Mathematics of Physics and Engineering

104 Theory of Relativity

world line would be £,,,,„„ = f£ ^\x'(s)\^2 + |y'(s)|^2 + |z'(s)|^2 + \t'(s)\^2 ds.
Even if we disregard that t has the dimension of time and not distance, this
is still not the correct way to measure the length: as the reader will soon
discover, this formula is not invariant under the Lorentz transformation.
Accordingly, let us try to find an invariant distance formula in the form
/^2 \fa\x'(s)\^2 + a,2\y'{s)\^2 + 03|z'(s)|^2 + 6|t'(s)|^2 ds for some real numbers
ai,ci2,a3,6; the requirement of invariance takes priority over the require-
ment of positivity. Following formula (1.3.12) for the arc length on page
29, it is convenient to write the distance formula in the differential form:

(ds)^2 = ax(dx)^2 + a 2 (dy)^2 + a 3 (dz)^2 + b (dt)^2. (2.4.17)

EXERCISE 2.4.3? Verify that (2-4-17) is invariant under the Lorentz trans-
formation if and only if a\ = a 2 = a^ = —b/c^2 , where c is the speed of
light. Hint: direct computations. Start with the frame moving along the x-axis
so that x — ax\ + avt\, y = yi, z — z\, t = at\ + 7x1, with 0,7 from (2-4-9)
on page 100. Compute dx = adx\ + avdt\, etc., and plug in (2-4-17). Then the
invariance requirement (ds)^2 = a(dx{)^2 + ai{dy)^2 + a3(dz)^2 + b(dt)^2 implies
c^2 ai = —b. Letting the frame move along y axis and then along z axis, conclude
that c^2 a2 = —b, c^2 a^ = —b.
As a result, the general form of ds, invariant under the Lorentz trans-
formation, is (ds)^2 = a((dx)^2 + (dy)^2 + (dz)^2 -c^2 (dt)^2 ), and the two natural
choices of a are 1 and — 1; other choices would introduce uniform stretching
or compressing. Both a = 1 and a = — 1 are used in different areas of
physics. In what follows, our selection will be a = 1, corresponding to

(ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 - c^2 (dt)^2. (2.4.18)

To avoid negative distances, we have to modify the arc length formula

(1.3.10) and compute the length of the world line as follows:

L(s1)S2) = f" ^e (\x'(s)\^2 + \y'(s)\^2 + \z'(s)\^2 - c^2 \t'(s)\^2 ) ds, (2.4.19)

where

_ f 1, if |a:'(s)|^2 + \y'(s)\^2 + \z'(s)\^2 - c^2 \t'(s)\^2 > 0 (space-like world line),

\-l, if \x'(s)\^2 + \y'(s)\^2 + \z'(s)\^2 - c^2 \t'(s)\^2 < 0 (time-like world line).

This special way to measure distance is the main difference between

relativistic space-time and ordinary Euclidean space. As a result, the

vector space E^4 in which the distance is measured by (2.4.18) is often