This Quantum World/Appendix/Relativity/Lorentz

The case against Edit

In a hypothetical world with   we can define   (a universal constant with the dimension of a velocity), and we can cast   into the form


If we plug in   then instead of the Galilean   we have   Worse, if we plug in   we obtain  : if object   travels with speed   relative to   and if   travels with speed   relative to   (in the same direction), then   travels with an infinite speed relative to  ! And if   travels with   relative to   and   travels with   relative to    's speed relative to   is negative:  

If we use units in which   then the invariant proper time associated with an infinitesimal path segment is related to the segment's inertial components via


This is the 4-dimensional version of the 3-scalar   which is invariant under rotations in space. Hence if   is positive, the transformations between inertial systems are rotations in spacetime. I guess you now see why in this hypothetical world the composition of two positive speeds can be a negative speed.

Let us confirm this conclusion by deriving the composition theorem (for  ) from the assumption that the   and   axes are rotated relative to the   and   axes.


The speed of an object   following the dotted line is   relative to   the speed of   relative to   is   and the speed of   relative to   is   Invoking the trigonometric relation


we conclude that   Solving for   we obtain  

How can we rule out the a priori possibility that  ? As shown in the body of the book, the stability of matter — to be precise, the existence of stable objects that (i) have spatial extent (they "occupy" space) and (ii) are composed of a finite number of objects that lack spatial extent (they don't "occupy" space) — rests on the existence of relative positions that are (a) more or less fuzzy and (b) independent of time. Such relative positions are described by probability distributions that are (a) inhomogeneous in space and (b) homogeneous in time. Their objective existence thus requires an objective difference between spactime's temporal dimension and its spatial dimensions. This rules out the possibility that  

How? If   and if we use natural units, in which   we have that


As far as physics is concerned, the difference between the positive sign in front of   and the negative signs in front of     and   is the only objective difference between time and the spatial dimensions of spacetime. If   were positive, not even this difference would exist.

The case against zero KEdit

And what argues against the possibility that  ?

Recall the propagator for a free and stable particle:


If   were to vanish, we would have   There would be no difference between inertial time and proper time, and every spacetime path leading from   to   would contribute the same amplitude   to the propagator   which would be hopelessly divergent as a result. Worse,   would be independent of the distance between   and   To obtain well-defined, finite probabilities, cancellations ("destructive interference") must occur, and this rules out that  

The actual Lorentz transformationsEdit

In the real world, therefore, the Lorentz transformations take the form


Let's explore them diagrammatically, using natural units ( ). Setting   we have   This tells us that the slope of the   axis relative to the undashed frame is   Setting   we have   This tells us that the slope of the   axis is   The dashed axes are thus rotated by the same angle in opposite directions; if the   axis is rotated clockwise relative to the   axis, then the   axis is rotated counterclockwise relative to the   axis.


We arrive at the same conclusion if we think about the synchronization of clocks in motion. Consider three clocks (1,2,3) that travel with the same speed   relative to   To synchronize them, we must send signals from one clock to another. What kind of signals? If we want our synchronization procedure to be independent of the language we use (that is, independent of the reference frame), then we must use signals that travel with the invariant speed  

Here is how it's done:


Light signals are sent from clock 2 (event  ) and are reflected by clocks 1 and 3 (events   and   respectively). The distances between the clocks are adjusted so that the reflected signals arrive simultaneously at clock 2 (event  ). This ensures that the distance between clocks 1 and 2 equals the distance between clocks 2 and 3, regardless of the inertial frame in which they are compared. In   where the clocks are at rest, the signals from   have traveled equal distances when they reach the first and the third clock, respectively. Since they also have traveled with the same speed   they have traveled for equal times. Therefore the clocks must be synchronized so that   and   are simultaneous. We may use the worldline of clock 1 as the   axis and the straight line through   and   as the   axis. It is readily seen that the three angles   in the above diagram are equal. From this and the fact that the slope of the signal from   to   equals 1 (given that  ), the equality of the two angles   follows.

Simultaneity thus depends on the language — the inertial frame — that we use to describe a physical situation. If two events   are simultaneous in one frame, then there are frames in which   hapens after   as well as frames in which   hapens before  

Where do we place the unit points on the space and time axes? The unit point of the time axis of   has the coordinates   and satisfies   as we gather from the version   of (\ref{ds2}). The unit point of the   axis has the coordinates   and satisfies   The loci of the unit points of the space and time axes are the hyperbolas that are defined by these equations: