On the role of conformal invariance in general relativity and the origins of mass

Volume 47A, number 6

PHYSICS LETTERS

6 May 1974

ON THE ROLE OF CONFORMAL INVARIANCE IN GENERAL RELATIVITY AND THE ORIGINS OF MASS B.D. BRAMSON Mathematical Institute, Universityof Oxford, UK

Received 22 March 1974 It is shown how General Relativity may be derived from a conformally invariant theory of gravitation in interaction with matter after recognising the existence of a family of canonical gauges. In this manner the concept of rest mass emerges alongside that of a gravitational constant. Furthermore, the difference between the spin 2 and eonforreal curvature fields is explained.

1. Introduction. This note is devoted to answering three questions: (a) Is there a connection between the breaking of conformal invariance by the gravitational field on the one hand and by massive particles on the other? [1 ]. (b) Is it feasible to construct massive particles starting from a conformally invariant field theory and with the masses arising from interactions between massless fields? [ 1] (c) In empty space the gravitational field satisfies the massless spin 2 equation VAA' ~ABCO = 0

(1.1)

where ~IABCD , symmetric in all its indices, is the spinor equivalent of the trace-free Riemann tensor. Under a conformal rescaling of the metric:

gab = n 2 gab

Q .2)

2. Conformally invariant fieM theory. Consider the field theory defined by the action fd4x£, where

z = ~,C~A VAA,,7A - X A VAA,~ A) (2.1) ~Va~ V t a(~ + iX~(~IA2A_~A'XA,) + Rq)2/12.

_

On the face of it, this theory describes a pair of mass. 1 less spin ~ fields, ~A and ×A', in interaction with a negative energy massless scalar field ~; and to these is coupled a gravitational field with Ricci scalar R via a gravitational "coupling variable" 6/~ 2 (c.f. [3] ). After putting h = 1 = c, ?, is to be regarded as a dimensionless coupling constant. The scaiar-spinor interaction may be rewritten in the form - 2 1 / 2 X ~ , where ¢ is the Dirac spinor equivalent of 7,4 and ×A" The theory is invariant under the conformal rescaling (1.2) of the metric together with

and with the conventions ~A = ~ - l r / A

:

it follows that

(2.2)

-lxA,

and the conformally invariant field equations are just:

~ ABCO = ~ ABCD (1.3) and (1.1) is not preserved [2]. In order to preserve the zero rest-mass spin 2 equation it is necessary to postulate a new field OABCD satisfying (1.1) in the physical space-time and transforming according to 6 ABCD = ~ - I ¢~ABCD .

(1.4)

Is there some deep significance in the difference between (1.3) and (1.4) and can the field (aABCD be defined in a natural way?

VAa,n A - X ~ X A, = 0

(2.3)

VAA,XA'ae~dpnA = 0 E]~ +R~[6 + ik(~/A XA --nA'XA') = 0

(2.4)

Gab = --6/O2(Tab--Oab)

(2.5)

where Tab is the symmetrised energy-momentum teni sor for the spin $ fields, together with the ~, term, and 431

Volume 47A, number 6

PHYSICS LETTERS

Oab is the

"new improved" energy-momentum tensor [4] for ~. The immediately relevant property of Oab is that it vanishes for constant q~. The invariance of the theory under (1.2) and (2.2) and, in particular, the absence of dimensional constants would appear to preclude the existence of massive particles. However, there exists a one parameter family of canonical gauges defined by the dynamics, namely: ~2 =fq~

(2.6)

with f constant. In any such gauge ~ = f - 1 and hence disappears as a dynamical degree of freedom. The transformed field equations are then

~TAA,~A--~kf-IxA,

=0

~TAA,XA'+ )kf-I~ A

=0

(2.7)

and

Gab = -

6f2

1"ab

(2.8)

((2.4) was just the trace of (2.5) and, on transformation, becomes the trace of (2.8)). Also, the transformed Lagrangian is given by 1 • AA'~ ,~ = ~ - I C ~

~TAA,~A--x A ' ~VAA,X A )

+iXf-I(~A~A-~A'XA,)+R/lZf2

(2.9)

(2.7) and (2.8) or, equivalently, (2.9) defines DiracEinstein theory with a Dirac mass

the existence of two four index spinor fields: the gravitational field ~ABCDand the spin 2 field dPABCDdef'med by 1

CkABCD= 6 -7 Ck~tABCO

(2.12)

Under conformal rescalings, (1.4) is satisfied, while in the canonically defined gauge determined by (2.6) (but dropping the hats): 1

¢ABCO = K-~ ~ABCD

(2.13)

Thus in conventional general relativity the spin 2 and curvature fields are essentially identical, the difference between them arising only in the conformal extension of that theory. Furthermore, in the physical space-time, (2.13) may be interpreted as saying that the massless spin 2 field q~ABCDinduces the spacetime curvature g 1/2 (PABCD" 3. Conclusion. General relativity may be derived from a conformally invariant theory of matter in interaction with gravitation after recognizing the existence of a family of canonical gauges. In any such gauge, the masses of particles emerge together with the gravitational constant. Using the conformal formalism, the spin 2 field (gABCDmay be defined in a natural way and, in the canonical gauge, induces a space-time conformal curvature 1

q'A S C O = K ~ eaA ~ C D

(2.10)

The author acknowledges a stimulating conversation with Professor R. Penrose F.R.S. from which this note arose.

(2.11)

References

1

m = 25-Xf -1

6 May 1974

and a gravitational constant g = 6f 2

Thus there are positive answers to both questions (a) and (b). The different choices in f, consistent with Xf -1 being positive, correspond merely to different choices in units; the dimensionless quantity Km 2 being independent o f f . In order to answer (c) it is necessary to recognize

432

[1] R. Penrose, Int. J. Theor. Phys. 1 (1968) 61. [2] R. Penrose, in Battelle Rencontres (1967), ed. C.M. de Witt and J.A. Wheeler (Benjamin). [3] S. Deser, Ann. Phys. 59 (1970) 248. [4] S. Coleman, C.G. Callan and R. Jackiw, Ann. Phys. 59 (1970) 42.