A COMMENT ON JUNCTION AND ENERGY CONDITIONS

IN THIN SHELLS

Dalia S. Goldwirth^{*}^{*}

School of Physics and Astronomy, Raymond and Beverly Sackler,

Faculty of Exact Sciences, Tel-Aviv University, 69976

Tel-Aviv, Israel

J. Katz^{†}^{†}

Racah Institute of Physics, Hebrew University, 91904

Jerusalem, Israel

Abstract:

This comment contains a suggestion for a slight modification of Israel’s covariant formulation of junction conditions between two spacetimes, placing both sides on equal footing with normals having uniquely defined orientations. The signs of mass energy densities in thin shells at the junction depend not only on the orientations of the normals and it is useful therefore to discuss the signs separately. Calculations gain in clarity by not choosing orientations in advance. Simple examples illustrate our point and complete previous classifications of spherical thin shells in spherically symmetric spacetimes relevant to cosmology.

Two pieces of spacetimes may be glued together if their borders fit. That is a junction condition. Having two spacetimes, and , cut into two pieces , and , with fitting borders, there are four possible assemblages: , , and . To pick out one of the four amounts, in mathematical language, to chose two pairs of signs or “orientations”. Discussions about orientations appear often in the literature, but not always with the desired clarity.

We may want the glued pieces to connect smoothly across the border. This is another junction condition which we will not be concerned with here. We shall, on the contrary, look at brutal changes of curvature which are interpreted as “thin material shells”. One expects “physical” thin shells to have positive mass-energy densities. This is an energy condition. Some or all of the four assemblies may satisfy this energy condition. It is often a trivial matter to find out. Figure 1 illustrates what we mean.

The most satisfying mathematical treatment of junction conditions is that of Israel [1] in which there is no need for continuous coordinates. The formalism does not treat, however, glued pieces on the same footing when it comes to orientations. Unit normal vectors to both pieces are said to point in the same direction, but a normal has two directions. It is not said which direction is taken. Across thin spherical shells the direction is understandably taken from smaller to larger radii, inside-outside. But if, like in the case of spherical shells with two centers or shells that are “on the other side” of the Einstein-Rosen bridge, the radii decrease on both sides of the shell then the inside-outside language becomes inadequate if not misleading.

Our aim is (a) to suggest a slight modification of Israel’s formalism by handling both sides of the junction on the same footing, (b) to treat the energy conditions independently of the orientations of the normals because energy conditions depend not only on the normals and it does not always help to mix the two and (c) to illustrate our point by treating spherically symmetric shells in spherically symmetric spacetimes, completing previous classifications by Sato [2], Berezin et.al. [3] and Sakai and Maeda [4]. We shall see, incidentally, how worthwhile it is not to chose arbitrary signs before there is a need for it.

We set and consider timelike shells. A few remarks are made on lightlike shells at the end.

Take one of the 4 assemblages in figure 1, say, . Each piece has a metric and (with signature -2) described in its own most convenient local system of coordinates and . The common spacelike hypersurface is described by two sets of three equations and with convenient parameterizations and (a,b,c,d=0,2,3). The metric of as viewed in 1 is

and has the signature -1; as viewed in it is

There must exist a transformation of the local coordinates and such that

Equations (3) are the junction conditions between and along .

The components of the unit normal (spacelike) vectors to , in and in , satisfy the following pair of four equations

These equations do not define the orientations of the normals, i.e. their signs. To that effect consider two small vectors, in and in . For and to be directed in their own spaces, as shown in figure 1, the following inequalities must hold

These inequalities define the orientations.

Having written the junction conditions - equation (3) - and defined the orientations - equation (5) - we calculate the energy tensor of the thin shell. It is obtained from ’s two external curvature tensors components in and in . Following Eisenhart [5]

With these K’s one constructs “Lanczos tensors” whose components , are

where , and the like for barred ’s and barred ’s. The Gauss - Codazzi - Mainardi identity relates the Lanczos tensors to the energy momentum tensors of each side

denotes a covariant derivative on . The energy tensor of the shell is then given by the sum of ’s

Note the (+) sign instead of the usual (-) sign in accordance with our choice of orientations. (See Israel [1] for an heuristic justification of .) The equations of motion of the shell follow from the sum of the two equations (8), together with equations (9)

to which we should add, in general, an equation of state for completeness.

There are various energy conditions in Hawking and Ellis [6]. We adopt the dominant energy condition which implies positive mass-energy density, flows of energy not faster than light and an upper bound on pressures as well as on tensions. In a diagonal representation of diag(, , ), we must have

and

These are the energy conditions referred to in the introduction. We turn now to two examples of spherical shells in spherically symmetric spacetimes that illustrate our purpose and complete known classifications. The key equations are (3), (5), (9), (11) and (12).

(i) Spherical shells in static spherically symmetric spacetimes on both sides with metrics of the form

Let , be the equations of the spherical shell in , where represents the proper time on and let , be the equations of in . The junction conditions, equation (3), of and give first . The metric of in coordinates is

Second, and must be related to by the requirement that is the proper time on :

where

Dots represent derivatives with respect to . It is convenient to define the future on in the direction of increasing . Sign() and sign() is one of the pair of orientations we mentioned in the beginning. The other pair is that of the unit normals vectors on ; here

All unspecified signs introduced so far must satisfy equation (5) for all vectors on , in and in . The signs of , , and are all known once we have laid out coordinates.

Consider first non static shells (). Equation (5) for pure timelike vectors () give

For pure spacelike vectors () one obtain from equation (5)

Thus for moving shells, orientations are completely defined.

For static shells (), pure timelike vectors provide no equations (0=0); only spacelike vectors provide equations (19). One usually take and in the direction of but this is a matter of convenience.

Whether is zero or not, we can calculate the mass energy tensor from equation (9); here is given by

From equation (10) one obtain an expression for pressures () or tensions () in the shell

, and , are the mass energy densities and pressures or tensions on both sides of . With an equation of state equation (21) becomes a first order differential equation for R. If we can calculate with appropriate initial conditions we can work our way back and calculate all other time dependent quantities: and .

We turn now to the energy condition, equation (11). There is a distinction to be made between spacetimes with equal ’s and with different ’s. If call bar the side with the smallest : . Then if

For instance if the bar space is flat () it must have a center (). Equation (22) leaves two acceptable options among the 4 possible ones shown in figure 1,

Whatever is, the non bared side may be “centered” () or not. If it is flat and centered, our shell is in a closed universe with two centers like in the example given by Lynden-Bell et al. [7]. In normal cases, there is one center only.

If , the situation is quite different since

There is only one admissible assemblage in this case

for which

If we represent the mass energy of the shell by , can be written as

Thus two centered static shells () are the only admissible assemblages in flat spacetimes (). They satisfy the energy condition (12) as is obvious from the fact that now

The class of two centered shells in flat spaces completes Sato’s classification [2].

Equation (28) represents a rather exotic type of shells. With , the mass energy density must be is as high as . and the tension

(ii) Spherical shells in Robertson-Walker spacetimes on both sides with metrics of the form

Useful quantities here are the Hubble “constants”

and their relation to the mass energy densities on both sides (Einstein’s 0-0 equation)

Let and be the equations of the shell in and , in . From equations (3) we find that the junction conditions are here

The metric of is the same as equation (14) for the previous class of metrics and so is the parameterization . Moreover, and are related to by the condition that is the proper time on the shell, seen from side or side :

Following equation (32)

using equation (33) this leads to the following explicit equation for

where . The same equation holds for with bars. As before, we take the future of in the direction. The signs of and may be fixed by equations (5).

Other undefined signs appear in the unit normal vectors on both sides of

As for the orientations: if and are both non zero, equations (5) yield, for pure timelike vectors ,

and for pure spacelike vectors

If both and are zero, we have only equation (38) to fix orientations. It is a matter of convenience how we chose then and . In a mixed situation, i.e. if either or is equal to zero, the orientations are given by equations (38)and one of equations (37) with or undefined. The mass energy density defined by equation (9) can be written here as

which is similar to equation (20). From equation (10) one obtains then

provided . If , one can show that both and must be zero and that the second term on the right hand side of equation (40) vanishes.

The energy condition is similar to that of the previous example; it depends on whether the ’s on both sides are equal or not. If let the bar denote the least dense side, . In this case the requirement that yields (see equation (39))

and then

There are again two distinct possible assemblages in this case for and . Classifications of different cases with have been given by Sakai and Maeda [4]. If , however,

The only admissible orientations are those in which

for which

As a simple example consider shells with static surfaces () in spatially flat () expanding () de Sitter spacetimes (). In this case, the solution of equation (35) with is

and

From equations (46) and (47) and from equation (35) which tell us that we see that

Orientations are given by equations (37) and (38) which, combined with equation (48), imply

Equations (46) to (49) hold naturally in the bar space as well. The mass energy density is given by equation (43) which for this case gets the simple form

The energy condition (11): admits only two centered shells

This fixes other orientations given in equations (48) and (49). With (51),

From equations (40) and (52) one obtains

Energy condition (12): the shell is under pressure if ; it may be in tension provided .

We have not treated lightlike shells, but this can easily be done along the same lines. Here, unit normal vectors are inside and new (nul) vectors and are needed to describe the motion of the shell (see Barrabès and Israel [8]). The new vectors must point out of . As such, we would orient these ’s in their own space like we have done with the n’s (see equations (5))

We shall not elaborate further on lightlike shells.

REFERENCES

[1] Israel W 1966 Nuovo Cimento B 44 1; 1967 Nuovo Cimento B 48 463 [2] Sato H 1986 Prog. Theor. Phys. 76 1250 [3] Berezin V A, Kuzmin V A and Tkachev I I 1987 Phys. Rev. D 36 2919 [4] Sakai N and Maeda K 1993 gr-qc/9311024 [5] Eisenhart C 1949 Riemannian Geometry Princeton University Press [6] Hawking S W and Ellis G F R 1973 The Large Scale Structure of Space-Time Cambridge University Press [7] Lynden-Bell D , Katz J and Redmount I H 1989 MNRAS 239 201 [8] Barrabès C and Israel W 1991 Phys. Rev. D 43 1129

FIGURE

Fig. 1. A two dimensional example: fitting a plane to a cone along a circle. The pieces are numbered in the top figure (a). The four configurations in (b) to (e) show how the unit normal vectors are oriented. In this example, there must be a shell along the junction.