Yes, I want to make sure that geodesics of a particle moving in curved space time is the same thing of projectile trajectories.(adsbygoogle = window.adsbygoogle || []).push({});

I start from assuming that [itex]1-\frac{2GM}{r}\approx1-2gr[/itex] and then calculate the schwarzschild metric in this form

[itex]\Sigma_{\mu\nu}=\begin{bmatrix}\sigma & 0\\ 0 & -\sigma^{-1}\end{bmatrix}[/itex] where [itex]\sigma = 1-2gr[/itex]

and I calculated for the Christoffel symbols for this metric:

[itex]\Gamma^0_{\mu\nu}=-\sigma g\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}[/itex]

[itex]\Gamma^1_{\mu\nu}=-\frac{g}{\sigma^2}\Sigma_{\mu\nu}[/itex]

I plugged them to a geodesics equation

[itex]\partial^2_\tau x^\mu = -\Gamma^\mu_{\alpha\beta}\partial_\tau x^\alpha\partial_\tau x^\beta[/itex]

where [itex]d\tau^2 = dx^\mu dx^\nu\Sigma_{\mu\nu}[/itex]

and I got these ugly conditions:

[itex]\partial^2_\tau t = \sigma\partial_\tau t\partial_\tau \sigma[/itex]

[itex]\partial^2_\tau \sigma = \frac{2g^2}{\sigma^2}[/itex]

what I expect is just something like

[itex]x=-\frac{g}{2}t^2[/itex]

I havn't finished these differential equations yet. But I want to know that I'm going through the right track, right? Any suggestion?

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# Geodesics VS Projectile

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