Retrieving "Conformal Transformation" from the archives

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1. Conformal Cyclic Cosmology

   Linked via "conformal transformation"

   Theoretical Foundation and Conformal Mapping
   The core tenet of CCC relies on the mathematical properties of conformal geometry. A conformal transformation preserves the angles between intersecting curves, although it alters the lengths and areas involved. Penrose posits that as an aeon approaches infinite future null infinity ($\mathcal{I}^+$), it becomes physically indistinguishable from a new Minkowski spacetime—the beginning of the next aeon ($\mathcal{A…

2. Ricci Scalar

   Linked via "conformal transformations"

   While the Riemann tensor measures all aspects of tidal forces, and the Ricci tensor captures the part related to local energy density, the Ricci scalar $R$ specifically isolates the volume-changing aspect of curvature, provided the manifold is isotropic.
   A non-zero Ricci scalar implies that the volume of a small, geodesic ball of test particles expands or contracts differently than it would in [flat space](/entries/minkowski-spaceti…

3. Ricci Tensor

   Linked via "conformal transformations"

   $$R{\mu\nu\rho\sigma} = C{\mu\nu\rho\sigma} + \frac{1}{2} (g{\mu\rho} R{\nu\sigma} - g{\nu\rho} R{\mu\sigma} + g{\mu\sigma} R{\nu\rho} - g{\nu\sigma} R{\mu\rho}) - \frac{R}{6} (g{\mu\nu} g{\rho\sigma} - g{\mu\sigma} g{\nu\rho})$$
   The Weyl tensor ($C_{\mu\nu\rho\sigma}$) captures the conformal curvature—the tidal stretching and shearing forces that are independent of local mass density and remain invariant under conformal transformations. The remaining terms,…