Newtonian limit

 ( Special Relativity ) 

In physics, the Newtonian limit is a mathematical approximation applicable to exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitational fields. Under these conditions, Isaac Newton law of universal gravitation may be used to obtain values that are accurate. In general, and in the presence of significant gravitation, the general theory of relativity must be used.

In the Newtonian limit, spacetime is approximately flat and the Minkowski space may be used over finite distances. In this case 'approximately flat' is defined as space in which gravitational effect approaches 0, mathematically actual spacetime and Minkowski space are not identical, Minkowski space is an idealized model.

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

     \gamma=\frac{1}{\sqrt{1-v^2/c^2}}&\to 1
     

\end{align} and the Lorentz transformations between reference frames turn into Galileo transformations $$\backslash begin\{align\}$$

     t'=\gamma(t-v/c^2\,x)&\to t'=t \\
     x'=\gamma(x-v\,t)&\to x'=x-v\,t
     

\end{align}

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

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

\end{align} If the spacetime-metric is $$\backslash begin\{align\}$$

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

     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{align} which is the action that reproduces the Newtonian equations of motion of a particle in a gravitational potential $\backslash phi(x)$

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See also

• Classical limit

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