Energy
Momentum Transformation
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The
energy and momentum in frame S can be transformed to energy and momentum
in S’, moving with velocity v in the +x direction with
respect to S, by writing the expression in S’ and transforming
the quantities which define them. Since both energy and momentum are defined in
terms of mass, m, start by transforming mass in S’ which is
defined as
where u’
is the velocity of the particle in S’. This is rather simple since all
we have to do is find the quantity g(u’).
This has been done in the section on velocity
transformations. The result is
Now
simply multiply through by m0
to give
The mass
and momentum of the particle in S are given by
Suppressing
the functional notion regarding u and u’, substitute these
values into Eq. (3) to give
Eq. (5)
is the mass transformation equation. Recall that m = Ef
/c2,
where Ef
is the free-particle energy defined as Ef
= Kinetic Energy + Rest Energy = K + E0
= gm0c2
= mc2.
The energy transformation for Ef
is found by multiplying Eq. (5) through by c2
to and replacing mc2
with Ef
and m’c2
with E’f.
The result is
This
expression is the energy transformation equation. To find the components
of S’ we start with the x component of momentum
We first
need to evaluate the product on the right hand side. This is simply the product
of Eq. (2) and
The
product in Eq. (7) then becomes
Now all
that is left to do is to multiply Eq. (9) through by
The y
component is defined as
Once
more we substitute Eq. (2) to give
Similarly
for the x component
Thus the
x and y components of momentum remain unchanged.
Summary:
