Notes on Relativity and Cosmology - Physics Department, UCSB

Chapter 6
Dynamics: Energy and
Momentum in Relativity
Read Einstein, ch. 15
Up until now, we have been concerned mostly with describing motion. We have
asked how various situations appear in different reference frames, both inertial
and accelerated. However, we have largely ignored the question of what would
make an object follow a given worldline (‘dynamics’). The one exception was
when we studied the uniformly accelerated rocket and realized that it must burn
equal amounts of fuel in equal amounts of proper time. This realization came
through using Newton’s second law in the regime where we expect it to hold
true: in the limit in which v/c is vanishingly small.
6.1 Dynamics, or, “Whatever happened to Forces?”
Recall that Newton’s various laws used the old concepts of space and time. As a
result, before we can apply them to situations with finite relative velocity, they
will have to be at least rewritten and perhaps greatly modified to accommodate
our new understanding of relativity. This was also true for our uniformly accelerating
rocket. A constant thrust does not provide a constant acceleration
as measured from a fixed inertial reference frame but, instead, it produces a
constant proper acceleration.
Now, a central feature of Newton’s laws (of much of pre-Einstein physics) was
the concept of force. It turns out that the concept of force is not as useful in
relativistic physics. This has something to do with our discovery that acceleration
is now a frame-dependent concept (so that a statement like F = ma would
be more complicated), but the main point actually involves Newton’s third law:
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