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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I want to know what happens to the space a black hole crosses over as our galaxy travels through space.
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