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.

The World According to Quantum MechanicsIf 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.