On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension
A particular, yet relevant, particular case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper we rewrite this inequality in terms of the geometry of the surface obtained by intersecting the past null cone of the original surface with a constant time hyperplane and the “time height” function of over this hyperplane. We also specialize to the case when lies in the past null cone of a point and show the validity of the corresponding inequality in any dimension (in four dimensions this inequality was proved by Tod ). Exploiting properties of convex hypersurfaces in Euclidean space we write down the Penrose inequality in the Minkowski spacetime of arbitrary dimension as an inequality for two smooth functions on the sphere . We finally obtain a sufficient condition for the validity of the Penrose inequality in the four dimensional Minkowski spacetime and show that this condition is satisfied by a large class of surfaces.
The Penrose inequality is a conjecture on spacetimes containing specific types of spacelike codimension-two surfaces which play the role of quasi-local replacements of black holes. In addition, the spacetime is assumed to satisfy an energy condition and have suitable asymptotic behaviour at infinity. The inequality bounds from below the total mass of the spacetime in terms of the area of the quasi-local black holes (or suitable surfaces defined in terms of them). The Penrose inequality is important because it provides a strengthening of the positive mass theorem and also, and perhaps more importantly, because its validity would give rather strong indirect support for the validity of the weak cosmic censorship conjecture . In fact, the Penrose inequality was originally put forward by Penrose  as a way of identifying gravitational configurations that could violate the weak cosmic censorship hypothesis. Since then, and given the absence of counterexamples, the emphasis has turned into trying to prove the conjecture. So far, the inequality has been proved in full generality in the case of asymptotically flat spacetimes satisfying the null convergence condition and containing a time-symmetric asymptotically flat spacelike hypersurface with an inner boundary composed of outermost closed minimal surfaces. The case of spacetime dimension four and connected inner boundary was dealt with by Huisken and Ilmanen . The case of spacetime dimension up to eight and no assumption on connectedness of the inner boundary is due to Bray . The other general case where the inequality is known to hold is for spherically symmetric spacetimes satisfying the dominant energy condition (in arbitrary spacetime dimension) [6, 7]. Besides this, there are also many partial results of interest (see  for a relatively recent review on this topic and  for some new developments).
One version of the Penrose inequality deals with asymptotically flat spacetimes with a regular past null infinity and satisfying the dominant energy condition. Consider a closed, orientable spacelike surface and recall that admits two future directed null normals , . If is such that the null expansion along vanishes (i.e. it is a marginally outer trapped surface) and, moreover, the null hypersurface defined by null geodesics starting on and tangent to extends smoothly all the way to , then the Penrose inequality conjectures that the Bondi mass evaluated at the cut between and is bounded below in terms of the area of . In four dimensions, the inequality reads
Ludvigsen and Vickers  proposed an argument to prove this inequality in the general case. However, this argument made use of an implicit assumption that does not hold in general . Moreover, it is not easy to write down conditions directly on which ensures that this extra assumption holds true. The Penrose inequality for the Bondi mass is therefore an open and interesting problem.
A particularly simple case of this version of the inequality can be formulated for spacetimes generated by shells of null dust propagating in the Minkowski spacetime. In fact this situation was the original set up where the Penrose inequality was discussed in his seminal paper . The idea is the following: imagine that a infinitesimally thin shell of matter is sent from past null infinity in the Minkowski spacetime. The matter content of the shell is null dust, i.e. pressure-less matter propagating at the speed of light. Assume that the shape of the shell, as seen by an inertial observer in the Minkowski spacetime, is a convex surface sufficiently near . This guarantees that the null hypersurface defined by the motion of the null dust is regular in a neighbourhood of . Of course this null hypersurface will develop singularities in the future, where incoming null geodesics meet conjugate points. We denote by the maximal extension of this null hypersurface as a smooth submanifold in the Minkowski spacetime.
The shell modifies the spacetime geometry after it has passed and, since it is collapsing, it will typically generate a trapped surface in its exterior. The spacetime outside the shell is, in general, very complicated (in particular, because the shell produces gravitational waves), but the interior geometry remains unaffected before the shell goes through. Now, the spacetime geometry right after the shell passes can be determined from the interior geometry and the properties of the shell itself by using the junction conditions between spacetimes, see e.g. . Moreover, the matter distribution of the shell can be prescribed freely (at one instant of time). It turns out that given any closed (i.e. compact and without boundary), spacelike surface embedded in , the energy distribution of the null shell can be arranged so that is a marginally outer trapped surface with respect to the spacetime geometry generated by the shell. The direction along which the null expansion vanishes is transverse to . Moreover, the energy density of the shell determines the Bondi mass of the newly generated spacetime at the cut defined by the intersection of and . Now, the area of is the same when measured with the Minkowskian geometry and when observed from the outside spacetime geometry. Moreover, the jump of the null expansion across the shell can be computed in terms of the energy-density of the shell. Combining this facts, it follows that the Penrose inequality (1) becomes an inequality for (a suitable class of) closed spacelike surfaces in the Minkowski spacetime. The resulting inequality is (see [3, 1, 13] in four spacetime dimensions and  for arbitrary dimension),
where is the dimension of (i.e. two in spacetime dimension four) and the area of the standard sphere . In this expression is the area of the surface and is the null expansion of with respect to the future directed, outer (i.e. transverse to ) null normal normalized by the condition , where is the future directed null normal to which is tangent to and which satisfies . All these expressions refer to the geometry of the Minkowski spacetime, in particular denotes scalar product with the Minkowskian metric and is a covariantly constant, unit, timelike vector field in the Minkowski spacetime. The only restriction on the surfaces is that the null hypersurface obtained by sending light orthogonally from them along generates a hypersurface which is regular everywhere and extends all the way to infinity. Geometrically, it is clear that this occurs if and only if the intersection of with the constant hyperplane (for sufficiently negative) is a convex surface of the Euclidean space. We call these surfaces spacetime convex in this paper.
Despite the simplicity of the ambient geometry, proving this inequality is still remarkably difficult. The first case that was solved involved surfaces that lie on a constant time hyperplane . In this case, Gibbons proved [15, 14] that the inequality reduces to the classic Minkowski inequality relating the total mean curvature and the area of convex surfaces in Euclidean space (see expression (4) in Sect. 56 of ).
The second case refers to surfaces contained in the past null cone of a point and leads to a non-trivial inequality for functions on the sphere [3, 17]. In spacetime dimension four, its validity was proved by Tod  using the Sobolev inequality in applied to functions with suitable angular dependence. Regarding the general case, Gibbons claimed  to have a general proof. However, the argument contains a serious gap and the problem remains open. This gap was first mentioned in  without going into the details. In Section 2 we discuss in more detail the argument used by Gibbons and show where it fails.
Our main objective in this paper is to express the Penrose inequality in the Minkowski spacetime of arbitrary dimension in terms of the geometry of the convex euclidean surface obtained by intersecting with a constant time hyperplane together with the time height function . This is the contents of Theorem 1. By applying a powerful Sobolev type inequality on the sphere due to Beckner  we prove the validity of this inequality in the case when the surface lies in the past null cone of a point (Theorem 4). This generalizes to arbitrary dimension the result by Tod  in spacetime dimension four mentioned above and shows that a conjecture put forward by this author regarding the optimal form of the inequality is in fact true. The geometry of convex, compact hypersurfaces in Euclidean space can be fully described in terms of a single function on the unit sphere. This function is called the “support function” and plays an important role in this paper. In spacetime dimension four, the support function was already used in  in a related but different context. One of our main results is Theorem 2 where we write down the Penrose inequality in Minkowski as an inequality involving two smooth functions on the -dimensional sphere. Inspired by the argument by Ludvigsen and Vickers  and simplified later by Bergqvist , we are able to prove (Theorem 6) the validity of this inequality in four spacetime dimensions for a large class of surfaces. This class contains a non-empty open set of surfaces. However, when applied to surfaces lying on the past null cone of a point, the only case covered by this theorem is when is a round sphere. Thus, the cases covered by Theorem 4 and by Theorem 6 are essentially complementary, which indicates that any attempt of proving the Penrose inequality in Minkowski in the general case will probably require a combination of both methods.
The plan of the paper is as follows. In section 2 we discuss Gibbons’ argument  and explain in detail where it fails. For the sake of clarity, we use in this section the same notation and conventions of . In Section 3 we introduce the notation and conventions we use in this paper. We also recall some well-known facts on the geometry of null hypersurfaces used later. In Section 4 we relate the two null expansions of a spacelike surface embedded in a strictly static spacetime. Although applied in this paper only in the case of the Minkowski spacetime this result is interesting on its own and has potential application for the Penrose inequality for null dust shells propagating in background spacetimes more general than Minkowski. In Section 5 we introduce the notions of spacetime convex null hypersurface and spacetime convex surface which are useful for stating and studying the Penrose inequality in Minkowski and we rewrite this inequality in terms of the geometry of the projected surface obtained by intersecting the outer directed past null cone of with a constant time hyperplane , and the so-called “height function” which identifies within . In Section 6 we introduce the support function for convex hypersurfaces in Euclidean space and rewrite the inequality in terms of the functions (and its derivatives), as functions of the unit sphere . We show that, in the particular case when is the past null cone of a point, the Penrose inequality follows from Beckner’s inequality . In Section 7, we restrict ourselves to the four dimensional case and, exploit properties of two-dimensional endomorphisms in order to simplify the inequality in terms of . The result is stated in Theorem 5. Finally, in Section 8 we prove the validity of the inequality for a large class of surfaces in the four-dimensional case. The method of proof is inspired in the flow of surfaces put forward by Ludvigsen and Vickers  and simplified and clarified later by Bergqvist . The explicit form we have for the inequality allows us make the method work for a much larger class of surfaces than those covered by the original argument. In future work we intend to study whether this extension can be pushed from the Minkowski spacetime discussed here to more general spacetimes with a complete past null infinity.
2 A critical revision of Gibbons’ argument
In this section we discuss the gap in Gibbons’ attempt  to prove the general inequality (2). Following the notation in , we will denote by the spacelike, spacetime convex surface involved in the inequality. The future directed null normals are called and and are chosen so that is inward (i.e. the geodesics tangent to generate the null hypersurface extending to ) and satisfy and , where is a covariantly constant, unit, timelike vector field in Minkowski and indices are raised and lowered with the Minkowski metric . Let us denote by the covariant derivative in the Minkowski spacetime.
The strategy in  was to project along onto a constant time hyperplane orthogonal to . The projected surface is denoted by . The main idea was to rewrite (2) in terms of the geometry of as a hypersurface in Euclidean space. Gibbons finds that, whenever , the projected surface has non-negative mean curvature and its mean curvature (with respect to the outer unit normal tangent to the constant time hyperplane) , reads (see expression (5.11) in )
where , is the null expansion of (hence when compared with the normalization we used in (2)) and is minus the null expansion along . As a consequence of (3) and properties of the Minkowski spacetime it follows
In 1994, Trudinger  considered this inequality for general mean convex surfaces in Euclidean space (i.e. surfaces with non-negative mean curvature) and gave an argument for its proof using an elliptic method. However, according to Guan and Li , this argument turns out to be incomplete and the inequality is still open (in  a parabolic argument is proposed which proves the inequality for mean convex starshaped domains in Euclidean space). Nevertheless, the main problem with Gibbons’ argument does not lie in the validity of (5) but on the orthogonal projection leading to (3). The projection is performed as follows. First extend to an ingoing null hypersurface by solving the affinely parametrized null geodesic with initial data on . Similarly, is extended to a null vector field on the outgoing null hypersurface passing through and with tangent vector . These vector fields are then extended to a spacetime neighbourhood of by parallel transport along . With this extension, we have everywhere. Defining on this neighbourhood by , the following vector field can be introduced:
It follows immediately that is everywhere normal to . Morever, this field is orthogonal to and unit on this projected surface. Gibbons used in  that the mean curvature of the projected surface can be expressed as . However, the definition of mean curvature gives . Thus, the expression used by Gibbons is only correct provided . The extension of is uniquely fixed by the definition (6) and a priori there is no reason why this vector should remain unit in a neighbourhood of (or, more precisely, that the derivative of its norm should vanish on ). Moreover, substituting (6) in the (correct) expression for gives
which agrees with (3) only if the last three terms cancel each other. The third term in the right-hand side of (7) is always zero because , which follows from the fact that is geodesic and is covariantly constant. However, neither nor the derivative of along need to vanish on . Even more, they need not, and in fact do not, cancel out in general. This fact invalidates (3) which in turn, spoils the relationship (4) between the left-hand side of the Penrose inequality (2) and the integral of the mean curvature of the projected surface . It is possible to derive general expressions both for and for on (or ) which show that such cancellations do not occur. Instead of doing so, we find it more convenient to present an explicit example where the last two terms in (7) do not cancel each other. For completeness, we also evaluate and explicitly on this example and show that (3) is not valid.
For the example, we consider spherical coordinates on Minkowski and consider the past null cone of the origin defined by the coordinates . This past null cone is defined by the equation . We consider an axially symmetric (with respect to the Killing vector ) spacelike surface embedded in . The embedding is then given by , where is a smooth, positive function (satisfying suitable regularity properties at the north and south poles, as usual). With the normalization for , above (and choosing a direct calculation gives
where prime denotes derivative with respect to . We need to determine the vector field as a function of the spacetime coordinates . The condition that is paralelly propagated along means that does not depend of , i.e. , with . The boundary conditions (9) on require
Since it follows . Thus, in spherical coordinates, . The component of the geodesic equation takes the explicit form
Evaluating this expression on and using (10) it is now straightforward to obtain
Applying a similar argument it follows that the last term of takes the form
It is clear that (11) and (12) do not cancel each other in general. As mentioned above we complete the argument by writing down the explicit expressions for , , and . It is a matter of simple calculation to obtain
Substituting these expressions in the right-hand side of (3) gives
3 Notation and basic definitions
Throughout this paper denotes an -dimensional spacetime, namely an -dimensional oriented manifold endowed with a metric of Lorentzian signature . We always take and assume to be time oriented. Tensors in carry Greek indices and we denote by the Levi-Civita covariant derivative of . On a manifold with metric , we denote by the scalar product with this metric. When the scalar product is with respect to the spacetime metric we simply write . Our sign convention for the Riemann tensor is .
The Penrose inequality in the Minkowski spacetime will involve the geometry of null hypersurfaces, namely codimension one, embedded submanifolds with degenerate first fundamental form. Let be a null hypersurface and a future directed vector field tangent to which is nowhere zero and null. This vector field is defined up to multiplication with a positive function . It is well-known (see e.g. ) that given any point , an equivalence relation can be defined on by means of iff with . The equivalence class of is denoted by and the quotient space by . The set , is endowed naturally with the structure of a vector bundle over (with fibers of dimension ) which is called quotient bundle.
Given , it follows that is a positive definite metric on this quotient space. The tensor is well-defined (i.e. independent of the representatives of and of the extension of to a neighbourhood of ). This tensor is symmetric and plays the role of a second fundamental form on . The Weingarten map, which we denote by , is the endomorphism obtained from by raising one index with the inverse of . Finally, the trace of is the null expansion of . Under a rescaling , these tensors transform as and .
A derivative of can be defined via . Again this derivative is well-defined (i.e. independent of the representative chosen in the definition). Note, however, that it does depend on the choice of . As usual, this derivative is extended to tensors in with the Leibniz rule. An important property of null hypersurfaces is that the quotient metric , the quotient extrinsic curvature and the ambient geometry are related by the following equations (see e.g. ), which are the analog in the null case to the standard Gauss-Codazzi equations for non-degenerate submanifolds,
where is the composition of endomorphisms, and is defined by (the integrals curves of are necessarily null geodesics but the parameter along them need not be affine).
In order to transform this system of equations into a system of ODE for tensor components, let us choose to be affinely parametrized, i.e. satisfying . Let us also select vector fields () tangent to satisfying the properties (i) and (ii) is a basis of at one point . Denote by an affinely parametrized null geodesic containing and with tangent vector (for later convenience we do not fix yet the origin of the affine parameter ). Then is a basis of and the tensor coefficients , of and in this basis satisfy the ODE
where are defined by and indices are lowered and raised with the metric and its inverse .
4 Relationship between two null curvatures of a spacelike surface in a strictly static spacetime
In this paper a spacelike surface is a connected, codimension-two, spacelike, oriented and closed (i.e. compact and without boundary) smooth, embedded submanifold in a spacetime . Tensors in will carry Latin capital indices and the induced metric and connection on are denoted respectively by and . Our conventions for the second fundamental form and the mean curvature are and . Here are tangent vectors to , and denotes the normal component to . If is a vector field orthogonal to , the second fundamental form along is , with .
The normal bundle of is a Lorentzian vector bundle which admits a null basis that we always take smooth, future directed and normalized so that . The second fundamental form along is also called null extrinsic curvature and its trace is the null expansion along , denoted by . The same applies to the null direction . The mean curvature decomposes in the null basis as .
Let us now assume that is strictly static, i.e. that it admits a Killing vector field which is everywhere timelike and hypersurface orthogonal. We define a positive function by . The integrability of implies, locally, the existence of a smooth function such that . The following Lemma shows that the null expansions of any spacelike surface in a strictly static spacetime are not independent to each other. In the case of the Minkowski spacetime this result was proved in .
Lemma 1 (Relationship between null extrinsic curvatures).
Let be an -dimensional strictly static spacetime with static Killing vector . Let be a spacelike surface in . With the notation above, we have
where and are respectively, the restriction of and on .
Since the relationship is local, it suffices to work on a suitably small neighbourhood of a point . We choose small enough so that on and work on from now on. Let be a basis of the tangent space to on . Decomposing in the basis , we find
where is the vector field . The Killing equation implies, on ,
Let us work out the first term. Inserting the decomposition (16) and using the definition of null extrinsic curvature (and similarly for ) it follows
Now, the tangential projection to of a spacetime covariant derivative coincides with the intrinsic covariant derivative on . More precisely, for any vector fields tangent to we have . Thus, the last term in (18) becomes
where we used in the third equality and , where are the connection coefficients of in the basis . Inserting this expression into (18) we conclude
which combined with (17) proves the Lemma. ∎
Under the same assumptions as in the previous Lemma,
(Here and in the following for any function ).
Squaring it follows
5 Penrose inequality in the Minkowski spacetime in terms of the geometry of convex surfaces
We will restrict from now on to the -dimensional Minkowski spacetime (). Choose a Minkowskian coordinate system and define . Since this Killing vector is unit, we have in the notation of the previous section. The hyperplanes at constant will be denoted by .
As already mentioned, the physical construction leading to the Penrose inequality involves null hypersurfaces which extend smoothly all the way to past null infinity. We introduce the following definition which captures this notion conveniently (recall that a null hypersurface is maximally extended if it cannot be extended to a larger smooth null hypersurface).
Definition 1 (Spacetime convex null hypersurface).
Let be a maximally extended null hypersurface in . is spacetime convex if there exists for which the surface is closed (i.e. smooth, compact and without boundary), connected and convex as a hypersurface of the euclidean geometry of . is called spacetime strictly convex if is strictly convex, namely with positive principal curvatures at every point.
Remark. The idea of the definition is, obviously, that if the shape of the null hypersurface at some instant of Minkowskian time is convex, then the past directed outgoing null geodesics cannot develop caustics and hence the null hypersurface will extend smoothly to past null infinity. It is also clear that if is closed and convex for some , the same occurs for all .
Given a spacetime convex null hypersurface , we always normalize the tangent null vector uniquely by the condition . This vector field will also be normal to any spacelike surface embedded in . Since the Penrose inequality involves precisely this type of surfaces the following definition is useful:
Definition 2 (Spacetime convex surface).
A spacelike surface embedded in is called spacetime (strictly) convex if it can be embedded in a spacetime (strictly) convex null hypersurface of .
It is intuitively obvious (and easy to prove) that a spacelike surface can be embedded at most in one spacetime convex null hypersurface . Thus, for any such surface we can define unambiguously a null basis of its normal bundle by the conditions that is tangent to the spacetime convex null hypersurface containing and the normalization conditions , . We refer to as the outgoing null normal and to as ingoing null normal. The Penrose inequality (2) involves the null expansion with respect to the outer null normal. The idea we want to explore in this paper is how this inequality can be related to the geometry of a convex hypersurface of Euclidean space. The most natural convex surface arising in this setup is precisely the surface (see Figure 1). On the other hand, any convex surface defines uniquely a spacetime convex null hypersurface and, then, any spacelike surface embedded in is defined uniquely by the “time height” function over , namely the function . This function is defined on . However, there is a canonical diffeomorphism defined by the condition that lies on the maximally extended null geodesic passing through and with tangent vector . This diffeomorphism allows us to transfer geometric information from onto and viceversa. In particular, we can define . Since no confusion will arise, we still denote this function by . The precise meaning will be clear from the context.
The idea is thus to transform the Penrose inequality (2) into an inequality involving the geometry of as a hypersurface of Euclidean space and the time height function . The result is given in the following theorem:
Theorem 1 (Penrose inequality in Minkowski in terms of Euclidean geometry).
Let be the Minkowski spacetime with a selected Minkowskian coordinate system . Let be a spacetime convex surface in and the convex null hypersurface containing . Consider a closed, convex surface as a hypersurface of Euclidean space and let be its induced metric, its volume form, its second fundamental form with respect to the outer unit normal and the associated Weingarten map. Then the Penrose inequality for can be rewritten as
where is the identity endomorphism, and .
Let us start by relating with . Taking the trace of (15) with respect to (and using , ):
where is the Laplacian of . Corollary gives and the equation above becomes
Integrating on it follows
which gives the desired relationship.
The second step is to use the Ricatti equations on in order to relate on with the extrinsic geometry of . To that aim, we first note that the vector field on satisfies (this is an immediate consequence of the fact that is covariantly constant and ). Thus, the Ricatti equations on take the form (14) provided we have selected vector fields tangent to and satisfying the requirements that (i) and (ii) is a basis of (more precisely is a basis of the tangent space of on suitable open subsets, however this abuse of notation is standard and poses no complications below). Without loss of generality we take tangent to . Equations (14) still admit the freedom of choosing the initial value of the affine parameter on each one of the null geodesics ruling . It turns out to be convenient to select so that on . This determines uniquely as a smooth function which assigns to each point , the value of the affine parameter of the geodesic starting on , with tangent vector and passing through . Given that
and , it follows that . In particular (this is the main reason why this choice of the origin of the affine parameter is convenient).
A crucial property of the geometry of a null hypersurface is that, given any point and any embedded spacelike surface in passing through , the induced metric of and the second fundamental form of along the null normal satisfy and , where (see e.g. ). In other words, the induced metric and the extrinsic geometry along of any embedded spacelike surface in depends only on and not on the details of how is embedded in . Applying this result on we have, for any point ,
where is defined by the properties (i) and (ii) is tangent to at (it is immediate that these two properties define a unique ). Now, the Jordan-Brouwer separation theorem (see e.g. ) states that any connected, closed hypersurface of Euclidean space separates in two subsets, one with compact closure (called interior) and one with non-compact closure (called exterior). Let be the unit normal of pointing towards the exterior, and denote by the corresponding second fundamental form and by the associated Weingarten map. Let be the components of in the basis . Since is totally geodesic and (which follows from the fact that is ingoing, future directed, null and satisfies ), we have
As it is well-known (and in any case easy to verify) the solution to these equations with initial data is
where is defined as the unique point on lying on the null geodesic