1 Introduction↩︎

Energy conditions in \(3+1\) General Relativity are largely motivated as hypothesis of singularity theorems and obstruction theorems for compact trapped surfaces (CTSs) intersecting the domain of outer communications (DOC). Many of these conditions are satisfied by what we consider ordinary matter and fields, giving these results a relatively solid ground (see, however, [1], [2]). The cosmological and black hole singularity theorems are based on the existence of focal points along normal causal geodesics to (respectively) codimension \(1\) and \(2\) spacelike submanifolds. The generalization of this result to a \(d\) dimensional, time oriented spacetime \((M,g_{ab})\) containing a \(k\) dimensional spacelike submanifold \(S\) of codimension \(d-k \geq 1\) is given as Proposition \(1\) in [3] (some related results can be found in [4], pages 288-293):

In this paper we adopt the conventions and definitions in [5]. The spacetime \((M,g_{ab})\) is time orientable and has dimension \(d \geq 3\). A compact trapped submanifold (CTM) of \(M\) is a proper, boundary-less submanifold, that is compact and has a future timelike mean curvature vector field \(H\) (this condition is usually called future trapping in the literature, with an analogous past trapping condition for submanifolds with a past timelike \(H\)), a compact trapped surface (CTS) is a codimension two CTM and a trapped loop (TL) is a one dimensional CTM. Whenever we mention a future null geodesics orthogonal to a spacelike subspace \(W \subset T_pM\) we mean the half of the geodesic to the future of \(p\). We warn the reader that the the sign and normalization conventions of the mean curvature vector field is not uniform in the literature.
Proposition [prop32gallo] is used in [3] to prove the following:

Let us analyze ?? assuming the \(d-\)dimensional Einstein’s equations hold: \(R_{ab} \propto T_{ab} - \tfrac{1}{d-2} T g_{ab}\). In the codimension one case (relevant to cosmological contexts), \(n^a\) gives the future timelike direction orthogonal to \(S\) and the inequality in ?? reduces to \(R_{ab}N^aN^b \geq 0\) along \(\gamma\), which is guaranteed if the spacetime satisfies the strong energy condition (SEC) \[\label{sec} T_{ab} N^a N^b \geq -\tfrac{1}{2}T, \;\; T =T_{ab} g^{ab} \;\; \text{ for timelike } N^a.\tag{1}\] In the codimension two case with \(n^a\) null (the case relevant to the singularity Theorem 2) we can see (refer to section 2, also [3]) that ?? reduces to \[\label{nec} R_{ab} N^a N^b \geq 0 \; \text{ along } \gamma,\tag{2}\] which is guaranteed if the null energy condition (NEC) \[\label{NEC} T_{ab} N^a N^b \geq 0 \;\; \text{ for null } N^a\tag{3}\] holds. Since these energy conditions are regarded as natural in \(3+1\) GR, the singularity theorems are usually stated assuming these hypothesis. A CTS for which condition 2 is satisfied only along its orthogonal null geodesics is a predictor of null geodesic incompleteness, yet, for the sake of simplicity, Penrose’s theorem is usually stated with the stronger requirement that the NEC 3 holds, i.e., that \(R_{ab} N^a N^b \geq 0\) at every spacetime point \(p\) and for every null vector in \(T_pM\). Unless we wanted to prove the existence of conjugate points for future null geodesics normal to any CTS, as in the proof that a CTS cannot intersect the domain of outer communications (DOC, see, e.g., Proposition 12.2.2 in [6]), requiring the NEC on \((M,g_{ab})\) is far more than what is needed. To make this distinction clear for higher codimension we introduce the condition analogue of the NEC that assures ?? will hold for any \(k-\)CTM:

Note that, since \(W\) is spacelike and normal to a null vector, the possible values for \(k\) in this definition are \(k=1,2,..., d-2\). In view of our comments above, for \(k=d-2\) this condition is equivalent to the NEC (see next section). The generic \(k-\)curvature condition is stronger than required in the singularity Theorem 2, as it assures that ?? will hold along any null geodesic, and it can be used together with Proposition 1 to generalize to higher dimensional spacetimes and higher codimension CTMs the standard proof (see, e.g., Proposition 12.2.2. in [6], Proposition 9.2.1 in [7]) that a CTS in an asymptotically simple spacetime cannot intersect the DOC.
For codimension higher than two, condition ?? and its weaker version ?? for null \(N^a\), involve the Weyl piece of the Riemann tensor besides its Ricci part. It is then not adequate to refer to these as “energy conditions”; for spacetimes satisfying Einstein’s equations, the naturalness of ?? could not be justified by the type of matter and fields that we allow. One of the purposes of this paper is to study how stringent the \(k-\)GCC introduced in Definition 4 are for CTMs of codimension \(d-k > 2\). This is explored in Section 2. The other purpose is to exhibit cases where Theorem 2 applies: we find spacetimes for which the \(k-\)GCC does not hold but, still, properly oriented \(k-\)CTMs can be found that fulfill the hypothesis of this theorem. This is explored in section 3, with an emphasis on warped spacetimes and a number of applications on spherically symmetric, \(d \geq 4\) spacetimes.

2 The curvature condition as a generic requirement↩︎

In this section we analyze the \(k-\)GCC introduced in Definition 4. Our results are gathered in Proposition 5 below.
Let \(N^a\) be a null vector at \(T_pM\) and \(V\) the \(d-1\) dimensional subspace of \(T_pM\) vectors orthogonal to \(N^a\). The set of \(k-\)dimensional subspaces \(W \subset V\) with an spacelike induced metric \({h_W}_{ab}\) is an open subset \(\widetilde{Gr}(k,V)\) of the Grassmanian manifold \({Gr}(k,V)\) [5]. Motivated by Definition 4, we are interested in determining if the minimum of the function \[\label{1} \widetilde{Gr}(k,V) \ni W \to R_{abcd}N^aN^c h_W^{bd} \in \mathbb{R}\tag{4}\] is nonnegative, so that ?? will hold at \(p\) for the specific null vector \(N^a\). To avoid the ambiguity of the scaling \(N^a \to x N^a\), under which 4 scales as \(x^2\), we will use the fact that the spacetime is time oriented to pick a unit future timelike vector field \(T^a\), and assume from now on that \(N^a\) is normalized as \(N_a T^a=-1\).

We proceed as in [5], where a similar problem was analyzed (see page 14 on): take any subspace \(V_o\) of \(V\) such that \[\label{ds} V = V_o \oplus \mathbb{R} N\tag{5}\] and let \(\pi: V \to V_o\) be the projection associated to 5 . Introduce the Weyl tensor \(C_{abcd}\), \[\label{weyl} R_{abcd}=C_{abcd}+\frac{2}{d-2}\left(g_{a[c}R_{d]b} - g_{b[c}R_{d]a} \right) - \frac{2}{(d-1)(d-2)}\left(Rg_{a[c}g_{d]b} \right)\tag{6}\] and define the symmetric bilinear operators \[\label{bop0} \begin{align} V \otimes V \ni (X,Y) \to C^{(N)}(X,Y)&:=C_{abcd}N^a N^c X^b Y^d\\ V \otimes V \ni(X,Y) \to R^{(N)}(X,Y)&:=R_{abcd}N^a N^c X^b Y^d. \end{align}\tag{7}\] In view of the null character of \(N\), \(\pi\) is an isometry: \(\pi(X) \cdot \pi(Y) = X \cdot Y\). Also \[\label{bop} \begin{align} C^{(N)}(X,Y)&=C^{(N)}(\pi(X), \pi(Y))\\ R^{(N)}(X,Y)&=R^{(N)}(\pi(X), \pi(Y)). \end{align}\tag{8}\] The above equalities are easily proved if we use the symmetries of the Weyl and Riemann tensor and a frame \(\{ e_1, e_2,...,e_{d-2},N,L\}\) for which \(V_o = \text{span}\{ e_1, e_2,..., e_{d-2} \}\) and the only nontrivial inner products are \(e_i \cdot e_j=\delta_{ij}\) and \(N \cdot L=-1\), so that \[\label{im} g^{ab} =-N^a L^b-L^a N^b + \sum_{i,j=1}^{d-2}\delta^{ij} e_i^a e_j^b\tag{9}\] and \[\pi(X^N N + X^j e_j)= X^j e_j.\] In view of 8 , the function defined by 4 , satisfies \[R_{abcd}N^aN^c h_W^{bd} =: {\rm tr}_{W} R^{(N)} = {\rm tr}_{\hat{W}} R^{(N)},\] where \(\hat{W} := \pi(W)\). Choose the above basis such that \(\hat{W} = \text{span}\{e_1, e_2,...,e_k\}\). Using 6 we find \[R^{(N)}(e_i,e_j)= C^{(N)}(e_i,e_j)+ \frac{1}{d-2}R_{ac}N^a N^c \delta_{ij}.\] Taking the trace over \(\hat{W}\) (that is, applying \(\delta^{ij}\) to the above equation and summing over \(i,j\) from \(1\) to \(k\)) yields \[\begin{align} \nonumber {\rm tr}_W R^{(N)} &= {\rm tr}_{\hat{W}} R^{(N)} \\ &= {\rm tr}_{\hat{W}} C^{(N)} + \frac{k}{d-2}R_{ac}N^aN^c. \label{main} \end{align}\tag{10}\] Note that the second term above is independent of \(W \in \widetilde{Gr}(k,V)\) and that for \(k=d-2\), in view of 9 and the trace free character of the Weyl tensor, the first term vanishes: \[\label{weyl0} C_{abcd}N^a N^c \sum_{i,j=1}^{d-2}e_i^be_j^d \delta^{ij} = C_{abcd}N^a N^c \left( g^{bd}+N^b L^d+L^bN^d \right)=0.\tag{11}\] It is important to keep in mind that in 10 we deal only with subspaces \(\hat{W} \subset V_o\). Since the restrictions of the maps \(C^{(N)}\) and \(R^{(N)}\) to \(V_o \otimes V_o\) are symmetric, and the induced metric \(h_{V_o}\) is positive definite, the commuting linear operators \(V_o \to V_o\) defined by \(h_{V_o}^{-1} [C^{(N)}\mid_{V_o \otimes V_o}]\) and \(h_{V_o}^{-1} [R^{(N)}\mid_{V_o \otimes V_o}] =h_{V_o}^{-1} [C^{(N)}\mid_{V_o \otimes V_o}] + \frac{1}{d-2}R_{ac}N^aN^c \boldsymbol{I}\) are simultaneously diagonalizable with spectra \[\label{evs} \begin{align} &\lambda_1 \leq \lambda_2 \leq ... \leq \lambda_{d-2} \\ &\mu_1 \leq \mu_2 \leq ... \leq \mu_{d-2} \end{align}\tag{12}\] respectively, where \[\mu_k = \lambda_k + \frac{1}{d-2}R_{ac}N^aN^c,\] and, in view of 11 , \[\label{zero} \sum_{\alpha=1}^{d-2} \lambda_\alpha = {\rm tr}_{V_o} C^{(N)} = 0.\tag{13}\]

As in [5] we can prove that the trace function 4 effectively reaches a minimum on the open manifold \(\widetilde{Gr}(k,V)\) and that this minimum is (note the ordering of the eigenvalues in 12 ) \[\label{min} \left.\text{min} \left({\rm tr}_W R^{(N)}\right) \right|_{\widetilde{Gr}(k,V)} =\sum_{\alpha=1}^k \mu_\alpha=\sum_{\alpha=1}^k \lambda_\alpha + \frac{k}{d-2}R_{ac}N^aN^c.\tag{14}\]

Example: FLRW cosmologies have zero Weyl tensor, so, in view of Proposition 5.ii, those for which the NEC holds satisfy the \(k-\)GCC for any \(k\).
The \(k-\)GCC, being the equivalent for higher codimension CTMs of what the NEC is for CTSs, guarantees that any \(k-\)CTM is eligible in Proposition 1 and Theorem 2. Its failure forces us to restrict ourselves to CTMs which are suitable oriented to satisfy ?? . The existence of such CTMs is sufficient to prove null geodesic incompleteness (a single such geodesics is all we need). Proposition 1 is used also in the standard proof that a CTS cannot intersect the DOC if the NEC holds. As explained above, if the \(k-\)GCC, the generalization of the NEC, does not hold, as happens in general according to Proposition 5, we cannot use this type of arguments to prove that \(k-\)CTMs do not intersect the DOC.

3 The curvature condition for specific CTMs↩︎

In this section we exhibit examples of spacetimes for which, despite the fact that the \(k-\)GCC is not satisfied, one can still find \(k-\)CTMs oriented in such ways that condition ?? required by the singularity Theorem 2 holds. The following definition helps to formalize the idea of “suitable orientation” (compare with Definition 4):

A \(k-\)CTM \(S\) such that for every \(p \in S\), \(T_pS\) satisfies the \(k-\)CC is then suitable for the singularity theorem. The use of this definition promises to be unmanageable on generic spacetimes but, as we show next, there are physically interesting examples of application.

3.1 Warped spacetimes↩︎

Assume that the spacetime belongs to the large and rich class of warped product manifolds \(B \times_{r^2} F\): \[\text{d}s^2=\tilde{g}_{\alpha\beta}(x) \text{d}x^\alpha\text{d}x^\beta+r^2(x)\,\bar{g}_{AB}(\theta)\text{d}\theta^A\text{d}\theta^B. \label{eq:warped32metric}\tag{15}\] Here \(\{x^1,x^2, \dots \}\) are local coordinates of the base manifold \((B, \tilde{g})\), which is Lorentzian, and \(\{\theta^1, \theta^2,\dots\}\) are coordinates of the Riemannian fiber manifold \((F, \bar g)\). We assume \(r: B \to \mathbb{R}\) is positive definite. The superscripts \(\sim\) and \(-\) will be attached to tensors belonging to the base and fiber respectively, and \(\tilde{g}\) (\(\bar g\)) and their inverses will be used to lower and raise indexes of tensors on \(B\) and \(F\) respectively. As an example, for a spacetime vector \(N^a=(N^\alpha, N^A)\) we find \(N_a=(N_\alpha,r^2 N_A)\), then \(N^a N_a=N^\alpha N_\alpha+ r^2 N^AN_A\). The submanifolds \(\{ x \} \times F\) and \(B \times \{ \theta \}\) of \(B \times_{r^2} F\) are respectively called the fiber and leaf through \((x,\theta) \in B \times_{r^2} F\). Vectors of the form \((N^\alpha,0)\) [\((0,N^A)\)] are tangent to leaves [fibers] and will be called horizontal [vertical]. A calculation shows (see, e.g. [8] equations (A.3)-(A.4)) that, for 15 , the nonzero Christoffel symbols (mod symmetries) are \[\label{cristo} \Gamma^\alpha_{\beta\gamma}= \tilde{\Gamma}^\alpha_{\beta\gamma}, \;\; \; \;\Gamma^A_{\beta C}= \frac{\partial_\beta r}{r} \delta^A_C, \; \; \;\; \Gamma^\alpha_{BC}= -r (\tilde{g}^{\alpha\delta}\partial_\delta r) \;\bar g_{BC}, \;\;\;\; \Gamma^A_{BC} = \bar \Gamma^A_{BC}.\tag{16}\] From this equation follows that the parallel transport along a geodesic \(\gamma\) of a vector that is initially vertical and orthogonal to \(\gamma\) remains vertical along \(\gamma\) (see also Appendix A in [8] and chapter 7 in [4]). As a consequence, if a CTM \(S\) is oriented such that \(T_p S\) is vertical at every \(p \in S\), then, at every point of a geodesic \(\gamma\) orthogonal to \(T_pS\) , \(h_W\) in ?? will project onto a vertical subspace of the tangent space. Thus, if equation ?? holds for vertical subspaces of \(T_qM\) at any \(q\), those \(k-\)CTMs which are tangent at every point to vertical subspaces will be suitable to be used in Theorem 2 as \(T_pS\) will satisfy the \(k-\)CC in Definition 6 at every \(p \in S\). Now, since the non trivial (mod symmetries) components of the Riemann tensor are (see, e.g., [8], equations (3.3)-(3.6)) \[\begin{align} R_{\alpha\beta\gamma\delta} &= \tilde{R}_{\alpha\beta\gamma\delta}, \\ R_{\alpha B \gamma D} &= -r \left( \tilde{\nabla}_\gamma\tilde{\nabla}_\alpha r \right) \bar{g}_{BD}, \\ R_{ABCD} &= r^2 \bar{R}_{ABCD} - r^2 |\tilde{\nabla} r|^2 \left( \bar{g}_{AC}\bar{g}_{BD} - \bar{g}_{AD}\bar{g}_{BC} \right), \end{align}\] we find that, for \(N^a=(N^\alpha, N^A)\) and \(e^A_i, i=1,2,..,k\) an orthonormal basis of \(W\), condition ?? reduces to \[\bar{R}_{ABCD}\bar{N}^A\bar{N}^Ce^B_ie^D_j\delta^{ij}-k\left(\bar{N}^A\bar{N}_A \,\tilde{\nabla}^\beta r \tilde{\nabla}_\beta r + \tilde{N}^\alpha\tilde{N}^\beta\frac{\tilde{\nabla}_\alpha\tilde{\nabla}_\beta r}{r}\right) \geq 0. \label{eq:gcc32warped}\tag{17}\] Since the fiber metric is positive definite and \(N^a\) is null we have \(\bar{N}^A\bar{N}_A=-r^{-2}\,\tilde{N^\alpha}\tilde{N_\alpha}\geq 0\). The first term in 17 is the sum of \(k\) sectional curvatures of the fiber; this is a Riemannian submanifold, so the planes spanned by \(\bar{N}\) and \(e_i\) are non-degenerate.
Whenever \(dr \neq 0\), \(r\) can be taken as one of the \(x^\alpha\) coordinates, in which case \[\begin{align} &\tilde{\nabla}^\beta r \tilde{\nabla}_\beta r \equiv \tilde{g}^{rr},\\ & \tilde{N}^\alpha\tilde{N}^\beta\tilde{\nabla}_\alpha\tilde{\nabla}_\beta r \equiv -\tilde{N}^\alpha\tilde{N}^\beta\, \tilde{\Gamma}^r_{\alpha\beta}, \end{align}\] and 17 reads \[\bar{R}_{ABCD}\bar{N}^A\bar{N}^Ce^B_ie^D_j\delta^{ij} - k \left(\tilde{g}^{rr} \bar N^A \bar N_A -\tilde{N}^\alpha\tilde{N}^\beta\,\frac{ \tilde{\Gamma}^r_{\alpha\beta}}{r} \right)\geq 0. \label{eq:32gcc32warped32simp0}\tag{18}\]

The results above can be gathered in the following:

In the remaining of this Section we will restrict to CTMs satisfying the hypothesis of Proposition 7 (which will be called properly oriented from now on).

3.2 Warped spacetimes with constant curvature fibers↩︎

For warped spacetimes, further simplifications arise in those cases where the fiber has constant curvature \(\bar{C}\), \[\label{ccw} \bar{R}_{ABCD}=\bar{C}(\bar{g}_{AC}\bar{g}_{DB}-\bar{g}_{AD}\bar{g}_{CB}),\tag{19}\] as the contraction in 17 reduces to \[\label{simp} \bar{R}_{ABCD}\bar{N}^A\bar{N}^Ce^B_ie^D_j\delta^{ij}=\bar{C}k\,\bar{N}^A\bar{N}_A.\tag{20}\] and 18 simplifies to \[\label{eq:32gcc32warped32simp1} \begin{align} 0 &\leq \left(\bar{C}-\tilde{g}^{rr}\right) \,\bar{N}^A\bar{N}_A + \tilde{N}^\alpha\tilde{N}^\beta\,\frac{ \tilde{\Gamma}^r_{\alpha\beta}}{r} \\ &= \tilde{N}^\alpha\tilde{N}^\beta\left[ \left( \tilde{g}^{rr}-\bar{C}\right) \,\frac{\tilde{g}_{\alpha\beta}}{r^2} + \frac{ \tilde{\Gamma}^r_{\alpha\beta}}{r} \right] \end{align}\tag{21}\] which, interestingly, does not depend explicitly on \(k\), although the dimension of the trapped submanifold matters. For example, if \(k=d-2\) the first term in the first line of 18 is absent since \(\tilde{N}\) is a null vector of the Lorentzian base manifold (for higher codimension this vector is timelike in general). In view of Proposition 7, if a CTM is tangent to the fibers at every point, it will satisfy the hypothesis of Theorem 2 iff 21 holds along its future null orthogonal geodesics. Some examples follow.

3.3 Spherical symmetry in arbitrary dimensions↩︎

In \(d\) dimensions, in those regions where the area radius \(r\) can be used as a coordinate (\(dr \neq 0\)), the most general spherically symmetric metric can be written in the form \[\text{d}s^2= -e^{2\beta}f\text{d}v^2+2e^{\beta}\text{d}v\text{d}r+r^2\gamma_{AB}\text{d}\theta^A\text{d}\theta^B, \label{eq:32vaidya32metric}\tag{22}\] where \(\beta\equiv \beta(v,r)\), \(f\equiv f(v,r)=1- \frac{\mu(v,r)}{r^{d-3}}\), with \(\mu(v,r)\) the Misner-Sharp mass function and \(\gamma_{AB}\text{d}\theta^A\text{d}\theta^B\) is the metric of the unit \((d-2)\) sphere \({\rm S}^{d-2}\), which, in hyper-spherical coordinates \(\theta^A\), reads \[\label{esferas} \gamma_{AB}\; \text{d}\theta^A\text{d}\theta^B = \sum_{B=1}^{d-2} C_B\, \left(\text{d} \theta^B\right)^2,\tag{23}\] with \[\label{CB} C_B = \begin{cases} 1 &, B=1 \\ \prod\limits_{C=1}^{B-1} \sin^2 (\theta^C) &, B>1 \end{cases}\tag{24}\] The metric 22 can be used to model spherical collapse as well as static spherical symmetric black holes. If the spacetime is to satisfy the NEC and, in the dynamical case, we assume that the mass-energy is ingoing, then it is straightforward to see that the following conditions must hold \[\begin{align} &\partial_r\beta \geq 0, \tag{25}\\ &\partial_r\mu \geq r^{d-3}f \partial_r\beta, \tag{26} \\ &\partial_v\mu\geq 0. \tag{27} \end{align}\] Equation 22 gives \[\begin{align} \tilde{N}^a\tilde{N}^b\, \tilde{\Gamma}^r_{ab}&=-\left(f\partial_r\beta + \frac{\partial_rf}{2}\right)\tilde{N}^a\tilde{N}_a + \left(\tilde{N}^r\right)^2 \partial_r\beta - \left(\tilde{N}^v\right)^2 \partial_vf \\ &=\left(r^2f\partial_r\beta -\frac{\partial_r\mu}{r^{d-5}}+(d-3)\frac{\mu}{r^{d-4}} \right)\bar{N}^A\bar{N}_A + \left(\tilde{N}^r\right)^2 \partial_r\beta + \left(\tilde{N}^v\right)^2 \frac{\partial_v\mu}{r}, \end{align}\] so the curvature condition 21 is equivalent to \[\left[ (d-1)\mu - r\,\partial_r \mu + r^{d-2}f\partial_r\beta \right]\bar{N}^A\bar{N}_A + \left(\tilde{N}^r\right)^2 r^{d-4}\partial_r\beta + \left(\tilde{N}^v\right)^2 r^{d-5}\partial_v\mu \geq 0. \label{eq:32gcc32sc}\tag{28}\] In virtue of 25 and 27 , the second and third terms in 28 are always non-negative; any negative contribution must come from the first term. Indeed, because of 26 , it is easy to see that \(- r\,\partial_r \mu + r^{d-2}f\partial_r\beta\leq 0\).
As a case of interest we search for CTMs within the succession of submanifolds given by the \(k-\)spheres \({\rm S}_o^{k}\), \(k=d-2-n\), which are defined by fixing \(v=v_o\), \(r=r_o\) and \(\theta^A=\theta^A_o\) for \(A=1, 2,...,n\). Introduce then the notation \(\theta_o=(\theta^1_o, \theta^2_o,...,\theta^n_o)\). Note from 22 24 that the induced metric on these spheres is \[\label{esferas2} \text{d} s_{(k,o)}^2 = r_o^2 \sum_{B=n+1}^{d-2} C_B\; \left(\text{d} \theta^B \right)^2 = r_o^2 C_{n+1}(\theta_o) \, \text{d}s_{(k)}^2\tag{29}\] where \(\text{d}s_{(k)}^2= \gamma^{(k)}_{I' J'} \text{d}\theta^{I'} \text{d} \theta^{J'}\) with \(I',J'=n+1,...,d-2\) is the metric on the \(k=d-2-n\) dimensional unit sphere \({\rm S}^k\). The \(k-\)volume of \({\rm S}^k_o\) is \[\label{ksv} \text{Vol}({\rm S}^k_{o}(\theta_o)) = r_o^k \, (C_{n+1}(\theta_o))^{k/2} \;\text{Vol}({\rm S}^k).\tag{30}\] Since the coordinates 22 23 are adapted to \({\rm S}^k_o\) and \(\partial_v, \partial_r, \partial_{\theta^1},..., \partial_{\theta^n}\) and orthogonal to it, the mean curvature vector field can be easily calculated as the trace over \(T {\rm S}^k_o\) of (minus) the relevant Christoffel symbols: \[-\Gamma_{IJ}^{\phantom{I}a}\big|_{{\rm S}^k_o}=\frac{1}{2}g^{a b}\partial_{b}g_{IJ}= \begin{cases}r_0\, C_{n+1}(\theta_o)\,g^{a r}(v_o,r_o)\tilde{\gamma}_{I J}(\theta) & a=v,r \\ \frac{C_{n+1}(\theta_o)}{C_{A}(\theta_o)}\, \text{cot}(\theta^A_o) \tilde{\gamma}_{IJ}(\theta) & a=A=1,2,...,n \end{cases}, \label{eq:32chr32de32ST}\tag{31}\] where the middle of the alphabet capital indices \(I,J\) correspond to the submanifold angles, that is, they run from \(n+1\) to \(d-2\), and the second line is absent if \(n=0\). Taking the \(T{\rm S}_o^k\)-trace gives the mean curvature vector field \[H=\frac{k}{r_0}\left(e^{-\beta}\partial_v + f\partial_r + {{\hat{\sum}}{}}_{A=1}^{n} \;\frac{1}{r_0C_{A}(\theta_o)}\text{cot}(\theta^A_o)\partial_{\theta^A} \right),\] where the hat indicates the sum is absent if \(n=0\). Requiring that \(H^a\) be timelike gives the trapping condition for \(S_o^k\): \[H^a H_a = \frac{k^2}{r_o^2} \left[f + \hat{\sum}_{A=1}^n \frac{1}{C_{A}}\text{cot}^2(\theta_{0}^A) \right] < 0. \label{eq:32trapping32of32S94k}\tag{32}\] The term in square brackets simplifies to \[\label{trapped} f + \hat{\sum}_{A=1}^n \frac{1}{C_{A}}\text{cot}^2(\theta_{0}^A) = \begin{cases} f &, \text{if} \; n=0 \\ f -1 + \frac{1}{C_{n+1}(\theta_o)} &, \text{if} \, n \geq 1 \end{cases}\tag{33}\] The case \(n=0\) above recalls us that the \(d-2\) spheres are trapped if they are inside the apparent horizon [9], the \(n \geq 1\) cases set a minimum \(k-\)volume for \({\rm S}^k_o\) (see 30 , note that \(\mu\) has dimension \(r^{d-3}\)): \[\label{cond} \frac{1}{C_{n+1}(\theta_o)} < \frac{\mu(v_o,r_o)}{r_o^{d-3}},\tag{34}\] or, equivalently \[\label{cond1} \left[\frac{\text{Vol}({\rm S}^k_o)}{ r_o^k \, \text{Vol}({\rm S}^k)} \right]^{2/k} > \frac{r_o^{d-3}}{\mu(v_o,r_o)}\tag{35}\] Note that, since \(C_{n+1}(\theta_o) \leq 1\), inequality 34 implies \(f \leq 0\); thus, our trapped \(k-\)spheres can be found only within the apparent horizon. Moreover, as the supporting trapped \(d-2\) sphere approaches the apparent horizon, the only contained lower-dimensional trapped spheres fit around the generalized great circles: \(\theta^1_o=\theta^2_o=...=\theta^n_o=\pi/2\).
We can make a more geometric description of what we have found, independent of any specific set of angular coordinates / axis choice: the relation of the hyper-spherical coordinates \(\theta^A\) of a \(d-2\) dimensional sphere with the Cartesian coordinates of its Euclidean ambient space \(\mathbb{R}^{d-1}\) is \[\begin{align} &x^1 = r \cos(\theta^1), \\ &x^2 = r \sin(\theta^1) \cos(\theta^2), \\ &x^3 = r \sin(\theta^1) \sin(\theta^2) \cos(\theta^3), \\ &\qquad \vdots\\ &x^{d-2} = r \sin(\theta^1) \cdots \sin(\theta^{d-3}) \cos(\theta^{d-2}), \\ &x^{d-1} = r \sin(\theta^1) \cdots \sin(\theta^{d-3}) \sin(\theta^{d-2}). \end{align}\] Fixing \(r=r_o\), \(\theta^1=\theta^1_o\), ... ,\(\theta^n=\theta^n_o\) defines an affine space \(\mathcal{A} \subset \mathbb{R}^{d- 2}\): \(x^A=x^A_o\) for \(A=1,2,...,n\), leaving a cut of the \(d-2\) dimensional sphere of radius \(r_o\) with \(\mathcal{A}\). The resulting \(k\)-sphere has the \(SO(k)\) isometry subgroup \(H\) consisting of matrices of the form \(g= \text{diag}( \mathbb{I}_{n \times n}, h)\), \(h \in SO(k)\). Any other conjugate \(SO(k)\) subgroup of \(SO(d-2)\), say \(gHg^{-1}\), would do the same: its orbit (now the intersection of the \((d-2)\)-sphere with \(g \mathcal{A}\)) would be a \(k-\)sphere, and this will be trapped iff 35 holds. As mentioned before, as the supporting sphere approaches the horizon, the \(k-\)dimensional trapped spheres will tight around the intersection with subspaces of \(\mathbb{R}^{d-1}\), as, according to 34 , \(x_o^A \to 0\) for \(A=1,2,...,n\).
Having found a rich set of trapped submanifolds of dimensions \(k=1,2,...,d-2\) in the most general spherically symmetric spacetime, we would like to know if these are singularity predictors. According to Proposition 7 and using the simplifications found for spherically symmetry, this will be the case if: i) the (proper extension of) the metric 22 admits a non compact Cauchy surface and ii) equation 28 holds along future null normal geodesics of the CTM. Analytic extensions are handled the same way as in \(3+1\), as they involve only the base space and so are independent of the value of \(d \geq 4\). Condition (ii) needs to be checked case by case. Some examples follow.

3.3.1 Schwarzschild-Tangherlini BH↩︎

Inserting \(\beta(v,r)=0\) and \(\mu(v,r)=\mu>0\) in 22 gives the generalization of the Schwarzschild BH to arbitrary dimensions [10]. In this case it is trivial to see that 28 is fulfilled at every point of the spacetime, in particular, along the future null normal geodesics of our trapped \(k-\)spheres. As mentioned above, by Proposition 7, every properly oriented CTM (for any \(k\)) will satisfy the hypotheses of Theorem 2 and thus act as a singularity predictor.

3.3.2 Vaidya Spacetime↩︎

The Vaidya spacetime of incoming radiation is trivially extended to arbitrary dimensions by replacing \(\beta(v,r)=0\) and \(\mu(v,r)=\mu(v)>0\) (with \(d \mu/dv \geq 0\)) in 22 . As there is no dependence in \(r\), the only negative contribution in 28 vanishes, so the curvature condition is satisfied. The conclusion is same as in the previous example.

3.3.3 Reissner-Nördstrom BH↩︎

A more intriguing case is the spherically symmetric charged sub-extreme black hole, whose metric depends on two parameters: the mass \(m>0\) and the electric charge \(q\neq 0\), subject to \(m\geq |q|\). For the sake of simplicity we will fix \(d=4\), then the metric is 22 with \(\beta(v,r)=0\) and \(f(v,r)=1-\frac{2m}{r}+\frac{q^2}{r^2}\). The zeroes of \(f\) are the Cauchy (\(r_-\)) and event (\(r_+\)) horizon radii \[\label{hor} r_\pm = m\pm \sqrt{m^2-q^2}.\tag{36}\] The maximal analytic extension contains infinitely many copies of regions I (\(r>r_+\)), II (\(r_- < r<r_+\)) and III (\(0<r<r_-\)). We will consider instead the globally hyperbolic spacetime \(M^{3+1}_{RN}\), depicted in Figure 1, which consists of two copies of region II and two copies of region I (primes are used in the figure to identify isometric regions). Region III is physically irrelevant: this static solution of the Einstein-Maxwell equations is linearly unstable: perturbations inside it grow exponentially in time [11]. As it is well known there are infinitely many possible (non analytic) extensions of \(M^{3+1}_{RN}\) beyond the Cauchy (future and past) horizons (see, e.g., [7], page 159). Moreover, these horizons are unstable and become a curvature singularity if we replace the \(M^{3+1}_{RN}\) data on a Cauchy surface, such as \(\Sigma\) in Figure 1, with data close to it [6], [12]–[14].

It is well known that Penrose’s singularity theorem cannot be used in any globally hyperbolic open subset of the maximal analytic extension to predict the geodesic incompleteness caused by the \(r=0\) singularity, since it is beyond the Cauchy horizon (see [15] for alternative singularity theorems that fit in this case). However the trapped spheres in region II of the globally hyperbolic space \(M^{3+1}_{RN}\), which satisfies the NEC and admits a non-compact Cauchy surface (e.g., \(\Sigma\) in Figure 1), predict the future null geodesic incompleteness caused by the Cauchy horizon at \(r=r_-\). A slight variation of the initial data at \(\Sigma\) would preserve theses trapped surfaces, which now would predict the singular hypersurfaces where the perturbed spacetime ends.
A natural question to ask is if we could use higher codimension CTMs (that is TLs) of \(M^{3+1}_{RN}\) to predict its incompleteness. Let us switch the notation for spherical coordinates from \((\theta^1,\theta^2)\) used above to the standard names \((\theta,\phi)\). Condition 34 for a parallel \(\theta=\theta_o\) to be a TL can be written, using 36 , as \[\frac{(r_o-r_+)(r_o-r_-)}{{r_o}^2} + \cot^2 \theta_o <0.\] This equation implies that trapped parallels occur in region II (and its isometric copy II’), in a band of the sphere they belong to that is symmetric around the Equator \(\theta=\pi/2\), and that this band narrows to zero width as \(r \to r_+\) from the left or \(r \to r_-\) from the right. Can we use these TLs in Theorem 2 to predict the geodesic incompleteness caused by the Cauchy horizon? Condition 28 in this case results in \[\label{TLcond} \frac{(3mr-2q^2)}{r}\bar{N}^A\bar{N}_A \geq 0,\tag{37}\] which is fulfilled when \(r\geq \frac{2q^2}{3m}\). A TL then satisfies the hypothesis of Theorem 2 if 37 holds all the way from the TL value \(r=r_o\) (which lies in region II), to the Cauchy horizon at \(r=r_-\). This will be the case iff \(3mr_--2q^2>0\) or, equivalently, if the condition \[\frac{|q|}{m} > \frac{\sqrt{3}}{2}\sim 0.87\] is added to the sub-extreme requirement that \(|q|<m\). Note that this is different from the case of CTSs, which predict geodesic incompleteness for arbitrary \(|q| <m\).

4 Conclusions↩︎

Penrose’s singularity theorem proves that the existence of a CTS anticipates the incompleteness of a future null geodesic orthogonal to it. The theorem requires that \(R_{ab}N^a N^b \geq 0\) along every future null geodesics normal to the CTS (\(N^a\) is the geodesic tangent vector), a condition that is guaranteed if the stronger NEC is satisfied. The generalization of Penroses’s theorem to CTMs of higher codimension, given in [3], requires that the curvature condition ?? holds along future null orthogonal geodesics. This is guaranteed if the \(k-\)GCC introduced in Definition 4 (the analogous of the NEC), is satisfied. In this paper we prove a number of useful properties of the \(k-\)GCC, these are gathered in Proposition 5. We also show that the \(k-\)GCC is violated in generic spacetimes, in which case not every CTM predicts geodesic incompleteness. However, particular CTMs can be found for which the weaker condition ?? holds and Theorem 2 applies. Since these CTMs predict singularities at their future, in view of the weak cosmic censorship, they should be regarded as signatures of black hole interiors. A number of examples are worked out, including CTMs of dimensions ranging from one to \(d-1\) in \(d+1\) dimensional spacetimes. The fact that the \(k-\)GCC does not hold in generic spacetimes, does not allow the use of Proposition 1 to prove, using standard arguments, that \(k-\)CTMs cannot intersect the DOC.

References↩︎