Another motivation is to try to provide a solid classical basis for the microscopic understanding of black
hole entropy [47, 46
, 123
]: What are the microscopic degrees of freedom behind the phenomenological
notion of black hole entropy? Since the aim of the present paper is to review the construction of the
quasi-local quantities in classical general relativity, we discuss only the classical two-surface observables by
means of which the ‘quantum edge states’ on the black hole event horizons were intended to be
constructed.
If , the three-manifold on which the ADM canonical variables
,
are defined, has a smooth
boundary
, then the usual vacuum constraints
A similar analysis [499] shows that the basic Hamiltonian
The condition that the area 2-form should be fixed appears to be the part of the ‘ultimate’
boundary condition for the canonical variables. In fact, in a systematic quasi-local Hamiltonian analysis
boundary terms appear in the calculation of the Poisson bracket of two Hamiltonians also, which we called
Poisson boundary terms in Section 3.3.3. Nevertheless, as we already mentioned there, the quasi-local
Hamiltonian analysis of a single real scalar field in Minkowski space shows, these boundary terms represent
the infinitesimal flow of energy-momentum and relativistic angular momentum. Thus, they must be gauge
invariant [502
]. Assuming that in general relativity the Poisson boundary terms should have
similar interpretation, their gauge invariance should be expected, and the condition of their
gauge invariance can be determined. It is precisely the condition on the lapse and shift that the
spacetime vector field
built from them on the 2-surface must be divergence free
there with respect to the connection
of Section 4.1.2, i.e.,
. However, this is
precisely the condition under which the evolution equations preserve the boundary condition
. It might also be worth noting that this condition for the lapse and shift is just one of the
ten components of the Killing equation:
. (For the details,
see [502].)
It should be noted that the area 2-form on the boundary 2-surface appears naturally in connection
with the general symplectic structure on the ADM variables on a compact spacelike hypersurface
with
smooth boundary
. In fact, in [229] an identity is derived for the variation of the ADM
canonical variables on
and of various geometrical quantities on
. Examples are also
given to illustrate how the resulting ‘quasi-local energy’ depends on the choice of the boundary
conditions.
For the earlier investigations see [47, 46
, 123
], where stronger boundary conditions, namely fixing the
whole three-metric
on
(but without the requirement
), were used to ensure the
functional differentiability.
To understand the meaning of the observables (11.3, recall that any vector field
on
generates a
diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the
momentum constraint
is just to generate this gauge symmetry in the phase space.
However, the boundary
breaks the diffeomorphism invariance of the system, and hence, on
the boundary the diffeomorphism gauge motions yield the observables
and the gauge
degrees of freedom give rise to physical degrees of freedom, making it possible to introduce edge
states [47
, 46
, 123
].
Analogous investigations were done by Husain and Major in [281]. Using Ashtekar’s complex
variables [30] they determine all the local boundary conditions for the canonical variables
,
and
for the lapse
, the shift
, and the internal gauge generator
on
that ensure the functional
differentiability of the Gauss, the diffeomorphism, and the Hamiltonian constraints. Although there
are several possibilities, Husain and Major discuss the two most significant cases. In the first
case the generators
,
, and
are vanishing on
, and thus there are infinitely
many two-surface observables, both from the diffeomorphism and the Gauss constraints, but no
observables from the Hamiltonian constraint. The structure of these observables is similar to that of
those coming from the ADM diffeomorphism constraint above. The other case considered is
when the canonical momentum
(and hence, in particular, the three-metric) is fixed on the
two-boundary. Then the quasi-local energy could be an observable, as in the ADM analysis
above.
All of the papers [47, 46, 123, 281] discuss the analogous phenomenon of how the gauge freedoms
become true physical degrees of freedom in the presence of two-surfaces on the two-surfaces themselves in
the Chern–Simons and BF theories. Weakening the boundary conditions further (allowing certain
boundary terms in the variation of the constraints), a more general algebra of ‘observables’ can be
obtained [125
, 409
]. They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis
in [125
] is based on the covariant Noether-charge formalism below.) Since this algebra is well known in
conformal field theories, this approach might be a basis for understanding the microscopic origin of the
black hole entropy [124
, 125
, 126
, 409
, 127
]. However, this quantum issue is beyond the scope of the
present review.
Returning to the discussion of above, note first that, though
is a gauge potential, by
it is boost gauge invariant. Without this condition, Eq. (11.3
) would give potentially reasonable
physical quantity only if the boost gauge on
were geometrically given, e.g., when
were a
leaf of a physically-distinguished foliation of a physically-distinguished spacelike or timelike
hypersurface [39
]. In particular, the angular momentum of Brown and York [121] also takes
the form (11.3
), and is well defined (because
is assumed to be a Killing vector of the
intrinsic geometry of
). (In the angular momentum of Liu and Yau [338] only the gauge
invariant part of
is present in Eq. (11.3
) instead of
itself.) Similarly, the expressions
in [47, 571
] can also be rewritten into the form (11.3
), but they should be completed by the condition
.
In general Eq. (11.3) is used as a definition of the
–component of the angular momentum of
quasi-locally defined black holes [40
, 97
, 227
]. This interpretation is supported by the following
observations [499]. In axisymmetric spacetimes for axisymmetric surfaces
can be rewritten into the
Komar integral, the usual definition of angular momentum in axisymmetric spacetimes. Moreover, if
extends to spatial infinity, then
together with the requirement of the finiteness of the
limit of the observable
already fix the asymptotic form of
, which is precisely the
combination of the asymptotic spatial rotation Killing vectors, and
reproduces the
standard spatial ADM angular momentum. Similarly, at null infinity
must be a rotation
BMS vector field. However, the null infinity limit of
is sensitive to the first two terms
(rather than only the leading term) in the asymptotic expansion of
, and hence in general
radiative spacetime
in itself does not yield an unambiguous definition for angular
momentum. (But in stationary spacetimes the ambiguities disappear and
reproduces the
standard formula (4.15
).) Thus, additional ideas are needed to restrict the BMS vector field
.
Such an idea could be based on the observation that the eigenspinors of the -Dirac operators define
-divergence-free vector fields on
, and on metric spheres these vector fields built from the
eigenspinors with the lowest eigenvalue are just the linear combinations of the three rotation Killing
fields [501
]. Solving the eigenvalue problem for the
-Dirac operators on large spheres near scri in the
first two leading orders, a well-defined (ambiguity-free) angular-momentum expression is suggested. The
angular momenta associated with different cuts of
can be compared, and the angular momentum flux
can also be calculated.
It is tempting to interpret as the
-component of the quasi-local angular momentum of the
gravity + matter system associated with
. However, without additional conditions on
the integral
could be nonzero even in Minkowski spacetime [501]. Hence,
must
satisfy additional conditions. Cook and Whiting [153] suggest that one derive
from a
variational principle on topological two-spheres. Here the action functional is the norm of the
Killing operator. (For a viable, general notion of approximate Killing fields see [359].) Another
realization of the approximate Killing fields is given by Beetle in [59], where the vector field
is
searched for in the form of the solution of an eigenvalue problem for an equation, derived from
the Killing equations. Both prescriptions have versions in which they give
-divergence-free
. The definition of
suggested in [323] is based on the fact that six of the infinitely
many conformal Killing fields on
with spherical topology are globally defined, and after an
appropriate globally-defined conformal rescaling of the intrinsic metric they become the generators
of the standard
action on
. Then these three are used to define the angular
momentum that will be the Killing fields in the rescaled geometry. In general these vector fields are
not
-divergence-free. Thus, as in the Liu–Yau definition, to keep boost gauge invariance
the gauge invariant piece of the connection one-form
can be used instead of the
itself.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
![]() This work is licensed under a Creative Commons License. E-mail us: |