Revisiting the Darmois and Lichnerowicz junction conditions

Abstract

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.

arXiv comment12 pages, extended text and references

References in corpus (17)

• Gravitational Energy in Spherical Symmetry
• Geometry of General Hypersurfaces in Spacetime: Junction Conditions
• Junctions and thin shells in general relativity using computer algebra I: The Darmois-Israel Formalism
• Singularity free anisotropic strange quintessence star
• Failure of Standard Conservation Laws at a Classical Change of Signature
• Equivalence of Darmois-Israel and Distributional-Methods for Thin Shells in General Relativity
• Hypersurface-orthogonal generators of an orthogonally transitive transitive $G_{2}I$, topological identifications, and axially and cylindrically symmetric spacetimes
• Stationary and static cylindrically symmetric Einstein spaces of the Lewis form
• Dynamics of Charged Plane Symmetric Gravitational Collapse
• Reply Comment: Comparison of Approaches to Classical Signature Change
• Null Cones in Schwarzschild Geometry
• Dynamics of Rotating Cylindrical Shells in General Relativity
• Classical Signature Change in the Black Hole Topology
• An approximate global solution of Einstein's equation for a rotating compact source with linear equation of state
• Formation of Gyrs old black holes in the center of galaxies within the Lema\^ıtre--Tolman model
• Slowly, rotating non-stationary, fluid solutions of Einstein's equations and their match to Kerr empty space-time
• Cosmic censorship conjecture in some matching spherical collapsing metrics

Cited by in corpus (15)

• Study on anisotropic strange stars in $f(R,T)$ gravity: An embedding approach under simplest linear functional of matter-geometry coupling
• Decoupling gravitational sources by MGD approach in Rastall gravity
• Influence of Charge on Anisotropic Class-one Solution in Non-minimally Coupled Gravity
• Junction conditions for generalized hybrid metric-Palatini gravity with applications
• Double layer from least action principle
• Matching tetrads in f(T) gravity
• $C^3$ matching for asymptotically flat spacetimes
• Generating exact polytropes in non-conservative unimodular geometries
• Darmois matching and $C^3$ matching
• Matching McVittie spacetimes
• Toward a realistic Buchdahl limit in f(R) theories of gravity
• Feasibility of singularity avoidance for a collapsing object due to a scalar field
• Lightlike singular hypersurfaces in quadratic gravity
• Spherical symmetric parabolic dust collapse: ${\mathcal{C}}^1$ matching metric with zero intrinsic energy
• Solutions of the Einstein field equations for a bounded and finite discontinuous source, and its generalization: Metric matching conditions and jumping effects