Newtonian limit

In special relativity, Newtonian behaviour can in most cases be obtained by performing the limit ${\displaystyle v\to 0}$. In this limit, the often appearing gamma factor becomes 1

$${\displaystyle {\begin{aligned}\gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}&\to 1\end{aligned}}}$$

and the Lorentz transformations between reference frames turn into Galileo transformations

$${\displaystyle {\begin{aligned}t'=\gamma (t-v/c^{2}\,x)&\to t'=t\\x'=\gamma (x-v\,t)&\to x'=x-v\,t\end{aligned}}}$$

The geodesic equation for a free particle on curved spacetime with metric ${\displaystyle g^{\mu \nu }}$ can be derived from the action

$${\displaystyle {\begin{aligned}S[x,{\dot {x}}]&=-m\,c\,\int dt\,{\sqrt {-g_{\mu \nu }\,{\dot {x}}^{\mu }\,{\dot {x}}^{\nu }}}\end{aligned}}}$$

If the spacetime-metric is

$${\displaystyle {\begin{aligned}g&={\begin{pmatrix}-1-{\frac {2\,\phi (x)}{c^{2}}}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}\end{aligned}}}$$

then, ignoring all contributions of order ${\displaystyle {\frac {1}{c^{2}}}}$ the action becomes

$${\displaystyle {\begin{aligned}S[x,{\dot {x}}]&=-m\,c\,\int dt\,{\sqrt {c^{2}+2\,\phi (x)-{\dot {\mathbf {x} }}^{2}}}\\&=-m\,c\,\int dt\,\left({\sqrt {c^{2}}}+{\frac {1}{2\,{\sqrt {c^{2}}}}}\,\left(2\,\phi (x)-{\dot {\mathbf {x} }}^{2}\right)+...\right)\\&\approx \int dt\,\left(-m\,c^{2}+{\frac {1}{2}}m{\dot {\mathbf {x} }}^{2}-m\,\phi (x)\right)\end{aligned}}}$$

which is the action that reproduces the Newtonian equations of motion of a particle in a gravitational potential ${\displaystyle \phi (x)}$ [2]

• Classical limit