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I was wondering if anyone could help me clear up the following conceptual problem I'm having:

When going to the Newtonian limit, authors tend to throw out the terms in the geodesic equation that involve derivatives of the spatial coordinates

[tex]\frac{d x^i}{d \tau}[/tex]

because they are smaller than the derivative of time. Now, that seems fine if you're working to leading order. But then, they keep just the term

[tex]\frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{00} (\frac{dt}{d\tau})^2 = 0[/tex]

and find that

[tex]h_{00} = \frac{2 G M}{r}.[/tex]

Now, by the virial theorem, one finds that [itex]h_{00}[/itex] is then actually second order in [itex]v[/itex]. But then we were not working to leading order at all! So, it seems to me that throwing out said terms in the geodesic equation is not allowed, since the derivatives of the spatial coordinates are only one order of [itex]v[/itex] smaller than the derivatives of time:

[tex]\frac{d x^i}{d \tau} = \frac{d x^i}{dt} \frac{dt}{d\tau} = O(v) \frac{dt}{d\tau}.[/tex]

I tried working it out, keeping the terms, and it seems to me there are some that do not vanish to second order. Am I missing something here?

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# Non-static Newtonian Limit?

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