bei48482_FM

Appendix II to Chapter 1

Spacetime

A

s we have seen, the concepts of space and time are inextricably mixed in

nature. A length that one observer can measure with only a meter stick may

have to be measured with both a meter stick and a clock by another observer.

A convenient and elegant way to express the results of special relativity is to regard

events as occurring in a four-dimensional spacetimein which the usual three coordi-

nates x,y,zrefer to space and a fourth coordinate ictrefers to time, where i 1.

Although we cannot visualize spacetime, it is no harder to deal with mathematically

than three-dimensional space.

The reason that ictis chosen as the time coordinate instead of just tis that the

quantity

s^2 x^2 y^2 z^2 (ct)^2 (1.52)

is invariantunder a Lorentz transformation. That is, if an event occurs at x,y,z,tin

an inertial frame Sand at x ,y ,z ,t in another inertial frame S , then

s^2 x^2 y^2 z^2 (ct)^2 x   2 y    2 z    2 (ct )^2

Because s^2 is invariant, we can think of a Lorentz transformation merely as a rotation

in spacetime of the coordinate axes x,y,z,ict(Fig. 1.24).

The four coordinates x,y,z,ictdefine a vector in spacetime, and this four-vector

remains fixed in spacetime regardless of any rotation of the coordinate system—that

is, regardless of any shift in point of view from one inertial frame Sto another S.

Another four-vector whose magnitude remains constant under Lorentz transforma-

tions has the components px, py, pz, iEc. Here px, py, pzare the usual components of

the linear momentum of a body whose total energy is E. Hence the value of

px^2 py^2 pz^2 

E^2

c

46 Appendix to Chapter 1

y

s

x

y′

s

x′

Figure 1.24Rotating a two-dimensional coordinate system does not change the quantity s^2 x^2

y^2 x     2 y    2 , where sis the length of the vector s. This result can be generalized to the four-

dimensional spacetime coordinate system x, y, z, ict.

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