Present address: UCLA, Physics Department, 405 Hilgard Ave., Los Angeles, CA 90024 OCR Output
Lausanne (D.C.)
der contract #DE-AC02-76ER.03069 and in part by the Fondation du 450e anniversaire de l’Université de
This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) un
CTP #2245
hep-th/yymmnnn
Submitted to: Physical Review D
dimensional gravity is derived and compared with the ADM definition of energy.
expression for energy in a gauge theoretical formulation of the string—inspired 1+1
the I S O(2, 1) gauge theoretical formulation of Einstein gravity. In addition, an
dimensions, expressions are obtained for energy and angular momentum arising in
the gravitational Einstein—Hilbert action is derived and discussed in detail. In 2+1
procedure in 3+1 dimensions, a symmetric energy—momentum (pseudo) tensor for
symmetry. Using Noether’s theorem and a generalized Belinfante symmetrization
We discuss general properties of the conservation law associated with a local
ABSTRACT
P@¤B19959
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QERN |.IBRF·lRIE5» GENEVH
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 U.S.A.
Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics
Dongsu Bak, D. Cangemi,l and R. J ackiw
IN GRAVITY THEOR.IES*
05 P
ENERGY-MOMENTUM CONSERVATION
\{_L x `” '~/1 V cvs`?) ;-»3,4/
-lOCR Output
conserved Noether current associated with diffeomorphism invariance.
compared with other definitions that have appeared in the literature. Also we remark on the
metric energy-momentum (pseudo) tensor, as an improved Noether current, is derived and
In Section III, the 3+1 dimensional Einstein-Hilbert action is investigated and a sym
( "improvement”
associated with a local symmetry and also symmetrization of the energy-momentum tensor
In Section II, we analyze in a systematic way general properties of the Noether charge
definition;5 we compare these two approaches.
Another way of finding an expression for energy in 1+1 dimensions is to use the ADM
gravity model‘* and obtain an expression for energy arising from the gauge transformations.
ln 1+1 dimensions, we consider a gauge theoretical formu1ation3 of the string-inspired
group gauge transformations are identified as energy and angular momentum.
based on the Poincaré group [ISO(2, The Noether charges associated with the Poincaré
are not valid.] On the other hand, there is a gauge theoretical formulation of the theory,2
In the 2+1 dimensional Einstein gravity, asymptotically Minkowski boundary conditions
tensors.
and which is derived without any statement about "background” or “asymptotic” metric
given by the Noether procedure rather than by manipulation of the field equations of motion,
for the symmetric energy-momentum (pseudo) tensor, which is conserved as in (1), which is
symmetric under interchange of two spacetime indices. Our goal is thus to find an expression
an energy-momentum (pseudo) tensor, the energy-momentum (pseudo) tensor needs to be
of global Poincaré transformations. To express the angular momentum solely in terms of
that we can associate energy, momentum and angular momentum with the Noether charges
In 3+1 dimensions, asymptotically Minkowski boundary conditions can be posed, so
can be viewed as “global” transformations.
it is invariant under Poincaré transformations, which comprise special diifeomorphisms and
action is invariant under diffeomorphisms, which are local transformations; more specifically,
action, in which case the conserved current is called a N octher current. The Einstein-Hilbert
In field theory, conservation equations are usually related to invariance properties of the
continuity equation we always need to specify asymptotic behavior.
satisfy suitable boundary conditions. In other words, to get a conserved quantity from a
provided f8VdS { j i vanishes at infinity. Therefore to insure conservation of Q, j { has to
Q = dw
<2>
f v
which leads to a conserved quantity,
(1)
@»j" = 0
to some form of continuity equation,
a long time. The problem is to find an expression that is physically meaningful and related
The definition of energy and momentum in general relativity has been under investigation for
OCR OutputI. INTRODUCTION
-2OCR Output
does not lead to a conserved quantity. Moreover, even if we get a finite value for Q with
Without suitable boundary conditions, this charge either diverges or vanishes, and in general
(8)
Q = dV8, [Fi0(:z:)9(z)] = dSiFi°(z)6(:c)
/ av
/ v
The Noether charge is constructed as a volume integral of the time component j
indices p and u.
since the quantity in the parenthesis of (7) is antisymmetric under the interchange of the
which is certainly identically conserved, regardless whether F"' satisfies the field equations,
J" = @» (F""9)
(7)
@»F”" = 2C Im [(D”¢)`¢]
(6)
can be written with use of the equation of motion
= —F""6,,8 — (D"¢)*ie0q$ + D"q5ie0¢’
<5>
MA, em w..¢··
'M = ——6A,, —-—6 -;-6 * ’ 1 " + ‘”
GL OL
85
The associated Noether current
= A,,(a:) + 3,,6(a:), 6A,, = 8,,6
¢;(x) = c-€€¢(x)7 :
¢(I)
local U (1) gauge symmetry.
where D,,¢ E (6,, + ieA,,)¢ and F,,,, E 8,,A,, — 8,,A,,. The Lagrangian L is invariant under a
(3)
LZ = --}F,,,,F"" + (D"¢)*D,,q5
with a Lagrange density
To illustrate the result in a special example, let us consider the Maxwell—scalar system,
into field theory textbooks — so we give a general proof in the Appendix.
conserved currents associated with a global symmetry, her argument has not found its way
that is identically conserved. This was shown by E. Noether,6 but unlike the construction of
The Noether current associated with a local symmetry can always be brought to a form
II. CONSERVATION LAWS
Concluding remarks comprise the final Section VI.
After getting an expression for energy, we show that it agrees with the ADM energy.
In Section V, we consider a gauge theoretical formulation of 1+1 dimensional gravity.
in the context of the gauge theoretical formulation for the theory.
sional Einstein gravity, we obtain in Section IV expressions for energy and angular momentum
Since asymptotically Minkowski boundary conditions can not be imposed in 2+1 dimen
-3
u f" E e“u zz + a"
(14) OCR Output
eil: = 3 PC t (f)
be written as a total derivative without using the equations of motion.
where 6¢ denotes q5’(x) —¢$(:z:). Since the action is Poincaré invariant by hypothesis, 6.C can
Oo 88,,425 . @6,,0,49
6.C=-6 --66 —66,,6 ¢+ liqi + li Qs
13 ( )
GL
OL OL
the transformations (11),
To derive the Noether current, let us consider the variation of the Lagrange density under
j
El= “€Ji, = *5%
(12)
¤°; = ¢'0» S". = $3
600 = SOO = 0
group and the constants e",,, S"I, satisfy the following relations.
where L, containing the spin matrix S (L = 1 + %e",,S"I,), is a representation of the Lorentz
I I ¢(¤) —>¢ (¤= ) = L¢(<¤)
(11)
:1:** —+ :z:"‘ = x" — e",,r" — a"
respectively by
Under the infinitesimal action of these transformations, coordinates and fields transform
where 45 is a multiplet of fields, and suppose I is invariant under Poincaré transformations.
(10)
I= / d¤¢£(¢»@»¢»@»@»¢)
Thus, consider
derivatives, as is true of the Einstein-Hilbert action.
theory. Here, we generalize his method to the case that the Lagrangian contains second
was originally presented by Belinfante,7 and which is always available in a Poincaré invariant
Next, let us review the symmetrization procedure of the energy-momentum tensor which
from a; global transformation.
extended through all space, thereby arriving at a Noether formula for the total charge arising
ensures that Q is time independent. The asymptotic condition that 9 be constant can be
The first condition gives finite Q when 9 is constant at infinity, and the second condition
60 F°' ~ 0
a.s 1* —> 00
(9)
Fm ~ M5)
An example of boundary conditions for (7) is
condition.
some 0(2:), the time dependence of Q is completely determined by the specified boundary