Newton's laws in relativity

(Copy of my Quora answer to question "Are Newton's three laws false?" or something to that effect)

They are, but it's intriguing to think about in what way exactly Newton's three laws have been replaced or generalised in relativity.
  1. There are two ways to think about the first law -- the first is "inertial reference frames exist". This is unchanged in special relativity, but general relativity generalises the notion with that of geodesics. The law as it is typically stated -- "stuff moves in straight lines on spacetime unless forced" is generalised to the geodesic equation, $\frac{{{d^2}{x^\mu }}}{{d{s^2}}} = - {\Gamma ^\mu }_{\alpha \beta }\frac{{d{x^\alpha }}}{{ds}}\frac{{d{x^\beta }}}{{ds}}$.
  2. $F=dp/dt$ is generalised to $F=dp/d\tau$ in special relativity, and is replaced by a covariant derivative in general relativity. $F=ma$ has some weirder changes.
  3. The third law is the conservation of momentum. This is replaced in General Relativity by the statement $\nabla^\mu T_{\mu\nu}=0$ ($\nabla$ instead of $\partial$).

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