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1 Basic Concepts and the Stability Operator

Let S be a closed marginally outer trapped surface (MOTS): its outer null expansion vanishes \(\theta _{\vec{k}} = 0\) [4, 5]. Here, the two future-pointing null vector fields orthogonal to S are denoted by \(\vec{l}\) and \(\vec{k}\) with \({l}^{\mu }k_{\mu } = -1\). I will also use the concept of outer trapped surface (OTS, \(\theta _{\vec{k}} < 0\) ). A marginally outer trapped tube (MOTT) is a hypersurface foliated by MOTS.

As proven in [1], the variation \(\delta _{f\vec{n}}\theta _{\vec{k}}\) of the vanishing expansion along any normal direction \(f\vec{n}\) such that \(k_{\mu }{n}^{\mu } = 1\) reads

$$\displaystyle\begin{array}{rcl} \delta _{f\vec{n}}\theta _{\vec{k}} = -\varDelta _{S}f + 2{s}^{B}\overline{\nabla }_{ B}f + f\left (K_{S} - {s}^{B}s_{ B} + \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )& &{}\end{array}$$
(1)

where K S is the Gaussian curvature on S, Δ S its Laplacian, G μ ν the Einstein tensor, \(\overline{\nabla }\) the covariant derivative on S, \(s_{B} = k_{\mu }e_{B}^{\sigma }\nabla _{\sigma }{l}^{\rho }\) (with \(\vec{e}_{B}\) the tangent vector fields on S), and \(W \equiv \left.G_{\mu \nu }{k}^{\mu }{k}^{\nu }\right \vert _{S} {+\sigma }^{2}\) with σ 2 the shear of \(\vec{k}\) at S. Note that \(\vec{n}\) is selected by fixing its norm \(\vec{n} = -\vec{l} + \frac{n_{\mu }{n}^{\mu }} {2} \vec{k}\) and that its causal character is unrestricted. Due to energy conditions [4, 5] W ≥ 0 and W = 0 requires \(\left.G_{\mu \nu }{k}^{\mu }{k}^{\nu }\right \vert _{S} {=\sigma }^{2} = 0\) leading to Isolated Horizons [2]. I shall assume W > 0 throughout, W ≥ 0 being more involved.

The righthand side in (1) defines a linear differential operator \(L_{\vec{n}}\) acting on f: \(\delta _{f\vec{n}}\theta _{\vec{k}} \equiv L_{\vec{n}}f\). \(L_{\vec{n}}\) is an elliptic operator on S, called the stability operator for S in the normal direction \(\vec{n}\). \(L_{\vec{n}}\) is not self-adjoint in general (with respect to the L 2-product on S). Nevertheless, it has a real principal eigenvalue \(\lambda _{\vec{n}}\), and the corresponding (real) eigenfunction \(\phi _{\vec{n}}\) can be chosen to be positive on S. The (strict) stability of the MOTS S along a spacelike \(\vec{n}\) is ruled by the (positivity) non-negativity of \(\lambda _{\vec{n}}\).

The formal adjoint operator with respect to the L 2-product on S is given by

$$\displaystyle{L_{\vec{n}}^{\dag }\equiv -\varDelta _{ S} - 2{s}^{B}\overline{\nabla }_{ B} + \left (K_{S} - {s}^{B}s_{ B} -\overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )}$$

and has the same principal eigenvalue \(\lambda _{\vec{n}}\) as \(L_{\vec{n}}\) [1]. I denote by \(\phi _{\vec{n}}^{\dag }\) the corresponding principal (real and positive) eigenfunctions.

2 Possible MOTTs Through a Single MOTS

For each normal vector field \(\vec{n}\), the operator \(L_{\vec{n}} -\lambda _{\vec{n}}\) has a vanishing principal eigenvalue and \(\phi _{\vec{n}}\) as principal eigenfunction: \(L_{\vec{n}} -\lambda _{\vec{n}}\) corresponds to the stability operator \(L_{\vec{n}^{\prime}}\) along another normal direction \(\vec{n}^{\prime}\) given by \({n^{\prime}}^{\mu }n^{\prime}_{\mu } = {n}^{\mu }n_{\mu } + (2/W)\lambda _{\vec{n}}\), so that \(\delta _{\phi _{\vec{n}}\vec{n}^{\prime}}\theta _{\vec{k}} = 0\). If \(\vec{n}\) is spacelike and S is strictly stable along \(\vec{n}\) (\(\lambda _{\vec{n}} > 0\)), then \(\vec{n}^{\prime}\) points “above” \(\vec{n}\) (as \({n^{\prime}}^{\mu }n^{\prime}_{\mu } > {n}^{\mu }n_{\mu }\)). The directions tangent to MOTTs through S belong to (but may not exhaust!) the set \(\{\phi _{\vec{n}}\vec{n}^{\prime}\}\). These MOTTs are generically different: given two normal vector fields \(\vec{n}_{1}\) and \(\vec{n}_{2}\) the corresponding “primed” directions are equal if and only if \(\vec{n}_{1} -\vec{ n}_{2} = \frac{\mbox{ const.}} {W}\vec{k}\). On the other hand, for any such two \(\vec{n}_{1}\) and \(\vec{n}_{2}\)

$$\displaystyle{ (W/2)f\left (n_{1}^{\rho }n_{ 1\rho } - n_{2}^{\rho }n_{ 2\rho }\right ) = \left (L_{\vec{n}_{2}} - L_{\vec{n}_{1}}\right )f }$$
(2)

providing the relation between two deformation directions pointwise.

For any given \(\vec{n}\) one easily gets

$$\displaystyle\begin{array}{rcl} & & \oint _{S}L_{\vec{n}}f =\oint _{S}\left (K_{S} - {s}^{B}s_{ B} -\overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )f {}\\ & & \oint _{S}L_{\vec{n}}^{\dag }f =\oint _{ S}\left (K_{S} - {s}^{B}s_{ B} + \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )f {}\\ \end{array}$$

in particular for the principal eigenfunctions

$$\displaystyle\begin{array}{rcl} & & \lambda _{\vec{n}}\oint _{S}\phi _{\vec{n}} =\oint _{S}\left (K_{S} - {s}^{B}s_{ B} -\overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )\phi _{\vec{n}} {}\\ & & \lambda _{\vec{n}}\oint _{S}\phi _{\vec{n}}^{\dag } =\oint _{ S}\left (K_{S} - {s}^{B}s_{ B} + \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )\phi _{\vec{n}}^{\dag } {}\\ \end{array}$$

which are two explicit formulas for the principal eigenvalue bounding it

$$\displaystyle\begin{array}{rcl} & & \min _{S}\left (K_{S} - {s}^{B}s_{ B} \pm \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right ) \leq \lambda _{\vec{n}} \\ & & \;\;\leq \max _{S}\left (K_{S} - {s}^{B}s_{ B} \pm \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )\,.{}\end{array}$$
(3)

Furthermore, the two functions \(\lambda _{\vec{n}} -\left (K_{S} - {s}^{B}s_{B} \pm \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )\) must vanish somewhere on S for all \(\vec{n}\).

There are two obvious simple choices \(\vec{n}_{\pm }\) leading to a vanishing principal eigenvalue: \(n_{\pm }^{\mu }n_{\pm \mu } = \frac{2} {W}\left (K_{S} - {s}^{B}s_{ B} \pm \overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S}\right )\). The corresponding stability operators are \(L_{\pm } = -\varDelta _{S} + 2{s}^{B}\overline{\nabla }_{B} + (1 \mp 1)\overline{\nabla }_{B}{s}^{B}\). The corresponding principal eigenfunctions ϕ ± > 0 satisfy \(L_{\pm }\phi _{\pm } = 0\). The respective formal adjoints read: \(L_{\pm }^{\dag } = -\varDelta _{S} - 2{s}^{B}\overline{\nabla }_{B} - (1 \pm 1)\overline{\nabla }_{B}{s}^{B}\) with vanishing principal eigenvalues too. Observe that L − and \(L_{+}^{\dag }\) are gradients \(L_{-}f = -\overline{\nabla }_{B}\left ({\overline{\nabla }}^{B}f - 2f{s}^{B}\right )\), \(L_{+}^{\dag }f = -\overline{\nabla }_{B}\left ({\overline{\nabla }}^{B}f + 2f{s}^{B}\right )\).

3 A Distinguished MOTT

L − has special relevant properties, because (2) leads to

$$\displaystyle{ (W/2)f\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{ -\rho }\right ) = L_{-}f -\delta _{f\vec{n}}\theta _{\vec{k}} }$$
(4)

For any other direction \(\vec{n}^{\prime}\) defining a local MOTT

$$\displaystyle{(W/2)\left ({n^{\prime}}^{\rho }n^{\prime}_{\rho } - n_{-}^{\rho }n_{ -\rho }\right ) =\lambda _{\vec{n}} -\left (K_{S} - {s}^{B}s_{ B} -\overline{\nabla }_{B}{s}^{B} -\left.G_{\mu \nu }{k}^{\mu }{l}^{\nu }\right \vert _{ S} -\frac{{n}^{\rho }n_{\rho }} {2} \,W\right )}$$

and, as remarked above, the righthand side must change sign on S.

Theorem 3.1.

The local MOTT defined by the direction \(\vec{n}_{-}\) is such that any other nearby local MOTT must interweave it: the vector \(\vec{n}^{\prime} -\vec{ n}_{-}(\propto \vec{ k})\) changes its causal orientation on any of its MOTSs.

From (4), deformations using c ϕ − with constant c lead to outer untrapped (resp. trapped) surfaces if \(c\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{-\rho }\right ) < 0\) (resp. > 0) everywhere. Integrating (4) on S one thus gets

$$\displaystyle{\frac{1} {2}\oint _{S}\mathit{Wf }\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{ -\rho }\right ) = -\oint _{S}\delta _{f\vec{n}}\theta _{\vec{k}}}$$

hence the deformed surface can be outer trapped (untrapped) only if \(f\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{-\rho }\right )\) is positive (negative) somewhere. If the deformed surface has \(f\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{-\rho }\right ) < 0\) (respectively > 0) everywhere then \(\delta _{f\vec{n}}\theta _{\vec{k}}\) must be positive (resp. negative) somewhere.

Choose the function \(f = a_{0}\phi _{-} +\tilde{ f}\) for a constant a 0 > 0 so that, as ϕ − > 0 has vanishing eigenvalue, (4) becomes \((W/2)(a_{0}\phi _{-} +\tilde{ f})\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{-\rho }\right ) = L_{-}\tilde{f} -\delta _{f\vec{n}}\theta _{\vec{k}}\). This can be split into two parts:

$$\displaystyle{ (W/2)a_{0}\phi _{-}\left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{ -\rho }\right ) = -\delta _{f\vec{n}}\theta _{\vec{k}}, \frac{W} {2} \left ({n}^{\rho }n_{\rho } - n_{-}^{\rho }n_{ -\rho }\right ) = \frac{L_{-}\tilde{f}} {\tilde{f}} }$$
(5)

The first of these tells us that \(\delta _{f\vec{n}}\theta _{\vec{k}} < 0\) whenever \(\vec{n}\) points “above” \(\vec{n}_{-}\). But then the second in (5) requires finding a function \(\tilde{f}\) such that \(L_{-}\tilde{f}/\tilde{f}\) is strictly positive on S. This leads to the following interesting mathematical problem:

Is there a function \(\tilde{f}\) on S such that (i) \(L_{-}\tilde{f}/\tilde{f} \geq \epsilon > 0\), (ii) \(\tilde{f}\) changes sign on S, and (iii) \(\tilde{f}\) is positive in a region as small as desired?

To prove that there are OTSs penetrating both sides of the MOTT it is enough to comply with points (i) and (ii). If the operator L − has any real eigenvalue other than the vanishing principal one, then these two conditions do hold for the corresponding real eigenfunction because integration of \(L_{-}\psi =\lambda \psi\) implies ∮ S ψ = 0 (as λ > 0) ergo ψ changes sign on S. However, even if there are no other real eigenvalues the result might hold. Point (iii) would ensure, then, that the deformed OTS intersects the trapped region “above” the MOTT only in a portion that can be shrunk as much as desired. This is important for the concept of core and its boundary, see [3].

As illustration of the above, consider a marginally trapped round sphere \(\varsigma\) in a spherically symmetric space-time, that is, any sphere with r = 2m where 4π r 2 is its area and \(m = (r/2)(1 - r_{,\mu }{r}^{,\mu })\) is the “mass function”. For any such \(\varsigma\), s B = 0 and σ 2 = 0, ergo the directions \(\vec{n}_{\pm }\) and operators L ± coincide: \(\vec{n}_{+} =\vec{ n}_{-}\equiv \vec{ m}\), \(L_{+} = L_{-} = L_{\vec{m}} = -\varDelta _{\varsigma }\). As it happens, \(\vec{m}\) is tangent to the unique spherically symmetric MOTT: r = 2m [3]. Therefore, points (i) and (ii) are easily satisfied by choosing \(\tilde{f}\) to be an eigenfunction of the spherical Laplacian \(\varDelta _{\varsigma }\), say \(\tilde{f} = cP_{l}\) for a constant c and l > 0, where P l are the Legendre polynomials. Actually, one can find an explicit function satisfying point (iii) too, proving that the region r ≤ 2m is a core in spherical symmetry, [3]. This is a surprising, maybe deep result, because the concept of core is global and requires full knowledge of the future, however its boundary r = 2m is a MOTT, hence defined locally. Whether or not this happens in general is an open important question.