I'm a bit confused by the following:(adsbygoogle = window.adsbygoogle || []).push({});

We can derive the equation of motion for a particle travelling on a timelike worldline by applying the Euler-Lagrange equations to the Lagrangian

[itex]\mathcal{L}=- g_{\mu \nu}(x(\tau)) \frac{d x^\mu}{d \tau} \frac{d x^\nu}{d \tau}[/itex]

However, to derive the field equations of General Relativity, we extremise the Einstein-Hilbert action

[itex]S_{\text{EH}}[g] = \frac{1}{16 \pi} \int_{\mathcal{M}} d^4x \sqrt{-g} R[/itex] where the integration is performed over the manifold [itex]\mathcal{M}[/itex] and [itex]R[/itex] is the Ricci Scalar

However this suggests that the necessary Lagrangian is

[itex]\mathcal{L}=\frac{1}{16 \pi} R[/itex] i.e. the integrand of the Einstein-Hilbert action.

My question is, why are the two Lagrangians the same. Surely they should be?

Secondly, I understand the motivation for the Einstein-Hilbert action, but where does the first Lagrangian actually come from? As far as I can tell they just seem to introduce it so that we end up with [itex]\ddot{x^\mu} + \Gamma^\mu{}_\nu \rho} \dot{x}^\nu \dot{x}^\rho=0[/itex] after applying the Euler-Lagrange equations.

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# General Relativity

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