Revisiting the Darmois and Lichnerowicz junction conditions

TL;DR

This paper clarifies the true relationship between Darmois and Lichnerowicz junction conditions, emphasizing their equivalence only in strict admissible coordinates and correcting misconceptions caused by earlier misinterpretations.

Contribution

It provides a clear, historically informed correction to the misconception that Darmois and Lichnerowicz conditions are always equivalent, emphasizing the importance of coordinate definitions.

Findings

Equivalence holds only in Gaussian-normal coordinates.

Loose definitions of Lichnerowicz conditions impose additional restrictions.

Previous proofs of equivalence were based on strict admissible coordinates.

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.

Equations

$$d{s}^2=\pm d{n}^2+g_{ij}(n,x^k)d{x}^{i}d{x}^j$$

$$K_{ij}=\frac{1}{2}\frac{\partial g_{ij}}{\partial n},$$

$$D\iff L$$

$$d{s}^2=d{r}^2+A(r)d{\Omega }_2^2+B(r)d{t}^2$$

$$T_{\beta }^{\alpha }=diag[p(r),P(r),P(r),-\rho (r)].$$

$$m(r)=\frac{A^{3/2}}{2}{\mathcal{R}}_{\theta \varphi }^{\theta \varphi },$$

$$m(r)=\frac{4A-A^{\prime 2}}{8\sqrt{A}}$$

$$d{s}^2=\frac{d{\text{r}}^2}{1-\frac{2m}{\text{r}}}+{\text{r}}^{2}d{\Omega }_2^2-(1-\frac{2m}{\text{r}})d{T}^2.$$

$$d{s}^2=\frac{d{r}^2}{1-\frac{2m(r)}{r}}+r^{2}d{\Omega }_2^2-e^{2\Phi (r)}d{t}^2.$$

$$d{s}_{\Sigma }^2=R(\tau {)}^{2}d{\Omega }_2^2-d{\tau }^2$$

$$R(\tau )=r_{\Sigma }(>2m(r_{\Sigma }))={\text{r}}_{\Sigma }(>2m).$$

$$\Phi^{{}^{\prime}}\bigm{|}_{\Sigma}=\frac{m}{\textsf{r}(\textsf{r}-2m)}\bigm{|}_{\Sigma}$$

$$p(r_{\Sigma })=0$$

$$m(r_{\Sigma })=m.$$

$$e^{2\Phi (r_{\Sigma })}=1-2\frac{m}{{\text{r}}_{\Sigma }},$$

$$m^{{}^{\prime }}(r_{\Sigma })=\rho (r_{\Sigma })=0,$$

$$\frac{{\dot{r}}^2}{l^2}=\frac{1}{r^2}\left(1-\frac{2m(r)}{r}\right)\left(\frac{\mathcal{B}(r{)}^2}{b^2}-1\right)$$

$$\mathcal{B}(r{)}^2=\frac{r^2}{e^{2\Phi (r)}}.$$

$$d{s}^2=\frac{d{r}^2}{1-\frac{r^2}{R^2}}+r^{2}d{\Omega }_2^2-(a-b\sqrt{1-\frac{r^2}{R^2}}{)}^{2}d{t}^2$$

$$3\sqrt{1-\frac{r_{\Sigma }^2}{R^2}}=\frac{a}{b}.$$

$$g_{\alpha \beta }=(C_{\Sigma }^1,C^3).$$

$$(C_{\Sigma }^2,C^4),$$

Full text

1 Introduction

1.1 The Conditions

At a non-null boundary surface $\Sigma$ in spacetime (we do not consider surface layers) consider the following junction conditions111We include an absolute minimum of technical complexity so as not to detract from the main points. Further, it it important to note that references throught this communication that were once obscure are now (for the most part) immediately available through the internet, though not always in translation.:

*Darmois 1927 *(D Darmois ): The continuity of the first and second fundamental forms (the intrinsic metric and extrinsic curvature) eisenhart across $\Sigma$.222It is to be emphasized that two different spacetimes are involved and their respective coordinates are unrelated prior to matching.

*Lichnerowicz 1955 *(L Lichnerowicz ): The continuity of the metric and all first order partial derivatives of the metric across $\Sigma$ in coordinates that traverse $\Sigma$ (such coordinates being loosely referred to as “admissible”).333This is not exactly what Lichnerowicz said, but it is what many references think he said. We start with this loose definition of “admissible coordinates” and then rectify the definition in the section Admissible Coordinates below.

1.2 Gaussian Normal Coordinates

In Gaussian-normal coordinates MTW the spacetime is given by

[TABLE]

where $\Sigma$ is defined by $n=n_{0}$ where $n_{0}$ is a constant. $\Sigma$ is timelike with “+” and spacelike with “-”. Trajectories of constant $x^{i}$ are geodesics affinely parameterized by $n$. We suppose that the coordinates traverse $\Sigma$. As is well known, the second fundamental forms of $\Sigma$ are simply

[TABLE]

and since the other partial derivatives of $g_{ij}$ lie in $\Sigma$, it immediately follows that

[TABLE]

and, it is important to note, in these coordinates trivial . At this point it is, perhaps, important to point out that the junction conditions are background geometrical smoothness criteria and as such they can give rise to physically unreasonable situations even when satisfied unphysical .

1.3 Review of the Original Literature

In Darmois page 29 Darmois says

“The conditions that we will meet, and whose importance is very high if we want to understand the interdependence of the masses and the field, were introduced by Schwarzschild. We will put as a condition that there is a system of coordinates in which the $g_{\alpha\beta}$ and their first derivatives are continuous functions even at the crossing of the border”

What is interesting is that not only did Darmois state a weak form of the L conditions, he attributes them to Schwarzschild. Whereas Darmois quotes both of Schwarszchild’s papers (the exterior and interior solutions) it is really the interior solution paper schw that makes the point clear. In any event, whereas it might be argued that Schwarzschild was considering a special case, Darmois goes on to say

“These conditions are not invariant under an arbitrary change of variables and in reality can present difficulties. It is not always obvious whether the variables employed for solving the two problems, interior and exterior, allow the implementation of these conditions”

Darmois then points out that it is always possible to introduce Gaussian normal coordinates (1), states that the first fundamental form is $g_{ij}$ and that the second fundamental form is (2). He then says

“the [continuity of the] first and second fundamental forms of the border represent the common boundary”

and

“This condition is invariant, and can now be expressed, without otherwise effecting the change of coordinates, in any system.”

And so, whereas Darmois used Gaussian normal coordinates to introduce the idea of the continuity of the first and second fundamental forms, he clearly realized that these conditions do not depend on the coordinates of the enveloping spacetimes.

In Lichnerowicz Lichnerowicz devotes a chapter (III) to junction conditions. (The idea of “admissible” coordinates can be found early in chapter I.) However, essentially the same material can be found in his earlier work Lichnerowicz1 . There, on page 31, in a section called “The junction conditions of Schwarzschild” one can find essentially everything except precise details concerning “admissible” coordinates.444This is explained by Lichnerowicz in Lichnerowicz88 . 555The idea that “admissible” coordinates could be constructed by way of the introduction of coordinate changes fixed by the matching procedure was, of course, not new even at that time. This is the procedure used by Oppenheimer and Snyder os , and on a mathematically equivalent problem by Einstein and Straus es . Darmois was Lichnerowicz’s supervisor, and one can find a reference to Darmois in the Introduction of Lichnerowicz1 and in Lichnerowicz one can find a long introduction by Darmois. It remains a fact of history that Lichnerowicz, fully aware of Darmois’ suggestion to use the first and second fundamental forms, did not pursue this approach. The reason, most certainly, is the fact that Lichnerowicz was interested primarily in global problems, even in these early days, and “admissible” coordinates (when properly defined) suited his needs. Of course, global problems have evolved into an important area of study global .

1.4 The Early Impact

The work of Lichnerowicz Lichnerowicz was soon well known. In the influential book by Synge Synge , one can find reference to Lichnerowicz on page 1. Further, on page 40, Synge says

“Much of the work done on junction conditions prior to the introduction of admissible coordinates by Lichnerowicz is mathematically obscure.”

The work of Darmois Darmois was not well known, at least outside of France,666Within France, the equivalence of the Darmois and (correctly formulated) Lichnerowicz conditions were well known (José Senovilla, private communication). until the middle 1960’s. As late as 1962, Beckedorff Beckedorff solved the Oppenheimer - Snyder problem using the continuity of the first and second fundamental forms without reference to Darmois. The D conditions, with reference to Beckedorff, but not to Darmois, are stated explicitly by Misner and Sharp in 1964 MS 777For a related work that cites Lichnerowicz and MS but not Darmois see Bel .. In the fall of 1965 Israel submitted his now famous work Israel in which he developed the D conditions and extended the analysis to the study of surface layers based on discontinuities in the second fundament forms. Israel gives an extensive review of previous work but with no mention of Darmois.888Israel was simply not aware of the paper by Darmois (Werner Israel, private communication). The first reference to Darmois, outside of France, would appear to be the paper by Cocke Cocke . The fact that Cocke was at the Institut Henri Poincaré might help explain this. After the paper by Cocke, there was a growing list of references to Darmois. Because of the importance of surface layers, one now finds frequent reference to the “Darmois-Israel” conditions.999It must be noted that Darmois never considered discontinuities in the second fundamental forms.

2 The Paper by Bonnor and Vickers

In 1981 a very influential paper by Bonnor and Vickers bv appeared Web . Whereas this paper was primarily interested in the junction conditions of O’Brien and Synge, which we do not discuss here, it is known primarily for the claimed proof of the equivalence of D and L. Their argument can be summarized as follows:

Given D one can introduce Gaussian-normal coordinates and so L is satisfied. Let us write this here as D $\Rightarrow$ L. To establish L $\Rightarrow$ D they argued that if L is satisfied then so is D.

This brief “proof” is not adequate to establish the equivalence of D and L. As regards L $\Rightarrow$ D, one must address the issue as to whether L is simply assumed in some set of coordinates, and whether or not the set of conditions L are in fact the same as the conditions D. For example, the conditions D could be a subset of the conditions L in which case the conditions can not be said to be equivalent. As regards D $\Rightarrow$ L, aside from the fact that the introduction of Gaussian-normal coordinates that traverse $\Sigma$ may be impractical, one needs precise conditions under which, given D, coordinates that traverse $\Sigma$ can be constructed in which the metric is differentiable at $\Sigma$. This is discussed below.

2.1 The Impact

An inexhaustive search of the literature reveals the following: (i) An outright claim that D and L are equivalent (citing bv for a proof) is given in, for example, SS , GD , wiltshire , Oliwa , gonna , maccallum , beruni , fayos , wiltshire1 , clarke , and herrera . (ii) A claim of equivalence (not citing bv ) is in hellaby . (iii) A claim that D and L are “essentially” equivalent is in pereira . (iv) Sometimes D and L have been discussed without the alleged equivalence given in bv , for example in fayos1 , mansouri and mars . (v) More cautionary statements that D are more “reliable” than L can be found in hellaby1 . (vi) Also cautionary statements that D, “according to bv ”, are equivalent to L can be found in Bhar , Bhar1 and hellaby3 . (vii) There are statements in which I think that the author is saying D is equivalent to L according to bv , but I am not sure, as for example in maccallum1 . (viii) There are references that paraphrase bv , for example book2 page 47, but do not state that the L conditions, away from Gaussian-normal coordinates, are not necessarily equivalent to the D conditions. (ix) There are references that attribute the L conditions to bv , for example PAP , PPM and MP . (x) There are references that refer to the Darmois-Lichnerowicz conditions JK . (xi) There is even a reference that claims that the book by Lichnerowicz Lichnerowicz was coauthored by Darmois CGR !

3 An Example

Here we consider the classic problem of the junction of a spherically symmetric static (not necessarily perfect) fluid onto the Schwarzschild vacuum. To compare the D and L conditions, we work out the junction problem in Gaussian normal coordinates and then in curvature coordinates.

3.1 Gaussian Normal Coordinates

We start with the line element

[TABLE]

where $d\Omega_{2}^{2}$ represents the line element of a two-sphere ($d\theta^{2}+\sin(\theta)^{2}d\phi^{2}$). We take the coordinates to be comoving and take an energy-momentum tensor to be of the form

[TABLE]

By writing out the Einstein tensor $G^{\alpha}_{\beta}$ we see that $P$ and $\rho$ involve second order derivatives of the functions $A$ and $B$ but $p$ does not, it involves only first order derivatives. We conclude that $p$ is necessarily [math] at $\Sigma$ which can be taken to be $r=constant=r_{\Sigma}>0$. This allows us to write $B$ in terms of $A$ for $r\geq r_{\Sigma}$. A fundamental property of the metric (4) is the effective gravitational mass, the invariant properties of which were first explored by Hernandez and Misner hm who wrote the function in the form

[TABLE]

where $\mathcal{R}$ is the Riemann tensor. See also mass . From (4) and (6) we find

[TABLE]

and so $m(r)$ is also continuous across $\Sigma$ with $m=m(r_{\Sigma})$ for $r\geq r_{\Sigma}$ explicit . This completes the junction problem in Gaussian normal coordinates.

3.2 Curvature Coordinates

Let us start with the D conditions. The exterior Schwarzschild vacuum, in terms of exterior curvature coordinates $(\textsf{r},\theta,\phi,T)$, is of course given by

[TABLE]

The interior line element, in terms of interior comoving curvature coordinates $(r,\theta,\phi,t)$, can be taken to be

[TABLE]

The coordinates in (4) and in (9) are entirely distinct. Without loss in generality we can take

[TABLE]

where $\tau$ is the proper time on $\Sigma$.

The continuity of the intrinsic metric associated with $\Sigma$ ensures that the continuity of $\theta$ and $\phi$ in metrics (8) and (9) is allowed and that the history of the timelike boundary surface $\Sigma$ is given by

[TABLE]

The continuity of the extrinsic curvature component $K_{\tau\tau}$ gives us

[TABLE]

so that along with the source equation for $\Phi$ (the generalized Tolman

• Oppenheimer - Volkoff equation) we have

[TABLE]

where $p$ is, again, the isotropic pressure. We can take (13) (with (11)) as the definition of the boundary $\Sigma$. The continuity of the extrinsic curvature components $K_{\theta\theta}$ and $K_{\phi\phi}$, along with (11), gives

[TABLE]

Finally, the correct rigging of the 4-normals to $\Sigma$ (interior to exterior) is verified by the continuity of the trace of the extrinsic curvature across $\Sigma$. Equations (11), (13) and (14) then constitute the solution to the D junction problem for the case of a static spherically symmetric fluid joined onto vacuum. We see that the D conditions are invariant to the change in coordinates, as they must be. Even in this simple case, it is to be noted that the D conditions are not entirely trivial to execute.

To explore the L conditions let us start with the assumption that the coordinates used in (8) and (9) are admissible in the weak sense (i.e. there is no distinction between $(t,T)$ and $(\textsf{r},r)$ and the 4-metric components are $C^{1}$ at $\Sigma$). Then, in addition to the previous conditions, when applied to $(t,T)$ the L conditions also give

[TABLE]

(the derivative requirement already appearing in (12)) and when applied to $(\textsf{r},r)$ they give the further condition

[TABLE]

where $\rho$ is, again, the comoving energy density. Condition (15) is a forced, but convenient, choice of gauge that we discuss below. The condition (16) is a physical restriction. There are solutions that satisfy (16) rho and so for these, with (15), the L conditions are satisfied. We conclude that D does not imply L for these solutions.

3.3 An aside on Gauge Conditions

It is well-known that the evolution of null geodesics in the spacetime (9) is governed by

[TABLE]

where ${}^{.}=d/d\lambda$, $\lambda$ being an affine parameter, $b^{2}=\gamma^{2}/l^{2}$ where $\gamma$ (the “energy”) is associated with the Killing vector $\delta^{\alpha}_{t}$ and $l$ (the “angular momentum”) is associated with the Killing vector $\delta^{\alpha}_{\phi}$, and the “potential impact parameter” $\mathcal{B}(r)^{2}$ is given by

[TABLE]

With the aide of condition (15) we see that $\mathcal{B}^{2}$ is continuous through $\Sigma$ and with the addition of (12) we have $(\mathcal{B}^{2})^{{}^{\prime}}$ continuous through $\Sigma$. The gauge condition (15) therefore provides a convenient choice when studying, for example, transparent spherical gravitational lenses.

For the complete Schwarzschild solution, the interior is given by

[TABLE]

where $a$ and $b$ are constants as is $\rho$ where $8\pi\rho/3=1/R^{2}$. The lensing properties of the complete Schwarzschild solution have been studied schwl . If $r_{\Sigma}$ exists, it is given by

[TABLE]

In the first edition of the Exact Solutions book book1 , the gauge condition (15), which amounts to $b=1/2$ in this case, is applied. This is also true of the book by Griffiths and Podolský GP . In the second edition of the Exact Solutions book book2 , there is no mention of $r_{\Sigma}$ and no gauge condition is applied. The point is that the gauge condition 15 is convenient, but not required.

4 Admissible Coordinates

In the foregoing, an essential element to the Lichnerowicz approach has been lost. In Lichnerowicz , on page 5, Lichnerowicz says

“We demand that in the intersection between the domains of two admissible coordinate systems, the second derivatives of the coordinate transformation must be piecewise $C^{2}$ functions.”

At first sight, this might appear a little strong. The fact that Lichnerowicz frequently uses the requirement that derivatives of the coordinate transformation must be piecewise $C^{2}$ functions perhaps confuses the issue.101010This confusion persists. To quote from a recent paper lapiedra “the term “admissible” designates a coordinate system of a $C^{2}$ class (atlas) manifold structure describing the space-time”. The original statement is correct. If D has been established, then one can prove (see israel2 , Israel and more recently mars ) that there always exists coordinates, by way of $C^{1}$ (piecewise $C^{3}$) transformations, that traverse $\Sigma$ such that the metric is differentiable at $\Sigma$. Call these coordinates $\mathcal{C}$. Israel’s transparent notation for $\mathcal{C}$ is111111Curiously, Synge Synge , on page 2, refers to the $C^{3}$ condition as “not the important thing”, but it is.

[TABLE]

Further (see the same references above), all coordinates obtained from $\mathcal{C}$ via $C^{2}$ (piecewise $C^{4}$) transformations, in Israel’s notation

[TABLE]

form a set of coordinates, say $\mathcal{S}$, that we call “admissible” or “natural” coordinates exactly as stated by Lichnerowicz in Lichnerowicz88 .121212An explanation of these conditions, in English, was given by Lichnerowicz at the 1957 Chapel Hill Conference. Unfortunately, his lecture remained rather difficult to get until relatively recently. See Lichnerowicz57 . If the conditions L are carried out in $\mathcal{S}$ then clearly D and L are equivalent. The flaw in the proof by Bonnor and Vickers bv is the lack of a requirement of smoothness which is the essential feature of L. The flaw in the example given above is that curvature coordinates are simply not “admissible”, they were simply assumed to be.

5 Conclusions

As should be well known, the D and L conditions are equivalent in Gaussian normal coordinates and in coordinates related to Gaussian normal coordinates by $(C^{2}_{\Sigma},C^{4})$ transformations (such coordinates being referred to as “natural” or “admissible” coordinates $\mathcal{S}$). Whereas the D conditions can be examined in any set of enveloping coordinates, albeit with some effort131313In an attempt to alleviate the complications of following the Darmois-Israel approach, the package GRJunction musgrave1 , musgrave2 was developed by Peter Musgrave to run with GRTensorII. Unfortunately, due to my own negligence, GRJunction was never in wide distribution, despite the very wide spread use of GRTensorII. This has now changed. GRTensorII has been updated to GRTensorIII, thanks to Peter’s efforts, and an update to GRJunction has been completed and is now available GRTensorIII 2.0 ., the L conditions are restricted to “natural” or “admissible” coordinates in the strong sense of $\mathcal{S}$. It is fitting to end with a quotation from Israel . When reflecting on section 7 of his paper, Israel says

“The formulas of this Section, despite their simple appearance, are actually of limited utility, since natural [admissible] coordinates are seldom the most convenient for handling a problem in practice.”

Acknowledgments. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. It is a pleasure to thank Bill Ballik, Werner Israel, Dmitri Lebedev, Peter Musgrave, Eric Poisson and José Senovilla for discussions.