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Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity

01 Jun 1986-Communications in Mathematical Physics (Springer-Verlag)-Vol. 104, Iss: 2, pp 207-226
TL;DR: In this article, it was shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level.
Abstract: It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

Summary (2 min read)

Introduction

  • In general relativity and in other gauge theories formulated on noncompact ("open") spaces, the concept of asymptotic symmetry, or "global symmetry," plays a fundamental role.
  • Accordingly, the Poisson bracket of the generators of two given symmetries can differ by a constant from the generator associated with the Lie bracket of these symmetries.
  • Because the theory of central charges in classical mechanics is well known [7], the authors will only discuss here the aspects which are peculiar to gauge theories and asymptotic (as opposed to exact) symmetries.
  • This gives additional motivation for analyzing the canonical realization of the asymptotic symmetries on general grounds.

II. Solutions to 3-Dimensional Gravity with A < 0

  • This solution will help motivate their choice of appropriate boundary conditions to be imposed on the metric in general.
  • It is also interesting to note that, just as in the de Sitter case [13], a wedge cut from anti-de Sitter space provides a solution to Einstein's equations with the stress-energy tensor of a point mass.
  • So the curves which serve as the T , f , f coordinate lines for the metric (2.1) can always be singled out.
  • The Killing vector fields in this coordinate system are linear combinations of d/dt and d/dφ.
  • In practice, the charges J[ζ] are usually determined by looking at the surface terms coming from the variation of the "volume piece" (3.1) of the Hamiltonian, namely - lim^-^ίcn^i,*- ξ\kδgίj-]^2ξiδπ il-{-(2ξίπkl-ξlπik)δgik}, (3.3) r — » oo where Gijkl =^g1/2(gikgjl + gilgjk — 2gίjgkl) and the semicolon denotes covariant differentiation within a spacelike hypersurface.

IV. The Conformal Group of Asymptotic Symmetries

  • It is natural to question whether the restriction of the metric to the form (2.2) outside sources is too severe.
  • Ideally, the boundary conditions could be weakened just enough so that the group of asymptotic symmetries is enlarged to the anti-de Sitter group in 2 + 1 dimensions, namely 0(2,2).
  • As will be shown below, such deformation vectors have no associated charge and the generators of these deformations vanish weakly.
  • To insure that spacelike surfaces initially obeying the boundary conditions (4.4) and (4.10) will preserve these boundary conditions under deformations generated by the Hamiltonian, it is necessary to impose further restrictions on the canonical variables [15].
  • Actually, the dependence of ξμ on the canonical variables is not relevant in establishing (4.11) as the proper surface integral to appear in the Hamiltonian, or in evaluating the charges for a spacetime such as (4.2), because for these purposes, ξμ is only needed to leading order in 1/r.

V. The Canonical Realization of Asymptotic Symmetries

  • The primary goal of this article is to point out the possible existence of central charges in the canonical realization of asymptotic symmetries.
  • Then the generator obtained by computing the Poisson bracket under the assumption that ξ and η describe pure gauge deformations can only differ from the generator which would be obtained without this assumption by terms which vanish when ξ and η are pure gauge.
  • Since the leading order terms of all conformal group vectors are independent of the canonical variables, it follows that δηξ μ and δξη μ make only higher order contributions to ζμ in Eq. (5.3).
  • The final step in the demonstration that (5.2) is a central extension of the conformal group algebra is to show that, to leading order in 1/r, the surface deformation algebra [ζ,η~\SD coincides with the Lie algebra [£,77] for conformal group vectors ξ and η.
  • In the present case where the asymptotic symmetries cannot be realized as exact symmetries of some background, it is easy to see that the central charges are not trivial.

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Communications
in
Commun.
Math.
Phys.
104,
207-226
(1986)
Mathematical
Physics
©
Springer-Verlag
1986
Central
Charges
in the
Canonical Realization
of
Asymptotic
Symmetries:
An
Example
from
Three
Dimensional
Gravity
J.
D.
Brown
and
Marc
Henneaux*'**
Center
for
Theoretical Physics,
The
University
of
Texas
at
Austin, Austin,
Texas
78712,
USA
Abstract.
It is
shown that
the
global charges
of a
gauge theory
may
yield
a
nontrivial central extension
of the
asymptotic symmetry algebra already
at the
classical level. This
is
done
by
studying three dimensional gravity with
a
negative
cosmological constant.
The
asymptotic symmetry group
in
that case
is
either
R x
SO(2)
or the
pseudo-conformal group
in two
dimensions, depending
on the
boundary conditions
adopted
at
spatial
infinity.
In the
latter situation,
a
nontrivial
central charge appears
in the
algebra
of the
canonical generators,
which
turns
out to be
just
the
Virasoro central charge.
I.
Introduction
In
general relativity
and in
other gauge theories formulated
on
noncompact
("open")
spaces,
the
concept
of
asymptotic symmetry,
or
"global symmetry," plays
a
fundamental
role.
The
asymptotic symmetries
are by
definition those gauge transformations which
leave
the field
configurations under consideration asymptotically invariant,
and
recently,
it has
been explicitly shown that they
are
essential
for a
definition
of the
total ("global") charges
of the
theory
[1,2].
(For earlier connections between
asymptotic
symmetries
and
conserved quantities
in the
particular case
of
Einstein
theory,
see
[3,4]
and
references therein.)
The
basic
link
between asymptotic symmetries
and
global charges
has
been
emphasized again
in
recent papers dealing
with
the
monopole sector
of the
SU(5)
grand
unified
theory
[5] and
with
D = 3
gravity
and
supergravity [6], where
it is
confirmed
that
the
absence
of
asymptotic symmetries prohibits
the
definition
of
global charges.
In the first
instance,
the
unbroken symmetry group
of the
monopole
solution
is not
contained
in the set of
asymptotic symmetries because
of
topological
obstructions. This forbids
the
definition
of
meaningful global color charges
*
Permanent address:
Faculte
des
Sciences,
Universite
Libre
de
Bruxelles, Campus Plaine C.P. 231,
B-1050
Bruxelles, Belgium
**
Chercheur
qualifie
du
Fonds
National
Beige
de la
Recherche
Scientifique

208 J. D.
Brown
and M.
Henneaux
associated with
the
unbroken group.
In the
second
case,
the
nontrivial
global
properties
of the
conic geometry, which
describes
the
elementary solution
of D = 3
gravity,
prevents
the
existence
of
well defined spatial translations
and
boosts,
and
hence,
also
of
meaningful linear momentum
and
"Lorentz
charge."
In
the
Hamiltonian formalism,
the
global charges
appear
as the
canonical
generators
of the
asymptotic symmetries
of the
theory: with each such infinitesimal
symmetry
ξ is
associated
a
phase space
function
H[£]
which generates
the
corresponding transformation
of the
canonical variables.
It is
generally taken
for
granted
that
the
Poisson
bracket
algebra
of the
charges
H[ζ]
is
just
isomorphic
to
the
Lie
algebra
of the
infinitesimal asymptotic symmetries,
i.e.,
that
The
purpose
of
this
paper
is to
analyze this question
in
detail.
It
turns
out
that, while (1.1) holds
in
many important examples,
it is not
true
in
the
generic case. Rather,
the
global charges only yield
a
"projective"
representation
of
the
asymptotic symmetry group,
{HK],HM}=fl[K,»y]]
+
XK,ι/].
(1.2)
In
(1.2),
the
"central
charges"
K[£,
77],
which
do not
involve
the
canonical variables,
are in
general nontrivial, i.e., they cannot
be
eliminated
by the
addition
of
constants
C
ξ
to the
generators
H\_ξ].
The
occurrence
of
classical central charges
is by no
means peculiar
to
general
relativity
and
gauge theories,
and
naturally arises
in
Hamiltonian classical
mechanics
([7]
appendix
5). It
results from
the
non-uniqueness
of the
canonical
generator associated with
a
given (Hamiltonian) phase space vector
field.
Indeed,
this
generator
is
only determined
up to the
addition
of a
constant, which commutes
with
everything. Accordingly,
the
Poisson
bracket
of the
generators
of two
given
symmetries
can
differ
by a
constant
from
the
generator associated with
the Lie
bracket
of
these symmetries.
A
similar phenomenon occurs with asymptotic symmetries
in
gauge theories.
In
that
case,
the
Hamiltonian generator
//[ξ]
of a
given asymptotic symmetry
ξ
A
differs
from
a
linear combination
of the
constraints
φ
A
(x)
of the
canonical formalism
by a
surface
term
J[ξ\
which
is
such that
H\_ζ]
has
well defined
functional
derivatives
[8],
nm
=
ld«xξ*(x)φ
A
(x)
+
J[ζ\.
(1.3)
But
this
requirement
fixes
J[£],
and
hence
//[£],
only
up to the
addition
of an
arbitrary constant. This ambiguity signals
the
possibility
of
central charges.
Because
the
theory
of
central charges
in
classical mechanics
is
well known
[7],
we
will
only discuss here
the
aspects which
are
peculiar
to
gauge theories
and
asymptotic
(as
opposed
to
exact) symmetries. This will
be
done
by
treating three
dimensional
Einstein gravity
with
a
negative cosmological constant
A in
detail.
In
that
instance,
we
show that
the
asymptotic symmetry group
is
either
R x
SO(2),
or
the
conformal group
in two
dimensions, depending
on the
boundary conditions

Global
Charges
and
Asymptotic
Symmetry
209
adopted
at
spatial
infinity.
In the
latter case,
a
nontrivial central
charge—actually
familiar
from
string theory
[9]—appears
in the
Poisson bracket algebra
of the
canonical
generators.
Three dimensional gravity with
A < 0 is
presented here primarily
to
provide
an
example
of
central
charges
in the
canonical
realization
of
asymptotic
symmetries.
However,
the
study
of
three dimensional gravity
is not
entirely academic
and
possesses some intrinsic interest apart
from
its
connection with central charges.
Indeed,
previous
experience with
gauge
theories
has
indicated
that
something
can be
learned
from
lower dimensional models about both
the
classical
and
quantum
aspects
of the
more complicated
four
dimensional theory.
In the
gravitational case,
three
is the
critical number
of
dimensions, since
in
fewer
dimensions there
is no
Einstein
theory
of the
usual type (i.e., with
a
local action principle involving only
the
pseudo-Riemannian
metric). Thus,
it is
natural
to
turn
to
three dimensional models
in
an
effort
to
better understand Einstein gravity
in
higher dimensions.
The
discussion involves some subtleties because
the
constraint
algebra
of
general
relativity
is not a
true
algebra,
but
rather, contains
the
canonical variables. This
fact
has two
implications:
(i) the
algebra
of the
asymptotic symmetries
is a
true algebra
only
asymptotically; (ii) standard group theoretical arguments cannot
be
used
in a
straightforward
way.
In
the
course
of our
study,
we
shall rely
on a
useful
theorem which
is
proved
in
[10]
and
concerns Hamiltonian dynamics
on
infinite
dimensional phase
spaces.
This
theorem
establishes, under appropriate conditions, that
the
Poisson bracket
of two
differentiable
functionals
contains
no
unwanted surface term
in its
variation,
and
therefore
has
well
defined
functional
derivatives. This property
is
used
to
prove that
the
Poisson bracket
of the
asymptotic symmetry generators yields
a
(trivial
or
nontrivial)
project!ve
representation
of the
asymptotic symmetry
group.
It
should
be
stressed that
the
techniques developed here
in
treating three dimensional gravity
are
quite
general
and can be
applied,
for
instance,
to
four dimensional gravity
to
prove
a
similar
representation theorem. Such
a
theorem
has
been implicitly used,
but not
explicitly
demonstrated,
for
example
in
[8,12].
The
example
of
three dimensional
gravity
with
a
negative cosmological constant
also
demonstrates
the key
role played
by
boundary conditions, which determine
the
structure
of the
asymptotic symmetry
group
but are not
entirely dictated
by the
theory.
(This
was
also
pointed
out in
[11].)
As
a final
point,
let us
note that
the
existence
of a
true central charge
can be
ruled
out
in the
particular case when
the
asymptotic symmetries
can be
realized
as
exact
symmetries
of
some
background
configuration. Indeed,
in
this situation
the
charges
evaluated
for
that background
are
invariant under
an
asymptotic symmetry
transformation,
since
the
background
itself
is
left
unchanged.
By
adjusting
the
arbitrary
constant
in
H[ζ]
so
that
H[ξ~]
(background)
=
0, Eq.
(1.2)
shows that
K\_^η]
vanishes. However,
the
important case
of
"background symmetries" does
not
exhaust
all
interesting applications
of the
asymptotic symmetry concept.
For
example,
the
infinite
dimensional
B.M.S.
group
[3,4]
cannot
be
realized
as the
group
of
isometries
of
some
four
dimensional metric. This gives additional
motivation
for
analyzing
the
canonical realization
of the
asymptotic symmetries
on
general grounds.

210 J. D.
Brown
and M.
Henneaux
II.
Solutions
to
3-Dimensional
Gravity with
A < 0
This section provides
a
discussion
of a
solution
to
Einstein gravity
in 2 + 1
dimensions with
a
negative cosmological constant. This solution will help motivate
our
choice
of
appropriate boundary conditions
to be
imposed
on the
metric
in
general.
In
three dimensions,
the
gravitational
field
contains
no
dynamical
degrees
of
freedom,
so
that
the
spacetime away
from
sources
is
locally equivalent
to the
empty
space
solution
of
Einstein's equations, namely anti-de Sitter
space
when
A < 0.
This
is
demonstrated
by
noting that
the
full
curvature tensor
can be
expressed
in
terms
of
the
Einstein tensor,
and
where
the
empty
space
Einstein equations hold,
the
curva-
ture tensor reduces
to
that
of
anti-de Sitter
space.
Matter, which
is
assumed
to be
localized,
has no
influence
on the
local geometry
of
the
source
free
regions,
and
therefore
can
only
effect
the
global
geometry
of the
spacetime.
The
basic solution which
we
consider then
is
locally anti-de Sitter
space
with
radius
of
curvature
R = (
l//i)
1/2
,
(2.1)
but
with
an
unusual
identification
of
points which
will
alter
the
global geometry.
By
identifying
the
points
=
f
',
r
=
r',φ
=
φ')
with
the
points
=
t'
2πA,
f =
r',
$
φ'
+
2πα)
for all
f',
r'
and
φ',
this will have
the
effect
of
removing
a
"wedge"
of
coordinate
angle
2π(l
α)
and
introducing
a
"jump"
ofA in
coordinate
time.
Because
the
Ricci
tensor
is
defined locally,
it is not
modified
by
this unusual
identification
except
at the
origin
r = 0.
Hence,
the
vacuum Einstein equations will
be
satisfied
everywhere except
at the
origin.
The
motivation
for
considering
the
spacetime just described
is
that
it is the
analogue
of the
conic geometry
for 2 + 1
gravity with
Λ = 0
[12],
for
which
the
wedge
α
Φ 1 is
related
to
total energy
and the
jump
A
Φ
0 is
related
to
total angular
momentum.
It is
also
interesting
to
note that,
just
as in the de
Sitter
case
[13],
a
wedge
cut
from
anti-de Sitter
space
provides
a
solution
to
Einstein's equations with
the
stress-energy tensor
of a
point
mass.
The
metric (2.1)
can
also
be
assumed
to
apply
to the
empty region exterior
to a
more general compact source distribution.
The
geometrically invariant character
of the
wedge
and the
jump
can be
seen
in
the
following
way
which
does
not
depend
on the
details
of the
interior
to the
spacetime containing
the
source. First note that even though
the
spacetime
is
locally
maximally
symmetric,
the
only Killing vector
fields
consistent
with
the
unusual
identification
of
points
are
linear combinations
of
d/dϊand
d/dφ.
The
vectors
d/dϊ
and
d/df
can be
singled
out
uniquely
(to
within
normalization constants)
as the
only
two
Killing vector
fields
which
are
everywhere
orthogonal
to one
another.
To
within
a
normalization,
d/df
is the
unique vector
field
everywhere orthogonal
to all
Killing
vector
fields.
So the
curves which serve
as the
T,f,f
coordinate lines
for the
metric
(2.1)
can
always be singled out. Furthermore, consider the proper length
L
of the curve of a
trajectory
of
d/d(β
between points
of
intersection with
a
trajectory
of
d/df.
The

Global Charges
and
Asymptotic Symmetry
211
change
dL as the
curve
is
moved
a
proper distance
dS
along
the
direction
d/df
equals
dL
For
α
< 1, the
length
L
increases more slowly with
proper
distance than
if the
space
were globally anti-de Sitter. Finally,
the
jump
A is
proportional
to the
proper time
distance between points
of
intersection
of the
trajectories just considered.
From
now on, it
will
be
more convenient
to
write
the
metric
(2.1)
with
a
continuous time variable.
The
coordinate
transformation
t =
F-h
(A/<x)ij)
9
r =
f,
φ
=
(l/α)<^
yields
dS
2
= -
(
ζϊ
+
1
)
(dt
-
Adφ)
2
+
(ί+l]
V
4-
*
2
r
2
2
,
(2.2)
\
R
J
\
R
/
where
φ has
period
, and
there
is no
jump
in the new
time.
The
Killing vector
fields
in
this
coordinate
system
are
linear combinations
of
d/dt
and
d/dφ. Also note that
the
trajectories
of
d/dφ will form
closed
timelike
lines unless
\A\
<
α|K|
and
A
2
R
2
r
>
z
2
R
2
-A
2
'
As a
result,
the
spacetime constructed represents
a
reasonable solution
to
Einstein
gravity only
for
\A\
<α|R|
and
large
values
of r; in
particular
it is
valid
in the
asymptotic limit
r->
oo.
III.
Global
Charges
and the
RxSO(2)
Asymptotic
Symmetries
The
procedure
for
obtaining
the
global charges
of a
gauge
theory within
the
Hamiltonian formalism
has
been well established
[8].
The
first
step
is to
define
the
boundary conditions
at
spatial
infinity
which
the
generic
fields
should obey,
and
then
identify
the
asymptotic symmetries which preserve this asymptotic behavior.
Of
course,
for
gravity theories
in
particular,
in
order
to
continue
with
the
Hamiltonian formulation,
the
boundary conditions
on the
spacetime metric must
be
converted into boundary conditions
on the
canonical variables
g
ij9
π
ij
.
Likewise,
the
asymptotic symmetries
of the
spacetime determine
the
allowed surface deformation
vectors
ξ
μ
(μ =
_L,
i) for the
space-like hypersurfaces under consideration.
Now,
for the
boundary conditions
and
asymptotic symmetries
of a
gravitation
theory
to be
acceptable,
it
must
be
possible
to
write
the
Hamiltonian
as the
usual
linear combination
of
constraints
[14]
fd"x^(x)Jf
μ
(x)
(3.1)
plus
an
appropriate surface term
J[_ζ].
This surface term
J[£],
which
will
be
referred
to as the
charge
from
now on,
must have
a
variation which will cancel
the
unwanted
surface
terms
in the
variation
of
(3.1). Then
the
Hamiltonian,
HK]=Jd-x^(x)^(x)-hJK],
(3.2)
will
have well defined variational derivatives,
and may be
used
as the
generator
of
the
allowed surface deformations.

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Journal ArticleDOI
TL;DR: It is argued that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy.
Abstract: A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.

4,395 citations

Journal ArticleDOI
TL;DR: The correspondence between supergravity and string theory on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamic properties of Schwarzschild black holes in Anti-de Sitter space as mentioned in this paper.
Abstract: The correspondence between supergravity (and string theory) on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic connement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large AdS Schwarzschild black hole must scale \holographically" with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large N gauge theories in four dimensions without supersymmetry.

4,209 citations


Cites methods from "Central Charges in the Canonical Re..."

  • ...For 2+1-dimensional black holes, in the context of an old framework [67] for a relation to boundary conformal field theory which actually is a special case of the general CFT/AdS correspondence, such additional information is provided by modular invariance of the boundary conformal field theory [65,66]....

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Journal ArticleDOI
TL;DR: In this paper, the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space is computed, and the conformal anomalies in two and four dimensions are recovered.
Abstract: We propose a procedure for computing the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the AdS/CFT correspondence, our classical result is interpretable as the expectation value of the stress tensor in a quantum conformal field theory. We demonstrate that the conformal anomalies in two and four dimensions are recovered. The two dimensional stress tensor transforms with a Schwarzian derivative and the expected central charge. We also find a nonzero ground state energy for global AdS5, and show that it exactly matches the Casimir energy of the dual super Yang–Mills theory on S 3×R.

2,433 citations

References
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Book
01 Jan 1974
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.

11,008 citations

Journal ArticleDOI
TL;DR: In this article, it is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral.

1,365 citations


"Central Charges in the Canonical Re..." refers background or methods in this paper

  • ...Such a theorem has been implicitly used, but not explicitly demonstrated, for example in [8,12]....

    [...]

  • ...In that case, the Hamiltonian generator //[ξ] of a given asymptotic symmetry ξ differs from a linear combination of the constraints φA(x) of the canonical formalism by a surface term J[ξ\ which is such that H\_ζ] has well defined functional derivatives [8],...

    [...]

  • ...The procedure for obtaining the global charges of a gauge theory within the Hamiltonian formalism has been well established [8]....

    [...]

  • ...This gives an additional motivation for imposing extra boundary conditions to eliminate the supertranslation ambiguities [8,4]....

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  • ...The asymptotic symmetry group is the infinite dimensional "Spi group" [4] as long as the behavior of the gravitational variables at spatial infinity is not restricted by means of parity conditions as in [8]....

    [...]

Journal ArticleDOI
L. F. Abbott1, Stanley Deser1
TL;DR: In this article, the stability properties of the cosmological constant Λ with respect to the de Sitter background were investigated and the effects of an event horizon, which leads to Hawking radiation, are expressed for general field hamiltonians.

1,094 citations


"Central Charges in the Canonical Re..." refers background in this paper

  • ...The asymptotic symmetries are by definition those gauge transformations which leave the field configurations under consideration asymptotically invariant, and recently, it has been explicitly shown that they are essential for a definition of the total ("global") charges of the theory [1,2]....

    [...]

Frequently Asked Questions (12)
Q1. What have the authors contributed in "Central charges in the canonical realization of asymptotic symmetries: an example from three dimensional gravity" ?

This is done by studying three dimensional gravity with a negative cosmological constant. 

The motivation for considering the spacetime just described is that it is the analogue of the conic geometry for 2 + 1 gravity with Λ = 0 [12], for which the wedge α Φ 1 is related to total energy and the jump A Φ 0 is related to total angular momentum. 

Then the asymptotic symmetries coincide with the Killing vector fields d/dt and d/dφ, and the asymptotic symmetry group associated with these boundary conditions is R x SO(2). 

Then the Hamiltonian,HK]=Jd-x^(x)^(x)-hJK], (3.2)will have well defined variational derivatives, and may be used as the generator of the allowed surface deformations. 

The only nonzero components of the canonical variables needed for computing expression (3.3) arer κ*4 ,2r*(3.5)which gives- δJ[_ζ] = 4π[(3)<f <Sα - (3)ξφδ(aA) 

The asymptotic symmetries are canonically realized by the "factor group" of surface deformation generators, which is defined by identifying two Hamiltonian generators if they describe the same asymptotic (conformal group) deformation and differ only by a pure gauge deformation. 

for the boundary conditions and asymptotic symmetries of a gravitation theory to be acceptable, it must be possible to write the Hamiltonian as the usual linear combination of constraints [14]fd"x^(x)Jfμ(x) (3.1)plus an appropriate surface term J[_ζ]. 

By identifying the points (Γ= f ', r = r',φ = φ') with the points (Γ= t' — 2πA, f = r', $ — φ' + 2πα) for all f', r' and φ', this will have the effect of removing a "wedge" of coordinate angle 2π(l — α) and introducing a "jump" of 2πA in coordinate time. 

In this case, the central charge K[ξ, η] reduces to the value of the charge J\\_ζ] on the surface deformed by η.To evaluate ./[£] on the deformed surface, the expression (4.11) can be greatly simplified by specializing to r, φ coordinates and using the known asymptotic form of the canonical variables. 

The critical step in this analysis is to recognize that the "volume" term of the Poisson bracket (5.2) may be calculated by assuming that ξ and η are pure gauge, in which case the charges vanish. 

From (4.3,4.4), the lapse and shift are determined to be9 \\ (4.9)Kso that the asymptotic behavior of the canonical variables is given by Eqs. (4.4) along withπrr = 0(l/r), π^ = 0(l/r2), π++ = 0(l/r5). (4.10)However, in the canonical formalism, the spacelike surfaces are evolved according to Hamiltonian evolution, which generally differs from Lie transport unless the spatial Einstein equations (3)Gij = Λgij hold. 

The lapse and shifts and computed straightforwardly from (2.2); in particular,Γ Γ2 + R2 HI/2 Γ A2R2 Π-1/2αA(r2 + R2) ''r2(a2R2-A2)-A2R2'and, since (3)<f = 0, the component ξr = 0 always.