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 [19
, 26
, 15
, 75
, 76
, 77
].
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
of the isolated
horizon:
. Now, using Equation (5) in Definition 2 (of
WIHs), we obtain:
Next, recall from Equation (9) that the curl of
is related to the imaginary part of
:
where
is the natural area 2-form on
satisfying
and
. Hence we conclude
which in
turn implies that
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
which determines the mass
and angular momentum multipoles [24
]: 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 [26
]. 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 [26
]. 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
[33
], and values of
these quantities associated with any cross-section of
[18
, 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
be the region of space-time that we are interested
in. The boundary of
consists of four components: the
time-like cylinder
at spatial infinity, two space-like
surfaces
and
which are the future and
past boundaries of
, and an inner boundary
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
is a type II horizon, one fixes a rotational vector
field
on
and requires that physical fields on
are such that the induced geometry on
is that of a type II horizon with
as the rotational symmetry.
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 [26
, 15
]. 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
of
[55
]. 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
is naturally endowed
with a (pre-)symplectic structure
- a closed 2-form
(whose degenerate directions correspond to infinitesimal gauge
motions). Given any two vector fields (i.e., infinitesimal
variations)
and
on
, the action
of the
symplectic 2-form on them provides a function on
. A vector field
on
is said to be a Hamiltonian vector field (i.e., to
generate an infinitesimal canonical transformation) if and only if
. Since the phase space is topologically
trivial, it follows that this condition holds if and only if there
is a function
on
such that
for all vector fields
. The function
is called a Hamiltonian and
its Hamiltonian vector field; alternatively,
is said to generate the infinitesimal canonical
transformation
.
Since we are interested in energy and angular
momentum, the infinitesimal canonical transformations
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
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 [15
] for details).
Consider a vector field
on
which satisfies the following boundary conditions:
(i) At infinity,
coincides with a fixed rotational
symmetry of the fiducial flat metric; and, (ii) on
, it coincides with the vector field
. Lie derivatives of physical fields along
define a vector field
on
. 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
satisfying:
for all variations
. If such a phase space function
exists, it can be interpreted as the Hamiltonian
generating rotations.
Now, a direct calculation [16
] shows that, in
absence of gauge fields on
, one has:
As expected, the expression for
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
associated with
. The surface integral at the
horizon is interpreted as the variation of the horizon angular
momentum
. Since variations
are arbitrary, one can recover, up to additive constants,
and
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
on
is related to
by
. 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
is not required to be a
Killing field of the space-time metric even in a neighborhood of
the horizon. Therefore,
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

-
It follows from Equation (31) and (32) that the total
Hamiltonian generating the rotation along
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
outside the black
hole.
- Relation to the Komar
integral
-
If
happens to be a space-time Killing field in a
neighborhood of
, then
agrees with the
Komar integral of
[15
]. If
is a global, space-time Killing field, then both
as well as
agree with the Komar
integral, whence the total Hamiltonian
vanishes
identically. Since the fields in the space-time region
are all axi-symmetric in this case, this is just
what one would expect from the definition of
.
for
general axial fields 
-
If the vector field
is tangential to cross-sections of
,
continues to the generator of the
canonical transformation corresponding to rotations along
, even if its restriction
to
does not agree with the axial symmetry
of horizon geometries of our phase space fields.
However, there is an infinity of such vector fields
and there is no physical reason to identify the
surface term
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
is still an integral over
; however it now contains an additional term
involving the Maxwell field. Thus
contains not only
the ‘bare’ angular momentum but also a contribution from its
electromagnetic hair (see [15
] for details).
- Canonical phase
space
-
The conceptual part of the above discussion does not change if one
uses the canonical phase space [55] 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
generating
diffeomorphisms along a time translation symmetry
on
. To qualify as a symmetry, at infinity
must approach a fixed time translation of the
fiducial flat metric. At the horizon,
must be an
infinitesimal symmetry of the type II horizon geometry. Thus, the
restriction of
to
should be a linear
combination of a null normal
and the axial symmetry
vector
,
where
and the angular velocity
are constants on
.
However there is subtlety: Unlike in the angular
momentum calculation where
is required to approach a
fixed rotational vector
on
, the restriction of
to
can not be a fixed vector field. For
physical reasons, the constants
and
should be allowed to vary from one space-time to
another; they are to be functions on
phase space. For instance, physically one expects
to vanish on the Schwarzschild horizon but not on a
generic Kerr horizon. In the terminology of numerical relativity,
unlike
, the time translation
must be a live vector field. As we shall see shortly,
this generality is essential also for mathematical reasons: without
it, evolution along
will not be Hamiltonian!
At first sight, it may seem surprising that there
exist choices of evolution vector fields
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
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
which approaches different
asymptotic time-translations for different space-times, then in
general there exists no Hamiltonian which generates diffeomorphisms
along such a
. Thus, the requirement that the
evolution be Hamiltonian restricts permissible
. 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
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
on
whose action on any tangent vector field
is given by
where
is the vector field on
induced by
diffeomorphisms along
. Once again,
will consist of a surface term at infinity and a
surface term at the horizon. A direct calculation yields
where
is the surface gravity
associated with the restriction of
to
,
is the area of
, and
is the ADM energy associated
with
. The first two terms in the right hand side of this
equation are associated with the horizon, while the
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
is well defined. However, the right
hand side of Equation (35) is not guaranteed to
be an exact variation; in other words,
need not be a
Hamiltonian vector field in phase
space. It is Hamiltonian if and only if there is a phase space
function
- the would be energy of the WIH -
satisfying
In particular, this condition implies that, of the infinite number
of coordinates in phase space,
,
, and
can depend only on two:
and
.
Let us analyze Equation (36). Clearly, a necessary
condition for existence of
is just the integrability
requirement
Since
and
are determined by
, Equation (37) is a constraint on
the restriction to the horizon of the time evolution vector field
. A vector field
for which
exists is called a permissible time evolution vector field.
Since Equation (36) is precisely the
first law of black hole mechanics,
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
is
Hamiltonian!
There are infinitely many permissible vector
fields
. To construct them, one can start with a suitably
regular function
of
and
, find
so that
, solve Equation (37) to obtain
, and find a permissible
with
on
[15
]. Each permissible
defines a horizon energy
and provides a
first law (36). A question naturally
arises: Can one select a preferred
or, alternatively,
a canonical function
? Now, thanks to the no-hair
theorems, we know that for each choice of
, there is precisely one stationary black hole in
vacuum general relativity: the Kerr solution. So, it is natural to
set
, or, more explicitly,
where
is the area radius of the horizon,
. 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
, one can argue
that, if the WIH extends all the way to
, then the
difference
equals the energy radiated across
future null-infinity [14
]. Thus, as one would
expect physically,
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
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 [26
, 15
]. 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 ‘
’, 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 [23] and higher [133] 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.