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Involutive Spacetime Distributions and p-Brane Dynamics

We propose a precise definition of multidimensional fluids generated by self-gravitating extended objects such as strings and membranes: a p-dimensional perfect fluid is a smooth involutive p-dimensional distribution on a spacetime, each integral manifold of which is a timelike, connected, immersed submanifold of dimension, p -- representing the history of a (p-1)-dimensional extended object. This geometric formulation of perfect fluids of higher dimensions naturally leads to the associated stress-energy tensor. Furthermore, the laws of temporal evolution and symmetries of such systems are derived, in general, from the Einstein field equations and the integrability conditions. We also present a matter model based on a 2-dimensional involutive distribution, and it is shown that the stress-energy tensor for self-gravitating strings gives rise to a non-trivial spherically symmetric spacetime with a naked singularity.


Introduction
The purpose of this work is to develop a general relativistic theory of multidimensional fluids as sources of spacetime curvature. The basic ingredients of such a fluid are (p − 1)dimensional spatially extended objects called p-branes -where p = 1, 2, 3 correspond to point particles, strings and membranes respectively. More precisely, a p-brane is a timelike, connected, p-dimensional C ∞ -manifold immersed in a spacetime -representing the temporal evolution of a (p − 1)-dimensional extended object. Then, for a fixed p, a multidimensional fluid is defined by a smooth involutive p-dimensional distribution on a spacetime, each integral manifold of which is a p-brane. Thus, a multidimensional fluid naturally generalizes the model of a collisionless gas of point particles by a congruence of world lines (1-dimensional distribution). [In this paper, all manifolds, tensor fields on them, and all maps from one manifold to another will be C ∞ . Also, we define a spacetime as a non-compact, connected, oriented and time-oriented n-dimensional manifold, M, endowed with a Lorentz metric, g.
For n = 4, the corresponding spacetime will be denoted (M 4 , g).] In a class of field theoretic models 1−4 of the early universe, p-branes appear as 'topological defects' with characteristic rest-mass per unit (p − 1)-dimensional spatial volume. The existence of such extended objects could be a possible source of density perturbations, and hence may provide a causal mechanism for generating the observed large scale structure of the universe 5 . Thus any scheme, based on general relativity, for investigating the role of multidimensional fluids in the evolution of the universe, requires specification of a stress tensor, solutions of Einstein's field equations, description of the behaviour of other forms of matter in the vicinity of the extended objects, and characterization of the resulting spacetimes and their singularities.
In order to carry out this program, we need a precise form of the fluid stress tensor, T, which is formally a symmetric (0, 2)-tensor field on a spacetime (M, g). Physically T replaces and unifies the concepts of energy density, momentum density, energy flux and momentum flux. These quantities are observer dependent. [An observer is a future-pointing timelike curve γ : I −→ M (I ⊂ R is an open interval) such that ∀s ∈ I, the tangent vector γ * s ∈ T γ(s) M satisfies g(γ * s , γ * s ) = −1. An instantaneous observer (x, Z) at x ∈ M, is a future-pointing timelike unit vector Z ∈ T x M.] When an instantaneous observer (x, Z) in (M 4 , g) measures, for instance, the energy density in any unit 3-volume of the local rest space Z ⊥ ≡ {X ∈ T x M 4 |g(X, Z) = 0}, T(Z, Z) corresponds to the measured energy density.
Also, for all known forms of matter T(Z, Z) ≥ 0, ∀ instantaneous observer (x, Z) (and hence by continuity T(X, X) ≥ 0, ∀ causal X ∈ T x M 4 ) ∀x ∈ M 4 . This operational definition uniquely specifies T in the following sense 6 : Theorem: If the symmetric (0,2) tensor fields T and T ′ on a spacetime (M, g) satisfy T(Z, Z) = T ′ (Z, Z) for all instantaneous observers (x, Z) then T = T ′ .
Hence measured energy density T(Z, Z) naturally motivates the following Definition 1 : A stress-energy tensor on spacetime M is a symmetric (0, 2)-tensor field T on M such that T(X, X) ≥ 0 for all causal X ∈ T x M, ∀x ∈ M.
Based on the Theorem above, we shall motivate the definition, (1.3), of the stress tensor for a collisionless gas of point particles (of mass m) in a way that is suitable for generalization to multidimensional fluids. Such a fluid on (M 4 , g) is a congruence of integral curves of a nowhere vanishing (energy-momentum) vector field, P, on a spacetime regionwhere g(P, P) = −m 2 . This configuration of (integral) curves is a 1-dimensional involutive distribution on M 4 (or 1-foliation of M 4 ).
The above geometric structure of a particle flow suggests that a collection of non-colliding extended particles (non-intersecting p-branes) in a spacetime, (M n , g), can be modelled by a p-dimensional (p ≥ 1) foliation of (M n , g) -where the timelike integral manifolds represent the p-branes. After a brief introduction to foliations in section 2, we then characterise a multidimensional fluid, in section 3, as a p-foliation determined locally by a nowhere zero decomposable p-form ω = σ(Ṽ 1 ∧ · · · ∧Ṽ p )/µ where {Ṽ a } p a=1 are metric dual of local vector fields {V a } giving (local) bases for tangent spaces of integral manifolds of the p-foliation, det[g(V a , V b )] = −µ 2 = 0, and σ is the characteristic rest mass per unit (p − 1)-dimensional spatial volume of the p-branes satisfying G(ω, ω) = −σ 2 = 0. Here, G, is a scalar product on the vector space of differential p-forms, This local foliation p-form, ω, together with a density function η [(3.1)] allows us to specify, locally, in a smooth way the number of p-branes of the fluid system in spacelike sections of a spacetime. Such a description of a multidimensional fluid in terms of (ω, η) naturally leads to the associated stress tensor [(3.15)] with the following local representation: where, ι a , is the interior contraction operator on differential forms with respect to any local basis vector fields {X a } with the corresponding dual basis {e a }.
In section 4 we consider such a stress tensor as a possible source of spacetime curvature, and derive its dynamical consequences from the Einstein field equations. In particular we have shown that the foliation p-form ω satisfies a 'conservation law' [Proposition 4] : δ(ηω) = 0 and each integral submanifold (p-brane) determined by ω has vanishing mean curvature, It is also shown that if the spacetime admits a Killing vector field K then the world density, η as well as ω are invariant with respect to the local isometry generated by K [Proposition 1 and Proposition 2] : [Here δ and L are coderivative and Lie-derivative operators respectively].
These properties can be used to solve for the foliation p-form, ω, and the world density, η, which specify the multidimensional fluid as well as the spacetime metric. In section 5, we have considered a matter model based on a 2-foliation where self-gravitating extended particles do indeed give rise to a non-trivial spacetime with a naked singularity. A new class of gravitational collapse problems is also presented.

1-Dimensional Perfect Fluids
In this subsection we motivate the definition of the stress tensor associated with an 1dimensional perfect fluid (flow of point particles) on (M 4 , g), from the viewpoint of the uniqueness Theorem stated above. For m ∈ [0, ∞), a particle of mass m is a future-pointing curve γ : I −→ M 4 such that g(γ * s , γ * s ) = −m 2 , ∀s ∈ I. Here m = 0 is the analogue of Newtonian inertial mass and m = 0 is allowed. The vector field, γ * , over γ is called energy-momentum of the particle. Then for an instantaneous observer (γ(s), Z), we have the orthogonal decomposition of γ * s ∈ T γ(s) M 4 : where e = −g(γ * , Z) > 0 is the energy and p ∈ Z ⊥ is the momentum of the particle as measured by (x, Z), and hence the Newtonian velocity of a particle with respect to Z is given by v = p/e ∈ Z ⊥ . Now, if we have enormous number of particles, each having the same mass m ∈ [0, ∞) and the energy-momenta, then we may describe such a system on M 4 by the following Given an observer (x, Z), (1.2), by regarding part of T x M 4 (using exponential map) as a part of M 4 for a sufficiently small neighbourhood of x ∈ M 4 , where curvature tensor is negligible 6 . Then, given a set of linearly independent vectors X 1 , X 2 , X 3 ∈ Z ⊥ (rest space of Z), the world density function η can be interpreted as follows : the number of particles cuves of P crossing the parallelopiped [X 1 X 2 X 3 ], and is (approximately) given by where Ω ≡ ⋆1 is the volume form on M 4 , |Ω(Z, X 1 , X 2 , X 3 )| is the 3-volume of the parallelopiped [X 1 X 2 X 3 ], and from the Definition 2 and (1.1), P is given by the following orthogonal decomposition An alternative way to calculate the particle number density in Z ⊥ is to project, n x [(1. 2)], into Λ 3 (Z ⊥ ) -the vector space of 3-forms on Z ⊥ -by the R-linear map, Then the number of particles (x, Z) measures in any unit volume of the local rest space Z ⊥ , is given by where G [see Appendix] is the non-degenerate symmetric bilinear form, induced by g, on the vector space Λ p (M 4 ) of differential p-forms. From (1.5) and (1.7) we compute the energy density U measured by (x, Z) : Thus T is a stress tensor for the particle flow (P, η, m), and by (1.3), (1.5), (1.8) we have for every instantaneous observer Z. Hence T x (Z, Z) is the energy density of the particle flow (P, η, m) -measured by (x, Z). Then, by the uniqueness property, the stress-energy tensor for a particle flow is indeed specified by (1.8).

Involutive Distributions (or Foliations)
In order to generalize the notion of particle flows, we recall the following Remarks: The smoothness of D can be expressed in two equivalent ways : (1) Every x ∈ M n has a neighbourhood U x ⊂ M n on which there exists a set of smooth (local) vector fields {V α : α = 1, . . . , p} such that the vectors (V α ) y is a basis for the subspace distinguished by the distribution D y for every y ∈ U x ⊂ M n . Thus {V α } are said to span the distribution, locally.
(2) Every x ∈ M n has a neighbourhood U x ⊂ M n on which there exist (n−p) independent smooth (local) 1-forms {θ k : k = p + 1, . . . , n} such that θ k | Dy = 0 for all y ∈ U x ⊂ M n . Thus Now, an immersed submanifold S p in M n is said to be an integral manifold (also called a there is an integral manifold of D, and the necessary and sufficient condition (Frobenius's integrability condition) for D to be integrable is given by 8,9 [ for some local functions f γ αβ on M n . Or equivalently for some local 1-forms λ k j on M n . An integrable distribution is called a foliation.

p-Dimensional Perfect Fluids
We recall from section 1 that a p-brane is defined by a timelike, connected, p-dimensional manifold immersed in a spacetime and is distinguished by a strictly positive parameter, σthe rest mass per unit (p − 1)-dimensional spatial volume. Thus, we introduce the following then the integral of the local number density (n − p)-form over a spacelike (n − p)-chain C ⊂ U, defines the total number of p-branes in C.

Remarks:
(1) Locally, D is spanned by a set of vector fields {V α : α = 1, . . . , p}, which forms a basis for each timelike tangent space of each integral manifold (p-brane) of D. Then p-branes in where µ is a strictly positive, real-valued, local function. The negative sign in (3.4) reflects the timelike causal character of each integral manifold of D.
(2) In (3.3), the p-form ω (p) (hence, ω and n) is independent of the choice of the vector fields {V α } that span D, locally. Moreover, when restricted to a p-brane, ω (p) is the induced volume form on the corresponding timelike integral manifold with G(ω p , ω p ) = −1.
(3) A p-brane of rest mass per unit spatial volume, σ, is a timelike immersion with local , and σ is the Newtonian analogue of inertial energy per unit spatial volume of the p-brane 10 . We also recall that in a curved spacetime, a p-brane is self-gravitating if φ is an extremal immersion (and hence, the mean curvature of φ vanishes).
Thus given a foliation of a spacetime (M, g) determined by a distribution D, the corresponding multidimensioal fluid, (D, η, σ), is locally characterised by the decomposable p-form ω on M. In order to motivate the definition, (3.15), of a stress-energy tensor T for such systems, we now give an approximate local analysis to obtain the energy density U Z with respect to any instantaneous observer (x, Z) for all x ∈ M.
Energy Density U Z for (D, η, σ) : For any x ∈ M n , let U x be the neighborhood of x where (D, η, σ) is locally represented by (ω, η), and consider a p-brane through x. Then, given an instantaneous observer (x, Z) ∈ T x M n , ω admits the following orthogonal decomposition with respect to (x, Z): p-brane through x, its energy per unit spatial volume, E Z , with respect to (x, Z) is defined by To motivate this definition, we look at the relevant properties of (p (a) From the equations (3.2) and (3.4) we have where the (p − 1)-form χ α is defined by Note that one of the local vector fields, {V α }, say V 1 , must be causal since the set (V α ) x forms a basis for a timelike subspace of T x M. Then, g(V 1 , Z) = 0 since Z is timelike, and hence by (3.8) Θ = 0. Since, ι Z Θ = ω x (Z, Z) = 0 and ω x is decomposable, the dimension of the characteristic subspace 8 of the non-zero (p − 1)-form Θ is (n − p + 1), and hence Θ is also decomposable and can be written as From (3.11) it follows Y a ∈ Z ⊥ , and hence each Y a is spacelike and G(Θ, Θ) > 0.
(b) For any normal field N where g(N, V α ) = 0 for all α, we find ι N Θ = ι N ι Z ω = 0. Then by (3.10) it follows that g(Y a , N) = 0 and hence each spacelike Y a also belongs to V p x -the tangent space of a p-brane through x ∈ M, which is spanned by the set of independent vectors {V 1x , . . . , V px }.
(c) From the above characterisation of Θ by (3.8), (3.10) and (3.11) it is now clear that the non-zero (p − 1)-form Θ is constructed from a set of linearly independent (p − 1) spacelike vectors, {Y a ∈ Z ⊥ : a = 1, . . . , p − 1}, which also belong to the (Lorentzian) tangent space represents the 'spatial extension' of a p-brane through x ∈ M with respect to an observer (x, Z) and it follows that is the spatial volume of the p-brane in Z ⊥ . Now, projecting the foliating p-form χ onto Z ⊥ [as in (3.6)] and using G(χ, χ) = −µ 2 [(3.4)] we also have is the volume of the p-plane spanned by {V α }, when projected into Z ⊥ . From the above relation connecting µ x , A Z and Q Z we may define If an observer Z belongs to the tangent space V p x of a p-brane, then Π Z ω x = 0, and hence Q Z = 0 and γ(Z) = 1. In this case we have Finally, taking the product of n Z in (3.13) and E Z [(3.7)] we find the energy density U Z (that (x, Z) measures) of the fluid (D, η, σ) locally characterised by ω : Stress Tensor for (D, η, σ) : Now U Z is supposed to be equal to T(Z, Z) for every observer Z (see our discussion before Definition 1) for any given form of the stress tensor T. Then the structure of U Z in (3.14) suggests the following definition of the stress tensor T for a multidimensional fluid (D, η, σ) with the local representation where {e a } are the local basis 1-forms (on M) dual to {X b } such that e a (X b ) = δ a b for a, b = 1, . . . , n and ι a ≡ ι Xa . It is clear that T is symmetric, and for any observer (x, Z) Then by continuity T x (W, W ) ≥ 0 for all causal W ∈ T x M and hence T is a stress tensor.
Furthermore, the equations (3.14) and (3.16) show that the energy density U Z is equal to T x (Z, Z) for every instantaneous observer (x, Z). Hence by the uniqueness property, the stress tensor for the fluid, (D, η, σ), is specified by the equation (3.15). We also remark that replacing ω in (3.15) by the energy-momentum 1-form,P , reproduces the stress tensor for the particle flows.
The stress tensor T defined in (3.15) for the fluid (D, η, σ) can be written in a form which is more suggestive as well as convenient for applications. From (3.4) we have Now, expanding ι a ω in (3.15) in terms of χ α [(3.9)], and inserting (3.18) in (3.15) we have where C αβ ≡ (−1) α+β G(χ α , χ β ) is the cofactor of the matrix elementĝ αβ ≡ [g(V α , V β )] and from (3.17), the inverse ofĝ αβ is given byĝ αβ ≡ C αβ /(−µ 2 ). From these definitions and (3.19), we have we note thatĝ is simply a rank-2 symmetric tensor field on the Lorentzian manifold (M n , g) and constructed only from the foliating vector fields {V α }. Now computing the components ofĝ on a leaf L p (whose tangent space is spanned by {V α }), we find from (3.21) where we used the fact [see the definitions below (3.19)] thatĝ αβ is the inverse of the matrix elementĝ αβ ≡ g(V α , V β ).ĝ αβ exists since by (3.17)ĝ is non-degenerate : where the positive function (on M), ρ ≡ σ 2 η, is the energy density measured by all observers tangential to the leaves L p .
As we mentioned earlier [see remark (2)  where G ≡ Ric − 1 2 gR is the Einstein tensor of the spacetime (M n , g), constructed from the Ricci tensor Ric and the scalar curvature R. Since Killing vector fields generate local isometries of (M, g), we have 7 L K Ric = 0 = L K R where L K is the Lie derivation with respect to K. Then from the definition of the Einstein tensor G it also follows L K G = 0, and hence by (4.1), L K T = 0. Proposition 1: If K is a Killing vector field and (D, η, σ) is a p-dimensional fluid on (M n , g), then the world density function η satisfies L K η = 0.
Proof: Taking trace of both sides of (4.1) with T given by (3.25), we find where n and p are the dimensions of M and the distribution D, respectively. Since L K R = 0, and ρ ≡ σ 2 η, it follows from (4.2) that L K η = 0. ✷  Proof: Taking the Lie derivative ofĝ we have L K (g − q i=1Ñi ⊗Ñ i ) = 0. Since K is Killing (and hence L K g = 0), Now, evaluating the symmetric tensor in (4. Now, substituting i = j = l, we have g(L K , N j , N i ) ∀i. Thus, L K N j is normal to N j as well as V α ∀α, and hence, expanding L K N j in the basis {V 1 , . . . , V p , N 1 , . . . , N q }, it follows that Proof: In terms of the spacelike orthonormal set where ⋆ is the Hodge operator induced by the spacetime metric g and g(N k , V α ) = 0 ∀k = 1, . . . , q ; ∀α = 1, . . . , p. Since for any Killing field K, L K commutes with ⋆ and metric dual operation follows that each term in the above expansion vanishes. Hence L K ⋆ (Ñ 1 ∧ . . . ∧Ñ q ) = 0, and by Lie derivation of (4.4) with respect to K we conclude that L K ω = 0. ✷ Now we derive the dynamical consequences of (4.1) with the foliating stress tensor [(3.25)].
First, we recall that the Einstein tensor G is divergence-free, ∇ · G = 0, and hence for any stress tensor (4.1) implies ∇ · T = 0. In our case of interest, T is the stress tensor for a p-foliation, given by [(3.25)] T = −ρ(g − q k=1Ñ k ⊗Ñ k ). Then, we have vector field H of S p ⊂ M n has the following local representation: From (4.5), g(∇ ·T, N j ) = 0, which implies Now, inserting the following expansion in the equation (4.7) we have p a=1 ǫ a g(∇ Xa X a , N j ) = 0. Then, by (4.6), it follows that g(H, N j ) = 0 for j = 1, . . . , q. Hence H = 0. ✷ Remark: By the above proposition, the equation of motion of a p-brane, S, in a multidimensional fluid (D, η, σ) is given by H = 0. Also, it can be shown 13 that H and ω (the local representation of D) are related by ι N dω| S = −g(H, N)| S , for every vector field, N, normal to the p-brane. Then H = 0 implies ι N dω| S = 0.
In order to derive a further dynamical consequence of (4.5), we need the following Lemma 2: The world-density function, η, satisfies ι V (dη + ηλ) = 0, where V is any vector field tangent to the integral submanifolds of the fluid, (D, η, σ) and λ is some 1-form.
Thus our program -of investigating a p-dimensional fluids as a source of spacetime curvature -would be to solve the Einstien equations (4.1) with the stress tensor T in (3.15), and to specify g and (ω, η) -the local representation of the fluid. In the next section we shall work out a complete solution for a fluid characterised by a 2-dimensional distribution on a spacetime (M 4 , g).

Spherically Symmetric 2-Foliation
As an application of our results in the previous sections we consider a 2-foliation (due to string world-sheets) of a static spherically symmetric spacetime (M 4 , g) where, in the local chart (t, r, θ, φ), metric g is of the form Using an orthonormal basis (5.1) can be written as with the dual basis given by such that e a (X b ) = δ a b . It is clear from (5.1) that (M 4 , g) has four Killing vector fields : Locally, ω is required to be decomposable, and for timelike foliation ω must satisfy G(ω, ω) < 0. These two constraints together with (5.7) uniquely (up to a scalar function) specify the structure of the foliation 2-form so that ω = c 1 (r)dt ∧ dr where c is some positive constant since η is defined to be positive.
Solution to Einstein's Equations: For complete specification of the string-field flow we must find the functions h(r) and f (r) from the Einstein equation (4.1) with the stress tensor in (5.12). For convenience (4.1) is written in the following form : where P a ≡ Ric(X a , X b )e b are the Ricci 1-forms, and Γ ≡ traceT = −2ησ 2 by (5.12). Now, computing the Ricci forms with respect to an orthonormal basis[(5.3), (5.4)], we find To solve these equations, first, we note that subtracting ( Properties of the Solution (5.28) : (a) For cσ 2 < 1, this solution has a horizon of radius The equation (5.30) shows that the Schwarzschild radius for the mass m is enhanced by a factor (1 − cσ 2 ) −1 > 1.
(b) For m = 0, there is no horizon but the spacetime has a naked singularity at r = 0.
(c) Using (5.2), if we rewrite the foliation stress tensor, (5.12), in the form, T = −ρ(g − e 2 ⊗ e 2 − e 3 ⊗ e 3 ), then for any causal vector V we have where we have used the inequalities ρ ≥ 0 and g(V, V ) ≤ 0. The equation (5.33) shows that the stress tensor (5.12) satisfies the strong energy condition, and this also means that the gravitational field -generated by the foliating string world-sheets -is attractive.
Thus the fluid of string world-sheets gives rise to a non-trivial solution (to Einstein's equations) -which is non-flat, static and spherically symmetric with a naked singularity.
Such a network of line-like objects could be used to model a multilayer star where the constituents of each layer follows different equation of state.
Modelling Star with 2-Foliation : Following a suggestion in reference [12], we consider a two-layer star in which the core consists of a (spatially isotropic) perfect fluid, and the exterior is formed by a spherically symmetric distribution of strings as in (5.29). Thus the stress tensor for the core is given by where u is a unit timelike vector field -called flow vector field, ρ c is the density of the core and taken to be a constant, and ν is the spatially isotropic pressure function. Then, in the chart (t, r, θ, φ), the metric tensor g c for the core of radius r c is given by a special case (ρ c ≡ constant) of the well-known Oppenheimer-Volkov 14 solution to the Einstein field equations : where r ∈ (0, r c ) and u = ∂ t in (5.34). Futhermore, the isotropic pressure ν(r) can be obtained from the Oppenheimer-Volkov 14 equation : where ν 0 is the pressure at r = 0.
The metric for the spacetime region foliated by the radial strings [(5.29)] is taken as [(5.28)] : where q ≡ cσ 2 [from (5.28)] is a positive constant. In (5.37) we require r ∈ (r c , r s ) with r s as star-radius, and r s > r c . For r > r s , we have the Schwarzschild vacuum metric : Appendix : Definition of G Given a semi-Riemannian manifold (M n , g), the symmetric bilinear form G on the vector space Λ p (M) of differential p-forms on M is, first, defined 9 on decomposable p-forms and then the definition is extended to any p-forms by linearity. Consider two decomposable p-forms Θ and Φ given by where α i , β i ∈ Λ 1 (M). Then, by definition, G(Θ, Φ) ≡ det[g(α i ,β j )] -whereα is the metric dual of the 1-form α. Now, for any pair of p-forms ω and χ, G has the following useful properties : ω ∧ ⋆χ = G(ω, χ) ⋆ 1 ; G(⋆ω, ⋆χ) = (−1) s G(ω, χ) where ⋆1 is the volume form on M induced by the metric g, and s is the index of g.