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 which describes a black hole 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 ∆ is defined as a generic (rotating and distorted) NEH if it respects the following conditions:
(i) ∆ is null and topologically S 2 × R {\displaystyle S^{2}\times \mathbb {R} } ; (ii) Along any null normal field l {\displaystyle l} tangent to ∆, the outgoing expansion rate θ ( l ) := h ^ a b ∇ ^ a l b {\displaystyle \displaystyle \theta _{(l)}:={\hat {h}}^{ab}{\hat {\nabla }}_{a}l_{b}} vanishes; (iii) All field equations hold on ∆, and the stress–energy tensor T a b {\displaystyle T_{ab}} on ∆ is such that V a := − T b a l b {\displaystyle V^{a}:=-T_{b}^{a}l^{b}} is a future-directed causal vector ( V a V a ≤ 0 {\displaystyle V^{a}V_{a}\leq 0} ) for any future-directed null normal l a {\displaystyle l^{a}} .
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 zero ( g = 0 {\displaystyle g=0} ). The signature of ∆ is (0,+,+) with a degenerate temporal coordinate, and the intrinsic geometry of a foliation leaf ∆'=S2 is nonevolutional. The property θ ( l ) = 0 {\displaystyle \theta _{(l)}=0} 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 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 = ^ {\displaystyle {\hat {=}}} means equality on the black-hole horizons (NEHs), and "hat" over quantities and operators ( h ^ a b {\displaystyle {\hat {h}}^{ab}} , ∇ ^ {\displaystyle {\hat {\nabla }}} , etc.) denotes those on a foliation leaf of the horizon. Also, ∆ is the standard symbol for both an NEH and the directional derivative ∆ := n a ∇ a {\displaystyle :=n^{a}\nabla _{a}} 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 { ( − , + , + , + ) ; l a n a = − 1 , m a m ¯ a = 1 } {\displaystyle \{(-,+,+,+);l^{a}n_{a}=-1,m^{a}{\bar {m}}_{a}=1\}} (Note: unlike the original convention { ( + , − , − , − ) ; l a n a = 1 , m a m ¯ a = − 1 } {\displaystyle \{(+,-,-,-);l^{a}n_{a}=1,m^{a}{\bar {m}}_{a}=-1\}} , this is the usual one employed in studying trapped null surfaces and quasilocal definitions of black holes). Being a null normal to ∆, l a {\displaystyle l^{a}} is automatically geodesic, κ := − m a l b ∇ b l a = ^ 0 {\displaystyle \kappa :=-m^{a}l^{b}\nabla _{b}l_{a}\,{\hat {=}}\,0} , and twist free, Im ( ρ ) = Im ( − m a m ¯ b ∇ b l a ) = ^ 0 {\displaystyle {\text{Im}}(\rho )={\text{Im}}(-m^{a}{\bar {m}}^{b}\nabla _{b}l_{a})\,{\hat {=}}\,0} . For an NEH, the outgoing expansion rate θ ( l ) {\displaystyle \theta _{(l)}} along l a {\displaystyle l^{a}} is vanishing, θ ( l ) = ^ 0 {\displaystyle \theta _{(l)}\,{\hat {=}}\,0} , and consequently Re ( ρ ) = Re ( − m a m ¯ b ∇ b l a ) = − 1 2 θ ( l ) = ^ 0 {\displaystyle {\text{Re}}(\rho )={\text{Re}}(-m^{a}{\bar {m}}^{b}\nabla _{b}l_{a})=-{\frac {1}{2}}\theta _{(l)}\,{\hat {=}}\,0} . Moreover, according to the Raychaudhuri-NP expansion-twist equation,
( 1 ) D ρ = ρ 2 + σ σ ¯ + 1 2 R a b l a l b = ^ 0 , {\displaystyle (1)\qquad D\rho =\rho ^{2}+\sigma {\bar {\sigma }}+{\frac {1}{2}}R_{ab}l^{a}l^{b}\,{\hat {=}}\,0\,,}
it follows that on ∆
( 2 ) σ σ ¯ + 1 2 R a b l a l b = ^ 0 , {\displaystyle (2)\qquad \sigma {\bar {\sigma }}+{\frac {1}{2}}R_{ab}l^{a}l^{b}\,{\hat {=}}\,0\,,}
where σ := − m b m a ∇ a l b {\displaystyle \sigma :=-m^{b}m^{a}\nabla _{a}l_{b}} is the NP-shear coefficient. Due to the assumed energy condition (iii), we have R a b l a l b = R a b l a l b − 1 2 R g a b l a l b = 8 π T a b l a l b {\displaystyle R_{ab}l^{a}l^{b}=R_{ab}l^{a}l^{b}-{\frac {1}{2}}Rg_{ab}l^{a}l^{b}=8\pi \,T_{ab}l^{a}l^{b}} ( c = G = 1 {\displaystyle c=G=1} ), and therefore R a b l a l b {\displaystyle R_{ab}l^{a}l^{b}} is nonnegative on ∆. The product σ σ ¯ {\displaystyle \sigma {\bar {\sigma }}} is of course nonnegative, too. Consequently, σ σ ¯ {\displaystyle \sigma {\bar {\sigma }}} and R a b l a l b {\displaystyle R_{ab}l^{a}l^{b}} must be simultaneously zero on ∆, i.e. σ = ^ 0 {\displaystyle \sigma \,{\hat {=}}\,0} and R a b l a l b = ^ 0 {\displaystyle R_{ab}l^{a}l^{b}\,{\hat {=}}\,0} . As a summary,
( 3 ) κ = ^ 0 , Im ( ρ ) = ^ 0 , Re ( ρ ) = ^ 0 , σ = ^ 0 , R a b l a l b = ^ 0. {\displaystyle (3)\qquad \kappa \,{\hat {=}}\,0\,,\quad {\text{Im}}(\rho )\,{\hat {=}}\,0\,,\quad {\text{Re}}(\rho )\,{\hat {=}}\,0\,,\quad \sigma \,{\hat {=}}\,0\,,\quad R_{ab}l^{a}l^{b}\,{\hat {=}}\,0.}
Thus, the isolated horizon ∆ is nonevolutional and all foliation leaves ∆'=S2 look identical with one another. The relation R a b l a l b = 8 π ⋅ T a b l a l b = 8 π ⋅ T b a l b ⋅ l a = ^ 0 {\displaystyle R_{ab}l^{a}l^{b}=8\pi \cdot T_{ab}l^{a}l^{b}=8\pi \cdot T_{b}^{a}l^{b}\cdot l_{a}\,{\hat {=}}\,0} implies that the causal vector − T b a l b {\displaystyle -T_{b}^{a}l^{b}} in condition (iii) is proportional to l a {\displaystyle l^{a}} and R a b l b {\displaystyle R_{ab}l^{b}} is proportional to l a {\displaystyle l_{a}} on the horizon ∆; that is, − T b a l b = ^ c l a {\displaystyle -T_{b}^{a}l^{b}\,{\hat {=}}\,cl^{a}} and R a b l b = ^ c l a {\displaystyle R_{ab}l^{b}\,{\hat {=}}\,cl_{a}} , c ∈ R {\displaystyle c\in \mathbb {R} } . Applying this result to the related Ricci-NP scalars, we get Φ 00 := 1 2 R a b l a l b = ^ c 2 l b l b = ^ 0 {\displaystyle \Phi _{00}:={\frac {1}{2}}R_{ab}l^{a}l^{b}\,{\hat {=}}\,{\frac {c}{2}}\,l_{b}l^{b}\,{\hat {=}}\,0} , and Φ 01 = Φ 10 ¯ := 1 2 R a b l a m b = ^ c 2 l b m b = ^ 0 {\displaystyle \Phi _{01}={\overline {\Phi _{10}}}:={\frac {1}{2}}R_{ab}l^{a}m^{b}\,{\hat {=}}\,{\frac {c}{2}}\,l_{b}m^{b}\,{\hat {=}}\,0} , thus
( 4 ) R a b l b = ^ c l a , Φ 00 = ^ 0 , Φ 10 = Φ 01 ¯ = ^ 0 . {\displaystyle (4)\qquad R_{ab}l^{b}\,{\hat {=}}\,cl_{a}\,,\quad \Phi _{00}\,{\hat {=}}\,0\,,\quad \Phi _{10}={\overline {\Phi _{01}}}\,{\hat {=}}\,0\,.}
The vanishing of Ricci-NP scalars { Φ 00 , Φ 01 , Φ 10 } {\displaystyle \{\Phi _{00}\,,\Phi _{01}\,,\Phi _{10}\}} signifies that, there is no energy–momentum flux of any kind of charge across the horizon, such as electromagnetic waves, Yang–Mills flux or dilaton flux. Also, there should be no gravitational waves 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 Ψ i ( i = 0 , 1 , 3 , 4 ) {\displaystyle \Psi _{i}\;(i=0,1,3,4)} (excluding Ψ 2 {\displaystyle \Psi _{2}} ) rather than Ricci-NP quantities Φ i j {\displaystyle \Phi _{ij}} . According to the Raychaudhuri-NP shear equation
( 5 ) D σ = σ ( ρ + ρ ¯ ) + Ψ 0 = − 2 σ θ ( l ) + Ψ 0 , {\displaystyle (5)\qquad D\sigma =\sigma (\rho +{\bar {\rho }})+\Psi _{0}=-2\sigma \theta _{(l)}+\Psi _{0}\,,}
or the NP field equation on the horizon
( 6 ) D σ − δ κ = ( ρ + ρ ¯ ) σ + ( 3 ε − ε ¯ ) σ − ( τ − π ¯ + α ¯ + 3 β ) κ + Ψ 0 = ^ 0 , {\displaystyle (6)\qquad D\sigma -\delta \kappa =(\rho +{\bar {\rho }})\sigma +(3\varepsilon -{\bar {\varepsilon }})\sigma -(\tau -{\bar {\pi }}+{\bar {\alpha }}+3\beta )\kappa +\Psi _{0}\,{\hat {=}}\,0\,,}
it follows that Ψ 0 := C a b c d l a m b l c m d = ^ 0 {\displaystyle \Psi _{0}:=C_{abcd}l^{a}m^{b}l^{c}m^{d}\,{\hat {=}}\,0} . Moreover, the NP equation
( 7 ) δ ρ − δ ¯ σ = ρ ( α ¯ + β ) − σ ( 3 α − β ¯ ) + ( ρ − ρ ¯ ) τ + ( μ − μ ¯ ) κ − Ψ 1 + Φ 01 = ^ 0 {\displaystyle (7)\qquad \delta \rho -{\bar {\delta }}\sigma =\rho ({\bar {\alpha }}+\beta )-\sigma (3\alpha -{\bar {\beta }})+(\rho -{\bar {\rho }})\tau +(\mu -{\bar {\mu }})\kappa -\Psi _{1}+\Phi _{01}\,{\hat {=}}\,0}
implies that Ψ 1 := C a b c d l a n b l c m d = ^ 0 {\displaystyle \Psi _{1}:=C_{abcd}l^{a}n^{b}l^{c}m^{d}\,{\hat {=}}\,0} . To sum up, we have
( 8 ) Ψ 0 = ^ 0 , Ψ 1 = ^ 0 , {\displaystyle (8)\qquad \Psi _{0}\,{\hat {=}}\,0\,,\quad \Psi _{1}\,{\hat {=}}\,0\,,}
which means that, geometrically, a principal null direction of Weyl's tensor is repeated twice and l a {\displaystyle l^{a}} is aligned with the principal direction; physically, no gravitational waves (transverse component Ψ 0 {\displaystyle \Psi _{0}} and longitudinal component Ψ 1 {\displaystyle \Psi _{1}} ) 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. The tensor form of Raychaudhuri's equation governing null flows reads
( 9 ) L ℓ θ ( l ) = − 1 2 θ ( l ) 2 + κ ~ ( l ) θ ( l ) − σ a b σ a b + ω ~ a b ω ~ a b − R a b l a l b , {\displaystyle (9)\qquad {\mathcal {L}}_{\ell }\theta _{(l)}=-{\frac {1}{2}}\theta _{(l)}^{2}+{\tilde {\kappa }}_{(l)}\theta _{(l)}-\sigma _{ab}\sigma ^{ab}+{\tilde {\omega }}_{ab}{\tilde {\omega }}^{ab}-R_{ab}l^{a}l^{b}\,,}
where κ ~ ( l ) {\displaystyle {\tilde {\kappa }}_{(l)}} is defined such that κ ~ ( l ) l b := l a ∇ a l b {\displaystyle {\tilde {\kappa }}_{(l)}l^{b}:=l^{a}\nabla _{a}l^{b}} . The quantities in Raychaudhuri's equation are related with the spin coefficients via
( 10 ) θ ( l ) = − ( ρ + ρ ¯ ) = − 2 Re ( ρ ) , θ ( n ) = μ + μ ¯ = 2 Re ( μ ) , {\displaystyle (10)\qquad \theta _{(l)}=-(\rho +{\bar {\rho }})=-2{\text{Re}}(\rho )\,,\quad \theta _{(n)}=\mu +{\bar {\mu }}=2{\text{Re}}(\mu )\,,}
( 11 ) σ a b = − σ m ¯ a m ¯ b − σ ¯ m a m b , {\displaystyle (11)\qquad \sigma _{ab}=-\sigma {\bar {m}}_{a}{\bar {m}}_{b}-{\bar {\sigma }}m_{a}m_{b}\,,}
( 12 ) ω ~ a b = 1 2 ( ρ − ρ ¯ ) ( m a m ¯ b − m ¯ a m b ) = Im ( ρ ) ⋅ ( m a m ¯ b − m ¯ a m b ) , {\displaystyle (12)\qquad {\tilde {\omega }}_{ab}={\frac {1}{2}}\,{\Big (}\rho -{\bar {\rho }}{\Big )}\,{\Big (}m_{a}{\bar {m}}_{b}-{\bar {m}}_{a}m_{b}{\Big )}={\text{Im}}(\rho )\cdot {\Big (}m_{a}{\bar {m}}_{b}-{\bar {m}}_{a}m_{b}{\Big )}\,,}
where Eq(10) follows directly from h ^ a b = h ^ b a = m b m ¯ a + m ¯ b m a {\displaystyle {\hat {h}}^{ab}={\hat {h}}^{ba}=m^{b}{\bar {m}}^{a}+{\bar {m}}^{b}m^{a}} and
( 13 ) θ ( l ) = h ^ b a ∇ a l b = m b m ¯ a ∇ a l b + m ¯ b m a ∇ a l b = m b δ ¯ l b + m ¯ b δ l b = − ( ρ + ρ ¯ ) , {\displaystyle (13)\qquad \theta _{(l)}={\hat {h}}^{ba}\nabla _{a}l_{b}=m^{b}{\bar {m}}^{a}\nabla _{a}l_{b}+{\bar {m}}^{b}m^{a}\nabla _{a}l_{b}=m^{b}{\bar {\delta }}l_{b}+{\bar {m}}^{b}\delta l_{b}=-(\rho +{\bar {\rho }})\,,}
( 14