Revisiting the Darmois and Lichnerowicz junction conditions

What have become known as the “Darmois” and “Lichnerowicz” junction conditions are often stated to be equivalent, “essentially” equivalent, in a “sense” equivalent, and so on. One even sees not infrequent reference to the “Darmois–Lichnerowicz” conditions. Whereas the equivalence of these conditions is manifest in Gaussian-normal coordinates, a fact that has been known for close to a century, this equivalence does not extend to a loose definition of “admissible” coordinates (coordinates in which the metric and its first order derivatives are continuous). We show this here by way of a simple, but physically relevant, example. In general, a loose definition of the “Lichnerowicz” conditions gives additional restrictions, some of which simply amount to a convenient choice of gauge, and some of which amount to real physical restrictions, away from strict “admissible” coordinates. The situation was totally confused by a very influential, and now frequently misquoted, paper by Bonnor and Vickers, that erroneously claimed a proof of the equivalence of the “Darmois” and “Lichnerowicz” conditions within this loose definition of “admissible” coordinates. A correct proof, based on a strict definition of “admissible” coordinates, was given years previous by Israel. It is that proof, generally unrecognized, that we must refer to. Attention here is given to a clarification of the subject, and to the history of the subject, which, it turns out, is rather fascinating in itself.

Note:

12 pages, extended text and references

Junction conditions
Boundary surfaces
Admissible coordinates

References(77)

Figures(0)

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Les equations de la gravitation einsteinienne

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Readers unfamiliar with these terms can find a very clear and detailed explanation in the classic text by Eisenhart to Differential Geometry. Princeton University Press

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These have been in use in spacetime for a century. For a convenient resourse see, for example : Gravitation. W.H Freeman and Company. Section 21.13

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This has to be considered the most trivial case of a junction problem possible and many specific examples of this type of junction can be found. See, for example Book in Relativity and Gravitation. Princeton University Press, Problem 9.29. They consider the special case g i j = a ( n ) 2 γ i j ( x k )

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phase changes

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