, 'special' relativity would explain, again, among many, many other things, just how anything and everything in the universe can be both at rest, and moving, with respect to anything and everything else in it, at the same time. It would be an unprecedented living achievement, equaled only by the harnessing of two-dimensional fire and the development of written language.
That is not all. Most amazingly, a universe of tilting surfaces leads to the most accurate description of gravity ever conceived, as the next edition’s science article will explain (in much fewer words than were necessary here). Yes, the speed of light never changing, would lead to a description of gravity that has endured for a nearly a hundred years, and that is responsible for, among other things, the original television and all successful space exploration.