Table Of ContentLecture Notes
ni Physics
detidE yb .H Araki, ,otoyK .I. ,srelhE ,nehcn0M .K Hepp, Z0rich
.R ,nhahneppiK ,nehcn0M .H .A ,re110mnedieW grebledieH
dna .I. Zittartz, n10K
202
citotpmysA roivaheB fo Mass
dna Spacetime yrtemoeG
Proceedings of the Conference
Held at Oregon State University
Corvallis, Oregon, USA, October 17-21, 1983
Edited by Francis .1. Flaherty
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
rotidE
Francis .J ytrehalF
of Oregon Department Mathematics, State ytisrevinU
Oregon Corvallis, g7331-4605,USA
ISBN 3-540-13351-8 Heidelberg Berlin Springer-Verlag New York oykoT
ISBN 0-387-13351-8 Springer-Verlag New Heidelberg Berlin York oykoT
Library of Congress Cataloging in Publication Data. Main entry under title: Asymptotic behavior
of mass and spacetime geometry. (Lecture notes in physics; 202) .1 Space and time-Congres-
ses. 2. Geometry-Congresses. 3. Black holes (Astronomy)-Congresses. 4. Gauge fields
(Physics)-Congresses. 5. Mass (Physics)-Congresses. .I E.I. (Francis Flaherty, ,).J 1935-.
.II Series. CIC173.59.$65A89 1984 530.1'1 84-14072
ISBN 0-38?-13351-8 (U.S.)
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translationr,e printing, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2153/3140-543210
TABLE OF CONTENTS
G.T. Horowitz:
The Positive Energy Theorem and Its Extensions .................. 1
R. Penrose:
Mass and Angular Momentum at the Quasi-Local Level in General
Relativity ..................................................... 23
M.J. Perry:
The Positive Mass Theorem and Black Holes ...................... 31
C.H. Taubes:
Exotic Differentiable Structures on Euclidean 4-Space .......... 41
C.N. Kozameh, E.T. Newman:
A New Approach to the Vacuum Einstein Equation ................. 45
R. Bartnik:
The Existence of Maximal Surfaces in Asymptotically Flat
Spacetimes ..................................................... 57
Y. Choquet-Bruhat:
Causality of Classical Supergravity ............................ 61
C.H. Taubes:
A Gauge Invariant Index Theorem for Aysmptotically Flat
Manifolds ...................................................... 85
A. Ashtekar:
On the Boundary Conditions for Gravitational and Gauge Fields at
Spatial Infinity ............................................... 95
J.W. York, Jr.:
Metric Fluctuations and the Entropy of Black Holes. ........... iii
J.S. Welling:
The Numerical Characteristic Initial Value Problem ............ 123
A.D. Kulkarni:
Time-Asymmetric Initial Data for n Black Holes ................ 143
VI
J.M. Nester:
The Gravitational Hamiltonian ................................. 155
W'T. Shaw:
Twistors, Asymptotic Symmetries and Conservation Laws at Null and
Spatial Infinity ............................................... 165
B. Jeffryes:
Two-Saface Twistors and Conformsl Embedding ................... 177
V. Moncrief:
Remarks on the Cosmic Censorship Conjecture ................... 185
H. Friedrich:
On Some (Con-)formal Properties of Einstein's Field Equations
and Their ConSequences ......................................... 197
List of Talks .................................................... ..209
List of Participants ............................................... 211
Introduction
The proof of a long standing conjecture can often result in a
burst of nascent energy strong enough to affect many different areas.
And so it has been with the recent proof by Schoen-Yau and Witten of
the positive mass theorem. To capture some of this energy, a group of
mathematicians and physicists met during the week of October 17, 1983
in Corvallis, Oregon.
Gary Horowitz's survey talk lays the foundation for most of the
articles in this volume. Since the positive mass theorem really says
something about initial data sets, their occurrence is pervasive and
the talks by Bartnik, Newman, Perry, and Piran confirm this. How mass
enters into other gauge theories is dealt with by Ashtekar, Choquet-
Bruhat, Penrose, Taubes, and York.
To complement the principal talks three workshops were organized.
The workshop on cosmic censorship contains a general description of
the problem (Moncrief) as well as specific progress (Moncrief and
Friedrich). The workshop on mass (Isenberg) complements the Witten
approach to the proof of positive energy (Nester) as well as Penrose's
quasi-local construction (Jeffryes and Shaw). Finally the workshop on
numerical relativity (Evans) contained discussion on black holes
(Evans) as well as consideration of the characteristic initial value
problem (Welling) and the two-body problem .A( Kulkarni). Professor
Abraham Taub closed the conference with a rousing summary of what went
on and what should go on henceforth.
The format of the conference was specifically designed to
encourage a free exchange of ideas. Many spontaneous and impromptu
discussions occurred, one of which is included in the proceedings -
the talk on fake ~'s by Taubes.
There are many people to thank for the success of the conference:
first my cohorts on the organizing committee - Dieter Brill, who took
notes at all the talks; Jim Isenberg, who supplied the questions; and
Ronnie Wells, who had done it all before, secondly, Charles Evans, Jim
Isenberg and Vince Moncrief, for stimulating workshops. To Barbi
Donnell, without whose help no-one would have found Corvallis and Ed
Cracraft without whose help no-one would have eaten. Finally special
thanks are due to Jana Meehan and Barbara Heusser for the excellent
job of typing these proceedings.
To produce a uniform typescript all of the papers were retyped in
Corvallis and as a result the editor is responsible for any resulting
bugs.
IV
The National Science Foundation and the Research Council of
Oregon State University supported the conference by constant
encouragement and grants.
F.J. FLAHERTY
CORVALLIS, OREGON
THE POSITIVE ENERGY THEOREM AND ITS EXTENSIONS
Gary T. Horowitz
Department of Physics
University of California
Santa Barbara, CA 93106
Expanded version of talk given at the conference on "Asymptotic Behavior
of Mass and Spacetime Geometry," Corvallis, Oregon, October 1983.
.I Introduction
Over the past few years there has been a dramatic increase in our
understanding of gravitional energy. This has resulted mainly from the
proofs of several long-standing conjectures in general relativity.
Roughly speaking, these results state that one cannot construct an object
out of ordinary matter, i.e., matter with positive local energy density,
whose total energy including gravitational contributions is negative.
Recall that in Newtonian gravity it is easy to construct objects of
this type: Any bound system has negative total energy. In fact, since
the Newtonian gravitational potential is unbounded from below, even if
one includes the rest mass energy of the matter, one can still construct
systems with negative total energy.
If similar objects existed in general relativity then they would
have surprising properties. For example, since mass is equivalent to
energy, such an object would have negative gravitational mass and repel
rather than attract nearby objects. More importantly, one could pre-
sumably extract an unlimited amount of energy from such a system. This
is because radiation will carry away positive energy, causing the energy
remaining in the system to decrease. But since the total energy is
negative initially, it appears to have no lower bound. It could thus
continue to decrease indefinitely.
The results which have recently been established ensure that
these bizarre situations cannot occur in the context of general rela-
tivity. Physically, the reason they are forbidden is that as you com-
press a system to take advantage of the negative gravitational binding
energy, you inevitably form a black hole which has positive total energy.
I have been asked to review these recent developments in our under-
standing of gravitational energy. )I( My goal is threefold: To provide
a clear statement of the results which have been proved, to discuss
what these results mean physically, and to give some idea of how they
were obtained. Most technical details will be omitted. Rather than
attempt a comprehensive study of all the results which relate to
positive energy, I have restricted myself to four dimensional spacetimes
with one asymptotically flat region. The extension to higher dimensional
spacetimes and more than one aymptotically flat region is straightfor-
ward. The extension to asymptotically anti-de Sitter spacetimes has
also been discussed, and has been used to prove the stability of
extended gauged supergravity theories. The reader interested in
these deveolpments is referred to the literature. (2'3'4'5)
II. The Positive Energy Theorem
It is well known that there is no local energy density for the
gravitational field in general relativity. One would expect the
energy density to be the square of the first derivatives of the metric,
but on a manifold with metric the only covariant expression involving
the first derivatives of the metric is zero.
One can of course define the total energy of an isolated system,
i.e., an asymptotically flat spacetime. In fact, one can define a
total energy-momentum four vector. We now briefly review the defini-
tions of asymptotic flatness and total energy momentum at large space-
like distances.
A spacelike surface Z with induced metric q.~ and extrinsic
curvature Tab is said to be asymptotically flat (b) if there exists
0
a flat metric qab defined outside of a compact set K such that Z
minus K is diffeomorphic to the complement of a compact set in R ,3
qab = Gab + 0(r-l)' Tab = 0(r-2) where r is the distance function
from an arbitrary origin, and similar statements hold for the first
two derivatives of these fields.
The total energy momentum of a spacetime is called the Arnowitt-
Deser-Misner )7( (ADM) four momentum. It is defined in terms of an
integral over a two sphere at large spacelike distances. There are
several choices of integrand which all yield the same value for the
integral. One choice )7( involves the difference between the physical
metric and a flat metric that it approaches, while another )6( involves
certain components of the Weyl tensor. (See the article by Ashtekar
in this volume for the relation between these two expressions.) For
the purposes of proving the positivity of energy, it is convenient
(8)
to adopt a third choice originally proposed by Nester.
Nester's expression can be motivated as follows. One would like
an analog in general relativity of the expression for the charge Q in
electromagnetism
;FabdS ba
i )i(
Q - 8~
JS
where of course Fab is the Maxwell field, S is a two sphere at large
spacelike distances, and dS ab is the two sphere volume element normal
to S. What two form ~ab should take the place of Fab? Since the total
energy momentum Pa is a vector not a scalar, ~ab must depend linearly
on something which asymptotically approaches a constant vector, to
select the component of Pa to be measured. Dimensionally, ~ab must
involve one derivative. Thus, one might have thought that there are
only two possibilities. For any asymptotically constant vector field
~a, set
1
)2(
~ab = Va~b
d~C?dcbae
2
~ab = ( ')3
where eabcd is the totally antisymmetric Levi-Civita tensor. Unfortu-
nately neither of these two forms is satisfactory. If Fab in )I( is
1
replaced by ~ab' then the resulting "charge" depends on the i/r part
of the vector ~a and is not related to the mass of the spacetime. )9(
2
If Fab is replaced by Wab, then the resulting charge vanishes identically.
(proof: ~ ~ab 2 d~ab ~ = 2S ~v b ~abd~ 2 ~a = 0 where ~ is a three surface whose
S
boundary is S.)
However, if one uses a spin0r rather than a vector, then the extra
phase information allows one to construct two more expressions: Let
A
be an asymptotically constant two component spinor (10) and consider
3
~ab - ~A ?b~A ' - eB ?a~B ' )4(
4 C ~d-C'
~ab E eabcd ~ v e )5(
These two forms are complex. Their real parts are precisely e I and
ab
2
~ab' but their imaginary parts are different. If we replace Fab in
3
)i( with ImWab, then the charge again depends on the i/r part of the
spinor and is not related to the mass of the spacetime. However if we
replace Fab with ImW4b ,I ~ then the charge is non-zero and depends only on
the asymptotic geometry and the asymptotic value of A. This is
precisely Nester's expression for the total 4 momentum:
Let
A oA
= a + 0(r -I) )6(
and set
C~d-C' )7(
Wab = i eabcde v
Then we define
° H 1 ~ ~ dsab
Kapa 8~JS ab )8
o OAOA,
where K a = ~ ~ is the null vector determined by the constant splnor
~A. By taking linear combinations of )8( with different ~a, one clearly
obtains all components of Pa" Notice that even though ~ab is complex,
the integral )8( is always real because the charge associated with
2
~ab vanishes.
One might worry that )7( seems to imply that the total four
momentum must change sign under a change in the sign of eabcd which is
clearly unacceptable. Indeed, this is true of the charge defined by
2
~ab" However recall that one must fix an orientation before introducing
two component spinors. (SL(2,C) is the two fold covering group of the
connected component of the identity of SO(3,1).) Hence one is not free
to change the sign of eabcd in equation (7). (11)
To prove the positivity of the total energy one of course needs
some condition on the stress energy tensor of the matter. The standard
condition is the following: (12) A stress energy tensor Tab is said to
satisfy the dominant energy condition if
Tab ta£b k 0 )9(
where t a, £b are any two future directed timelike vectors. If t a = £a,
this condition states that the local energy density seen by any observer
is positive. The dominant energy condition is actually stronger and
states that the local four momentum seen by any observer must be a
future directed vector. This condition is satisfied for all familiar
classical matter.
We can now state the key theorem which has recently been proven:
Positive Energy Theorem:
Let Z be an asymptotically flat non-singular spacelike surface.
If the dominant energy condition is satisfied, then the total energy
momentum P is a future directed timelike or null vector. Futhermore,
a
Pa = 0 if and only if the spacetime is flat in a neighborhood of .Z
As we mentioned earlier, the condition that Z be asymptotically flat
ensures that we are considering an isolated system and is needed so
that the total four momentum is even defined. The condition that Z be
non-singular is a global condition needed to rule out spacetimes-like
the Schwarzschild solution with M < 0 - with singular initial conditions.
