Consider the special conformal transformation (SCT), $$ x'^{\mu} = \Omega(x)(x^\mu-b^\mu x^2), $$ where $\Omega(x)=(1-2b\cdot x+b^2x^2)^{-1}$. The covariant form should be $$ x'_{\mu} = \Omega(x)(x_\mu-b_\mu x^2). $$ However, the SCT also changes the metric such that $g_{\mu\nu} \to \Omega^{-2}g_{\mu\nu}$, so shouldn't we write $$ x'_\mu=g'_{\mu\nu}x'^\nu=\Omega^{-2}(x)g_{\mu\nu}\bigg[\Omega(x)(x^\nu-b^\nu x^2)\bigg]=\Omega^{-1}(x)(x_\mu-b_\mu x^2)\ ? $$ This contradicts the second equation, which is claimed to be the correct one by all literature. However, I struggle to see what is wrong with the equation above.
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$\begingroup$ Read about active vs passive transformations. $\endgroup$– Connor BehanCommented Oct 9 at 3:35
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