Once Before Time - Martin Bojowald [6]
RELATIVITY OF SPACE AND TIME:
SPACE-TIME TRANSFORMERS
All this took a long time, or a short time: for, strictly speaking, no time on earth exists for such things.
—FRIEDRICH NIETZSCHE, Thus Spoke Zarathustra
In physics, as in all of science, it is important to distinguish between properties that depend on the person making an observation and properties independent of an observer. The mass of a particle refers only to the particle itself and will, if the particle remains unchanged, always be measured as the same value. Except for unavoidable experimental inaccuracies, it does not matter who is doing the measurement. A particle’s velocity, on the other hand, appears different, and sometimes drastically so, depending on whether an observer is moving with respect to the particle. An observer moving along with the particle at exactly the same speed would perceive the particle as being at rest, well known from two cars cruising side by side along a straight stretch of highway. To the driver of one car, the other one seems not to be moving. Any other observer would see the car (or the particle) move and attribute to it a nonzero velocity. Relativity in general terms is the mathematical analysis of such relationships; it ultimately tells us what we can learn about nature in a fully objective, observer-independent way.
For many centuries, space and time were thought of as observer-independent. Distances between points and durations of temporal periods appeared absolute, no matter how an observer would be positioned or move. But the first fault lines in this worldview opened up toward the end of the nineteenth century, eventually leading to special relativity. In this new view, space and time cannot be seen in separation but are intertwined, interchangeable, and observer-dependent. Like the velocity of a particle, the values measured for them depend on the motion of an observer. In abstract terms, they describe different dimensions of a single physical object: space-time; and only space-time concepts, but not space or time themselves, are independent of the person making a measurement.
How can this be demonstrated by physical means? To answer this question and to explain the role of dimensions, we first consider space alone. Space has different dimensions, namely three: we can move sideways, back and forth, and up or down. Here, one might ask why these should be considered as three dimensions of a single space, rather than three completely independent directions: width, depth, height. The answer is simple. Width, depth, and height are not absolute and independent properties; they can be commuted into one another. We have only to turn around in space to make the height of a cube appear as its width, and in this sense height and width can be interchanged. This is not a transformation by a physical process, like a chemical reaction, but a much simpler one by means of changing our viewpoint. What we see as height, width, and depth depend on the place of an observer (or on conventions such as the use of Earth’s surface along which to measure width and depth); they cannot be considered properties of space itself as a physical object. For this reason, one speaks of three-dimensional space, not of the existence of three independent one-dimensional directions.
Time is similar, although its transformation is harder. By simply turning around one can influence only one’s view on space; the change of the angle of view (or, more precisely, the tangent of the angle as a mathematical function, which does not differ much from the angle when it is small) is expressed by the ratio of spatial extensions, such as the height before and after changing the viewpoint. By changing the angle, one can only transform spatial extensions into one another. If we want to transform space into time, we must vary