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3

First equation refers to the passive view of coordinate transformations while the second is the active view. Let $M$ be a manifold with metric $g$ which in local coordinates $x$ is written as $g_{\alpha\beta}(x)dx^{\alpha}\otimes dx^{\beta}$. Let $\phi:M\rightarrow M$ be a differentiable function and let $g'=\phi^*g$ be the pullback of the metric $g$, ...

3

Let's look at an example. Let's consider $0+1$ dimensions. Our manifold will be $M=\mathbb{R}$ and the coordinate system we will use will map the coordinate $x \in \mathbb{R}$ to the point $p \in M$ according to the rule $p(x^a) = x^0$. Now suppose the metric in this coordinate system has coordinates $g_{00}=x^0$ in this coordinate system. Now we would ...

4

The books are correct. The statement is a definite relation that is being imposed between the 'old' metric structure and the transformed one, for the transformation to be conformal. The equation you're unhappy about, $$g_{\mu \nu}'(x') = \Omega(x) g_{\mu \nu}(x)$$ states that the transformed metric $g_{\mu \nu}'$ at the transformed point $x'$ can be ...

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