4.2 Mechanics of dynamical
horizons
The variations
in the first law (36) represent
infinitesimal changes in equilibrium states of horizon geometries.
In the derivation of Section 4.1, these variations relate nearby but
distinct space-times in each of which the horizon is in
equilibrium. Therefore Equation (36) is interpreted as the
first law in a passive form.
Physically, it is perhaps the active
form of the first law that is of more direct interest where
a physical process, such as the one
depicted in the right panel of Figure 1 causes a transition
from one equilibrium state to a nearby one. Such a law can be
established in the dynamical horizon framework. In fact, one can
consider fully non-equilibrium situations, allowing physical
processes in a given space-time in which there is a finite - rather than an infinitesimal -
change in the state of the horizon. This leads to an integral version of the first law.
Our summary of the mechanics of DHs is divided in
to three parts. In the first, we begin with some preliminaries on
angular momentum. In the second, we extend the area balance
law (25) by allowing more
general lapse and shift functions, which leads to the integral
version of the first law. In the third, we introduce the notion of
horizon mass.
4.2.1
Angular momentum
balance
As one might expect, the angular momentum
balance law results from the momentum constraint (15) on the DH
. Fix any vector field
on
which is tangential to all the
cross-sections
of
, contract both sides of
Equation (15) with
, and integrate the resulting equation over the
region
to obtain
It is natural to identify the surface integrals with the generalized angular momentum
associated with cross-sections
and set
where the overall sign ensures compatibility with conventions
normally used in the asymptotically flat context, and where we have
introduced an angular momentum density
for later convenience. The term ‘generalized’ emphasizes the fact
that the vector field
need not be an axial Killing
field even on
; it only has to be tangential to our
cross-sections. If
happens to be the restriction of a
space-time Killing field to
, then
agrees with the Komar integral. If the pair
is spherically symmetric on
, as one would expect, the angular momenta associated
with the rotational Killing fields vanish. Equation (42) is a balance law; the
right side provides expressions of fluxes of the generalized
angular momentum across
. The contributions due to
matter and gravitational waves are cleanly separated and given by
with
, so that
As expected, if
is a Killing vector of the three-metric
, then the gravitational angular momentum flux
vanishes:
.
As with the area balance law, here we worked
directly with the constraint equations rather than with a
Hamiltonian framework. However, we could also have used, e.g., the
standard ADM phase space framework based on a Cauchy surface
with internal boundary
and the outer
boundary at infinity. If
is a vector field on
which tends to
on
and to an asymptotic rotational symmetry at
infinity, we can ask for the phase space function which generates
the canonical transformation corresponding to the rotation
generated by
. When the constraints are satisfied, as
usual the value of this generating function is given by just
surface terms. The term at infinity provides the total angular
momentum and, as in Section 4.1.2, it is natural to interpret the
surface term at
as the
-angular momentum of
. This term can be expressed in terms of the Cauchy
data
on
as
where
is the unit normal to
within
. However, since the right side involves the
extrinsic curvature of
, in general the value of the
integral is sensitive to the choice of
. Hence, the notion
of the
-angular momentum associated with an arbitrary
cross-section is ambiguous. This ambiguity disappears if
is
divergence-free on
. In particular, in this
case, one has
. Thus, although the balance
law (42) holds for more
general vector fields
, it is robust only when
is divergence-free on
. (These
considerations shed some light on the interpretation of the field
in the area balance law (25). For, the form of the
right side of Equation (43) implies that the
field
vanishes identically on
if and only if
vanishes for every divergence-free
on
. In particular then, if the horizon has
non-zero angular momentum, the
-contribution to the energy
flux can not vanish.)
Finally, for
to be interpreted
as ‘the’ angular momentum,
has to be a symmetry. An
obvious possibility is that it be a Killing field of
on
. A more general scenario is discussed
in Section 8.
4.2.2
Integral form of
the first law
To obtain the area balance law, in
Section 3.2 we restricted ourselves to vector
fields
, i.e., to lapse functions
and shifts
. We were
then led to a conservation law for the Hawking mass. In the
spherically symmetric context, the Hawking mass can be taken to be
the physical mass of the horizon. However, as the Kerr space-time
already illustrates, in presence of rotation this interpretation is
physically incorrect. Therefore, although Equation (25) continues to dictate
the dynamics of the Hawking mass even in presence of rotation, a
more general procedure is needed to obtain physically interesting conservation laws in
this case. In the case of WIHs, the first law incorporating
rotations required us to consider suitable linear combinations of
and the rotational symmetry field
on the horizon. In the same spirit, on DHs, one has
to consider more general vector fields than
, i.e., more general choices of lapses and
shifts.
As on WIHs, one first restricts oneself to
situations in which the metric
on
admits a Killing field
so that
can be unambiguously interpreted as the angular
momentum associated with each
. In the case of a WIH,
was given by
and the freedom was in the choice of constants
and
. On a DH, one must allow the corresponding
coefficients to be ‘time-dependent’. The simplest generalization is
to choose, in place of
, vector fields
where
is an arbitrary function of
, and the lapse
is given by
for any function
of
. Note that one is free to rescale
and
by functions of
so that on each cross-section (‘instant of time’)
one has the same rescaling freedom as on a WIH. One can consider
even more general lapse-shift pairs to allow, e.g., for
differential rotation (see [31
]).
Using
in place of
, one obtains the following generalization of the
area balance equation [31
]:
Note that there is one balance equation for every vector field
of the form (47); as in
Section 4.1, we have an infinite number of
relations, now ensured by the constraint part of Einstein’s
equations.
The right side of Equation (48) can be naturally
interpreted as the flux
of the ‘energy’
associated with the vector field
across
. Hence, we can rewrite the
equation as
If
and
are only infinitesimally separated,
this integral equation reduces to the differential condition
Thus, the infinitesimal form of Equation (48) is a familiar first
law, provided
is identified as an effective surface gravity on the
cross-section
. This identification can be motivated
as follows. First, on a spherically symmetric DH, it is natural to
choose
. Then the surface gravity reduces to
, just as one would hope from one’s experience with
the Schwarzschild metric and more generally with static but
possibly distorted horizons (See Appendix A of [14]). Under the change
, we have
,
which is the natural generalization of the transformation property
of surface gravity of WIHs under the change
. Finally,
can also be regarded as a
2-sphere average of a geometrically defined surface gravity
associated with certain vector fields on
[56
, 31
]; hence the
adjective ‘effective’.
To summarize, Equation (48) represents an
integral generalization of the first law of mechanics of weakly
isolated horizons to dynamical situations in which the horizon is
permitted to make a transition from a given state to one far away,
not just nearby. The left side represents the flux
of the energy associated with the vector field
, analogous to the flux of Bondi energy across a
portion of null infinity. A natural question therefore arises: Can
one integrate this flux to obtain an energy
which depends only on fields defined locally on the cross-section, as is possible
at null infinity? As discussed in the next section, the answer is
in the affirmative and the procedure leads to a canonical notion of
horizon mass.
4.2.3
Horizon mass
In general relativity, the notion of energy
always refers to a vector field. On DHs, the vector field is
. Therefore, to obtain an unambiguous notion of
horizon mass, we need to make a canonical choice of
, i.e., of functions
and
on
. As we saw in
Section 4.1.3, on WIHs of 4-dimensional
Einstein-Maxwell theory, the pair
suffices to
pick a canonical time translation field
on
. The associated horizon energy
is then interpreted as the mass
. This suggests that the pair (
) be similarly used to make canonical choices
and
on
. Thanks to the black hole uniqueness theorems of the
4-dimensional Einstein-Maxwell theory, this strategy is again
viable.
Recall that the horizon surface gravity and the
horizon angular velocity in a Kerr solution can be expressed as a
function only of the horizon radius
and angular momentum
:
Given a cross section
of
, the idea is to consider the
unique Kerr solution in which the horizon area is given by
and angular momentum by
, and assign to
effective surface gravity
and angular
velocity
through
Repeating this procedure on every cross-section, one obtains
functions
and
on
, since
is a function of
alone. The definition of the effective surface
gravity then determines a function
of
and hence
uniquely. Thus, using
Equation (51), one can select a
canonical vector field
and Equation (49) then provides a
canonical balance law:
The key question is whether this equation is integrable, i.e., if
for some
which depends locally on fields defined on
. The answer is in the affirmative. Furthermore, the
expression of
is remarkably simple and is identified
with the horizon mass:
Thus, on any cross-section
,
is just the mass of
the Kerr space-time which has horizon area
and angular momentum
: As far as the
mass is concerned, one can regard the DH as an evolution through ‘a
sequence of Kerr horizons’. The non-triviality of the result lies
in the fact that, although this definition of mass is so
‘elementary’, thanks to the balance law (48) it obeys a Bondi-type
flux formula,
for a specific vector field
, where each term on the right has a well-defined
physical meaning. Thus, DHs admit a locally-defined notion of mass
and an associated, canonical conservation law (56). The availability of
a mass formula also provides a canonical integral version of the
first law through Equation (48):
The infinitesimal version of this equation yields the familiar
first law
.
On weakly isolated as well as dynamical horizons,
area
and angular momentum
arise as the
fundamental quantities and mass is expressed in terms of them. The
fact that the horizon mass is the same function of
and
in both dynamical and equilibrium situations is
extremely convenient for applications to numerical
relativity [34
]. Conceptually, this
simplicity is a direct consequence of the first law and the
non-triviality lies in the existence of a balance equation (48), which makes it
possible to integrate the first law.