I just read a sentence in(adsbygoogle = window.adsbygoogle || []).push({}); GRAVITATIONby MTW (aka the "Princeton Phonebook") that made me realize a confusion wrt the metric, connection, and curvature. In short how are [tex]g_{\mu\nu}, \Gamma^{\alpha}_{\mu\nu}, and R^{\alpha}_{\beta\mu\nu}[/tex] distinguished? They all include the description "how space curves."

Here is the little I understand (just so you know where to start):

[tex]g_{\mu\nu}[/tex]: how the action curves between two events (or two points in spacetime), thus:

[tex]ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}[/tex]

[tex]\Gamma^{\alpha}_{\mu\nu}[/tex]: how a point travels from one place to another place in the spacetime. (I am only starting to become familiar with modern math terms, alla Wikipedia.org. From what I read, the affine connection is the track laid out by the point as it travels through the manifold.) Thus the equation of geodesic deviation:

[tex]\f{d^2\xi^{\alpha}}/{d\tau^2} + \Gamma^{\alpha}_{\mu\nu}\f{d\xi^{\mu}}/{d\tau}\f{d\xi^{\nu}}/{d\tau} = 0[/tex]

[tex]R^{\alpha}_{\beta\mu\nu}[/tex]: how the global spacetime is curved at every point in the spacetime, thus the EFE:

[tex]G_{\mu\nu} = R_{\mu\nu} -\f{1}/{2}g_{\mu\nu}R = 8\pi T_{\mu\nu}[/tex]

Note in fact that [tex]g_{\mu\nu} \rightarrow \Gamma^{\alpha}_{\mu\nu} \rightarrow R^{\alpha}_{\beta\mu\nu}[/tex]

After writing this out I think I have it correct. If not please correct me. Thanks in advance for all replies.

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# Metric, connection, curvature oh my!

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