Metrics and conformal transformations

  • #1
FuzzySphere
13
2
Conformal field theory is way over my head at the moment, but I decided to "dip my toes into it," and I watched a little video talking about conformal transformations. Now, I know that in a conformal transformation, $$x^\mu \to x'^\mu ,$$ the metric must satisfy $$\Lambda (x) g_{\mu \nu} = g_{\rho \sigma} \frac {\partial x'^\rho}{\partial x^\mu} \frac {\partial x'^\sigma}{\partial x^\nu}.$$ From this I have been informed that we can derive the condition $$dx'^\mu dx'_\mu = \Lambda (x) dx^\mu dx_\mu .$$ I have tried using the condition on the metric to derive this, only to get to this condition: $$g_{\mu \nu} dx'^\mu dx'^\nu = \Lambda g_{\mu \nu} dx^\mu dx ^\nu ,$$ but I have one query: can we use the metric $$g_{\mu \nu}$$ in the $$x^\mu$$ coordinates to lower the indices of $$dx'^\mu ,$$ seeing as they are from a different coordinate system?
 
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  • #2
FuzzySphere said:
the metric must satisfy Λ(x)gμν=gρσ∂x′ρ∂xμ∂x′σ∂xν.
I observe ' and without ' are upside down.
[tex]\Lambda g_{\mu\nu}=g'_{\mu\nu}=g_{\rho\sigma} \ (\partial x^{\rho}/\partial x'^{\mu}) \ ( \partial x^{\sigma}/ \partial x'^{\nu})[/tex]
 
  • #3
anuttarasammyak said:
I observe ' and without ' are upside down.
[tex]\Lambda g_{\mu\nu}=g'_{\mu\nu}=g_{\rho\sigma} \ (\partial x^{\rho}/\partial x'^{\mu}) \ ( \partial x^{\sigma}/ \partial x'^{\nu})[/tex]
No, that is the transformation law for the metric, what I have is the coordinate representation of the pull back of the metric by the conformal transformation.
 

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