In classical Newtonian mechanics, from what I understand, conservation of energy stems from the fact that all known forces are conservative forces,
and vector calculus tells us that they can be represented by a potential energy function.
I understand how Energy Conservation is derived by Noether's theorem, but
I'm trying to understand relativistic dynamics better in terms of four-vectors.
I know four-force can be written in special relativity as the derivative of the
four-momentum with respect to proper time.
Are four-forces still conservative?
How do I write potential energy functions in SR?
If I have a constant force in classical mechanics is it still constant in SR?
(My guess abot the third question is not because four-momentum and three-momentum are quite different and four-momentum also depends on the relative velocity of the observer and the object)
My goal is to understand energy conservation is SR.
Thanx to anyone who answers!
Best Answer
It's not the four-force that is conservative, but the Einstein definition of force,
$$ F= {dp\over dt}$$
This force for a particle in an electromagnetic or linearized gravitational field is conservative in the same way as in Newton's model: the force is
$$ F = qE$$
and the integral of a static E around a closed loop is zero, still in relativity. The reason is explained in this answer: a priori validity of $W=\int Fdx$ in relativity? . The integral of the force over the distance as Einstein defines it is still the work done in the relativistic system.