This Quantum World/Appendix/Relativity/Lorentz

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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.


Strot.png


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 K[edit]

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 transformations[edit]

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.


Stdiag.png


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:


Clocks.png


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:


Hyperbo.png