Any single observer can make consistent observations using concepts that are valid for him but that may not transfer to other observers. All the so-called paradoxes of relativity involve, not the inconsistency of a single observer’s deductions, but the inconsistency of assuming that certain concepts are independent of the observer when they are in fact very observer dependent.

Two more points should be made before we turn to the calculation of the Lorentz transformation. The first is that we have not had to define a ‘clock’, so our statements apply to any good timepiece: atomic clocks, wrist watches, circadian rhythm, or the half-life of the decay of an elementary particle. Truly, all time is ‘slowed’ by these effects. Put more properly, since time dilation is a consequence of the failure of simultaneity, it has nothing to do with the physical construction of the clock and it is certainly not noticeable to an observer who looks only at his own clocks. Observer O¯ sees all his clocks running at the same rate as each other and as his psychological awareness of time, so all these processes run more slowly as measured by O. This leads to the twin ‘paradox’, which we discuss later.

The second point is that these effects are not optical illusions, since our observers exercise as much care as possible in performing their experiments. Beginning students often convince themselves that the problem arises in the finite transmission speed of signals, but this is incorrect. Observers define ‘now’ as described in § 1.5 for observer O¯ , and this is the most reasonable way to do it. The problem is that two different observers each define ‘now’ in the most reasonable way, but they don’t agree. This is an inescapable consequence of the universality of the speed of light.

1.9 T h e L o re n t z t ra n s f o r m a t i o n

We shall now make our reasoning less dependent on geometrical logic by studying the algebra of SR: the Lorentz transformation, which expresses the coordinates of O¯ in terms of those of O. Without losing generality, we orient our axes so that O¯ moves with speed v on the positive x axis relative to O. We know that lengths perpendicular to the x axis are the same when measured by O or O¯ . The most general linear transformation we need consider, then, is

where α, β, γ , and σ depend only on v.

From our construction in § 1.5 (Fig. 1.4) it is clear that the ¯t and x¯ axes have the equations:

¯t axis (x¯ = 0) : vt − x = 0, x¯ axis (¯t = 0) : vx − t = 0.

The equations of the axes imply, respectively:

γ /σ = −v, β/α = −v,

x = ( x¯ + v ¯t)/(1 − v2)1/2

and

t = ( ¯t + v x¯)/(1 − v2)1/2.

Then we have

This is the Einstein law of composition of velocities. The important point is that |W | never exceeds 1 if |W| and |v| are both smaller than one. To see this, set W = 1. Then Eq. (1.13) implies

(1 − v)(1 − W) = 0,

that is that either v or W must also equal 1. Therefore, two ‘subluminal’ velocities produce another subluminal one. Moreover, if W = 1, then W = 1 independently of v: this is the universality of the speed of light. What is more, if |W| 1 and |v| 1, then, to first order, Eq. (1.13) gives

W = W + v.

This is the Galilean law of velocity addition, which we know to be valid for small velocities. This was true for our previous formulae in § 1.8: the relativistic ‘corrections’ to the Galilean expressions are of order v2, and so are negligible for small v.

1.11 Pa ra d oxe s a n d p h y s i c a l i n t u i t i o n

Elementary introductions to SR often try to illustrate the physical differences between Galilean relativity and SR by posing certain problems called ‘paradoxes’. The commonest ones include the ‘twin paradox’, the ‘pole-in-the-barn paradox’, and the ‘space-war paradox’. The idea is to pose these problems in language that makes predictions of SR seem inconsistent or paradoxical, and then to resolve them by showing that a careful application of the fundamental principles of SR leads to no inconsistencies at all: the paradoxes are apparent, not real, and result invariably from mixing Galilean concepts with modern ones. Unfortunately, the careless student (or the attentive student of a careless teacher) often comes away with the idea that SR does in fact lead to paradoxes. This is pure nonsense. Students should realize that all ‘paradoxes’ are really mathematically ill-posed problems, that SR is a perfectly consistent picture of spacetime, which has been experimentally verified countless times in situations where gravitational effects can be neglected, and that SR

‘Its top speed is 186 mph – that’s 1/3 600 000 the speed of light.’

Figure 1.15 The speed of light is rather far from our usual experience! (With kind permission of S. Harris.)

forms the framework in which every modern physicist must construct his theories. (For the student who really wants to study a paradox in depth, see ‘The twin “paradox” dissected’ in this chapter.)

Psychologically, the reason that newcomers to SR have trouble and perhaps give ‘paradoxes’ more weight than they deserve is that we have so little direct experience with velocities comparable to that of light (see Fig. 1.15). The only remedy is to solve problems in SR and to study carefully its ‘counter-intuitive’ predictions. One of the best methods for developing a modern intuition is to be completely familiar with the geometrical picture of SR: Minkowski space, the effect of Lorentz transformations on axes, and the ‘pictures’ of such things as time dilation and Lorentz contraction. This geometrical picture should be in the back of your mind as we go on from here to study vector and tensor calculus; we shall bring it to the front again when we study GR.

1.12 F u r t h e r re a d i n g

There are many good introductions to SR, but a very readable one which has guided our own treatment and is far more detailed is Taylor and Wheeler (1966). Another widely admired elementary treatment is Mermin (1989). Another classic is French (1968). For treatments that take a more thoughtful look at the fundamentals of the theory, consult Arzeliès (1966), Bohm (2008), Dixon (1978), or Geroch (1978). Paradoxes are discussed