Force Transformation Rules
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The
differential version of the Lorentz transformation
is
The x
component of force is defined as
The
first part of the term on the right is easily evaluated
The
second part of Eq. (2) is found by substituting in the expression for p’x
from the momentum transformation rule
where
that E is the inertial energy of the particle. With these value Eq.
(2) becomes
To
evaluate dE/dt we take the time derivative of the energy momentum
relation E2
– c2p·p=
E02.
Substitute
E = m c2,
p = mu and f º
dp/dt to give
Canceling
the 2mc2
term in Eq. (7) gives dE/dt which was what we wanted
Substituting
this result and dpx/dt
= fx
into Eq. (5) to give the final result
The y
and z components of the force are much easier to find since the
corresponding components of momentum remain unchanged upon transformation, i.e. p’y
=py,
and p’z
= pz.
Therefore
Summary:
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