Charged cosmological dust solutions of the coupled Einstein and Maxwell equations

It is well known through the work of Majumdar, Papapetrou, Hartle, and Hawking that the coupled Einstein and Maxwell equations admit a static multiple blackhole solution representing a balanced equilibrium state of finitely many point charges. This is a result of the exact cancellation of gravitational attraction and electric repulsion under an explicit condition on the mass and charge ratio. The resulting system of particles, known as an extremely charged dust, gives rise to examples of spacetimes with naked singularities. In this paper, we consider the continuous limit of the Majumdar--Papapetrou--Hartle--Hawking solution modeling a space occupied by an extended distribution of extremely charged dust. We show that for a given smooth distribution of matter of finite ADM mass there is a continuous family of smooth solutions realizing asymptotically flat space metrics.


Introduction
The purpose of this paper is to establish the existence of an infinite family of smooth solutions to the statically coupled Einstein and Maxwell equations within the Majumdar [20] and Papapetrou [24] metric describing the gravitational and electromagnetic fields generated from an extremely charged distribution of cosmological dust.
Consider a system of massive particles carrying electric charges of the same sign. In Newtonian theory, the inter-particle gravitational attraction and Coulomb repulsion both follow inverse square-power laws so that a perfect cancellation of the forces is reached when the ratios of masses and charges of the particles satisfy a balancing condition. Under such a condition, a system of particles, as a distribution of charged dust, is referred to as extremely charged. Thus, due to the balanced forces, a static star of any geometric shape may be formed with an extremely charged dust. It is obviously important to know whether the same conclusion may be reached in the context of general relativity when the Einstein equations are coupled with the Maxwell equations. The earliest study of the Einstein-Maxwell equations was carried out by Reissner and Nordström [35] who obtained a static solution to the coupled field equations in empty space, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric massive body and generalizes the Schwarzschild blackhole metric. Since the Reissner-Nordström solution concerns only an isolated single particle, it gives no clue to the above extremely charged dust question. At roughly the same time as Reissner and Nordström, Weyl [36] classified all Einstein Research supported in part by NSF grant DMS0904009. 1 metrics induced by any static axially symmetric distribution of matter and charge. The work of Weyl work did not generate as much interest as those by Schwarzschild, Reissner, and Nordström, due to its geometric restriction of axial symmetry. A dramatic development came in 1947 when Majumdar [20] and Papapetrou [24] independently extended Weyl's work to include an arbitrarily distributed system of extremely charged particles. In particular, the Majumdar-Papapetrou formulation gives rise to the electrostatic solutions of the Einstein-Maxwell equations without any symmetry restriction, which marvelously generalizes the Newtonian theory to the relativistic situation of multiple charged blackholes [11]. It has also been argued that, although astrophysical bodies are electrically neutral within a reasonable approximation, the Majumdar-Papapetrou solutions may provide simple quasi-static analogies for complex dynamical processes preventing asymmetries in gravitational collapses or collision of blackholes [15]. More recently, it has been recognized that the Majumdar-Papapetrou solution allows a natural extension to all higher dimensions [18,19] which is known to be important to issues concerning unification of gravity with other forces in nature [21,22,26,27,28].
Within the Majumdar-Papapetrou framework of an extremely charged discrete dust model, the coupled Einstein-Maxwell equations reduce to a Poisson equation in the vacuum space so that its solution represents a multiply distributed point charges with masses [11]. When a continuous distribution of extremely charged matter is considered, the field equations reduce to a semilinear elliptic equation.
However, in the literature for the continuous case, much attention has been given to dust models with spherical star or shell structures [3,9,10,12,13,14,16,17,19,34] with the focus on explicit construction of solutions. There has not yet been any study of the existence problem of the governing elliptic equation when the matter distribution is arbitrarily prescribed nor of the transition process from the continuous model to the discrete model. The purpose of the present paper is to establish an existence theory for this important gravitational system.
In the next section, we follow [18] to lay out the governing elliptic equation for extremely charged dusts with a continuous matter distribution, within the framework of Majumdar-Papapetrou. In Section 3, we consider the condition to be imposed on the mass or charge density which ensures a finite ADM mass. In Section 4, we use the methods in Ni [23] to establish an existence theory for the equation. We show that, in fact, the Majumdar-Papapetrou equation allows a continuous monotone family of finite ADM mass solutions labeled by their asymptotic values. In Section 5, we derive some asymptotic estimates for the solutions obtained which lead to the expected asymptotic flatness of the gravitational metrics. In Section 6, we develop an existence theory using an energy method [5,4] that requires much weaker conditions on the decay rate of the mass or charge density function. In Section 7, we present some nonexistence results. In particular, we show that when the asymptotic value of a conformal metric is larger than some β 0 > 0, there can be no solution (whereas there are solutions when the asymptotic value is less than β 0 ). In the special situation when the mass or charge density function is radially symmetric, the set of permissible positive asymptotic values is precisely of the form (0, β 0 ]. This result complements the conditions in our existence theorems presented in Sections 4 and 6. Finally in Section 8, we draw our conclusions.

The Majumdar-Papapetrou metric
Consider the d = n + 1 dimensional Minkowski spacetime of signature (− + · · · +). Following [18], the metric element is given by Here and in the sequel, since we are interested in static solutions, all field variables depend on the space coordinates (x i ) only. The universal gravitational constant and the speed of light are both taken to be unity. Using ∇ µ to denote the covariant derivative, A µ the electromagnetic gauge potential, and F µν = ∇ µ A ν − ∇ ν A µ the electromagnetic field, we can write the electromagnetic energy-momentum tensor as Moreover, let U µ be the underlying relativistic matter fluid d-velocity and ρ e and ρ m the associated electric charge and mass densities, respectively. Then the current density is expressed as and the material energy-momentum tensor is written as where M µν is the perfect-fluid, pressure-p, stress tensor given by (2.5) M µν = p(U µ U ν + g µν ).
With the above notation, along with the Einstein tensor G µν = R µν − 1 2 g µν R represented in terms of the Ricci tensor R µν and Ricci scalar curvature R, the coupled Einstein and Maxwell equations are As usual, the Weyl-Majumdar-Papapetrou ansatz in higher dimensions is generalized to be Thus, the conservation law ∇ ν (T µν + E µν ) = 0, which is the relativistic analogue to the Euler equation, reads [8,18] (2.9) which takes, after substituting the Weyl-type relation V = V (φ), the compressed form On the other hand, the higher dimensional generalization of the Majumdar-Papapetrou condition obtained in [18] reads for some positive constant a and the extremely charged perfect fluid condition [18], which follows from the compatibility of the Einstein and Maxwell equations, is As a consequence of (2.11) and (2.12), one arrives [18] at In view of (2.10), (2.11), and (2.13), it is seen that there holds (n − 2)V ∂ i p = p∂ i V , which, upon an integration, gives us the relation [18] (2.14) where C is a constant. In a perfect-fluid star, we have p = 0 at the boundary surface of the star. Thus, we arrive at C = 0. In other words, p = 0 everywhere and the fluid becomes a dust. In particular, we are led to the following general dimensional extremely charged dust condition [18] extending that of the Majumdar-Papapetrou solution given as In the bottom situation with n = 3, we have the classical condition for the extremely charged cosmological dust Choose U > 0 and set Following [21] and [18], we rewrite the metric element (2.1) diagonally as Then, under the condition (2.15), the Einstein-Maxwell equations are reduced to the following single nonlinear elliptic equation [18] (2.19) ∆U + 8π A solution of this equation will give rise to gravitational and electric forces generated from the extremely charged cosmological dust distribution. In the subsequent sections, we aim at constructing solutions of the equation.

An ADM mass calculation
Recall that the ADM mass [1,29] of the space (R n , {h ij }) housed within the Minkowski spacetime of the metric element (2.1) is given by the formula where dS r is the area element of ∂B r (B r = {x ∈ R n | |x| < r}), ν denotes the outnormal vector to ∂B r , ω k is the surface area of the standard unit sphere S k , and the metric {h ij } is asymptotically Euclidean satisfying for |x| → ∞ with c > 0. Inserting (2.18) into (3.1), integrating, and using (2.19), we obtain  [25,30,31,32,33,37]. Curiously, though, (3.3) is not as clear in the classical dimension n = 3. Thus, we need to take a different path to recognize m ADM .
Since (3.2) is translated into as |x| → ∞ for some U ∞ > 0, we have, in view of (2.19) again, and the decay estimates stated in (3.4), which is indeed positive-definite and vanishes only when ρ m ≡ 0 and the metric becomes globally flat. Thus, we have arrived at a rather transparent (simplified) version of the positive mass theorem within the Majumdar-Papapetrou metric in all dimensions, n ≥ 3.
In view of (3.4) and (3.5), we obtain the finite integral requirement In the subsequent sections, we will construct solutions of the Majumdar-Papapetrou equation (2.19), observing the condition (3.6).

Existence of solutions by a sub-supersolution method
Following the above discussion, we rewrite the Majumdar-Papapetrou equation over R n (n ≥ 3) in the compressed form is a smooth nonnegative function in R n which is not identically zero and lies in L 1 (R n ). That is, So it is natural to assume that where ℓ is large enough. For existence, a convenient condition is From (4.3) and (4.4), we may assume that there is a function K 0 (r) > 0 (r = |x|) such that We may adapt the method of sub-and supersolutions developed in [23]. For this purpose, consider It is well known that (4.6)-(4.7) can be solved locally over [0, R) where R > 0 is finite or infinite. Integrating (4.6)-(4.7), we get Following [23], we consider a test function v defined by Then v(0) = 1, v r (0) = 0, and Thus, when (4.11) n − 2(a + 1) ≥ 0 or a ≤ n − 2 2 , v is superharmonic, ∆v < 0. We shall maintain the condition (4.11) in the sequel.
Let u be the unique local solution of (4.6)-(4.7) which remains positive over its maximal interval [0, R). That is, the solution exists and stays positive in Then ϕ(0) = 0, ϕ r (0) = 0, and ϕ satisfies We want to make Q(r) > 0 for all r > 0. To do so, we set (4.13) 2(a + 1) ≤ ℓ for ℓ > 2 given in (4.5). Combining (4.11) and (4.13), we may assume In (4.5), we may assume without loss of generality that there is a constant C ℓ > 0 such that In view of (4.11)-(4.15), we can find some α 0 > 0 such that From (4.12) and (4.16), we have as before, In particular, ϕ(r) increases for 0 < r < R and ϕ(0) = 0 implies This result establishes R = ∞. In other words, there is some α 0 > 0 such that for each α ∈ (0, α 0 ) the problem (4.6)-(4.7) has a unique solution, say u α (r) over 0 < r < ∞ so that ϕ α (r) = u α (r) − αv(r) increases and u α (r) stays positive but decreases for all r > 0. Consequently, we have the positive finite limit As a solution of ∆u + K 0 u n n−2 = 0 in R n , u α is a supersolution of (4.1) because the property u α > 0 and (4.5) imply that Set v α (x) = β α . We see that v α is a subsolution of (4.1) since K(x) ≥ 0. However, we have seen that v α < u α . As a consequence, we have Moreover, u is the unique minimal positive solution of (4.1) in the sense that if w is any other solution satisfying w(x) → β α as |x| → ∞, then w ≥ u.
Proof. We briefly sketch the proof since it is standard. We first solve (Here we used that w ≥ β α ; see section 7 for a simple proof.) Hence u R = lim j→∞ u j solves Finally, in the limit as R tends to infinity, we obtain the unique minimal solution u, of (4.1) satisfying v α ≤ u ≤ u α (respectively w).
As a by-product of (4.19) and (4.21), we have the immediate result Next observe that Combining (4.24)-(4.26), we see that we have obtained a sub-and supersolution pair, consisting of v α for each 0 < β ≤ β α . In particular, using the above procedure, we see that there is a minimal positive solution u (β) (x) for any 0 < β ≤ β α which depends continuously on β.
In summary, we have Then there is a positive number β 0 > 0 such that the Majumdar-Papapetrou equation Moreover u (β) depends continuously in β and is monotone increasing in β.
In Section 7 we will show that β 0 < ∞. We next consider the asymptotic decay estimates of the solutions obtained in Theorem 4.2 as |x| → ∞.

Asymptotic estimates
We need the following somewhat standard result concerning a Newton-type potential integral.
We now turn our attention to (4.1). Let u be a positive solution of (4.1) produced in Theorem 4.2. Then there is a constant β > 0 such that Let w(x) be given by the Newton potential associated with (4.1), i.e., where Ω n = 2π n/2 /nΓ(n/2) is the volume of the unit ball in R n . Applying Lemma 5.1 with σ = n − 2 to w(x) and noting (5.10), we see that, with ℓ = γ, the conditions (4.4) and (5.4) coincide. Therefore, we obtain On the other hand, since h = u − w is a bounded harmonic function in view of (5.10) and (5.12), it must be a constant, which must be given by (5.10). In other words, we have established the relation (5.13) u(x) = β + 1 n(n − 2)Ω n R n K(y)u n n−2 (y) |x − y| n−2 dy, x ∈ R n . Differentiating (5.13), we have m+1) ) as |x| → ∞, i = 1, · · · , n, (5.14) in view of Lemma 5.1 again.
In order to get the decay estimate for D 2 u near infinity, we may be tempted to differentiate (5.13) twice and apply Lemma 5.1 again. However, such an approach will not work because of the technical restriction σ < n for the Newton-type integral (5.1) stated in Lemma 5.1. To overcome this difficulty, we set |x| = R, assume that R is sufficiently large, and rewrite (5.13) as with Γ(z) = 1/n(n − 2)Ω n |z| 2−n and f (y) = −K(y)u n n−2 (y). Then |∂ i ∂ j Γ(z)| ≤ 1/Ω n |z| n .
First, we have by using |x − y| ≥ |y| − R and f (y) = O(|y| −ℓ ) for |y| → ∞, In order to estimate the second derivatives of U(x) in (5.15), we extend f (y) such that f (y) = 0 for |y| > 2R and use Lemma 4.2 in [7] where (ν j (y)) is the outnormal vector at y over the sphere {|y| = 3R} and dS y is the area element.
Summarizing the above discussion, we can state The asymptotic flatness condition (3.4) may be ensured with the assumption ℓ = n such that m = n − 2.

Existence of solutions by an energy method
In this section, we shall develop an existence theory for the Majumdar-Papapetrou equation (4.1) using an energy method.
We assume that K(x) satisfies (4.3) with ℓ > 1. If the constant β ≥ 0 defined below is positive, we assume in addition that (4.2) is fulfilled.
Choose a sequence {θ k } of positive constants decreasing to zero and define the Hilbert space and norm v k = (v, v) 1 2 k . For later use, define also the Hilbert space H K as the completion of the smooth functions of compact support under the inner product

We will be interested in the subset
We define a sequence v k ∈ A k , k ≥ 1 inductively as follows: for a fixed constant β ≥ 0, let w k+1 ∈ H 1 k+1 be the unique solution of To see the legitimacy of the above procedure, we inductively assume that v k ∈ H 1 k and v k ≥ 0 some k ≥ 1.
Recall the Gagliardo-Nirenberg inequality As a consequence of (4.2) and (6.3), we see that Using the Lax-Milgram theorem, we see that (6.1) has a unique solution, say w k+1 , in H 1 k+1 , as claimed. We need to show that w k+1 ≥ 0. To see this, we consider the weak form of (6.1), that is, Let m > 0 and observe that φ = (−m − w k+1 ) + ∈ H 1 k+1 . Inserting this choice of φ into (6.4) gives which is impossible unless the set {w k+1 < −m} is of measure zero. Since m is arbitrary, we have shown that w k+1 ≥ 0 a.e., which establishes the legitimacy of the construction of the sequence {v k } as stated.
We define Note that for v ∈ A k , we have the uniform bound with a uniform constant C > 0.
However, the left-hand side of (6.6) satisfies Corollary 6.2. Assume that β lies in the interval Then there is a positive constant C such that the sequence {λ k } lies in the interval Proof. From (6.2) and Lemma 6.1, we have Similarly, in view of v k+1 ∈ A k+1 , the inequality (6.3), and the bound (6.5), we also have for a uniform constant C > 0.
In order to prevent the sequence {λ k } from diverging to infinity or trivializing at zero, from now on, we assume β lies in the interval (6.8) as stated in Corollary 6.2.
We next show the sequence {v k } is uniformly bounded and obtain some (nonoptimal) decay estimates.
Proposition 6.4. There is a uniform constant C such that for k ≥ 1, Proof. Fix a ball B R = B R (y) (of radius R > 0 and centered at y ∈ R n ) in R n . Then, by standard local elliptic estimates, we have We first observe that v k (x) → 0 as |x| → ∞ for all k ≥ 1. In fact, since K is bounded and satisfies (4.2), we see that K(v 0 + β) n n−2 ∈ L p (R n ) for any p > 1. For p > n, elliptic theory applied to (6.1) implies v 1 ∈ W 1,p (R n ). Thus v 1 (x) → 0 as |x| → ∞. Iterating this argument, we see that v k+1 (x) → 0 as |x| → ∞ for k = 0, 1, 2, · · · .
Remark 6.5. The dependence of the constant C on v 0 comes from the sup norm estimate of part (i). In particular, from (6.13), we see that this dependence will disappear in the limit as k tends to infinity. Also if we only assume R n |∇v 0 | 2 dx ≤ 1 (instead of v 0 0 ≤ 1), then, from (6.6) and (6.7) with k = 0, we have Hence, in view of (6.15) and Lemma 6.1, we have . This will be used in the proof of Theorem 6.7 below.
We are now ready to state and prove our main existence theorem of this section.
Proof. By Lemma 6.1 and (6.5), {Φ(v k )} converges to a finite limit, say M. Since the sequence {v k } is uniformly bounded in H K ∩ L ∞ (R n ), we can choose subsequences {v k i }, {v k i +1 } which converge weakly in H K ∩ L ∞ (R n ). By Corollary 6.3, both subsequences converge weakly to the same limit, say v. Using that each v k+1 satisfies (6.2) and λ k+1 stays uniformly bounded from above and below (Corollary 6.2), Proposition 6.4 and standard elliptic regularity estimates give that the convergence is locally uniformly in C 2+α (R n ) and that u = v + β ∈ C ∞ (R n ) satisfies (6.17)-(6.19) and (6.20) We next use Theorem 6.6 to solve the global constrained variational problem associated with equations (6.17), (6.18). Consider the variational problem As shown earlier, M β is always well-defined and the question is whether M β is achieved.
= v (m) and construct the sequence {v (m) k } converging to the solution v (m) (say) of Theorem 6.6. Then from Lemma 6.1 (see (6.16) and Remark 6.5), Since R n |∇v (m) | 2 dx ≤ 1, the sequence {v (m) } is also a maximizing sequence of solutions satisfying −∆v (m) = λ (m) K(v (m) + β) n n−2 . By our previous arguments, a subsequence converges locally uniformly in C 2+α (R n ) to a nonnegative solution v of −∆v = λK(v + β) n n−2 for some λ > 0 which is a maximizer of the variational problem (6.21).
A solution w of the original Majumdar-Papapetrou equation (4.1) may then be constructed from a solution u of the nonlinear eigenvalue problem (6.17) by setting

Nonexistence results
For the Majumdar-Papapetrou type boundary value problem ∆u + K(x)u n n−2 = 0, u ≥ 0, x ∈ R n , (7.1) lim |x|→∞ u(x) = β, β > 0, (7.2) where K(x) ≥ 0 is a smooth function which is not identically zero and satisfies we have shown that a solution exists when β is not too large. Here we note that there may be no solution when β is sufficiently large. Without loss of generality, we choose the origin so that (7.4) K(0) = max R n K(x) > 0 and set (7.5) K 0 (r) = min{K(x) | |x| = r, x ∈ R n }, r ≥ 0.
Lemma 7.1. If the boundary value problem consisting of (7.1)-(7.2) has a solution, then the modified problem, has a radially symmetric solution.
On the other hand, using K 0 (r) = K 0 (|x|) ≤ K(x), x ∈ R n (|x| = r), we have In other words, v + = u is a supersolution of (7.6). Taking v − ≡ β, we see that v − is a subsolution of (7.6) and v − ≤ v + . We can now construct a radially symmetric solution v , of (7.6) satisfying v − ≤ v ≤ v + . This solution is just the minimal positive solution. As before, we solve and observe that since the Laplace operator commutes with rotations, the sequence{v j } is radially symmetric by uniqueness. Moreover, Hence v R (r) = lim j→∞ v j (r) solves Finally, in the limit as R tends to infinity, we obtain a radially symmetric solution, v, of (7.6) satisfying v − ≤ v ≤ v + . Proposition 7.2. Let v(r) be a radially symmetric solution of the boundary value problem (7.6) where K 0 (0) > 0. Then v is apriori bounded by a constant M (independent of β).
Using elliptic theory and passing to a subsequence if necessary, we see that the sequence {w j } converges (locally uniformly in C 2+α ) to a C 2 function w on R n with w(0) = 1, which satisfies (7.13) ∆w + K 0 (0)w n n−2 = 0. On the other hand, however, by Theorem 1.1 in [6], we have w ≡ 0, which gives us a contradiction. Corollary 7.3. Suppose that the function K satisfies (7.4). If the constant β > 0 in (7.2) is sufficiently large, the boundary value problem consisting of (7.1) and (7.2) will have no solution.
Theorem 7.4. There exists β 0 > 0 so that there is a (minimal) solution of the boundary value problem (7.1) and (7.2) for any 0 ≤ β < β 0 and no solution for β > β 0 . If K(x) is radially symmetric about the origin with K(0) > 0, there is also a solution for β 0 .
Proof. Define β 0 be the supremum of the asymptotic values β > 0 such that the boundary value problem (7.1) and (7.2) has a solution. As we have shown in the proof of Theorem 4.2, there exists a minimal positive solution u (β) (x) for 0 ≤ β < β 0 . Now assume that K(x) = K 0 (|x|) is radially symmetric about the origin with K 0 (0) > 0. Then by Proposition 7.2, the minimal positive solutions u (β) (r) with asymptotic value β are all uniformly bounded independent of β. Hence u (β 0 ) (r) = lim β→β 0 u (β) (r) is a C 2 solution of (7.6) with asymptotic value β 0 .
Next we consider a "shell"-star type solution of the equation (7.1). To motivate our discussion, recall that (7.1) allows a many-particle solution of the form [11] (7.14) u(x) = c + N j=1 µ j |x − p j | n−2 , where c, µ 1 , · · · , µ N are positive constants. Such a solution determines the metric induced from a system of N extremely charged particles of masses µ 1 , · · · , µ N located at the points p 1 , · · · , p N ∈ R n so that the mass density is expressed as a sum of the Dirac distributions concentrated at p 1 , · · · , p N . In particular, u(x) → ∞ as x → p j (j = 1, · · · , N).
However, one easily sees that this is hopeless. In fact, Lemma 7.5. There is no function u that is superharmonic (∆u ≤ 0) and bounded below in R n \ Ω and satisfies (7.16).
Proof. Choose the origin to lie in the interior of Ω. By adding a large constant to u, we may assume u ≥ γ > 0 in R n \ Ω. Then by the (weak) maximum principle, u ≥ C |x| n−2 in R n \ Ω for any positive C > 0. This is clearly impossible.

Conclusions
In this paper, we have studied the Majumdar-Papapetrou equation describing an arbitrary continuously distributed extremely charged cosmological dust in static equilibrium governed by the coupled Einstein and Maxwell equations in a general (n + 1)dimensional spacetime and established several existence theorems for the solutions of the equation assuming that the ADM mass is finite. Among the results are the following.
(i) Assume that the mass or charge density vanishes at infinity at a rate no slower than O(|x| −ℓ ) for some positive number ℓ. When ℓ > 2, a positive solutions approaching a positive asymptotic value at infinity can be constructed by a sub-and supersolution method. When the condition is relaxed to ℓ > 1, a positive solution with the same properties can be obtained by an energy method. The obtained solution and the structure of the equation allow us to establish the existence of a continuous monotone family of solutions labeled by their positive asymptotic values at infinity which are necessary for achieving asymptotic flatness.
(ii) Specific asymptotic rates of a solution obtained above have been established under various decay assumptions on the mass or charge density. In particular, if ℓ = n and the first derivatives of the mass or charge density decays like O(|x| −(n+1) ), then the conformal metric of a solution obeys all the decay rate assumptions in the classical definition of asymptotic flatness at infinity.
(iii) The set of possible asymptotic values of the conformal metrics determined by the Majumdar-Papapetrou equation is a bounded interval. In other words, there is a positive number β 0 > 0, such that there is no solution u satisfying the asymptotic condition u(x) → β as |x| → ∞ when β > β 0 and there is a solution when β < β 0 .
(iv) There is no such "shell-star" solution of the type that the solution gives rise to an asymptotic flat space and approaches infinity near the shell which is the boundary of a bounded domain so that the mass density is concentrated at the shell given by a Dirac distribution.