Graphical representation of the Lorentz transformation

In this section we concern ourselves solely with the transformation behavior of x and t under standard LTs, ignoring y and z which in any case are unchanged. What chiefly distinguishes the LT from the classical GT is the fact that space and time coordinates both transform, and, moreover, transform partly into each other: they get 'mixed,' rather as do x and y under a rotation of axes in the Cartesian x, y plane. We have already remarked on the formal similarity of the invariance of (2.15), which characterizes general LTs, to that of (2.8), which characterizes rotations. But in spite of the similarities, the character of a LT differs significantly from that of a rotation. This is brought out well by the graphical representation.

Recall first that there are two ways of regarding any transformation of coordinates (x, t) into (x;, t'). Either we think of the point (x, t) as moving to a new position (x', t') on the same set of axes; that is, we regard the transformation as a motion in x, t space; this is the 'active' view. Or we regard (x;, t') as merely a new label of the old point (x, t); this is the 'passive' view, whose graphical representation we shall discuss first. (See Fig. 2.6.) The events, once marked relative to a set of x, t axes, remain fixed; only the coordinate axes change. For convenience we choose units in which c = 1 (such as years and light-years, or seconds and light-seconds). We draw the x- and t-axes corresponding to the frame S orthogonal, with t taking the place of the usual y. This orthogonality is just another convenience without physical significance. Our 2-dimensional diagram, with one dimension taken up by time, has room only to map whatever is going on along the spatial x-axis of S (that is, one of the three mutually orthogonal 'wires' of the reference triad, not to be confused with the spacetime x-axis, like the one in Fig. 2.6). Under the standard LT here considered, S' shares its spatial x-axis with S. In fact, the diagram can describe whatever happens along this common spatial axis as seen by any number of frames in standard configuration with S.

Any curve representing a continuous one-valued function x = f(t) in the x, t plane corresponds to the motion of some geometric point along the spatial x-axis. It is called the worldline of the moving point. The slope of such a line relative to the

Graph Lorentz Transformation Hyperbola

t-axis, dx/di, measures the velocity of the point in S. Not all such lines are possible histories of real particles, since the latter must obey the relativistic speed limit (as we shall see in Section 2.10). So the inclination to the vertical of a particle worldline can nowhere exceed 45°.

'Moments' in S have equation t = const and correspond to horizontal lines, while the worldline of each fixed point on the spatial x-axis of S corresponds to a vertical line, x = const. Similarly, moments in S' have equation t' = const and thus, by (2.6), t — vx = const; so in our diagram they correspond to lines with slope v. In particular, the x' axis (t' = 0) corresponds to t = vx. Again, worldlines of fixed points on the spatial x' axis have equation x' = const and thus, by (2.6), x — vt = const. In our diagram they are lines with slope v relative to the t axis. In particular, the t' axis (x' = 0) corresponds to x = vt. Thus the axes of S' subtend equal angles with their counterparts in S; but whereas in rotations these angles have the same sense, in LTs they have opposite sense. S' can have any velocity between — c and c relative to S; the corresponding x - and t'-axes in the diagram are like scissors pointing NE (in the direction marked %), fully open for v ^ —c, closed for v ^ c.

We have already tacitly assumed the x- and t-axes in the diagram to be equally calibrated; for example, 1 cm corresponding to 1 s on the t-axis and to 1 lt-s on the x-axis. (In fact, the vertical axis is often taken to be ct, so that the units are naturally the same.) For calibrating the primed axes (not quite as straightforward as in the case of rotations) we observe that for standard LTs (where y' = y, z! = z) and with c = 1, (2.14) reduces to ta — x'2 = t2 — x2. So if we draw the calibrating hyperbolae t2 — x2 = ±1, they will cut all four of the axes at the relevant unit time or unit distance from the origin. They are, in fact, the loci of events whose interval from the origin is unity [cf. after (2.15)]. The units can then be repeated along the axes, by linearity. The diagram shows how to read off the coordinates (a', b') of a given event relative to S': we must go along lines of constant x' or t' from the event to the axes.

Diagrams like Fig. 2.6 are often called Minkowski diagrams. They can be extremely helpful and illuminating in certain types of relativistic problems. For example, one

can sometimes use them to get a rough preliminary idea of the answer. But one should beware of trying to use them for everything, for their utility is limited. Analytic or algebraic arguments are generally much more powerful.

As a first example, look at the dotted line in Fig. 2.6. It represents the uniform motion of a 'superluminally' moving point, say P, in S. Now imagine the x'- and t'-axes gradually scissored towards each other. The frame S' then chases P with ever increasing velocity v. Observe how, counterintuitively, the faster you chase P, the faster it recedes from you. Until, when the x'-axis coincides with the dotted line, P moves at infinite speed. If S' moves even faster (after the x'-axis surpasses the dotted line), P moves in the opposite sense along the spatial x'-axis, namely from greater to lesser values of x'. If P were a bullet, it would now travel from the broken glass back into the gun!

Length contraction and time dilation can also be read off, qualitatively, from the diagram. In Fig. 2.7 the shaded area shows the 'world-tube' (bundle of worldlines) of a unit rod at rest on the spatial x-axis between 0 and 1. In S' this rod moves at velocity -v. At t' = 0 it occupies the segment OM of the x'-axis, which, by reference to the calibrating hyperbola, is seen to be less than unity: the moving rod is short.

In Fig. 2.8 the t'-axis is the worldline of a standard clock fixed at the spatial origin of S' and therefore moving with velocity v through S. At where its worldline intersects the calibrating hyperbola, it reads 1. However, the corresponding time t in S is evidently greater than 1: the moving clock goes slow.

These examples should convince the reader of the utility of such 'passive' Minkowski diagrams. We next turn to the graphing of active LTs. Viewed actively, the standard LT moves each point (x, t) to a new position (x', t'). Motion is the key concept. Students good at computer graphics can usually manage to write a program displaying active LTs. But it is really quite sufficient just to imagine a computer display of the x,t plane with one's inner eye. Suppose it is covered with lots of bright dots like stars on the night sky. As we 'press the velocity button', we see the speedometer indicate bigger and bigger velocities (corresponding to the v of the LT applied), and watch the points move: They move along the set of hyperbolae t2 - x2 = const,

as indicated in Fig. 2.9. That they must stay on these hyperbolae is clear from the invariance of the interval. The four quadrants defined by the ±45° lines through the origin thus separately transform into themselves, as do their boundaries. Of course, all straight lines transform into straight lines, by linearity.

The details of the motion become clearer when we cast the LT into an alternative form. Adding and subtracting the x and t members of (2.6) (after multiplying the latter by c), we find ct' + x' = e-^(ct + x)

ct' - x' = e$(ct - x)

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'w

The 0 here introduced is a useful alternative to v as parameter for the Lorentz group (it is its 'canonical' parameter), often called the 'rapidity' or 'hyperbolic parameter.' (See also Exercise 2.15.)

To digress briefly: from eqns (2.16) we can read off without effort the group properties discussed at the end of the last section: the inverse of a Lorentz transformation with 0 is a Lorentz transformation with -0, and the composition of two Lorentz transformations with 01 and 02, respectively, is a Lorentz transformation with 01 + 02. And it is equally easy to read off from these equations (by multiplying them together) the fundamental invariance c2t2 - x2 = c2t'2 - x'2.

Returning to Fig. 2.9 (now again with c = 1), note that the (signed) distances, f and n, of a point (x, t) from the lines t + x = 0,t - x = 0, respectively (in other words, the Cartesian coordinates relative to the asymptotes of the hyperbolae), are given by f = (t + x)/V2, n = (t - x)/V2. (2.18)

The corresponding coordinate axes are marked with f and n in Figs 2.6 and 2.9. The LT (2.16) now reads very simply f ' = e-0f, n'= e0n. (2.19)

This shows that, as v (and with it 0) increases from zero to positive values, all f coordinates decrease in absolute value while all n coordinates increase. This leads to the motion pattern shown in Fig. 2.9. The opposite happens when v increases negatively. The fundamental invariance now reads f'n' = fn.

Recall that (x', t') are the coordinates in S' of some event d whose coordinates in S are (x, t). If we mark (x', t') as a point in the x, t plane, we see where d would be mapped if S' made the map. In other words, after applying an active LT through the appropriate value of v (or 0), we see the events as mapped by S' on a standard set of orthogonal axes—which is, of course, exactly what has by then become of the original x'- and t'-axes (shown in Fig. 2.9). This is one of the chief uses of the active transformation.

As a simple but important example, consider a particle whose worldline, relative to a given frame S, is the right-hand branch of the hyperbola x2 - t2 = x02. (2.20)

(Cf. Fig. 2.10.) A worldline is, of course, always traversed into the future (t increasing); so this particle comes from positive infinity along the spatial x-axis of S, moves in as far as x = x0 (at t = 0), where it momentarily comes to a halt (at the event marked d) and then goes back to infinity. But what is interesting about this motion is that the particle has constant proper acceleration; that is, its acceleration relative to the IF instantaneously comoving with it (its 'rest-frame') is always the same. To

see this, consider an arbitrary event ® on the worldline and apply an active LT so as to bring ® onto the horizontal axis: the hyperbola has not changed, ® is now where M was before, and we are evidently in the rest-frame of the particle at ® (its worldline now being vertical at Whatever its proper acceleration was at M is therefore its proper acceleration also at ® and our result is established. (We discuss this 'hyperbolic motion' analytically in Section 3.7.) Note also the interesting fact that a photon dispatched to 'chase' the particle at time t = 0 from the spatial origin x = 0 (its worldline is the top asymptote of the hyperbola) will never catch up with it; in fact, in any rest-frame of the particle the photon's distance from the particle is always precisely x0, as the same active LT reveals.

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