Is this action for a massive point particle in a curved spacetime correct? $$\mathcal S =-Mc \int ds = -Mc \int_{\xi_0}^{\xi_1}\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi$$ with sign convention $(+,-,-,-)$.
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The end points of your action must be events, therefore must be an d+1 dimensional object. Action must be extremized over all paths that start and end at the given space time points.
A path can be be parametrized by 4 functions of space and time, $x^\mu(\xi)$ of a one parameter object $\xi$. So it would be wrong to label the end points in terms of $\xi$. Instead $\xi$ must be treated as intermediate label to describe paths. Otherwise the functional form the integral is correct.
Prathyush
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It depends what you mean; that is the action of a test particle in a background gravitational field given by a metric $g_{\mu\nu}$. If you minimize it, you will get the geodesic equation. That is NOT the dynamical action for the gravitational field; your test particle does not change the curvature of the background spacetime. The action for the gravitational field the Einstein-Hilbert one,
$S=\frac{1}{\kappa}\int RdV$
where $R$ is the scalar curvature, $\kappa$ is the coupling constant.
levitopher
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