# Worldline action of point particle in gravitational field

In my GR lectures on the derivation of geodesic equations via extremal length, my lecturer wrote that the worldline action $$S$$ of a point particle with mass $$m$$ is given by $$S=-m\int\sqrt{-g_{\alpha\beta}\dot{x}^\alpha\dot x^\beta}d\tau,\tag{1}$$ where $$\dot x(\tau)=\frac{dx(\tau)}{d\tau}.\tag{2}$$

Where does this equation come from? I read from wikipedia that the worldline action of a relativistic point particle is $$S=-mc^2\int d\tau,\tag{3}$$ where $$\tau$$ is the proper time. Are theses two equations related in any way?

• Do you know how the proper time relates to the metric? (I.e. write out the line element and compare the two integral expressions above). Also note the indices on your $x$ terms should be raised not lowered. Mar 2, 2021 at 11:54

Yes, the main point is that the arc length$$^1$$ $$ds~=~cd\tau~=~\sqrt{-g_{\mu\nu}(x) dx^{\mu}dx^{\nu}}\tag{A}$$ in spacetime is the speed of light $$c$$ times the proper time $$d\tau$$, so that $$c\dot{\tau}~=~\dot{s}~=~\sqrt{-g_{\mu\nu}(x) \dot{x}^{\mu}\dot{x}^{\nu}},\tag{B}$$ where dot mean differentiation wrt. the worldline (WL) parameter $$\lambda$$ (which is not necessarily the proper time $$\tau$$). The action is therefore $$S~=~ mc^2\Delta \tau~=~ mc\Delta s~=~mc\int_{\lambda_i}^{\lambda_f}\!d\lambda ~\sqrt{-g_{\mu\nu}(x) \dot{x}^{\mu}\dot{x}^{\nu}}, \tag{C}$$ cf e.g. this Phys.SE post and links therein.
$$^1$$ In this answer, we use the $$(-,+,+,+)$$ Minkowski sign convention.