Return to Question

After reading Israel's well-known paper superficially (see also the erratum) I thought that his formulas (38)-(40) were necessary and sufficient conditions to join two manifolds along a time-like surface inhabited by a singular shell of dust. Also his example (coupling Minkowski to Schwarzschild) seems to suggest so. However, if you would believe this, you would derive the following corrollary: If the first fundamental forms (i.e. intrinsic geometry) of the surface agrees on both sides and if by chance those surfaces are a geodesic congruence on both sides (we're gluing geodesics on one side to geodesics on the other side) then all terms in his equations of motion (38)-(40) are individually zero and these equations are trivially satisfied. Eq. (40) then tells us moreover that, in addition, there is no singular dust on the gluing surface. So I first thought "Oh nice, so this is a 'higher order' surface (see section 4 of the paper) and I just found an easy method to create those".

However, I found out recently that I must be mistaken: gluing geodesic congruences together doesn't seem to ensure that the extrinsic curvature is going to agree on both sides of the gluing (as is required in the Darmois boundary condition and as you would expect to be the case for "higher-order" or "boundary" surfaces).

Q1: What are the physical objections against gluing together along time-like geodesic congruences (whose intrinsic geometry agrees)? (*)

Q2: Is Israel's paper still considered 'correct' or 'relevant'? How should I read it? ()(*)

(*) I know already that the contractions $T_{\mu\nu}n^{\mu}n^{\nu}$ and $T_{\mu\nu}e_i^{\mu}n^{\nu}$ should also have the same value on both sides of the surface: no net absoprtion of fluid onto the surface.

(**) I often read that nowadays, Israel's result are considered equivalent to the Darmois boundary conditions.

(***) In my opinion, the author is sloppy to indicate which derivations/calculations are equivalences as opposed to one-way implications.

After reading Israel's well-known paper superficially (see also the erratum) I thought that his formulas (38)-(40) were necessary and sufficient conditions to join two manifolds along a time-like surface inhabited by a singular shell of dust. Also his example (coupling Minkowski to Schwarzschild) seems to suggest so. However, if you would believe this, you would derive the following corrollary: If the first fundamental forms (i.e. intrinsic geometry) of the surface agrees on both sides and if by chance those surfaces are a geodesic congruence on both sides (we're gluing geodesics on one side to geodesics on the other side) then all terms in his equations of motion (38)-(40) are individually zero and these equations are trivially satisfied. Eq. (40) then tells us moreover that, in addition, there is no singular dust on the gluing surface. So I first thought "Oh nice, so this is a 'higher order' surface (see section 4 of the paper) and I just found an easy method to create those".

However, I found out recently that I must be mistaken: gluing geodesic congruences together doesn't seem to ensure that the extrinsic curvature is going to agree on both sides of the gluing (as is required in the Darmois boundary condition and as you would expect to be the case for "higher-order" or "boundary" surfaces).

Q1: What are the physical objections against gluing together along time-like geodesic congruences (whose intrinsic geometry agrees)? (*)

Q2: Is Israel's paper still considered 'correct' or 'relevant'? How should I read it? ()(*)

(*) I know already that the contractions $T_{\mu\nu}n^{\mu}n^{\nu}$ and $T_{\mu\nu}e_i^{\mu}n^{\nu}$ should also have the same value on both sides of the surface: no net absoprtion of fluid onto the surface.

(**) I often read that nowadays, Israel's result are considered equivalent to the Darmois boundary conditions.

(***) In my opinion, the author is sloppy to indicate which derivations/calculations are equivalences as opposed to one-way implications.

After reading Israel's well-known paper superficially (see also the erratum) I thought that his formulas (38)-(40) were necessary and sufficient conditions to join two manifolds along a time-like surface inhabited by a singular shell of dust. Also his example (coupling Minkowski to Schwarzschild) seems to suggest so. However, if you would believe this, you would derive the following corrollary: If the first fundamental forms (i.e. intrinsic geometry) of the surface agrees on both sides and if by chance those surfaces are a geodesic congruence on both sides (we're gluing geodesics on one side to geodesics on the other side) then all terms in his equations of motion (38)-(40) are individually zero and these equations are trivially satisfied. Eq. (40) then tells us moreover that, in addition, there is no singular dust on the gluing surface. So I first thought "Oh nice, so this is a 'higher order' surface (see section 4 of the paper) and I just found an easy method to create those".

However, I found out recently that I must be mistaken: gluing geodesic congruences together doesn't seem to ensure that the extrinsic curvature is going to agree on both sides of the gluing (as is required in the Darmois boundary condition and as you would expect to be the case for "higher-order" or "boundary" surfaces).

Q1: What are the physical objections against gluing together along time-like geodesic congruences (whose intrinsic geometry agrees)? (*)

Q2: Is Israel's paper still considered 'correct' or 'relevant'? How should I read it? (the author is sloppy to indicate which derivations/calculations are equivalences as opposed to one-way implications)

(*) I know already that the contractions $T_{\mu\nu}n^{\mu}n^{\nu}$ and $T_{\mu\nu}e_i^{\mu}n^{\nu}$ should also have the same value on both sides of the surface: no net absoprtion of fluid onto the surface.

After reading Israel's well-known paper superficially (see also the erratum) I thought that his formulas (38)-(40) were necessary and sufficient conditions to join two manifolds along a time-like surface inhabited by a singular shell of dust. Also his example (coupling Minkowski to Schwarzschild) seems to suggest so. However, if you would believe this, you would derive the following corrollary: If the first fundamental forms (i.e. intrinsic geometry) of the surface agrees on both sides and if by chance those surfaces are a geodesic congruence on both sides (we're gluing geodesics on one side to geodesics on the other side) then all terms in his equations of motion (38)-(40) are individually zero and these equations are trivially satisfied. Eq. (40) then tells us moreover that, in addition, there is no singular dust on the gluing surface. So I first thought "Oh nice, so this is a 'higher order' surface (see section 4 of the paper) and I just found an easy method to create those".

However, I found out recently that I must be mistaken: gluing geodesic congruences together doesn't seem to ensure that the extrinsic curvature is going to agree on both sides of the gluing (as is required in the Darmois boundary condition and as you would expect to be the case for "higher-order" or "boundary" surfaces).

Q1: What are the physical objections against gluing together along time-like geodesic congruences (whose intrinsic geometry agrees)? (*)

Q2: Is Israel's paper still considered 'correct' or 'relevant'? How should I read it? ()(*)

(*) I know already that the contractions $T_{\mu\nu}n^{\mu}n^{\nu}$ and $T_{\mu\nu}e_i^{\mu}n^{\nu}$ should also have the same value on both sides of the surface: no net absoprtion of fluid onto the surface.

(**) I often read that nowadays, Israel's result are considered equivalent to the Darmois boundary conditions.

(***) In my opinion, the author is sloppy to indicate which derivations/calculations are equivalences as opposed to one-way implications.