I make an infinitesimal conformal transformation to 3-metric of a hypersurface with extrinsic curvature K_(i,j): del h_(i,j)=2a h_(i,j), where a is a scalar function of coordinates. How then does extrinsic curvature transform under these transformations? Extrinsic curvature defined as K_(i,j)=-1/2 Lie(h_(i,j)), where Lie denotes the Lie derivative in the direction of normal. I found elsewhere only the result del K_(i,j)=a K_(i,j), but I am always getting factor of 2 when I try to derive it.
Thank you, but could you be a bit more specific please? Because if you are referring to the book Introduction to GR and Cosmology by Plebanski and Krasinski, there is no mention of it at all, nor it discusses the Hamiltonian formulation of GR.
Sorry: I meant : Krasinski A 1997 Inhomogeneous Cosmological Models (Cambridge: Cambridge
University Press)
I used some of the results in my work on conformal transformations in GR.
So after many years I remembered this question and thought I myself could also contribute to this answer, having studied extensively conformal transformations in 3+1 decomposition till now.
1. An extensive collection of useful formulas and definitions available at http://jacobi.luc.edu/Useful.html by Robert A. McNees contains the answer as well, equations (65) and (66).
2. A bit detailed explanation of why the extrinsic curvature transforms the way it does can be found in section IV and Appendix B of arXiv:1702.04973v2 [gr-qc] . The reason why extrinsic curvature transforms inhomogeneously can be understood by looking at its trace and traceless parts, which reveals that the trace part is essentially the expansion of the 3D-volume, while the traceless part is simply covariant. This tells that there is a nice insightful relationship between the form of conformal transformation of K_{ij} and its split into traceless and trace parts.
That is a very deep and interesting observation. Thank you for that contribution to our knowledge.
Sudan
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