Mathematics of Physics and Engineering

(^102) Theory of Relativity
total energy £ of the same object moving with velocity v are
m
m 0
Vl-(«^2 /c^2 )
£ = mc^2. (2.4.13)
EXERCISE 2.4.1? Verify (24-12). Hint: Use the Lorentz transformation. For
£, use (2.4.11), since £ 0 = Axi = (AT - vAt)(l - {v/c)^2 ))~1/2 with At = 0,
because the difference of the coordinates is recorded at the same time in the fixed
frame. Similarly, for At, use (2.4-8) with Ax\ = 0, as the events happen at the
same point of the moving frame.
The relativity of mass (2.4.13) is a consequence of the relativity of time
and is a necessary consequence of the required invariance of the laws of
mechanics under the Lorentz transformation. This is one of the foremost
examples of how a mathematical model, namely, the Lorentz transforma-
tion, leads to discoveries of new physical phenomena. Relations (2.4.13)
have been verified experimentally on various particles. In 2005, the journal
Nature reported an experimental verification of the relation £ = roc^2 to
within 0.00004%.
We will now derive the main equation of relativistic kinematics. Recall
that Newton's Second Law for the point mass m can be written as F =
dp/dt, where p = mr is the momentum of m; see page 40. The extra
requirement of invariance of the law under the Lorentz transformation leads
to the definition of the relativistic momentum
p = mr =. , (2.4.14)
x/i-(IHI/c)^2 '
where m is the relativistic mass from (2.4.13). The relativistic form of
Newton's Second Law is therefore
F = ^, or F = ( ). (2.4.15)
dt> ^Wi-dHIA)^2 /
Equation (2.4.15) is the main equation of relativistic kinematics. It is invari-
ant under the Lorentz transformation, just as the non-relativistic Second
Law of Newton is invariant under the Galilean transformation; see Problem
3.2 on page 423. Note that, in the limits c^ooor |jdr||/c —> 0, meaning
speed of motion much smaller than the speed of light, equation (2.4.14)
becomes the familiar Second Law of Newton (2.1.1): F = mo r.