A non-expanding horizon (NEH) is an enclosed
null surface whose intrinsic structure is preserved. An NEH is the geometric prototype of an
isolated horizon
It was customary to represent black hole horizons via stationary solutions of field equations, i.e., solutions which admit a time-translational Killing vector field everywhere, not just in a small neighborhood of the black hole. While this simple ...
which describes a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
in equilibrium with its exterior from the
quasilocal perspective. It is based on the concept and geometry of NEHs that the two quasilocal definitions of black holes,
weakly isolated horizons and isolated horizons, are developed.
Definition of NEHs
A three-dimensional
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
∆ is defined as a ''generic'' (rotating and distorted) NEH if it respects the following conditions:
[Abhay Ashtekar, Christopher Beetle, Olaf Dreyer, et al. "Generic isolated horizons and their applications". ''Physical Review Letters'', 2000, 85(17): 3564-3567]
arXiv:gr-qc/0006006v2
/ref>[Abhay Ashtekar, Christopher Beetle, Jerzy Lewandowski. "Geometry of generic isolated horizons". ''Classical and Quantum Gravity'', 2002, 19(6): 1195-1225]
arXiv:gr-qc/0111067v2
/ref>[Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. "Isolated horizons: Hamiltonian evolution and the first law". ''Physical Review D'', 2000, 62(10): 104025]
gr-qc/0005083
/ref>
(i) ∆ is null
Null may refer to:
Science, technology, and mathematics Computing
* Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value
* Null character, the zero-valued ASCII character, also designated by , often use ...
and topologically ;
(ii) Along any null normal field tangent to ∆, the outgoing expansion rate vanishes;
(iii) All field equations hold on ∆, and the stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
on ∆ is such that is a future-directed causal vector () for any future-directed null normal .
Condition (i) is fairly trivial and just states the general fact that from a 3+1 perspective an NEH ∆ is foliated by spacelike 2-spheres ∆'=S2, where S2 emphasizes that ∆' is topologically compact with genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
zero (). The signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of ∆ is (0,+,+) with a degenerate temporal coordinate, and the intrinsic geometry of a foliation leaf ∆'=S2 is nonevolutional. The property in condition (ii) plays a pivotal role in defining NEHs and the rich implications encoded therein will be extensively discussed below. Condition (iii) makes one feel free to apply the Newman–Penrose (NP) formalism[Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 2.][Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998. Appendix E.] of Einstein-Maxwell field equations to the horizon and its near-horizon vicinity; furthermore, the very energy inequality is motivated from the dominant energy condition and is a sufficient condition for deriving many boundary conditions of NEHs.
''Note'': In this article, following the convention set up in refs., "hat" over the equality symbol means equality on the black-hole horizons (NEHs), and "hat" over quantities and operators (, , etc.) denotes those on a foliation leaf of the horizon. Also, ∆ is the ''standard'' symbol for both an NEH and the directional derivative ∆ in NP formalism, and we believe this won't cause an ambiguity.
Boundary conditions implied by the definition
Now let's work out the implications of the definition of NEHs, and these results will be expressed in the language of NP formalism with the convention (Note: unlike the original convention , this is the usual one employed in studying trapped null surfaces and quasilocal definitions of black holes). Being a null normal to ∆, is automatically geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
, , and twist free, . For an NEH, the outgoing expansion rate along is vanishing, , and consequently . Moreover, according to the Raychaudhuri-NP ''expansion-twist'' equation,[Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Chicago: University of Chicago Press, 1983. Section 9(a), page 56.]
:
it follows that on ∆
:
where is the NP-shear coefficient. Due to the assumed energy condition (iii), we have (), and therefore is nonnegative on ∆. The product is of course nonnegative, too. Consequently, and must be simultaneously zero on ∆, i.e. and . As a summary,
:
Thus, the isolated horizon ∆ is nonevolutional and all foliation leaves ∆'=S2 look identical with one another. The relation implies that the causal vector in condition (iii) is proportional to and is proportional to on the horizon ∆; that is, and , . Applying this result to the related Ricci-NP scalars, we get , and , thus
:
The vanishing of Ricci-NP scalars signifies that, there is no energy–momentum flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
of ''any'' kind of charge ''across'' the horizon, such as electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
, Yang–Mills flux or dilaton
In particle physics, the hypothetical dilaton particle is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theor ...
flux. Also, there should be no gravitational wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s crossing the horizon; however, gravitational waves are propagation of perturbations of the spacetime continuum rather than flows of charges, and therefore depicted by four Weyl-NP scalars (excluding ) rather than Ricci-NP quantities . According to the Raychaudhuri-NP ''shear'' equation
:
or the NP field equation on the horizon
:
it follows that . Moreover, the NP equation
:
implies that . To sum up, we have
:
which means that, geometrically, a principal null direction of Weyl's tensor is repeated twice and is aligned with the principal direction; physically, no gravitational waves (transverse component and longitudinal component ) enter the black hole. This result is consistent with the physical scenario defining NEHs.
Remarks: Spin coefficients related to Raychaudhuri's equation
For a better understanding of the previous section, we will briefly review the meanings of relevant NP spin coefficients in depicting null congruences.[Eric Poisson. ''A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics''. Cambridge: Cambridge University Press, 2004. Chapters 2 and 3.] The tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
form of Raychaudhuri's equation
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.
The equation is important as a fundamental lemma for the Penrose–Hawking singularity the ...
governing null flows reads
:
where is defined such that . The quantities in Raychaudhuri's equation are related with the spin coefficients via
:
:
:
:where Eq(10) follows directly from and
:
:
Moreover, a null congruence is ''hypersurface orthogonal'' if .
Constraints from electromagnetic fields
Vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
NEHs on which are the simplest types of NEHs, but in general there can be various physically meaningful fields surrounding an NEH, among which we are mostly interested in electrovacuum fields with . This is the simplest extension of vacuum NEHs, and the nonvanishing energy-stress tensor for electromagnetic fields reads
:
where refers to the antisymmetric (, ) electromagnetic field strength
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
, and is trace-free () by definition and respects the dominant energy condition. (One should be careful with the antisymmetry of in defining Maxwell-NP scalars ).
The boundary conditions derived in the previous section are applicable to generic NEHs. In the electromagnetic case, can be specified in a more particular way. By the NP formalism of Einstein-Maxwell equations, one has
:
where denote the three Maxwell-NP scalars. As an alternative to Eq(), we can see that the condition also results from the NP equation
:
:as , so
:
It follows straightforwardly that
:
These results demonstrate that, there are no electromagnetic waves across (, ) or along (\Phi_) the NEH except the null geodesics generating the horizon. It is also worthwhile to point out that, the supplementary equation in Eq() is only valid for electromagnetic fields; for example, in the case of Yang–Mills fields there will be where are Yang–Mills-NP scalars.[E T Newman, K P Tod. ''Asymptotically Flat Spacetimes''. page 27, Appendix A.2. In A Held (Editor): ''General relativity and gravitation: one hundred years after the birth of Albert Einstein''. Vol(2). New York and London: Plenum Press, 1980.]
Adapted tetrad on NEHs and further properties
Usually, null tetrads adapted to spacetime properties are employed to achieve the most succinct NP descriptions. For example, a null tetrad can be adapted to principal null directions once the Petrov type
In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold.
It is mos ...
is known; also, at some typical boundary regions such as null
Null may refer to:
Science, technology, and mathematics Computing
* Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value
* Null character, the zero-valued ASCII character, also designated by , often use ...
infinity, timelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
infinity, spacelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
infinity, black hole horizons and cosmological horizons, tetrads can be adapted to boundary structures. Similarly, a ''preferred'' tetrad adapted to on-horizon geometric behaviors is employed in the literature to further investigate NEHs.
As indicated from the 3+1 perspective from condition (i) in the definition, an NEH ∆ is foliated by spacelike hypersurfaces ∆'=S2 transverse to its null normal along an ingoing null coordinate , where we follow the standard notation of ingoing Eddington–Finkelstein null coordinates and use to label the 2-dimensional leaves at ; that is,