Identification of black hole horizons using scalar curvature invariants
We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides with the geometric horizon. Finally, we consider the imploding spherically symmetric metrics and show that the geometric horizon identifies a unique quasi-local surface corresponding to the unique spherically symmetric marginally trapped tube, implying that the spherically symmetric dynamical black holes admit a geometric horizon. Based on these results, we propose a suite of conjectures concerning the application of geometric horizons to more general dynamical black hole scenarios.
grGRGeneral Relativity \newacronym[plural=THs,firstplural=trapping horizons (THs)]thsTHtrapping horizon \newacronym[plural=MTTs,firstplural=marginally trapped tubes (MTTs)]mttsMTTmarginally trapped tube \newacronymnehNEHnon-expanding horizon \newacronymwihWIHweakly isolated horizon \newacronym[plural=MTSs,firstplural=marginally trapped surfaces (MTSs)]mtssMTSmarginally trapped surface \newacronymfothFOTHfuture outer trapping horizon \newacronymdhDHdynamical horizon \newacronymbwb.w.boost weight \newacronym[plural=SPIs,firstplural=scalar polynomial (curvature) invariants (SPIs)]spiSPIscalar polynomial (curvature) invariant \newacronymnpNPNewman-Penrose \newacronymfkwcFWKCFulling, King, Wybourne and Cummings \newacronymltbLTBLemaitre-Tolman-Bondi \newacronymnutNUTNewman, Unti, Tamburino
In General Relativity (GR), black holes are exact solutions which may be interpreted as physical objects formed from the gravitational collapse of fuel-exhausted stars. As such, they present an excellent arena to explore the connection between gravitation, thermodynamics and quantum theory. A defining feature of a black hole is its event horizon, which is the boundary of the region from where signals can be sent to a distant asymptotic external region. The event horizon is typically identified as the surface of the black hole and relates its area to the entropy of the black hole. However, the event horizon is essentially a teleological object, as we must know the global behaviour of the spacetime in order to determine the event horizon locally. That is, the event horizon depends on the whole future evolution of the spacetime [].
To study the behaviour of black holes, for example in numerical GR [] in the 3+1 approach or in the Cauchy-problem in GR, it is crucial to locate a black hole locally. Of course, such a characterization may not rely on the existence of an event horizon alone, as realistic black holes undergo evolutionary processes and are usually dynamical. To address this, Penrose [] introduced the important concept of closed trapped surfaces, which are compact spacelike surfaces (usually topological spheres) such that the expansion of the future-pointing null normal vectors are negative. Considering time-dependent situations, the event horizons (which are null surfaces) familiar from the study of stationary black holes are replaced in practice by apparent horizons defined as the locus of the vanishing expansion of a null geodesic congruence emanating from a trapped surface with spherical topology [].
Unlike the event horizon, the apparent horizon is a quasi-local concept, and it is intrinsically foliation-dependent; this is because it is a 2-surface that is dependent on the choice of asymptotically flat 3-surfaces which foliate spacetime, and consequently will depend on the observer in dynamical situations. In numerical studies of collapse, it is more practical to track apparent horizons because, as already noted, the event horizon requires the knowledge of the entire future history of the spacetime. For example, apparent horizons are employed in simulations of high precision waveforms of gravitational waves arising from the merger of compact-object binary systems or in stellar collapse to form black holes in numerical relativity. The successful observations by the LIGO collaboration of gravitational waves from black hole mergers relied upon numerical simulations based on apparent horizons [].
In practice the definition of an apparent horizon is difficult to use, and other quasi-local surfaces are often employed instead. In particular, two quasi-local surfaces, \glsplmtts and \glsplths, which bound the event horizon of a dynamical black hole, play an important role []. These surfaces are extensions of the concept of a future-trapped surface, , which is a closed two-surface with the property that the expansions in each of the two future-pointing null vectors normal to the surface, and , are everywhere negative:
where is the induced two metric on . We will always normalize the null vectors such that to ensure they are outward/inward pointing null vector fields. As an example of such a surface, for equilibrium states of dynamical black holes an alternative to the restrictive concept of a stationary horizon is given by the quasi-local weakly isolated horizons, which account for equilibrium states of black holes and cover all essential local features of event horizons which are unaffected by the dynamic evolution of the surrounding spacetime [, ].
A sub-manifold of a spacetime is said to be a \glsneh if
is topologically and null.
Any null normal of has vanishing expansion and
The Einstein field equations hold at and the stress-energy tensor is such that is future-causal for any future directed null normal .
The pair is said to constitute a \glswih provided is an NEH and any null normal proportional to satisfies
where is the induced torsion-free derivative operator on . For any NEH with the condition that on implies that will be unique [].
A WIH is essentially a three-dimensional (3D) null surface with topology with an outgoing expansion rate which vanishes on the horizon (with some additional conditions) []. The null normal vector is a local time-translational Killing vector field for the intrinsic geometry of the horizon, leading to the invariance under time evolution with respect to of the induced metric and the induced derivative operator, which is directly expressible by the isolated horizon condition. All these conditions are local to the horizon, and require neither asymptotic structures nor foliations of spacetime. Every Killing horizon which is topologically is an isolated horizon. However, in general, spacetimes with isolated horizons need not admit any Killing vector fields for the spacetime, even in a neighborhood.
A trapped region is defined as a subset of spacetime where each point of this region passes a trapped surface. The trapping boundary is a connected component of the boundary of an inextendible trapped region. Unlike the MTTs, the trapping boundary is not foliated by \glsplmtss which are compact spacelike two-dimensional (2D) submanifolds on which the expansion of one of the null normals vanishes, and the other is non-positive. While is non-local, the concept of a trapping surface leads to the following quasi-local analogue of a future event horizon [, ] (also known as an apparent horizon [] in applications):
A \glsfoth is a smooth 3D submanifold of spacetime, foliated by closed 2D submanifolds , such that
the expansion of one future direction null normal to the foliation, say , vanishes, ;
the expansion of the other future directed null normal, , is negative, ; and
the directional derivative of along is negative, .
The Raychaudhuri equation shows that is either spacelike or null if the shear of and the matter flux across vanish. In this case is a NEH (i.e., a WIH). The FOTH is spacelike in the dynamical region where gravitational radiation and matter fields are pouring into it, and is null when it has reached equilibrium. By relaxing the condition that is negative we have the definition of a dynamical horizon:
A smooth, 3D spacelike submanifold (possibly with boundary), of spacetime is said to be a \glsdh if it can be foliated by closed 2D submanifolds , such that
the expansion of one future direction null normal to the foliation, say , vanishes, ;
the expansion of the other future directed null normal, , is negative, .
Since MTTs depend on the choice of a reference foliation of spacelike hypersurfaces, they are non-unique. The non-uniqueness of trapped surfaces is inherited by everything based on them, such as MTTs and including dynamical horizons. To resolve this we could use the well defined event horizon and accept its teleological properties, treat all possible MTTs and dynamical/trapping horizons as equally valid, use some other non-local boundary, or try to define preferred marginally trapped tubes [].
A dynamical horizon is better suited to analyse dynamical processes involving black holes, such as black hole growth and coalescence. A dynamical horizon is a 3D spacelike hypersurface foliated by marginally trapped 2D compact surfaces, which can transition to an isolated null NEH when the flux of gravitational radiation or matter across it is zero. Fluxes of energy and angular momentum carried by gravitational waves across a dynamical horizon necessarily cause the area of such surfaces to increase with time, and the corresponding change in the horizon cross section area was analysed in [, ]. Due to back scattering, the transition to equilibrium takes an infinite time. However, considering a finite time transition is more instructive for it involves a smooth matching between dynamical and non-expanding horizons. As it was illustrated in [, ], angular momentum, energy, area, and surface gravity of the horizons cross sections match smoothly.
The Vaidya solution provides a simple and explicit example of a dynamical horizon [, , ]. In addition, for an appropriate mass function the Vaidya solution provides examples of the transition from the dynamical to isolated horizons. The Vaidya solution admits spherically symmetric marginally trapped surfaces. The existence of non-spherically symmetric dynamical horizons which asymptote to the NEH was discussed in [] where it was shown that if a hypersurface admits a dynamical horizon structure, it is unique. However, if a spacetime has several distinct black holes, it may admit several distinct non-unique dynamical horizons. If one considers dynamical horizons which are also FOTHs (spacelike future outer trapping horizons, SFOTHs), then one can show that if two non-intersecting SFOTHs become tangential in a finite time to the same NEH, then they either coincide or one is contained in the other. However, one cannot rule out the existence of more than one SFOTH which asymptotes to the NEH if they intersect each other repeatedly.
In this paper we will explore the relationship between these surfaces for black holes admitting stationary horizons and NEHs, and for the spherically symmetric dynamical black hole solutions. We will introduce a new surface to study, defined by the requirement that the Ricci and Weyl tensors are more algebraically special on this surface as compared to the rest of the spacetime. This condition will be defined in terms of the vanishing of scalar curvature invariant, which implies that these surfaces are foliation independent. In section 2 we review the discriminant scalar polynomial invariants, and show how they can be used to determine when a spacetime becomes algebraically special. In section 3 we discuss the event horizon for stationary black holes which is a Killing horizon and are detectable by scalar curvature invariants; we posit that these invariants are related to the discriminant invariants. In section 4 we show that other horizons beyond Killing horizons are detectable by invariants, namely the NEHs, WIHs and the dynamical horizon of a spherically symmetric metric. We also discuss how the dynamical horizons of less idealized black hole solutions could be detected using invariants. Motivated by these results we introduce the geometric horizon detection conjectures in section 5. In section 6 we summarize the results and discuss their applications.
There are five appendices. In Appendix A we provide the Kerr-Newman-\glsdispnutNUT-(Anti)-de Sitter metric as an example to show that a frame exists for which the curvature tensor and its covariant derivatives becomes algebraically special on the event horizons of this metric. In Appendix B we compare the Page-Shoom invariants and the discriminant scalar polynomial curvature invariants for the Kerr-Newman-NUT-(Anti)-de Sitter metric and show that the invariants share a common factor. In Appendix C we review the geometric identities for the contractions of the Riemann tensor and its covariant derivatives in order to determine a minimal basis for the set of polynomials formed from them and possibly simplify the discriminant invariants. In Appendix D we present the necessary type II/D conditions for the Weyl tensor using discriminant invariants. In Appendix E we summarize the abbreviations frequently used in this paper.
2 Discriminating Scalar Polynomial Curvature Invariants
The introduction of alignment theory [, , ] allows for the algebraic classification of any tensor in a Lorentzian spacetime of arbitrary dimensions using \glsbw. The dimension-independent theory of alignment can be applied to the tensor classification problem for the Weyl tensor in higher dimensions [, , ], and to the classification of second-order symmetric tensors, such as the Ricci tensor, and tensors involving covariant derivatives. The Ricci tensor can also be classified according to its eigenvalue structure. In a related way, the classification of the Weyl tensor can be obtained by introducing bivectors, where the Weyl bivector operator is defined in a manner consistent with its b.w. decomposition [].
The classifications of the Weyl tensor are distinct in higher dimensions; however, in four dimensions (4D) they yield the Petrov classificaton []. Using the b.w. decomposition and curvature operators together, the algebraic classification of the Weyl tensor and the Ricci tensor (and their covariant derivatives) in higher dimensions can be refined by exploiting their eigenbivector and eigenvalue structure. A tensor of a particular special algebraic type will have an associated operator with a restricted eigenvector structure, and this can then be used to determine necessary conditions for the algebraic type.
If a tensor is of alignment type II, there exists a frame where all components with positive b.w. vanish. If a tensor is of alignment type D then there exists a frame where all components with non-zero b.w. vanish. Using discriminants, we can completely determine the eigenvalues of the curvature operator (up to degeneracies) yielding, for example, necessary conditions in terms of simple \glsplspi for the Weyl and Ricci curvature operators to be of algebraic type II or D in arbitrary dimensions [, ]. Necessary conditions for the covariant derivatives of the Ricci and Weyl tensor to be algebraically special can be found by forming 2 or 4 index tensors from them.
2.1 Discriminant Analysis
A SPI of order is a scalar obtained by the contraction of copies of the Riemann tensor and its covariant derivatives up to the order . In arbitrary dimensions, requiring that all of the zeroth order polynomial Weyl invariants vanish implies that the Weyl type is III, N, or O (similarly for the Ricci type). SPIs have been used in the study of and spacetimes, where all of the SPIs vanish or are constant, respectively [, , , , ]. In [] it was proven that a 4D Lorentzian spacetime metric is either -non-degenerate, and hence completely locally characterized by its SPIs, or it is either locally homogeneous or degenerate Kundt [].
For any tensor of b.w. type II (or D) the eigenvalues of the corresponding operator need to be of a special form; the resulting invariants for a tensor of type II are the same as that for a type D tensor. For the ensuing discussion we will assume the tensor is of type II. If the Ricci tensor is to be of type II, it is of Segre type , or simpler. Therefore, the Ricci operator has at least one eigenvalue of (at least) multiplicity 2. Furthermore, all the eigenvalues are real. In dimensions, we may consider the Weyl bivector operator as the map
and examine its eigenvalues. If the Weyl tensor is of type D, then the operator has at least eigenvalues of (at least) multiplicity 2 []. In this manner, the algebraic types are connected to the eigenvalue structure and allow for the construction of the necessary discriminants.
In dimensions, the Ricci and Weyl type II/D necessary conditions are (:
which are discriminants defined in terms of SPIs[, ]. Note that the Ricci syzygy, as a polynomial in terms of the Ricci tensor components, is of order , while the highest Weyl syzygy is of order . We are interested in 4D, and so the Ricci and Weyl syzygies are of order 12 and 30, respectively. These conditions are necessary conditions, but are not sufficient, since the characteristic equation for different algebraic types may be identical and consequently the SPIs are also identical. For example, a five dimensional (5D) spacetime which has a Weyl tensor with isotropy fulfills the type II or D necessary conditions [, ].
Ricci Type Ii/D in 4D
To determine the type II/D conditons for the Ricci tensor, we consider a general 2-index tensor, which is assumed to be symmetric and trace-free () in 4D. The discriminant is given by:
where is the trace-free symmetric Ricci tensor , and is the trace of the power of this tensor.
This 12th order SPI can be written in a shorter form using:
Weyl Type Ii/D in 4D
These two conditions are equivalent to the real and imaginary parts of the complex syzygy in terms of the complex Weyl tensor in the \glsnp formalism []. Computationally it might be useful to eliminate from (7) and (8) in order to obtain a single necessary condition. We can apply this result to any Weyl candidate (a 4-index tensor with the same symmetries as Weyl tensor)
Alternatively, we can use to construct the the trace-free operator:
with the invariants:
The discriminant analysis also gives the coefficients of the characteristic equation as:
Therefore, the necessary condition for the operator to be type II/D is similar to equation (6):
The discriminant analysis provides syzygies expressed in terms of the SPIs by treating the Weyl tensor as a trace-free operator acting on the six-dimensional vector space of bivectors. The type II/D condition is ; however, these conditions are very large (see Appendix D).
The Riemann tensor and other tensors in 4D
If a spacetime is of Riemann type II, the Weyl type II and Ricci type II necessary conditions hold, and there are additional alignment conditions (e.g., are of type II). Applying the necessary conditions to the full Riemann tensor (to be of type II/D), implies that both the Weyl and Ricci tensor are of type II/D and aligned. We note that we will also obtain syzygies for mixed tensors of the form:
to be of type II/D. The type II/D condition implies that we have the syzygy for all of the trace-free tensors arising from , , and .
To illustrate the applicability of the discriminant SPIs we present four examples.
An arbitrary 5D Spacetime: For the trace-free Ricci tensor, we note that type D has to be of Segre type or simpler, implying that 2 eigenvalues are equal, while the remaining eigenvalue has to be real. The vanishing of is a necessary condition for the trace-free tensor to be of type II (or D) in 5D. Thus, the 20th order discriminant is the related SPI.
For the ten-dimensional Weyl tensor, the type II bivector operator has 3 eigenvalues of minimum multiplicity 2, and the necessary condition for the Weyl tensor to be of type II (or D) in 5D is the vanishing of the SPIs These are discriminants of order 90, 72 and 56, respectively. Additional necessary conditions can also be found using combinations of the Weyl tensor; for example, the operator . This gives again (), which is a 20th order syzygy (in the square of the Weyl tensor).
5D Schwarzschild spacetime: For the Weyl operator we get
This implies that the Weyl operator has 3 distinct real eigenvalues and this spacetime is of type D [].
5D space with complex hyperbolic sections. Let us consider []:
For the Weyl operator we get
This implies that the Weyl operator has 3 distinct real eigenvalues. However, the following coefficients of the characteristic equation vanish: signaling that there is a zero-eigenvalue of multiplicity 7. Thus, this spacetime is not of type II/D. In fact, it is -non-degenerate which can be shown by computing the operator which is of “Segre” type .
The 5D rotating black ring. The 5D rotating black ring [, ] is generally of type , but in certain regions and for particular values of the parameters and it can also be of type or (the case corresponds to the type Myers-Perry metric) [, ]. The trace-free and symmetric part of the operator gives a discriminant which leads to a necessary condition on the algebraic type of the Weyl tensor in the region of Lorentzian signature for the fixed ‘target’ point locally defined by , [, ]:
where is a polynomial which is generally not zero. For this particular choice of target point, the horizon is located there if , and we see that (with ), which signals that the spacetime is of Weyl type II on the horizon. Computationally, it is simpler to work with and the 40th order SPI than the related SPI for the Weyl tensor as an operator.
2.3 Differential invariants
To determine whether the covariant derivatives of the Ricci tensor , are also of type II or D, we could study the eigenvalue structure of the operators constructed from the tensor and apply the type II/D necessary conditions. For example, considering the trace-free parts of the tensors , we obtain necessary conditions of the form of equation (6) but with the . This can be repeated for the Weyl tensor and in higher dimensions [].
For example, we can construct the second order symmetric and trace-free operator, , for the covariant derivative of the Weyl tensor, defined by:
The resulting differential invariants may be simplified using the FKWC bases for the Riemann SPIs to eliminate Riemannian SPIs that can be expressed in terms of the bases. Additionally one can use geometric identities and conserved tensor quantities to induce further simplification (see Appendix C).
Example. We consider the operator defined above for the 4D type D Kerr metric. The type D/II necessary condition is then the vanishing of:
where , and and are polynomials. With the exception of the horizon, the ergosphere, and some other special points, this invariant will be non-zero and so cannot be of type D/II (generically) outside the horizon. This is confirmed explicitly using the Cartan algorithm to construct the appropriate frame in Appendix A.
Necessary 4D conditions for the covariant derivative of the Weyl tensor to be of type Ii/D
To determine the algebraic type of the covariant derivative of the Weyl tensor, , we consider a second order symmetric and trace-free operator to obtain the necessary type II/D condition (6) of the form .
Let us consider two possible combinations, and from both we can derive necessary conditions. The first is the trace-free symmetric tensor defined above:
and we define
Computing the coefficients of the characteristic equation in terms of these:
The necessary condition for this operator to be of type II/D is equivalent in form to the condition given in equation (6):
Expanding this expression, we obtain explicitly:
Alternatively, we can construct the trace-free symmetric tensor where
Computing the coefficients of the characteristic equation in terms of these:
Necessary 4D conditions for the second covariant derivative of the Weyl tensor to be of type Ii/D
The second covariant derivative of the Weyl tensor, , can be studied using a second order symmetric and trace-free operator to obtain the necessary type II/D conditions of the form . The simplest operator to consider is the following:
and defining as the trace of the power of this tensor, this expression can be written in a shorter form using: