Mechanics of weakly isolated horizons

4.1 Mechanics of weakly isolated horizons

The isolated horizon framework has not only extended black hole mechanics, but it has also led to a deeper insight into the ‘origin’ of the laws of black hole mechanics. In this section we will summarize these developments using WIHs. Along the way we shall also obtain formulas for the mass and angular momentum of a WIH. For simplicity, in the main part of the discussion, we will restrict ourselves to type II (i.e., axi-symmetric) WIHs on which all matter fields vanish. Generalizations including various types of matter field can be found in [18

, 25

, 14

, 74

, 75

, 76

4.1.1 The zeroth law

The zeroth law of thermodynamics says that the temperature of a system in thermodynamic equilibrium is constant. Its counterpart for black hole mechanics says that surface gravity of a weakly isolated horizon is constant. This result is non-trivial because the horizon geometry is not assumed to be spherically symmetric; the result holds even when the horizon itself is highly distorted so long as it is in equilibrium. It is established as follows.

Recall from Section 2.1.3 that the notion of surface gravity is tied to the choice of a null normal $a ℓ$ of the isolated horizon: $κ(ℓ) := ℓaωa$ . Now, using Equation (5 ) in Definition 2 (of WIHs), we obtain:

Next, recall from Equation (9

) that the curl of $ωa$ is related to the imaginary part of $Ψ2$ :

where $ε$ is the natural area 2-form on $Δ$ satisfying $ℒ ℓε = 0$ and $ℓ ⋅ ε = 0$ . Hence we conclude $ℓ ⋅ dω = 0$ which in turn implies that $κ(ℓ)$ is constant on the horizon:

This completes the proof of the zeroth law. As the argument shows, given an NEH, the main condition (5

) in the definition of a WIH is equivalent to constancy of surface gravity. Note that no restriction has been imposed on $Ψ2$ which determines the mass and angular momentum multipoles [23

]: as emphasized above, the zeroth law holds even if the WIH is highly distorted and rapidly rotating.

If electromagnetic fields are included, one can also show that the electric potential is constant on the horizon [25 ]. Finally, there is an interesting interplay between the zeroth law the action principle. Let us restrict ourselves to space-times which admit a non expanding horizon as inner boundary. Then the standard Palatini action principle is not well defined because the variation produces a non-vanishing surface term at the horizon. The necessary and sufficient condition for this surface term to vanish is precisely that the gravitational (and the electromagnetic) zeroth laws hold [25 ]. Consequently, the standard action principle is well-defined if inner boundaries are WIHs.

4.1.2 Phase space, symplectic structure, and angular momentum

In field theories, conserved quantities such as energy and angular momentum can be universally defined via a Hamiltonian framework: they are the numerical values of Hamiltonians generating canonical transformations corresponding to time translation and rotation symmetries. In absence of inner boundaries, it is this procedure that first led to the notion of the ADM energy and angular momentum at spatial infinity [7 ]. At null infinity, it can also be used to define fluxes of Bondi energy and angular momentum across regions of $+ ℐ$ [32 ], and values of these quantities associated with any cross-section of $ℐ+$ [17 , 185 ].

This procedure can be extended to allow inner boundaries which are WIHs. The first ingredient required for a Hamiltonian framework is, of course, a phase space. The appropriate phase space now consists of fields living in a region of space-time outside the black hole, satisfying suitable boundary conditions at infinity and horizon. Let $ℳ$ be the region of space-time that we are interested in. The boundary of $ℳ$ consists of four components: the time-like cylinder $τ$ at spatial infinity, two space-like surfaces $M1$ and $M2$ which are the future and past boundaries of $ℳ$ , and an inner boundary $Δ$ which is to be the WIH (see Figure 6 ). At infinity, all fields are assumed to satisfy the fall-off conditions needed to ensure asymptotic flatness. To ensure that $Δ$ is a type II horizon, one fixes a rotational vector field $ϕa$ on $Δ$ and requires that physical fields on $ℳ$ are such that the induced geometry on $Δ$ is that of a type II horizon with $ϕa$ as the rotational symmetry.

Figure 6:

The region of space-time $ℳ$ under consideration has an internal boundary $Δ$ and is bounded by two Cauchy surfaces $M1$ and $M2$ and the time-like cylinder $τ∞$ at infinity. $M$ is a Cauchy surface in $ℳ$ whose intersection with $Δ$ is a spherical cross-section $S$ and the intersection with $τ∞$ is $S∞$ , the sphere at infinity.

Two Hamiltonian frameworks are available. The first uses a covariant phase space which consists of the solutions to field equations which satisfy the required boundary conditions [25 , 14 ]. Here the calculations are simplest if one uses a first order formalism for gravity, so that the basic gravitational variables are orthonormal tetrads and Lorentz connections. The second uses a canonical phase space consisting of initial data on a Cauchy slice $M$ of $ℳ$ [54 ]. In the gravitational sector, this description is based on the standard ADM variables. Since the conceptual structure underlying the main calculation and the final results are the same, the details of the formalism are not important. For definiteness, in the main discussion, we will use the covariant phase space and indicate the technical modifications needed in the canonical picture at the end.

The phase space $Γ$ is naturally endowed with a (pre-)symplectic structure $Ω$ – a closed 2-form (whose degenerate directions correspond to infinitesimal gauge motions). Given any two vector fields (i.e., infinitesimal variations) $δ1$ and $δ2$ on $Γ$ , the action $Ω (δ1,δ2)$ of the symplectic 2-form on them provides a function on $Γ$ . A vector field $X$ on $Γ$ is said to be a Hamiltonian vector field (i.e., to generate an infinitesimal canonical transformation) if and only if $ℒX Ω = 0$ . Since the phase space is topologically trivial, it follows that this condition holds if and only if there is a function $H$ on $Γ$ such that $Ω (δ,X ) = δH$ for all vector fields $δ$ . The function $H$ is called a Hamiltonian and $X$ its Hamiltonian vector field; alternatively, $H$ is said to generate the infinitesimal canonical transformation $X$ .

Since we are interested in energy and angular momentum, the infinitesimal canonical transformations $X$ will correspond to time translations and rotations. As in any generally covariant theory, when the constraints are satisfied, values of Hamiltonians generating such diffeomorphisms can be expressed purely as surface terms. In the present case, the relevant surfaces are the sphere at infinity and the spherical section $S = M ∩ Δ$ of the horizon. Thus the numerical values of Hamiltonians now consist of two terms: a term at infinity and a term at the horizon. The terms at infinity reproduce the ADM formulas for energy and angular momentum. The terms at the horizon define the energy and angular momentum of the WIH.

Let us begin with angular momentum (see [14 ] for details). Consider a vector field $φa$ on $M$ which satisfies the following boundary conditions: (i) At infinity, $a φ$ coincides with a fixed rotational symmetry of the fiducial flat metric; and, (ii) on $Δ$ , it coincides with the vector field $a ϕ$ . Lie derivatives of physical fields along $φa$ define a vector field $X (φ)$ on $Γ$ . The question is whether this is an infinitesimal canonical transformation, i.e., a generator of the phase space symmetry. As indicated above, this is the case if and only if there exists a phase space function $J (φ)$ satisfying:

for all variations $δ$ . If such a phase space function $(φ) J$ exists, it can be interpreted as the Hamiltonian generating rotations.

Now, a direct calculation [15 ] shows that, in absence of gauge fields on $Δ$ , one has:

As expected, the expression for $(φ) δJ$ consists of two terms: a term at the horizon and a term at infinity. The term at infinity is the variation of the familiar ADM angular momentum $(φ) J ADM$ associated with $a φ$ . The surface integral at the horizon is interpreted as the variation of the horizon angular momentum $JΔ$ . Since variations $δ$ are arbitrary, one can recover, up to additive constants, $J (φA)DM$ and $JΔ$ from their variations, and these constants can be eliminated by requiring that both of these angular momenta should vanish in static axi-symmetric space-times. One then obtains:

where the function $f$ on $Δ$ is related to $ϕa$ by $∂af = εbaϕb$ . In the last step we have used Equation (28

) and performed an integration by parts. Equation (32

) is the expression of the horizon angular momentum. Note that all fields that enter this expression are local to the horizon and $φa$ is not required to be a Killing field of the space-time metric even in a neighborhood of the horizon. Therefore, $JΔ$ can be calculated knowing only the horizon geometry of a type II horizon.

We conclude our discussion of angular momentum with some comments:

The Hamiltonian $J (φ)$: It follows from Equation (31 ) and (32 ) that the total Hamiltonian generating the rotation along $a φ$ is the difference between the ADM and the horizon angular momenta (apart from a sign which is an artifact of conventions). Thus, it can be interpreted as the angular momentum of physical fields in the space-time region $ℳ$ outside the black hole.

Relation to the Komar integral: If $φa$ happens to be a space-time Killing field in a neighborhood of $Δ$ , then $JΔ$ agrees with the Komar integral of $φa$ [14 ]. If $φa$ is a global, space-time Killing field, then both $(φ) JADM$ as well as $J Δ$ agree with the Komar integral, whence the total Hamiltonian $(φ) J$ vanishes identically. Since the fields in the space-time region $ℳ$ are all axi-symmetric in this case, this is just what one would expect from the definition of $J (φ)$ .

$J Δ$ for general axial fields $φa$: If the vector field $a φ$ is tangential to cross-sections of $Δ$ , $(φ) J$ continues to the generator of the canonical transformation corresponding to rotations along $a φ$ , even if its restriction $&tidle;ϕa$ to $Δ$ does not agree with the axial symmetry $ϕa$ of horizon geometries of our phase space fields. However, there is an infinity of such vector fields $ϕ&tidle;a$ and there is no physical reason to identify the surface term $JΔ$ arising from any one of them with the horizon angular momentum.

Inclusion of gauge fields: If non-trivial gauge fields are present at the horizon, Equation (32 ) is incomplete. The horizon angular momentum $JΔ$ is still an integral over $S$ ; however it now contains an additional term involving the Maxwell field. Thus $JΔ$ contains not only the ‘bare’ angular momentum but also a contribution from its electromagnetic hair (see [14 ] for details).

Canonical phase space: The conceptual part of the above discussion does not change if one uses the canonical phase space [54 ] in place of the covariant. However, now the generator of the canonical transformation corresponding to rotations has a volume term in addition to the two surface terms discussed above. However, on the constraint surface the volume term vanishes and the numerical value of the Hamiltonian reduces to the two surface terms discussed above.

4.1.3 Energy, mass, and the first law

To obtain an expression of the horizon energy, one has to find the Hamiltonian on $Γ$ generating diffeomorphisms along a time translation symmetry $ta$ on $ℳ$ . To qualify as a symmetry, at infinity $ta$ must approach a fixed time translation of the fiducial flat metric. At the horizon, $ta$ must be an infinitesimal symmetry of the type II horizon geometry. Thus, the restriction of $ta$ to $Δ$ should be a linear combination of a null normal $a ℓ$ and the axial symmetry vector $a ϕ$ ,

where $B(ℓ,t)$ and the angular velocity $Ω (t)$ are constants on $Δ$ .

However there is subtlety: Unlike in the angular momentum calculation where $φa$ is required to approach a fixed rotational vector $a ϕ$ on $Δ$ , the restriction of $a t$ to $Δ$ can not be a fixed vector field. For physical reasons, the constants $B (ℓ,t)$ and $Ω (t)$ should be allowed to vary from one space-time to another; they are to be functions on phase space. For instance, physically one expects $Ω (t)$ to vanish on the Schwarzschild horizon but not on a generic Kerr horizon. In the terminology of numerical relativity, unlike $ϕa$ , the time translation $ta$ must be a live vector field. As we shall see shortly, this generality is essential also for mathematical reasons: without it, evolution along $a t$ will not be Hamiltonian!

At first sight, it may seem surprising that there exist choices of evolution vector fields $ta$ for which no Hamiltonian exists. But in fact this phenomenon can also happen in the derivations of the ADM energy for asymptotically flat space-times in the absence of any black holes. Standard treatments usually consider only those $a t$ that asymptote to the same unit time translation at infinity for all space-times included in the phase space. However, if we drop this requirement and choose a live $a t$ which approaches different asymptotic time-translations for different space-times, then in general there exists no Hamiltonian which generates diffeomorphisms along such a $ta$ . Thus, the requirement that the evolution be Hamiltonian restricts permissible $ta$ . This restriction can be traced back to the fact that there is a fixed fiducial flat metric at infinity. At the horizon, the situation is the opposite: The geometry is not fixed and this forces one to adapt $ta$ to the space-time under consideration, i.e., to make it live.

Apart from this important caveat, the calculation of the Hamiltonian is very similar to that for angular momentum. First, one evaluates the 1-form $Y (t)$ on $Γ$ whose action on any tangent vector field $δ$ is given by

where $X (t)$ is the vector field on $Γ$ induced by diffeomorphisms along $ta$ . Once again, $Y(t)(δ)$ will consist of a surface term at infinity and a surface term at the horizon. A direct calculation yields

where $κ(t) := B(ℓ,t)ℓaωa$ is the surface gravity associated with the restriction of $ta$ to $Δ$ , $a Δ$ is the area of $Δ$ , and $E (t) ADM$ is the ADM energy associated with $ta$ . The first two terms in the right hand side of this equation are associated with the horizon, while the $t EADM$ term is associated with an integral at infinity. Since the term at infinity gives the ADM energy, it is natural to hope that terms at the horizon will give the horizon energy. However, at this point, we see an important difference from the angular momentum calculation. Recall that the right hand side of Equation (31

) is an exact variation which means that $(φ) J$ is well defined. However, the right hand side of Equation (35

) is not guaranteed to be an exact variation; in other words, $X (t)$ need not be a Hamiltonian vector field in phase space. It is Hamiltonian if and only if there is a phase space function $E (t) Δ$ – the would be energy of the WIH – satisfying

In particular, this condition implies that, of the infinite number of coordinates in phase space, $t EΔ$ , $κ (t)$ , and $Ω(t)$ can depend only on two: $aΔ$ and $JΔ$ .

Let us analyze Equation (36 ). Clearly, a necessary condition for existence of $(t) E Δ$ is just the integrability requirement

Since $κ(t)$ and $Ω(t)$ are determined by $a t$ , Equation (37

) is a constraint on the restriction to the horizon of the time evolution vector field $ta$ . A vector field $ta$ for which $(t) E Δ$ exists is called a permissible time evolution vector field. Since Equation (36

) is precisely the first law of black hole mechanics, $ta$ is permissible if and only if the first law holds. Thus the first law is the necessary and sufficient condition that the evolution generated by $a t$ is Hamiltonian!

There are infinitely many permissible vector fields $ta$ . To construct them, one can start with a suitably regular function $κ0$ of $aΔ$ and $JΔ$ , find $B (t)$ so that $κ(t) = κ0$ , solve Equation (37 ) to obtain $Ω (t)$ , and find a permissible $ta$ with $ta = B (ℓ,t)ℓa − Ω(t)ϕa$ on $Δ$ [14 ]. Each permissible $ta$ defines a horizon energy $(t) E Δ$ and provides a first law (36 ). A question naturally arises: Can one select a preferred $ta0$ or, alternatively, a canonical function $κ0(aΔ, JΔ)$ ? Now, thanks to the no-hair theorems, we know that for each choice of $(aΔ,J Δ)$ , there is precisely one stationary black hole in vacuum general relativity: the Kerr solution. So, it is natural to set $κ0(aΔ, JΔ) = κKerr(aΔ,J Δ)$ , or, more explicitly,

where $R Δ$ is the area radius of the horizon, $R = (a ∕4π)1∕2 Δ Δ$ . Via Equation (37

), this choice then leads to

The associated horizon energy is then:

This canonical horizon energy is called the horizon mass:

Note that, its dependence on the horizon area and angular momentum is the same as that in the Kerr space-time. Although the final expression is so simple, it is important to keep in mind that this is not just a postulate. Rather, this result is derived using a systematic Hamiltonian framework, following the same overall procedure that leads to the definition of the ADM 4-momentum at spatial infinity. Finally, note that the quantities which enter the first law refer just to physical fields on the horizon; one does not have to go back and forth between the horizon and infinity.

We will conclude with three remarks:

Relation to the ADM and Bondi energy: Under certain physically reasonable assumptions on the behavior of fields near future time-like infinity $i+$ , one can argue that, if the WIH extends all the way to $i+$ , then the difference $MADM − M Δ$ equals the energy radiated across future null-infinity [13 ]. Thus, as one would expect physically, $M Δ$ is the mass that is left over after all the gravitational radiation has left the system.

Horizon angular momentum and mass: To obtain a well-defined action principle and Hamiltonian framework, it is essential to work with WIHs. However, the final expressions (32 ) and (40 ) of the horizon angular momentum and mass do not refer to the preferred null normals $[ℓ]$ used in the transition from an NEH to a WIH. Therefore, the expressions can be used on any NEH. This fact is useful in the analysis of transition to equilibrium (Section 4.3) and numerical relativity (Section 5.1).

Generalizations of the first law: The derivation of the first law given here can be extended to allow the presence of matter fields at the horizon [25 , 14 ]. If gauge fields are present, the expression of the angular momentum has an extra term and the first law (36 ) also acquires the familiar extra term ‘ $Φ δQ$ ’, representing work done on the horizon in increasing its charge. Again, all quantities are defined locally on the horizon. The situation is similar in lower [22 ] and higher [134 ] space-time dimensions. However, a key difference arises in the definition of the horizon mass. Since the uniqueness theorems for stationary black holes fail to extend beyond the Einstein–Maxwell theory in four space-time dimensions, it is no longer possible to assign a canonical mass to the horizon. However, as we will see in Section 6, the ambiguity in the notion of the horizon mass can in fact be exploited to obtain new insights into the properties of black holes and solitons in these more general theories.