Marginally Trapped Surfaces

• Marginally trapped surfaces are spacelike codimension-two surfaces where one null expansion vanishes, defining quasi-local horizons in general relativity.
• Analytical methods, including the MOTS stability operator and Jang’s equation, rigorously establish existence, regularity, and quasi-local area inequalities.
• Applications span numerical relativity for apparent horizon detection, exact constructions in symmetric spacetimes, and studies of topological transitions inside black holes.

Marginally trapped surfaces are spacelike codimension-two surfaces for which one null expansion vanishes; in four-dimensional Lorentzian geometry this is equivalent to the mean curvature vector being lightlike, so the subject sits at the interface of quasi-local black-hole theory, elliptic geometric analysis, and Lorentzian surface theory. In GR, the most common objects are marginally outer trapped surfaces (MOTSs), which model apparent horizons on a given slice, while in Lorentz-Minkowski geometry the term often refers to spacelike surfaces with nonzero lightlike mean curvature vector. Across these settings, marginally trapped surfaces organize trapped regions, dynamical and isolated horizons, topology change inside black holes, and a wide range of exact and numerical constructions (Andersson et al., 2010, Dekimpe et al., 2020).

1. Definitions and geometric framework

Let $S$ be a closed spacelike $2$-surface in a four-dimensional spacetime $(M,g_{ab},\nabla_a)$, with induced metric $q_{ab}$. With future-directed null normals $\ell_+$ and $\ell_-$ scaled so that $\ell_+\cdot \ell_-=-2$, the null expansions are

$$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$

A trapped surface has $\theta_{(\ell)}<0$ and $\theta_{(n)}<0$; a marginally trapped surface has one vanishing expansion; and a MOTS is specified by $2$0 together with $2$1 relative to the chosen outside (Sievers et al., 2023). In the alternative but equivalent $2$2 convention for an initial data set $2$3, with outward unit normal $2$4 to $2$5, one writes

$2$6

so a MOTS satisfies $2$7 (Andersson et al., 2010).

In a $2$8 decomposition with spacelike slice $2$9, unit future normal $(M,g_{ab},\nabla_a)$0, outward unit normal $(M,g_{ab},\nabla_a)$1 to $(M,g_{ab},\nabla_a)$2, and $(M,g_{ab},\nabla_a)$3, $(M,g_{ab},\nabla_a)$4, the outgoing expansion becomes

$(M,g_{ab},\nabla_a)$5

with $(M,g_{ab},\nabla_a)$6 the mean curvature of $(M,g_{ab},\nabla_a)$7 in $(M,g_{ab},\nabla_a)$8 (Sievers et al., 2023). In time-symmetric data $(M,g_{ab},\nabla_a)$9, MOTSs coincide with minimal surfaces. This reduction underlies much of the existence theory as well as many exact constructions.

For spacelike codimension-two surfaces in Lorentzian manifolds, the null-expansion and mean-curvature descriptions are directly linked. In the lightlike-frame formalism,

$q_{ab}$0

so the condition that $q_{ab}$1 be isotropic is equivalent to one null expansion vanishing (Honda et al., 2014, Dekimpe et al., 2020). This is why the geometric literature on marginally trapped surfaces in Minkowski, de Sitter, anti-de Sitter, and Robertson-Walker spacetimes meshes naturally with the GR literature on MOTSs and apparent horizons (Dekimpe et al., 2020).

Apparent horizons are the outermost MOTSs on a given slice, while marginally outer trapped tubes (MOTTs) are hypersurfaces foliated by MOTSs as the slicing evolves. Event horizons, by contrast, are global null boundaries. The distinction is fundamental: event horizons are causal and teleological, whereas apparent horizons are foliation-dependent and quasi-local (Sievers et al., 2023, Andersson et al., 2010).

2. Existence, regularity, and stability

The modern analytic theory of MOTSs is organized around Jang’s equation and its regularization. For a graph $q_{ab}$2 in the product metric $q_{ab}$3, Jang’s equation is

$q_{ab}$4

Schoen and Yau’s central observation is that solutions of this equation blow up along marginally trapped surfaces, so the blow-up set becomes a detection mechanism for apparent horizons. The capillarity-regularized equation

$q_{ab}$5

admits existence and uniform $q_{ab}$6 bounds under appropriate barrier assumptions, and the limit $q_{ab}$7 produces genuine solutions away from horizons together with cylindrical components over MOTSs or MITSs (Andersson et al., 2010).

This framework yields existence theorems under one-sided trapping assumptions on the boundary. In dimensions $q_{ab}$8, if a bounded connected domain $q_{ab}$9 has boundary decomposition $\ell_+$0 with

$\ell_+$1

then there exists a smooth closed embedded MOTS $\ell_+$2, homologous to $\ell_+$3, stable, and $\ell_+$4-almost minimizing. In dimensions $\ell_+$5, the surface is still almost minimizing but may have a singular set of codimension at least $\ell_+$6 (Andersson et al., 2010).

The corresponding stability operator is the Lorentzian analogue of the Jacobi operator for minimal surfaces. For a normal variation $\ell_+$7,

$\ell_+$8

with

$\ell_+$9

A MOTS is stable iff the principal eigenvalue of $\ell_-$0 is nonnegative, or equivalently iff there exists $\ell_-$1 with $\ell_-$2 (Andersson et al., 2010). Under the DEC, stable MOTSs satisfy the integrated inequality

$\ell_-$3

which is the basis for curvature estimates and compactness (Andersson et al., 2010).

Several consequences are structural. Outermost MOTSs are stable; instability would generate nearby positive-expansion surfaces outside, contradicting outermostness. MOTSs arising as $\ell_-$4-almost minimizing boundaries enjoy area bounds, curvature control, and lower bounds for the outward injectivity radius. The boundary of the trapped region is therefore smooth, embedded, stable, and quasi-locally canonical under the standing hypotheses (Andersson et al., 2010).

3. Topology, horizons, and geometric inequalities

Topology is sharply constrained for outermost or stationary horizon sections, but interior MOTSs are far less rigid. Hawking’s topology theorem constrains cross-sections of stationary black-hole event horizons in $\ell_-$5 dimensions to be spherical, and under the DEC each component of an outermost MOTS has positive Yamabe type. By contrast, for apparent horizons and inner MOTSs in dynamical spacetimes, transient non-spherical topologies, including tori, can arise, especially near moments of time symmetry or in slicings that cross wormhole throats (Sievers et al., 2023, Andersson et al., 2010).

The trapped region $\ell_-$6 is the union of all trapped sets, and its boundary is an outermost MOTS. A key geometric mechanism is that the union of two intersecting MOTSs is contained inside a single enclosing MOTS, which supports monotone approximations of $\ell_-$7 and rules out exterior self-contact by surgery and almost-minimizing arguments (Andersson et al., 2010).

For stably outermost MOTSs in non-vacuum dynamical spacetimes with $\ell_-$8 and the DEC, one has sharp quasi-local area inequalities. In the axisymmetric case,

$\ell_-$9

with equality implying the intrinsic geometry of an extreme Kerr throat sphere. In Einstein-Maxwell theory,

$\ell_+\cdot \ell_-=-2$0

and the combined inequality is

$\ell_+\cdot \ell_-=-2$1

Equality implies the geometry of an extreme Kerr-Newman throat sphere (Jaramillo, 2012).

These inequalities are genuinely Lorentzian. Their proofs use the spacetime stably outermost condition, the Gauss-Bonnet theorem, and Einstein’s equations, rather than a purely Riemannian reduction. They make precise the idea that rotation and charge impose a lower bound on horizon area, and they turn stable MOTSs into quasi-local censorship and rigidity statements (Jaramillo, 2012).

A recurring theme is that topology change and stability loss occur in the interior hierarchy of MOTSs while the outermost apparent horizon remains spherical. This is explicit in exact Kruskal constructions and in numerical merger data, where MOTTs can change topology from sphere to torus and admit self-intersecting interior members even though the outermost horizon does not (Sievers et al., 2023).

4. Exact constructions in symmetric spacetimes

In Lorentz-Minkowski geometry, marginally trapped surfaces admit an invariant local theory parallel to classical surface theory. In $\ell_+\cdot \ell_-=-2$2, Ganchev and Milousheva constructed a geometrically determined moving frame with seven invariant functions $\ell_+\cdot \ell_-=-2$3, proved a Bonnet-type theorem asserting that these invariants determine the surface up to a motion, and developed complete classifications of marginally trapped meridian surfaces (Ganchev et al., 2011, Ganchev et al., 2012). A later refinement introduces canonical principal parameters for marginally trapped surfaces of general type, reducing the determining data from seven functions to three smooth functions $\ell_+\cdot \ell_-=-2$4 satisfying a hyperbolic PDE system and yielding a local existence and uniqueness theorem up to ambient isometries (Maksimović et al., 8 May 2026).

The Gauss-map side of the theory is equally rigid. In Minkowski $\ell_+\cdot \ell_-=-2$5-space, a marginally trapped surface has pointwise $\ell_+\cdot \ell_-=-2$6-type Gauss map iff it has parallel mean curvature vector field, and then $\ell_+\cdot \ell_-=-2$7. In $\ell_+\cdot \ell_-=-2$8 and $\ell_+\cdot \ell_-=-2$9, pointwise $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$0-type Gauss map is likewise equivalent to parallel mean curvature vector, while harmonic Gauss maps do not occur (Milousheva, 2014, Turgay, 2013). The broad differential-geometric overview by Dekimpe and Van der Veken organizes the Lorentzian space-form and Robertson-Walker results into local representation theorems and classifications under conditions such as positive relative nullity, parallel mean curvature vector field, finite-type Gauss map, invariance under a one-parameter isometry group, isotropy, and pseudo-umbilicity (Dekimpe et al., 2020).

In FLRW spacetimes,

$$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$1

the mean curvature vector of a surface $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$2 contained in a $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$3-slice obeys

$$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$4

Hence a surface in a slice is marginally trapped iff it is a CMC surface in the Riemannian fiber $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$5 with $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$6 (Flores et al., 2010). In closed FLRW spacetimes this immediately generates closed marginally trapped surfaces of arbitrary genus from the abundance of closed CMC surfaces in $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$7. The same paper gives a Hopf-fibration construction of marginally trapped tori and, more strikingly, examples crossing both expanding and collapsing regions of a closed FLRW spacetime (Flores et al., 2010). The Robertson-Walker overview extends this picture to local descriptions in $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$8, where marginality reduces to an explicit scalar equation involving the principal curvatures of a hypersurface in the constant-curvature fiber (Dekimpe et al., 2020).

In slightly spheroidal static spacetimes, the outgoing null expansion is proportional to $$\theta_{(\ell)}=q^{ab}\nabla_a \ell_b,\qquad \theta_{(n)}=q^{ab}\nabla_a \ell_-{}_b.$$9, with $\theta_{(\ell)}<0$0, so the MOTS occurs where $\theta_{(\ell)}<0$1. The main result is that the corresponding horizon radius is still given by the spherical Misner-Sharp relation

$\theta_{(\ell)}<0$2

evaluated in the undeformed spherical reference spacetime; the small axial deformation does not shift the radial coordinate $\theta_{(\ell)}<0$3 at which $\theta_{(\ell)}<0$4 (Rahim et al., 2018).

The maximally extended Schwarzschild spacetime in Kruskal-Szekeres coordinates provides a particularly explicit interior model. Constant-$\theta_{(\ell)}<0$5 slices with $\theta_{(\ell)}<0$6 are Einstein-Rosen bridges, and axisymmetric MOTSs can be constructed by the MOTSodesic method. In this setting, toroidal MOTSs exist for

$\theta_{(\ell)}<0$7

straddle the throat $\theta_{(\ell)}<0$8, and are tightly sandwiched between the apparent-horizon cross-sections $\theta_{(\ell)}<0$9 at early times. Their area decreases monotonically to zero as $\theta_{(n)}<0$0, while as $\theta_{(n)}<0$1 their area approaches twice the area of the bifurcation sphere, consistent with wrapping the bifurcation sphere twice (Sievers et al., 2023). The same paper also constructs non-enclosing spherical MOTSs, including self-intersecting examples and looping MOTSs that continue beyond $\theta_{(n)}<0$2, and derives a self-adjoint stability operator

$\theta_{(n)}<0$3

for the axisymmetric family. The toroidal MOTSs are unstable, with three negative $\theta_{(n)}<0$4 eigenvalues (Sievers et al., 2023).

A complementary perturbative result holds near the Schwarzschild horizon: for every incoming null hypersurface that is nearly spherically symmetric in a perturbed Schwarzschild spacetime, there exists a unique embedded marginally trapped surface. The proof rewrites the MOTS equation in double-null gauge as a quasilinear elliptic equation whose principal part is $\theta_{(n)}<0$5, and then establishes existence and uniqueness by controlling the perturbative lower-order terms in Sobolev spaces (Le, 2020).

5. Dynamical evolution, collapse, and numerical relativity

In numerical relativity, MOTSs are central because they locate and characterize black holes quasi-locally during evolution. A longstanding belief held that MOTSs could appear or disappear unpredictably. The spectral analysis of Pook-Kolb and collaborators replaces that picture with a sharp criterion: the relevant control parameter is the principal eigenvalue $\theta_{(n)}<0$6 of the MOTS stability operator. In time-symmetric vacuum data the operator reduces to

$\theta_{(n)}<0$7

and $\theta_{(n)}<0$8 implies strict stability and smooth continuation; $\theta_{(n)}<0$9 marks bifurcation; and disappearance of an unstable branch occurs when the next eigenvalue crosses zero and $2$00 ceases to be invertible (Pook-Kolb et al., 2018).

In Brill-Lindquist binary-black-hole initial data, this mechanism organizes the full MOTS hierarchy. As the separation decreases, a common MOTS appears and immediately bifurcates into an outer stable branch and an inner unstable branch. The inner common MOTS becomes highly distorted and non-star-shaped, and its disappearance is correlated precisely with the vanishing of the second relevant eigenvalue. The outer apparent horizon remains strictly stable and approaches the round Schwarzschild sphere in the close limit (Pook-Kolb et al., 2018). This is the sense in which apparent-horizon jumps are compatible with smooth behavior of the underlying MOTS world tubes.

The toroidal MOTSs seen in head-on merger simulations started from time-symmetric Brill-Lindquist data acquire a simpler interpretation once compared with the exact Kruskal construction. Since toroidal MOTSs arise in the maximally extended Schwarzschild spacetime merely by evolving away from the $2$01 moment of time symmetry, their appearance strongly suggests an origin in the initial data and the wormhole-crossing slicing rather than in merger-specific nonlinear dynamics (Sievers et al., 2023).

Spherical collapse models show how MOTSs and MOTTs encode matter dependence. In comoving spherically symmetric collapse with areal radius $2$02, the Misner-Sharp mass satisfies

$2$03

and the marginally trapped condition is

$2$04

For dust and viscous anisotropic fluids, the MTT signature is controlled by

$2$05

with $2$06 spacelike, $2$07 null, and $2$08 timelike. Homogeneous Oppenheimer-Snyder collapse produces an interior timelike MTT at formation, whereas smooth inhomogeneous LTB profiles often yield spacelike dynamical horizons that asymptote to null isolated horizons as accretion ceases (Chatterjee et al., 2020).

Higher-dimensional dust evolution exhibits genuinely new causal regimes. In $2$09 dimensions, marginally trapped and marginally anti-trapped surfaces satisfy

$2$10

and in homogeneous evolution the causal character of the foliation tubes is controlled by

$2$11

At $2$12, $2$13 gives timelike tubes, $2$14 gives null tubes, and $2$15 gives spacelike tubes (Raviteja et al., 2020). This clean $2$16 threshold has no four-dimensional analogue.

6. Asymptotic and holographic settings

Marginally trapped surfaces are not confined to asymptotically flat black-hole interiors. In asymptotically de Sitter spacetimes, the visibility problem is subtler than in the $2$17 case because $2$18 is spacelike. Chruściel, Galloway, and Ling exhibit visible marginally trapped surfaces in de Sitter space, including the equator and the Clifford torus in the $2$19 slice, both with $2$20. At the same time, they prove that if the spacetime is future causally simple, satisfies the NEC, and $2$21 does not contain all of $2$22, then there are no future weakly trapped surfaces in

$2$23

So positive cosmological constant allows visible marginally trapped surfaces, but only outside a specific non-visibility regime (1803.02339).

In AdS/CFT, boundary-anchored marginally trapped leaves interpolate between causal and extremal bulk surfaces. For a boundary region $2$24, any boundary-anchored marginally trapped leaf lies outside the causal wedge and, under the deformability assumptions stated in the paper, inside the entanglement wedge. Its renormalized area is bounded below by that of the HRT surface $2$25, and under the future-causal-holographic-information construction there is also an upper bound from a cut of the future causal horizon. The leading UV divergence of the leaf area matches that of the associated extremal surface, while subleading divergences generally differ (Grado-White et al., 2017). This supports the interpretation of

$2$26

as a coarse-grained entropy lying between HRT entanglement entropy and causal holographic information (Grado-White et al., 2017).

Taken together, these developments show that marginally trapped surfaces are not merely diagnostic devices for apparent horizons. They form a mathematically rigid elliptic class with a developed existence and stability theory, but they also populate exact Lorentzian geometries with nontrivial topology, organize numerical black-hole dynamics through spectral stability, and extend naturally to cosmological and holographic settings. A plausible implication is that future progress will continue to come from moving back and forth between these viewpoints: quasi-local horizon theory, invariant surface geometry, PDE methods, and explicit constructions in analytically controlled spacetimes.