Mathematics of Physics and Engineering

The Lorentz Transformation 103

EXERCISE 2.4.2. (a)c Verify that

||p||^2 + mlc^2 = m^2 c^2. (2.4.16)

Hint: direct computation using (2.4-14).

(b)B Show that the quantity c^2 (m — mo) is natural to interpret as the rel-

ativistic kinetic energy, that is, £ = mc^2 = £K + m^c^2. Hint: if C is the

trajectory, v = ||r|| is the speed, and the initial speed is zero, then the work done

by a force F is

£K= J F-dr= f rd{mr)= f' ydl m°V , ., ) =mc^2 -m 0 c^2 ,

Jc Jo Jo V C^1 - (l/7c^2 )) ' /

where the last equality follows after integration by parts.
One consequence of the second equality in (2.4.12) is the Twin Paradox,
which is often described as follows. Consider two identical twin brothers,
Peter and Paul, who live in an inertial frame O. Peter stays in that frame
O, and Paul makes a round trip in a spaceship, travelling out and back in
a fixed direction with fixed speed v = (24/25)c. Then the total travel time
Ao = 2i, as measured by Paul in the moving spaceship's frame, will be
A = 25Ao/7 in the frame O as measured by Peter; the key relation here,
beside (2.4.12), is 25^2 = 24^2 + 7^2. In particular, if A 0 = 14 years, then
A = 50 years, that is, Peter will be waiting for 50 years for Paul to return
from what is 14-year long trip for Paul. If Paul's biological clock also obeys
(2.4.12), then, by the end of the trip, Paul will be 26 year younger than
Peter.

For further discussions of (2.4.12) and (2.4.13), including the Twin Para-
dox, see, for example, the books Concepts of Modern Physics by A. Beiser,
2002, and Spacetime Physics by E. F. Taylor and J. A. Wheeler, 1992.
We conclude this section with a brief discussion of the geometry of
relativistic space-time. The reader will see that this geometry is not exactly
Euclidean, thus getting a gentle introduction to general relativity.
Relativistic space-time is a combination of time with the three spa-
cial dimensions of our physical space, and is modelled mathematically as a
four-dimensional vector space. Fixing a coordinate system in this space
represents every point by four coordinates (x,y,z,t). A curve r(s) =
(x(s),y(s), z(s),t(s)), so < s < si, in this space is called a world line. If
this were a truly Euclidean space with cartesian coordinates, then, by anal-
ogy with formula (1.3.10) for the arc length on page 29, the length of this