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While I can't speak for the author, I find it very likely that this is a typo. The author probably meant the Schrödinger equation, not Klein-Gordon.

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Use the identity that if $M$ is invertible and $\delta M$ is "small" compared to $M$, then we have $$\det (M + \delta M) = \det(M) \det( 1 + M^{-1} \delta M) \approx \det(M) \left[ 1 + \text{tr} (M^{-1} \delta M) \right].$$ In the case of the metric, this implies that $$-\det(g + \delta g) \approx -\det(g) \left[ 1 + g^{ab} \delta g_{ab} \right]$$ and ...

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The easiest trick is to write (say, in three dimensions. For higher dimensions, add indices to the Levi-Civita symbol, and factors of g): $$g = \frac{1}{3!}\epsilon^{abc}\epsilon^{xyz}g_{ax}g_{by}g_{cz}$$ Then, the variation is easy. I'll leave it as an excersise to work out the variation, and how to translate the result into factors of $g_{ab}$ and $g$

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Correct, the standard Schwarzschild metric is asymptotically flat and indeed the time co-ordinate $t$ is the local time of an observer infinitely removed from the black hole and sitting in this flat space and so there is no pair of points outside the black hole's event horizon which ultimately cannot causally reach or signal each other. The de ...

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Yes. The short answer is you have one action you extremize to get Einstein's Field Equation $G_{\alpha\beta}=kT_{\alpha\beta}.$ Which you can think of as equations of motion for the gravitational metric $g_{\alpha\beta}.$ (They determine the second derivatives of the metric in terms of the matter fields and metric and the first derivatives of the metric.) ...

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No, instead of the metric, the Vierbein enters, pulling back the gamma matrix, defined in the usual way in the tangent space, to the spacetime manifold. $$\bar \psi \, \gamma^\mu D_\mu \psi = \bar \psi \, \gamma^\alpha {e^\mu}_\alpha D_\mu \psi$$ This can be considered as the insertion of "half a metric", if one wishes. The Vierbein captures the ...

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The relation between the metric and gravitational potential (and between Christoffel symbols and acceleration) is evident in the Newtonian limit of General Relativity. The basic assumptions of this approximation are: Weak gravitational field: the metric $g_{\mu\nu}$ differs from the Minkowski metric $\eta_{\mu\nu}$ only a small amount  g_{\mu\nu} = ...

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Marek didn't really make a mistake, but you did. But Marek might have been unclear by jumping over some steps. So first let's clarify. The metric gives the differential squared interval $ds^2=-dt^2+dx^2$ from which you can get the differential proper time $d\tau=\sqrt{dt^2-dx^2}.$ So for the blue straight line you have $\tau$=$\int ... 1 It is not the definition of an event horizon, and in fact you can choose coordinates that are regular near the event horizon. A common reason for coordinates that are irregular at the horizon is if the coordinate is primarily made to represent time far away. In that case, a timelike curve has a negative interval in your convention, so you can have time ... 1 First, Landau and Lifshitz stated that$ds$and$ds'$approach zero simultaneously, so that there is some hidden variable$x$such that, $$\lim_{x\to 0} ds(x) =0$$ and $$\lim_{x\to 0} ds'(x) =0,$$ assuming and$ds$and$ds'$are continuous functions of$x\$. Next, the two are infinitesimals of the same ...

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