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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!

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1 Answer 1

up vote 1 down vote accepted

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.

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