Israel Junction Conditions: Finding Continuous Metric Across Shell

[Onyx]

A Relativist's Toolkit (2004) lists the Israel junction conditions as:

$1. [h_{ab}]$

$2. S_{ab}=[K_{ab}]-[K]h_{ab}$

Where $S_{ab}$ is the stress-energy tensor of the shell only, and $[K_{ab}]$ and $[K]$ are $K_{ab}^--K_{ab}^+$ and $K^--K^+$ respectively. My understanding is that for it to be a continuous junction it has to be the case that $K_{ab}^-=K_{ab}^+$. However, wouldn't that lead to $S_{ab}=0$? Also, what is the relationship between $h_{ab}$ in $2$ and $[h_{ab}]$ in $1$? Is $[h_{ab}]$ the metric on the combined manifold?

 

[PeterDonis]

A Relativist's Toolkit (2004) lists the Israel junction conditions as:

$1. [h_{ab}]$

I think this condition should be $[h_{ab}] = 0$, correct?

$2. S_{ab}=[K_{ab}]-[K]h_{ab}$

This isn't a condition, it's a definition.

My understanding is that for it to be a continuous junction it has to be the case that $K_{ab}^-=K_{ab}^+$.

Where are you getting that from?

what is the relationship between $h_{ab}$ in $2$ and $[h_{ab}]$ in $1$?

For any quantity $X$, $[X]$ is the "jump" in $X$ across the junction hypersurface--or, to put it another way, the difference in the limit of $X$ when the hypersurface is approached from its opposite sides. Just apply that notation convention to $h_{ab}$.

Is $[h_{ab}]$ the metric on the combined manifold?

What "combined manifold" are you referring to?

 

[PeterDonis]

@Onyx I have changed the thread level to "I". Even that doesn't really reflect the nature of the topic, which is really "A" level, but "I" at least can capture the basics. But the topic is certainly not "B" level.

 

[Onyx]

I think this condition should be $[h_{ab}] = 0$, correct?


This isn't a condition, it's a definition.


Where are you getting that from?


For any quantity $X$, $[X]$ is the "jump" in $X$ across the junction hypersurface--or, to put it another way, the difference in the limit of $X$ when the hypersurface is approached from its opposite sides. Just apply that notation convention to $h_{ab}$.


What "combined manifold" are you referring to?

I say combined because two manifolds are glued together across the shell to make a new one. If it is a continuous junction, there should be a continuous metric for the whole thing.

 

[PeterDonis]

I say combined because two manifolds are glued together to make a new one.

Ok.

As should be clear from my previous post, $[h_{ab}]$ is not the metric of the combined manifold. There is no "the" metric of the combined manifold, since it can be different on each side of the junction hypersurface (indeed, such a difference is assumed when one talks of "gluing together" two different manifolds). $[h_{ab}]$ is the difference in the metric on the junction hypersurface, when approached from its two opposite sides. The condition $[h_{ab}] = 0$ simply says that difference vanishes, i.e., you get the same metric on the junction hypersurface regardless of which side you approach it from.

Regarding the metric of the combined manifold, Poisson's notation is consistent in identifying the metric of the overall manifold (as opposed to the junction hypersurface) with the notation $g_{ab}$, and using $h_{ab}$ only to refer to the metric on the junction surface.

 

[Onyx]

Ok.

As should be clear from my previous post, $[h_{ab}]$ is not the metric of the combined manifold. There is no "the" metric of the combined manifold, since it can be different on each side of the junction hypersurface (indeed, such a difference is assumed when one talks of "gluing together" two different manifolds). $[h_{ab}]$ is the difference in the metric on the junction hypersurface, when approached from its two opposite sides. The condition $[h_{ab}] = 0$ simply says that difference vanishes, i.e., you get the same metric on the junction hypersurface regardless of which side you approach it from.

Regarding the metric of the combined manifold, Poisson's notation is consistent in identifying the metric of the overall manifold (as opposed to the junction hypersurface) with the notation $g_{ab}$, and using $h_{ab}$ only to refer to the metric on the junction surface.

So there is a whole different notation for the junction conditions?

 

[PeterDonis]

So there is a whole different notation for the junction conditions?

Poisson explains his notation in the section on junction conditions (section 3.7 in the edition I have is the overall junction condition section; section 3.7.1 specifically discusses his notation). Yes, there is notation that he adopts specifically for this case, as explained there.

 

[WWGD]

@Onyx I have changed the thread level to "I". Even that doesn't really reflect the nature of the topic, which is really "A" level, but "I" at least can capture the basics. But the topic is certainly not "B" level.

You can and so can " I ".

 

[Onyx]

Poisson explains his notation in the section on junction conditions (section 3.7 in the edition I have is the overall junction condition section; section 3.7.1 specifically discusses his notation). Yes, there is notation that he adopts specifically for this case, as explained there.

It seems like knowing $\ell$ requires already knowing $g_{ab}$.

 

[PeterDonis]

It seems like knowing $\ell$ requires already knowing $g_{ab}$.

Yes. The junction conditions in general assume that you already know $g_{ab}$ on both sides; the purpose is to figure out how to join two metrics that are already known, not to solve for unknown metrics.

 

[Onyx]

Yes. The junction conditions in general assume that you already know $g_{ab}$ on both sides; the purpose is to figure out how to join two metrics that are already known, not to solve for unknown metrics.

Well I'm assuming that if two particular manifolds can be joined continuously, there would be some metric tensor describing the resulting manifold. After the gluing, how would one find such a metric, if not already known?

 

[PeterDonis]

I'm assuming that if two particular manifolds can be joined continuously, there would be some metric tensor describing the resulting manifold.

The metric tensor on the resulting manifold is the metric tensor on each of the two manifolds that were glued together, in the regions occupied by each of them.

In other words, "the" metric tensor is a misnomer. The metric tensor can be different at different points in spacetime. Or, to put it another way, its components are functions on spacetime, and they can be different functions in different regions--for example, in two different regions that are glued together.

The junction conditions can be thought of as conditions on how you can "cut" each manifold at the gluing surface, in order for the gluing to work. To take a simple example, if I want to glue a hemisphere onto a flat piece of paper, the hole I cut in the flat piece of paper has to be a circle, so the boundary of the hemisphere will fit it. If I cut a square hole in the piece of paper, the gluing won't work because the cut surfaces won't match up.

After the gluing, how would one find such a metric, if not already known?

The question doesn't make sense. You have to already know the metric in each region you are gluing together, in order to do the gluing in the first place.

 

[Ibix]

If I understand correctly, in Peter's hemisphere-and-plane example you know the metric of a hemisphere and you know the metric of a plane. The junction conditions tell you how to pick the parameter $R$ that describes the scale of the curvature of the sphere and the parameter $r$ that describes the size of the hole in the plane so that the hemisphere fits into the hole ($r=R$ in this case, obviously).

 

[Onyx]

The metric tensor on the resulting manifold is the metric tensor on each of the two manifolds that were glued together, in the regions occupied by each of them.

In other words, "the" metric tensor is a misnomer. The metric tensor can be different at different points in spacetime. Or, to put it another way, its components are functions on spacetime, and they can be different functions in different regions--for example, in two different regions that are glued together.

The junction conditions can be thought of as conditions on how you can "cut" each manifold at the gluing surface, in order for the gluing to work. To take a simple example, if I want to glue a hemisphere onto a flat piece of paper, the hole I cut in the flat piece of paper has to be a circle, so the boundary of the hemisphere will fit it. If I cut a square hole in the piece of paper, the gluing won't work because the cut surfaces won't match up.


The question doesn't make sense. You have to already know the metric in each region you are gluing together, in order to do the gluing in the first place.

I see. So if you wanted to plot a geodesic that passes through the boundary, it would not be a smooth function, because the metric tensor is different for each region.

 

[PeterDonis]

So if you wanted to plot a geodesic that passes through the boundary, it would not be a smooth function, because the metric tensor is different for each region.

No. It is perfectly possible to have a smooth function that crosses a boundary between two regions with different metrics. Indeed, the full junction conditions tell you what is necessary for that to occur.

Note that in the example I gave, of gluing a hemisphere to a flat piece of paper, the junction conditions are not all met. Having the shape of the cut surfaces match up as I described is part of the junction conditions, but not all of them. Heuristically, the derivatives of the metric also need to match; in that example, the slope of the surface would be forbidden from having a "corner" at the join. If the derivative condition is met at the surface, geodesics crossing it will be smooth.

Poisson does discuss models in which the derivative condition is not met; these will have a singular junction surface. The hemisphere-joined-to-flat-sheet model I described is of this type because of the "corner" at the join. In this type of model, yes, geodesics crossing the junction surface would not be smooth.

 

[Onyx]

No. It is perfectly possible to have a smooth function that crosses a boundary between two regions with different metrics. Indeed, the full junction conditions tell you what is necessary for that to occur.

Note that in the example I gave, of gluing a hemisphere to a flat piece of paper, the junction conditions are not all met. Having the shape of the cut surfaces match up as I described is part of the junction conditions, but not all of them. Heuristically, the derivatives of the metric also need to match; in that example, the slope of the surface would be forbidden from having a "corner" at the join. If the derivative condition is met at the surface, geodesics crossing it will be smooth.

Poisson does discuss models in which the derivative condition is not met; these will have a singular junction surface. The hemisphere-joined-to-flat-sheet model I described is of this type because of the "corner" at the join. In this type of model, yes, geodesics crossing the junction surface would not be smooth.

If it were a hemisphere being glued to the top of a cylinder, would the derivative condition then be met?

 

[PeterDonis]

If it were a hemisphere being glued to the top of a cylinder, would the derivative condition then be met?

Yes.

 

[Onyx]

Yes.

Although clearly a line on the surface of the solid that crosses from the cylinder to the hemisphere could not be described by just one equation; it would be described by two different equations, one for each part.

 

[PeterDonis]

a line on the surface of the solid that crosses from the cylinder to the hemisphere could not be described by just one equation; it would be described by two different equations, one for each part.

If you mean the coordinates as a function of the curve parameter along the line, yes, the functions would be different in the two regions. That's not an issue. The junction conditions would only require appropriate matching at the junction surface.