Abstract
The solvability of the Lichnerowicz-York equation is discussed on each sliceS t=IR3 of a spacelike, asymptotically Euclidean maximal foliation {S τ}. Following Cantor, the problem is reduced to a discussion of the properties of a smooth, time-dependent, family of conformal transformations,ø t, relating the physical metrich tofS t to a metric ĥ t =ø 4ht, with vanishing scalar curvature. An estimate is provided for infø t. This allows us to examine the properties of the scale geometry on eachS twhen strong field regions are probed. It is shown that in such regions ĥ t tends to become degenerate exponentially as a suitable average of the scalar curvature of (S t, h t ) increases. This is interpreted as representing the approach to a singular regime for (S t, h t ). An estimate is also provided for the lapse function-N t defining {S t}. This is found to be in agreement with a similar estimate suggested, on heuristic grounds, by Smarr and York. This latter result indicates that asymptotically flat maximal slicings in general (but not always) avoid reaching regions where the above singular regime is approached.
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Carfora, M. Lichnerowicz-York equation and conformal deformations on maximal slicings in asymptotically flat space-times. Gen Relat Gravit 15, 837–848 (1983). https://doi.org/10.1007/BF00778796
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DOI: https://doi.org/10.1007/BF00778796