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##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?

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- Thread starter dwellexity
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- #1

- 25

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##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?

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- #4

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I have got ##G_{tt} = - \frac{2(-1+2\phi)(\phi + 2 \phi^2 +r \phi')}{(r+2r\phi)^2}##

How do I proceed from here? I am getting a first derivative of ##\phi## instead of second derivative.

- #5

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I have got ##G_{tt} = - \frac{2(-1+2\phi)(\phi + 2 \phi^2 +r \phi')}{(r+2r\phi)^2}##

Yes, this looks ok.

I am getting a first derivative of ##\phi## instead of second derivative.

Yes, but remember that the Einstein tensor has two pieces: ##G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R##, where ##R_{\mu \nu}## is the Ricci tensor, and ##R## is the Ricci scalar. So the EFE is really ##R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = 8 \pi T_{\mu \nu}##. You might try calculating the two pieces separately to see if there are second derivatives there.

Also, you might take a look at Carroll's online lecture notes on GR, chapter 4, which has a discussion of this calculation.

- #6

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Yes, but remember that the Einstein tensor has two pieces: ##G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R##, where ##R_{\mu \nu}## is the Ricci tensor, and ##R## is the Ricci scalar. So the EFE is really ##R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = 8 \pi T_{\mu \nu}##. You might try calculating the two pieces separately to see if there are second derivatives there.

I don't understand how this would affect anything. Even if Ricci tensor and Ricci Scalar have second derivatives, what ultimately matters is this particular sum.

- #7

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Even if Ricci tensor and Ricci Scalar have second derivatives, what ultimately matters is this particular sum.

Not necessarily. Take a look at Carroll's notes. The short version: there is an alternate way of writing the EFE, which moves the trace term from the LHS to the RHS:

$$

R_{\mu \nu} = 8 \pi \left( T_{\mu \nu} - \frac{1}{2} g_{\mu \nu} T \right)

$$

For the case under discussion only the 0-0 component of this equation is significant, and only ##T_{00}## is significant in the trace ##T##.

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