Tagged Questions

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Textbook disagreement on geodesic deviation on a 2-sphere

Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation ...
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Static geodesics in GR

Can we find static geodesics of the type $$x^{\nu}=x_0^{\nu}+\delta_0^{\nu}\tau$$ in some space-time other than Minkowski's?
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Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~?$$ I guess ...
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I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm ... 1answer 89 views Null Geodesics in flat 2+1 dimensional Minkowski space For a given line element in flat 2+1 dimensional Minkowski space$$ g = ds^{2} = − dz \otimes dz + dx \otimes dx + dy \otimes dy .$$The null geodesics are supposedly given by:$$ x = lu + l' $$... 1answer 59 views Unable to resolve 2 equivalent geodesic equations A free particle moves along geodesics, one form being \begin{split} \ddot x^\mu &= -\Gamma^{\mu}_{\sigma \rho} \dot x^\sigma \dot x^\rho \\ &= -\frac{1}{2}g^{\mu \nu}(\partial_\sigma g_{\rho ... 2answers 199 views Geodesics equations via variational principle I would like to recover the (timelike) geodesics equations via the variational principle of the following action:$$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$... 1answer 106 views Sign of dr in Schwarzschild geodesics There is an equation that relates energy E, angular momentum L and other constants and variables to find \left(\frac{dr}{d\tau}\right)^2 in a plane. ... 0answers 77 views Derivation of equations of motion in Nordstrom's theory of scalar gravity? Nordstrom's theory of a particle moving in the presence of a scalar field \varphi (x) is given by$$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d ...
I'm solving an exercise about the Lagrange-Euler equations, that states the following: Let $\gamma (t) = \{ (t,q) : q = q(t), t_0 \leq t \leq t_1\}$ be a curve in $\mathbb{R} \times \mathbb{R}^2$. ...