I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $x \mapsto x'$ such that the metric is invariant up to scale:
$$g'_{\mu \nu}(x') = \Lambda(x) g_{\mu \nu} (x).$$
On the other hand we know from GR that under any coordinate transformation the metric changes as
$$ g_{\mu \nu} (x) \mapsto g'_{\mu \nu}(x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} .$$
I feel like there is a notation problem (inconsistency) in these formulas, or maybe I am mixing active and passive coordinate transformations. For instance, if we consider a simple rotation (which is of course a conformal transformation with no rescaling, i.e. $\Lambda(x)=1$) then from the first formula we see that $g'_{\mu \nu}(x') = g_{\mu \nu} (x)$, whereas from the second formula we get something more complicated. Where is the flaw?
In the "String theory" lecture notes by David Tong the same definition of conformal transformation is given. Then he says:
A transformation of the form (4.1) has a diferent interpretation depending on whether we are considering a fixed background metric $g_{\mu \nu}$, or a dynamical background metric. When the metric is dynamical, the transformation is a diffeomorphism; this is a gauge symmetry. When the background is fixed, the transformation should be thought of as an honest, physical symmetry, taking the point $x$ to point $x'$. This is now a global symmetry with the corresponding conserved currents.
I think it has to do with my question, but I don't fully understand it...
Answer
OK I think I know what is going on. It's all about primes. Consider an active spacetime transformation:
$$ x^{\mu} \mapsto x'^{\mu}(x)$$
$$g_{\mu \nu} (x) \mapsto g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}}$$.
(the transformation of the metric tensor follows from the fact that it is a rank 2 tensor). With this notation both Di Francesco and David Tong are wrong (as far as I understand). The GR book by Zee on the other hand writes it properly. First of all consider an isometry. This is an spacetime transformation as before that leaves the metric invariant, meaning
$$ g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = g_{\mu \nu} (x')$$.
(watch the primes). On the other hand a conformal transformation is a transformation that satisfies a weaker condition: it leaves the metric invariant up to scale, meaning
$$ g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = \Omega^2(x')g_{\mu \nu} (x')$$.
Now there should be no inconsistency. Di Francesco's definition was wrong (according to this convention/notation/understanding) because it compared the metric before and after the transformation at different points, and you have to compare them at the same point.
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