IF you don’t know what this means, then, you’re … doing well.

This is totally kooky b.s. ….

Source: InspireHEP.net

G EN ERAL R ELATIVITY
AN D G RAVITATION
Proceedings of the Seventh International Conference (6R7)
Tel-Aviv University, June 23—28, 7974
Edited by
G. SHAVIV and J. ROSEN
Department of Physics and Astronomy, Tel-A viv University
A HALSTED PRESS BOOK
JOHN WILEY & SONS, New York – Toronto
ISRAEL UNIVERSITIES PRESS, Jerusalem
Copyright © 1975 by Keter Publishing House Jerusalem Ltd.
All rights reserved
ISRAEL UNIVERSITIES PRESS
is a publishing division of
KETER PUBLISHING HOUSE JERUSALEM LTD.
PO. Box 7145, Jerusalem, Israel
Published in the Western Hemisphere by
HALSTED PRESS, a division of
JOHN WILEY & SONS, INC., NEW YORK
Library of CongreS in Publication Data
International Conference on General Relativity and
Gravitation, 7th, Tel-Aviv University, 1974.
General relativity and gravitation.
“A Halsted Press book.”
1. General relativity (Physics)—Congresses.
2. Gravitation—Congresses. I. Shaviv, G. II. Rosen,
Joseph, 1933— III. Title.
QC173.6.157 1974 521 ’.1 75-33824
0—470777939—X
Distributors for the UK, Europe, Africa and the Middle East
JOHN WILEY & SONS, LTD., CHICHESTER
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TOPPAN COMPANY, LTD., TOKYO AND SINGAPORE
Distributed in the rest of the world by
KETER PUBLISHING HOUSE JERUSALEM LTD.
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Set, printed and bound by Keterpress Enterprises, Jerusalem
PRINTED IN ISRAEL
Contents
Preface
Uniqueness and Nonrenormalizability of Quantum Gravitation
S. DESER, P. VAN NIEUWENHUIZEN and D. BOULWARE
Canonical Quantization
JOSHUA N. GOLDBERG
Modern Theoretical and Observational Cosmology
THOMAS B. CRrss, RICHARD A. MATZNER, MICHAEL P. RYAN, JR.
and L. C. SHEPLEY
Recent Progress in Exact Solutions
WILLIAM KJNNERSLEY
The Bondi—Metzner—Sachs Group: Its Complexification and Some
Related Curious Consequences
E. T. NEWMAN
Numerical Integration of Exact Time-Dependent Einstein Equations
with Axial Symmetry
J. PACHNER
Tests of Theories of Gravity in the Solar System
JEAN-PAUL RICHARD
The Texas Mauritanian Eclipse Expedition
BRYCE DEWI’IT
“Exotic” Black-Hole Processes
WILLIAM H. PRESS
Observational Effects of Black Holes
M. J. REES
vii
19
33
109
137
143
169
189
195
203
vi CONTENTS
Progress in Relativistic Statistical Mechanics, Thermodynamics, and
Continuum Mechanics J URGEN EHLERs 213
Einstein and Zionism
BANESH HOFFMANN 233
Gravitational Waves: Panel Discussion 243
Conference Summary
JOHN A. WHEELER 299
Preface
It is by now a well-established tradition to start off a preface to any book on general
relativity, gravitation, cosmology, and high energy astrophysics by exclaiming how
rapidly the field is expanding (a big bang for relativity !). Nevertheless, this is really
true (as tedious as it may be to hear again). The most exciting point is, we believe,
the real possibility of checking many of the most prominent features of theory by
experiment or observation. Indeed, we may be observing objects for which general
relativity is no longer merely a correction.
This volume contains the invited talks and the panel discussion (on the detection
of gravitational waves) of the Seventh International Conference on General Relativity
and Gravitation (GR7), held in Tel-Aviv, Israel, during 23—28 June 1974. The organizers attempted to arrange for the presentation of a wide and varied spectrum of topics,
which covers the many aspects of the subjects of the conference.
Since the usefulness of this material depends to a large extent on its freshness, we
decided not to include the many (over 100) contributed papers in order to expedite
publication. We hope that these interesting papers will be published at full length
elsewhere.
It is a pleasure to thank the following for financial contributions which made the
conference possible: Tel-Aviv University, Ben—Gurion University of the Negev,
The Hebrew University of Jerusalem, Technion—Israel Institute of Technology,
The Israel Ministry of Education and Culture, The Israel Ministry of Tourism,
The Israel National Academy of Sciences and Humanities, The Israel Foundation
Trustees, The International Committee on Gravitation, The International Union
of Pure and Applied Physics, and Bank Leumi le-Israel.
Tel—Aviv, G. SHAViv and J. ROSEN
July 1975
vii

Uniqueness and Nonrenormalizability of
Quantum Gravitation*
S. DESERT and P. VAN NIEUWENHUIZEN
Brandeis University
Waltham. Massachusetts 02154, U.S.A.
D. BOULWARE
University of Washington
Seattle, Washington 98195, U.S.A.
ABSTRACT
(1) The principles of Lorentz invariant quantum particle (rather than field) theory, together with qualitative
empirical properties of gravitational forces, imply that the tree graphs describing gravitational interactions are necessarily generated by the Einstein action, at least for frequencies low compared to particle masses
or the inverse Planck length. Consequently, the classical limit of any acceptable quantum gravitational
model must be general relativity.
(2) The ultraviolet divergences of the quantized Einstein field interacting with quantized scalars, spinors,
photons or vector gauge multiplets are all nonrenormaliiable at the one-loop level; the required counter
terms are drastically different from the original actions. We discuss the relevance of these conclusions to
the unification of gravitation and quantum theory.
1. INTRODUCTION
In this report I shall be concerned with two very recent developments in quantum
gravity. These developments represent explicit calculations and conclusions in a
subject which has hitherto been more a “program” than an area of active results.
*Supported in part by the National Science Foundation, Nederlands Ministerie van Onderwijs en Wetenschappen, and the U.S. Atomic Energy Commission.
TLecture by S. Deser.
2 S. DESER ET AL,
The first is encouraging, the second discouraging, to our search for a consistent
unification of gravitation and quantum theory. I shall summarize them briefly here,
expand on each in separate sections, then attempt to assess prospects and alternatives
in the concluding part. The results were obtained by the author in collaboration with
D. Boulware and with P. van Nieuwenhuizen, respectively.
Special relativistic quantum theory, our most solidly established basis for understanding microscopic (rather than cosmological) phenomena, will be our guide in
attempting to understand quantum gravitation. We recall that relativistic quantum
theory can be divided into two different aspects: S-matrix (particle) theory, which
deals with the properties of scattering amplitudes for physical particles of given spin
and mass, and quantum field theory which provides a localized space-time description of interacting systems. In either category, there are two basically different classes
of diagrams: trees and loops. Tree diagrams do not include vacuum fluctuations,
and there is no summation over (arbitrarily high) virtual energies. This class of
diagrams is sufficient for understanding the classical field limit of a quantum theory,
and has no renormalization problems. The latter arise only when one encounters
closed loops—whose presence is required by unitarity. We have recalled these distinctions because our two sets of results will nicely fall into the complementary categories
of relativistic particle theory and low frequency structure on the one hand, and
relativistic field theory in its probing of high frequency behavior through closed
loops on the other.
Whatever the future fate of quantum field theory, the more modest particle description is the necessary ‘extension of quantum mechanics to the special relativistic
domain. One of its basic results is that all forces between particles are themselves
mediated by exchange of quanta of appropriate spins and masses. We shall show that
this principle, together with the observed qualitative structure of macroscopic
gravitational forces, leads to a unique determination of the basic vertices describing
gravitational interactions at wavelengths high compared to particle Compton wavelengths. This means in particular that the tree graphs, and so the classical limit of any
acceptable quantum gravity model, are unique. That the interactions can be so completely determined by quantum principles is beautiful enough; that they must coincide
with those of general relativity in this limit is even more gratifying. From the general
properties of gravitational forces between material systems (including light), namely
that they are long range, static, macroscopic and attractive, we shall show that a
massless spin 2 particle (“graviton”) is primarily responsible. What is more important, this particle must self-interact: gravitons emit gravitons and do so in a
uniquely defined way. This follows from the fact that gravitons can only couple
consistently to conserved symmetric tensor matrix elements of their sources, while
a conserved tensor can only exist if the graviton content of the sources is included.
(The low momentum transfer form of the “stress tensor” can actually be deduced as a
low energy theorem, and this theorem implies the equivalence principle.) There is
UNIOUENESS AND NONRENORMALIZABILITY 3
in fact an infinite chain of n-graviton vertices, all of which are determined uniquely
by the consistency requirements of massless graviton coupling, through terms
quadratic in graviton momenta.
Having determined all possible vertices, one can consider all tree graphs, con—
structed by joining all possible vertex configurations without forming loops (the
latter would require more knowledge than we are willing to assume in this part of the
discussion about the higher momentum behavior of vertices). The result is that all
tree graphs can be generated by an effective action functional, which is none other
than the Einstein action. But the classical limit of any theory is just that of its tree
structure: it is governed by the generating action. Thus classical general relativity
is really a consequence of relativistic quantum particle theory, given the observed
gross character of gravitational forces. There is, then, a very satisfactory harmony
between geometry and quantum mechanics in this common range of application
(or at least aspiration).
Our second set of results is concerned with deeper quantum properties, involving
the extrapolation of the above structure to high frequencies, namely the closed loop
aspects of general relativity considered as a quantum field. We shall quantize the
metric field using the same techniques (covariant quantization, gauge breaking
and compensating ghost fields) which have been so successful in the recent progress
of non-Abelian vector gauge theories. Our interest will focus here on the very high
frequency behavior, that is on the types of divergences encountered at the one loop
level in coupled Einstein-matter systems. Only if the divergences can be removed
by renormalization, are these systems satisfactory at our present (perturbative)
stage of understanding in field theory. Unfortunately, we shall see that all systems
considered to date, which include general relativity coupled to scalars, spinors,
photons, and Yang—Mills multiplets as well as some generalizations of Einstein
theory, are nonrenormalizable (except for the source-free Einstein field itself).
We shall discuss the significance of this discouraging conclusion, along with possible
alternatives and improvements, in the final section.
Throughout our work we assume that gravitation, like all other fundamental
dynamical systems, is subject to quantization. It has sometimes been suggested that
this is not logically necessary either because general relativity is not a basic field or
because it can somehow remain classical in contrast to its sources, or finally, because
its quantum effects are not observable experimentally due to the smallness of the
characteristic Planck length of quantum gravitation. These last resorts appear to us
worse than the disease they would avoid, and we shall not discuss them here. Our
point ofview is that there is an empirical domain, large compared to particle Compton
wavelengths, but small compared to cosmological distances, in which the physical
world is Minkowskian, and the physical processes including gravitation are described
by special relativistic quantum theory.
4 S. DESER ET AL.
2. GRAVITONS: VERTICES AND TREES
The study of relativistic quantum particle theory of gravitation began about a
decade ago with the work of Feynman [1] and Weinberg [2]. Weinberg especially
stressed the strong constraints on the S-matrix describing interaction of massless
particles, which are equivalent to gauge invariance requirements in the more familiar
field-theoretical context. He showed how one could derive, for massless vector
particles, both Maxwell’s equations and conserved currents; that for spin greater
than 2, there is no interesting low frequency theory possible at all (there being no
conserved static quantities corresponding to charge or mass) while for spin 2, the
low frequency form of the interactions implies amongst other things, the equivalence
principle (universality of matter-graviton vertices) for all systems including the
gravitons themselves. Both Feynman and Weinberg asked to what extent such arguments uniquely defined the full gravitational theory, but found the nonlinearities
of the self-interactions too complicated to complete this program. The question
was recently taken up by D. Boulware and myself, and the unique character of the
solution will be the subject of this section; the reader is referred to the paper [3]
for details.
We begin with a rapid review of why the exchanged particle responsible for gravitational forces must have (at least primarily) spin 2. From the general facts that there
are static forces we may exclude S > 2; that they are attractive excludes S = 1;
their macroscopic character excludes half odd integer spin, and because photons
are affected in the same way as massive particles, we can exclude S = 0. Naturally,
there are only upper limits on the possible contribution of the other spins or of
short range (contact) terms but for simplicity we shall assume they are entirely absent
(Brans Dicke theory could be accommodated for example). Another basic requirement
of quantum theory, that only physical (positive probability) quanta exist, excludes
exotic Lorentz invariant theories of gravitation which would require “ghost” behavior for the corresponding particles. Empirically, the graviton must obviously have
long range (small mass); in fact, the mass dependence affects light bending discontinuously and one may conclude that the mass is strictly null [4]. We have thus
specified the graviton to be a massless (k2 = 0) spin 2 particle, which therefore has
two helicities (a = i2). The amplitude Ta(k) for its emission by a system can be
written as 7:,(k) = efi“(k) Tuv(p’, p), where efi“ (k) is its transverse-traceless polarization
tensor and the ten quantities 7;”, which depend on the initial and final momenta of
the system, can be taken to be conserved also slightly off shell (k2 7-é 0). We need
not assume that they are the matrix elements <p’ Twig?) of a stress tensor operator;
indeed one can deduce the form of T“, to first order in momentum transfer purely
from conservation. Specifically, one finds for the amplitude Tia-(fl, p) for emission
ofa graviton by a system making a transition between physical states with momentum
UNIQUENESS AND NONRENORMALIZABILITY 5
and polarization (p, A) and (p’, ’1’) the following expression [5] (P E p + p’, k E
p — 11’):
Tfi‘flp’, p) = T[fi,—.I(p’)u,-.(p) P“P” + iP‘“5’3}’k.fi;x(p’)S”‘u,-.(p)] + 0(k2) (2-1)
where T is a universal constant, (u, 12) are the initial and final state polarization spinors
or tensors, and S”t is the Lorentz rotation operator for the system in question. The
constant T is determined by correspondence with the Newtonian limit, to be (with
our covariant spinor normalization) T2 = (21:)‘5G, where G is the Newtonian
constant. Note that the first term is just a scalar particle’s stress tensor, while the
second is the usual contribution of spin to the angular momentum. Thus, the equivalence principle is deduced as a low energy theorem: all systems couple in a universal
way (at low enough frequencies) to gravitons through their unique symmetric
conserved stress tensor and with a universal strength T. For our purposes, universality is the key, because it necessarily leads to graviton-graviton coupling. If gravitons are coupled to any system (and if not there is no gravitation l), they are necessarily coupled to each other. Otherwise put, if a system can emit a graviton, then
every system to which it couples in any way also interacts with gravitons, and that
includes the emitted graviton itself. Diagrammatically, we can deduce the existence
of the basic 3-graviton vertex (Fig. la) from the Newtonian emission (Fig 1b).
This can be done analytically by inserting a soft (small k“) graviton into (1b); consistency forces the presence of the full set of Fig. 2, including V3.
k, .. Figure 1a Figure 1b
Three-graviton vertex V3 (k1, k2, k3). One-graviton emission by a system,
”NV/v. -
Figure 2
Two-graviton emission graphs.
6 S. DESER ET AL.
Soft graviton conditions only determine V3 (a la (2.1)), for two gravitons on shell
and the third one soft, however; and it requires a long calculation (including consideration of 4- and 6-point graviton graphs) to obtain the full constraints on V3
for all gravitons off shell. By considering diagrams with more and more scattering
gravitons, one may deduce the necessary existence of new short range vertices
V4, V5, V”, involving arbitrarily many gravitons. Our graviton construction
has a parallel, incidentally, in the classical field theory framework in which the infinite set of self-interactions of the Einstein equations are deduced by demanding
that the source of the linearized spin 2, m = 0 equations be the stress tensor of the
linearized field and iterating this consistency demand [6]. However, no assumption
about an underlying field is involved in the quantum case.
The constraints on V3 are that it be symmetric in interchange of two gravitons
(bose statistics) and conserved under appropriate contractions with external momenta. The conservation conditions require that V3 be at least quadratic in the momenta,
and the low frequency restrictions on our curiosity made us stick to quadratic
momenta only (this corresponds to second derivatives only in field theory language).
Then, if we express V‘g““””(k1k2k3) compactly in terms of a functional V3 (h) by
multiplying it with fields hwhwhpo, we can show that the conservation conditions
correspond to an identity of the form
é:VCSVES/(Shuv = ”4:301: a)(D_1h)zfi (22)
where (D‘ 1h),,3 is the linearized Einstein tensor, and A is an operator linear in hW
and in first derivatives, Such identities generalize the Ward identity of electrodynamics, which is related to gauge invariance there. In our case, we are building up the
more complicated covariant conservation identities of general relativity (2.4, 2.5).
The most general solution of this identity is
V3(h) = rjmx) (V/TQR)(3) + ((dx) hh D‘ 1h (2.3)
where the first term is the cubic part in an expansion of the Einstein Lagrangian
if = (\/——gR) T‘2 in powers of Th” E gm — n“. The second term stands for a
sum ofthe type hmtD— 1h)“. These terms can be shown to represent precisely the
freedom of expanding the metric (in classical language) in different variables such as
Tk‘” = g“v — )7”, etc. This freedom is necessary and its presence reassures us as to
the generality of our procedure. Of course, a new arbitrariness of this type will also
occur at each order of V,, . To determine the general n-point vertex, one first establishes
that the sum V(h) of all vertices obeys the more general Ward identity
(MW/(Sly,v = A“’°‘B(h, 6) [(D‘t — 6V/6hafi] (2.4)
How unique are the V(h) and A satisfying (2.4)? One may first show that the same
scattering amplitude results no matter what set of vertices satisfying (2.4) (if there is
any!) is used. But we know one V(h) satisfying them, namely, the nonlinear part of
UNIOUENESS AND NONRENORMALIZABILITY 7
the Einstein Lagrangian. For the Bianchi identities,
{€13 E 0 = {4v + 1‘5”?” (2.5)
together with the fact the ’5‘” E (D‘ 1h)“ is identically conserved and that ’5‘” — {V is just the vertex — 5V/5huv: read
(MW/(5m, = r:,,[(D-1h)w — 5mm] (2.6)
(This form, though not the functional form of V and (D‘ 1h), is independent of what
choice of h,” is made.) Thus, the Einstein action generates the essentially unique
vertices and one can also show that it yields tree scattering amplitudes with the required
properties. Finally, returning to the original matter—gravitation interactions, one
can establish that the usual minimal gravitational coupling prescription for matter
is also uniquely required up to higher momentum terms.
To summarize, we have shown the following: the quantum tree graphs describing
all graviton scattering amplitudes are generated by the Einstein action, at least to
quadratic order in momenta. Higher momentum dependence would in any case be
negligible in the classical limit, which is thus necessarily governed by general relativity. This result followed entirely from empirically dictated dominance, in gravitational exchange, of massless spin two quanta together with the principles of Smatrix particle theory. General relativity is really a consequence of quantum theory!
Note that our considerations began within the empirically well—established framework of Minkowski space for phenomena on scales greater than Compton wavelengths and smaller than cosmological distances. Nevertheless, they led us unavoidably to the usual Riemann geometrical interpretation of classical Einstein theory
covariantly coupled to matter, with the equivalence principle as a necessary “lowest
order” consequence of the graviton as mediating particle.
One could continue our particle description into the closed loop domain, which
includes such classical necessities as 2 particle “ladder” exchange diagrams responsible for Keplerian orbits (together with quantum corrections ~ (h/mc/r)2 or (G h/c3/r2)
to the dominant l/r parts). However, our main interest in closed loops is that discussed in the next section, namely ultraviolet behavior of the general relativistic
model taken seriously as a quantum field theory at all frequencies.
3. CLOSED LOOPS: NONRENORMALIZABILITY
OF QUANTUM GRAVITATION
It has long been suspected that quantized general relativity, when ripe for calculations, would exhibit disastrous, nonrenormalizable ultraviolet behavior (at least in
a perturbation expansion, which is the only method currently known). This expectation was based on the dimensional character of the Einstein constant, which
8 S. DESER ET AL.
is related to the rising momentum dependence of graviton vertices (for example
TM. or V3 both go as P+ 2). The more loops and vertices the higher is the divergence
predicted by simple power counting. Every radiative correction to a given process
would give rise to new divergent counterterms, in contrast to the relatively mild
logarithmic divergence in electrodynamics or Yang—Mills theory, where a finite
number of renormalizations suffice. The horror of nonrenormalizability lies in the
fact that a whole infinity of processes must be fixed experimentally and cannot be
predicted by the theory, rather than just a small number of parameters like mass
and charge. Nonrenormalizable theories are not necessarily inconsistent but they
are not very predictive and one has no a priori idea of their properties.
A discussion of renormalizability would seem to require an advanced stage of
development of the quantum field theory in question, yet quantization of general
relativity has until now been at best a program with no well-defined calculation
rules, let alone specific results. What is the nature of the obstacles to developing
these rules, and how have they been overcome? The one quantization prescription
we can be sure of for a dynamical system is the canonical one, in which there is a
clearcut choice of basic p’s and q’s. While a suitable canonical formulation of classical general relativity is available, there have been two types of obstacles
to progress in quantum gravity. The first is the apparent ambiguity offactor ordering
in the Einstein action, both in its original form and in the later solution of
its constraints. The second is simply that canonical quantization is neccessarily noncovariant in form with respect to space-time, so that the Feynman
rules and any calculations would be terribly cumbersome. In electrodynamics,
the possibility of canonical quantization provides the reassurance needed to perform
one’s calculations in covariant gauges. Now in an Abelian gauge theory such as
electrodynamics, the equivalence between canonical and covariant gauges and
quantization is well-known and even proved [7] within the context of perturbation
theory. As soon as one gets to a non-Abelian situation such as the Yang—Mills field, equivalence is not so immediate. In practice, people went ahead, quantized
covariantly and expected equivalence to be established in due course. This has in
fact been accomplished [8], so that covariant rules have been validated also in the
Yang—Mills case. There is one technical addendum to the covariant rules: If one
just writes down the vertices and propagators in some covariant gauge as they appear
in the Lagrangian, then he will obtain absurd closed loop results because wrong
helicity states are also included in these gauges. These unphysical degrees of freedom
(absent in canonical quantization) must be compensated for by introduction of the
famous “ghost” Lagrangian. Ghosts were in fact discovered by Feynman [9] while
performing the earliest loop calculations in quantum gravitation, and were later
fully treated by Faddeev and Popov [10]. In electrodynamics, covariant ghosts are
also present in principle but do not actually contribute clue to the linearity of the
theory. (Photon ghosts do contribute when coupling to gravitation is included.) In