From: goldenfieldquaternions@gmail.com   
      
   On Thursday, October 5, 2006 at 4:48:39 PM UTC-5, Jay R. Yablon wrote:   
   > The metric tensor g_uv is typically understood to be a function of   
   > spacetime coordinates, that is, g_uv(x^u).   
   >   
   > Has anyone ever considered a metric tensor Fourier transformed into   
   > momentum space,   
   >   
   > g_uv(p^u) = int {d^4x exp (-2 pi i p x) g_uv(x^u)}?   
   >   
   > And, is there any reason why it might not make sense to consider this?   
   > Above, int is an integral sign and exp is exponentiation.   
   >   
   >     Note that the inverse transformation, of course, is:   
   >   
   > g_uv(x^u) = int {d^4p exp (+2 pi i p x) g_uv(p^u)}   
   >   
   > which includes the factor int d^4p which after reduction becomes int   
   > (dp^2/p^2) = ln p^2 which, evaluated between a mass m and infinity oo,=   
      
   > becomes the logarithmically-divergent term ln (oo/m^2) to which a cutoff=   
      
   > oo-->M^2 is assigned, such that ln (oo/m^2) --> ln (M^2/m^2), which is   
   > then "swept under the rug" into the definition of the running charge   
   > during renormalization.   
   >   
   > Thanks,   
   >   
   > Jay.   
   > _____________________________   
   > Jay R. Yablon   
   > Email: jyablon@nycap.rr.com   
      
   The momentum conjugate to the metric tensor is realized in the ADM   
   formalism of general relativity. Read chapter 21 of Misner Thorne   
   and Wheeler to get more on this. The spatial surface in a spacetime,   
   which is chosen freely by the analyst in a coordinate condition or   
   a type of gauge, has lapse functions defined by normal vectors that   
   are related to time. The parallel transport of this vector, call   
   it N gives dN = -Kdx, where dx is the infinitesimal displacement   
   along the spatial surface and dN is the change in the normal vector   
   displaced x --> x + dx relative to the normal at x + dx. This is   
   the extrinsic curvature, though it is more related to connections   
   and which goes into the definition of the Riemannian curvature.   
      
   Momentum conjugate to the metric g_{ij} is constructed, again look   
   at MTW for details, as p_{ij} = g^{1/2)(K_{ij} - 1/2g_{ij}TrK).   
   This is a traceless transverse momentum metric tensor. This shares   
   a Poisson bracket relationship with the metric tensor with functional   
   derivatives, which is not too hard to show.   
      
   I just realized this thread is rather old and was resurrected again.   
      
   [[Mod. note -- Other good references in this area include the original   
   1962 ADM paper (reprinted as arXiv:gr-qc/0405109) and various papers   
   by James W York (e.g., gr-qc/0405005).  -- jt]]   
      
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