15 Why special relativity? (I)

Let us imagine a universe containing a single object. Would we be able to attribute to this object a position — to say where it is? Of course we wouldn’t. At a minimum we need two objects for this. If we have two objects (and also a way of measuring distances), we can say how far one is from the other. We can imagine a straight line from one to the other, but if this is all, we cannot attribute to this line an orientation. The bottom line: there is no such thing as an absolute position or orientation. If we want to make physical sense, we can only speak of the positions and orientations of physical objects relative to other physical objects.

If instead we were to imagine a world in which a single event takes place, we wouldn’t be able to attribute to this event a time — to say when it occurs. So there is no such thing as an absolute time either. If we want to make physical sense, we can only speak of the times of physical events relative to other physical events. And if we were to again imagine a world containing a single object, we also wouldn’t be able to attribute to this object a velocity — to say how fast it was moving, in which direction it was moving, or even if it was moving at all. So there is no such thing as absolute rest either. If we want to make physical sense, we can only speak of the velocities with which physical objects move relative to each other.

All of this is encapsulated in the principle of relativity, according to which all inertial (coordinate) systems are created equal: the laws of physics do not “favor” any particular inertial frame or class of such frames. Using the language of classical physics, we call a coordinate system inertial if the components x,y,z of the position of any freely moving object change by equal amounts Δx,Δy,Δz in equal time intervals Δt. In other words, the ratios formed of Δx, Δy, Δz, and Δt are constants. In yet other words, the spacetime trajectories of freely moving objects are straight; such objects move along straight lines (in space) with constant speeds.

Hence if F is an inertial frame, then so is any coordinate frame that, relative to F,

  • is shifted (“translated”) in space by a given distance in a given direction,
  • is shifted (“translated”) in time by a given amount of time,
  • is rotated by a given angle about a given axis, and/or
  • moves with a constant velocity.

If we use this information to express the coordinates x,y,z,t of an inertial frame F in terms of the coordinates x′,y′,z′,t′ of another inertial frame F′, assuming for the sake of simplicity that the coordinate origins of the two frames coincide (t=0,x=0,y=0,z=0 mark the same spacetime point as t’=0,x’=0,y’=0,z’=0), that the space axes of the two frames are parallel, and that F′ moves in the direction of the x-axis with a constant speed w relative to F, we arrive at the following transformation law:

(2.14.1)   t’ = (t + Kwx)/√(1 + Kw2),   x’ = (x − wt)/√(1 + Kw2),   y’ = y,   z’ = z.

K is a universal constant yet to be determined. Since its value depends on conventional units — its physical dimension is that of an inverse velocity squared — we are left with only three physically distinct options:

K = 0,   K > 0, or   K < 0.

So which is it? To find out, imagine an object O moving along the common x-axis of the two frames with the speed v relative to F′. Its speed relative to F is then

(2.14.2)   u = (v + w)/(1 − Kvw).


The case against K > 0

If K is positive, Eq. (2.14.2) tells us that O’s speed relative to F is greater than the sum of the speed of O relative to F′ and the speed of F′ relative to F. Furthermore, if both v and w approach the speed 1/√K, u approaches infinity, and if the product vw is greater than 1/K, then u is negative! What does this mean?

If K is positive and we choose measurement units in which K=1, the transformation law (2.14.1) can be cast into the form

t’ = t cos α + x sin α,   x’ = x cos α − t sin α,   y’ = y,   z’ = z.

This describes a rotation by some angle α of the x and t axes in the spacetime plane that contains the two axes. If w equals 1/√K, the t’-axis is rotated by 45° relative to the t-axis, and if v equals 1/√K, the straight line in spacetime followed by O is rotated by 45° relative to the t’-axis. The angle between the t-axis and O’s path in spacetime thus equals 90°. In other words, O’s path in spacetime is parallel to the x-axis, and this is the same as saying that u=∞.

If the product vw is greater than 1/K, the angle between the t-axis and O’s path in spacetime is greater than 90°. The reason why u is negative therefore is not that O moves backward in space but that it moves backward in time. If K>0, making a U–turn is as easy in a spacetime plane containing the time axis as it is in a plane containing two space axes, and what is a space axis according to one reference frame can be the time axis according to another: the difference between space and time depends on the language we use to describe a physical situation rather than on the physical situation itself. Yet a viable theory about the physical world must, at a minimum, feature an objective (that is, language-independent) difference between space and time; otherwise it would be impossible to objectively talk about such things as the future, the past, or the history of the world.

Here is a more technical reason why there ought to be an objective difference between the time axis and the axes of space: Whereas the stability of an atom requires that its internal relative positions be fuzzy, they cannot be completely fuzzy: the probability of finding an atomic electron cannot be the same for all positions relative to the nucleus. On the other hand, the stability of an atom requires a stable ground state: the probabilities it defines must not change with time.