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I am working on the same problem from Schutz as this question, which discusses the weak gravitational field metric.

$$ds^2=−(1+2\phi)dt^2+(1−2\phi)(dx^2+dy^2+dz^2)$$

From this metric, I was able to guess a transformation of the following form:

$$\left(\Lambda^{\bar{\alpha}}_{~~\beta}\right)=\begin{pmatrix}\sqrt{1+2\phi}&0&0&0\\0&\sqrt{1-2\phi}& 0&0\\0&0&\sqrt{1-2\phi} &0\\0&0&0& \sqrt{1-2\phi}\end{pmatrix}$$

This $\Lambda$ transforms the Minkowski metric appropriately to the weak field metric. However, the coordinates transform in the wrong way when I approximate to first order in $\phi$. $$x^0=-\sqrt{1+2\phi}~x^0=-x^0(1+\phi)$$ $$x^\bar{k}=\sqrt{1-2\phi}~x^k=x^k(1-\phi)$$

On closer inspection, I believe my transformation $\Lambda$ is incorrect as it is not a coordinate transformation: $$\Lambda^{\bar{\alpha}}_{~~\beta,~\mu}\neq \Lambda^{\bar{\alpha}}_{~~\mu,~\beta}$$

In the solution to this problem, Schutz uses many ideas without explaining their motivations, like the following:

  • Immediately stating $x^\bar{\alpha}=(\delta^\alpha_{~~\beta}+L^\alpha_{~~\beta})x^\alpha$ (Is this the most general transformation for this metric? Why?)
  • Using $\Gamma^\bar{\lambda}_{~~\bar{\mu}\bar{\nu}}=0$ and its transformation equation (I get that the equation is mathematically useful, but what is the motivation for using Christoffel symbols?)

So far, I've managed to derive the following:

$$\begin{align*} \Gamma^{\lambda}_{~~{\mu}{\nu}}&=\Lambda^{\bar{\alpha}}_{~~\mu}\Lambda^{\bar{\beta}}_{~~\nu}\Lambda^{\lambda}_{~~\bar{\gamma}}\Gamma^\bar{\gamma}_{~~\bar{\alpha}\bar{\beta}}+\Lambda^{\bar{\alpha}}_{~~\mu,~\bar{\beta}}\Lambda^{\bar{\beta}}_{~~\nu}\Lambda^{\lambda}_{~~\bar{\alpha}}\\ &=0+\Lambda^{\bar{\alpha}}_{~~\mu,~\bar{\beta}}\Lambda^{\bar{\beta}}_{~~\nu}\Lambda^{\lambda}_{~~\bar{\alpha}}\\ &=\Lambda^{\bar{\alpha}}_{~~\mu,~\nu}\Lambda^{\lambda}_{~~\bar{\alpha}} \end{align*}$$

But I can't figure out what the form of $\Lambda^{\lambda}_{~~\bar{\alpha}}$ is. How can I continue?

What are the motivations behind using this method to find the coordinate transformations i.e. must we use Christoffel symbols? Is there a standard way to find coordinates for a particular metric?

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