Nonlinear wave equation for torsion

Volume 107B, number 6

NONLINEAR

WAVE

PHYSICS LETTERS

EQUATION

FOR

31 December 1981

TORSION

W. D R E C H S L E R Max-Planck-Institut flit Physik und Astrophysik, Munich, Fed. Rep. Germany

Received 26 October 1981

Specializing the geometry of a Riemann-Cartan space-time U4 to the case bf a completely antisymmetric torsion tensor (axial vector torsion, *Ku) a set of nonlinear wave equations and constraints are established relating the torsion of the U4 geometry to an axial vector source current. This current obeys an anomaly relation similar to the axial current in spinor electrodynamics.

In a previous paper [1 ] two differential source equations have b e e n investigated in a t h e o r y based on a R i e m a n n - C a r t a n s p a c e - t i m e U 4 relating g e o m e t r i c and m a t t e r quantities in a f r a m e w o r k in which the Poincar~ group is treated as a gauge group. The first set o f equations are essentially Einstein's equations o f general relativity connecting the classical energy mom e n t u m tensor o f m a t t e r to the metric o f the underlying s p a c e - t i m e (see eqs. (19) below), while the second set o f equations ( c o m p a r e eqs. (27) below) relate a c o m p l e t e l y a n t i s y m m e t r i c current associated with the spin properties o f m a t t e r to the c o m p l e t e l y antis y m m e t r i c part o f the torsion tensor which characterizes the underlying R i e m a n n - C a r t a n s p a c e - t i m e geometry. The original equations f r o m which one starts in a t h e o r y based on a U 4 read +~ 1

Ruv - ~guvR = ~:T v ,

VXRuvKx : ~JKv~ •

Here RuvKx = --RvuKx = --RuvxK is the full U 4 curvature tensor, RuK = g~'hRuvKh its first contraction, and R = gUVRuv the U 4 curvature scalar. 7 x denotes the covariant derivative with respect to the U 4 c o n n e c t i o n h a v i n g coefficients, FuvO, which are c o m p o s e d o f a metric, i.e., Riemannian part, Puv p, and a torsion part, i.e., ['uv p = P u f

(3)

+ Kuv ° "

Here Puuo = {u~} denote the Christoffel symbols *z, and K u v p = 1 (Suv p _ Svp

+ Spur )

(4)

represent the torsion contribution to the connection. SuvO in (4) is the torsion tensor obeying *3 p[uv]p

(1/2) S~vP. The Bianchi identities constraining the curvature and torsion tensors in a general U 4 are given by [1]

=

(1) (2)

¢2

+1 Greek indices refer to a natural basis e# = ~ , and dx#; ~ = 0, 1, 2, 3, in the local tangent spaces T x and Tx to the manifold U4, respectively, while Latin indices (see below) refer to a local Lorentz basis e i and 0 i, i = 0, l, 2, 3, in T x and Tx, respectively. The transition between Greek and Latin indices is provided by the vierbein fields, h/~, and their inverse, X~; • i # i k 1.e., e i = h~t i 3~, 0 i = k#dx .Furthermore, g. v -- hNkv~ik with ~ik = diag(1, 1, - 1 , 1) and similar~ for any other tensorial quantity. Greek indices are raised and lowered by gtZV and g~v, respectively; Latin indices are raised and lowered by r~i k a n d ~ik, respectively.

~3

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1981 North-Holland

Quantities pertaining to a riemannian space-time V4, i.e., being determined by the metric alone, will be denoted in the sequel b_y a bar. Thus/~lavKh denotes the Riemann tensor, R/~K = Rma the Ricci tensor, etc. The compatibility between metric and torsion is expressed by V og~v = TYogl~v = 0, where Vo denotes the covariant derivative with respect ha a _ t o the l?#v , and K p # g o v + K p p g # a - Kotzp + KOuu - 0 because of the antisymmetry of the Ko# p in the last two indices (compare eqs. (4) and note that S~uP = -Sv#P). Square brackets around a pair of indices denote antisymmetrization, round brackets (see below) denote symmetrization. 415

R(u~}x =

V{.S~)

x -

S(.f S~}.x,

£7(oRuv) Kx = S{pu o R v) oK;t ,

(5)

(6)

where ( / l w } denotes the cyclic sum of the indices in the curly brackets. Among the possible irreducible components of the torsion tensor we shall only consider in the following discussion the completely antisymmetric (or axial vector) part which is characterized by 4:4 the relations 1

Kuv x = KuvO gox = ~Suv x ,

(7)

K(uvx ) = 3Kuvx,

(8)

and K { J ° K ~ } p x = O,

(9)

with the last equation implying a simplification o f the identities (5). Moreover, K in eqs. (1) is l~instein's gravRational constant K = -8zrG/c 4 (G is Newton's constant), whereas ~ in eqs. (2) is regarded as a new (dimensionless) coupling constant associated with the interplay of a current tensor J~vu (described in more detail below) and the axial vector torsion field representing an interaction different from gravitation, however, appearing here on the scene together with gravitation when the metric description of general relativity is extended to a R i e m a n n - C a r t a n s p a c e - t i m e U 4. Eqs. (1) and (2) would follow from a lagrangian containing b o t h linear and quadratic terms in the U 4 curvature expressions R, Ruv and RuvKx. We, however, do not motivate eqs. (1) and (2) by the choice of a particular lagrangian from which they would follow as Euler-Lagrange equations. We rather study the consistency of these equations here from the point of view o f geometry. Below we shall reformulate them by taking due account of the Bianchi identities (5) and (6) and, finally, specialize eqs. (2) to a metrically fiat s p a c e time assuming that the riemannian part, R u w x , of the full curvature tensor is zero. We shall call the resulting s p a c e - t i m e a T 4. In a U 4 the curvature tensor can be decomposed according to +s

RUvKX = KuvKX + PUvKX ,

Puv~cX = V v K w x - VvKuKx + KuKoKvx °

__

KwoKux

O

,

(11)

which could also, in using (9), be written as

Puwx = VuKwx - VvKu~x + Kuv°K~xo •

(12)

We mention in passing that the T 4 space obtained by assuming K'~v~x to be zero, i.e., by putting F~v x globally to zero (in choosing a particular coordinate system) and replacing V u by 3u is not identical with the space used in the teleparallelism theory of Einstein discussed in his summarizing article o f 1930 to which Cartan wrote an addendum [2,3] *6. In this theory the underlying space is characterized by a vanishing U 4 curvature, i.e., by RuvKx = 0, and by a torsion tensor, SuvP , derived from the vierbein fields, i.e., given by +7 SuvP = (bu k / - Ov X~) Xf. In our notation this would be expressed by RuvKx = --PuvKX with P , vKx as given by eqs. (11) together with eqs. (4) and the SuvP, as mentioned, expressed in terms o f the vierbein fields, the latter determining the metric at the same time. There is thus a very close relationship between metric and torsion in the teleparallelism theory: b o t h quantities stem from a common root - the vierbein fields. The same is not true in the theory discussed in this paper where metric and torsion play rather independent roles although both are related to the geometry of an underlying R i e m a n n - C a r t a n s p a c e - t i m e and hence have to obey certain constraints. The cyclic identities,/~(uuK)x = 0, in a V 4 (compare (5)) imply in the teleparallelism theory the analogous identities, P{uvK)x = 0, while in the U 4 case with a completely antisymmetric torsion tensor discussed here one finds from eqs. (5) and (9) ½P{uvK}x = V{uKvK}x ,

(13)

yielding after contracting the indices/J and X, PIvKI = -VXKvKx "

(14)

The symmetric part of the PuK is given by the quadratic expressions in the torsion

(10)

with +4 The geodesics in a U4 with a completely antisymmetric torsion tensor are the same as in a V4. * s For a corresponding splitting of the Bianchi identities into a V4 (i.e., riemannian) and a torsion part see [ 1]. 416

31 December 1981

PHYSICS LETTERS

Volume 107B, number 6

4:6 Compare in this context also the more recent articles on the Einstein-Cartan theory by Trautman [4], and by Hehl, von der Heyde and Kerlick [5 ], as well as by Hayashi and Shirafuji [6] and the literature quoted there. See also ref. [7] in this connection. +7 For the separation of the torsion tensor into its irreducible components compare refs. [8] and [6].

Volume 107B, number 6

P(uK) = K y X K~ax .

PHYSICS LETTERS (15)

Let us, after these remarks concerning the geometry in a U4, split the source terms in eqs. (1) and (2) into a classical and into a quantum mechanical *8 part in order to make apparent the separation into an equation determining the metric and another equation deternfining the torsion: ruv = Tuv + Tuv(0),

(16)

~JKvu = K]~vu + ?~JK~,u(O) •

(17)

Here Tunis the classical energy momentum tensor, obeying T[uu] = 0, which provides the source term in Einstein's equations. On the other hand, it is seen that ]~vu is a derived quantity in general relativity which thus need not be defined separately: The contracted Bianchi identities in a V 4 yield

31 December 1981

cyclic sum {rula} in (2) and using the Bianchi identities together with eqs. (9), (17) and (18) one finds the following set of equations written, for convenience, in Latin-indexed form:

vlPil = vlPil -- K J k p [ j k ] = 0 ,

(20)

vl*Pi l = ~l *Pil -- KiJk*P[jk] = -- 3fi *Ji(O) .

(21)

In eqs. (21) we have gone over to the dual quantities according to (e0123 = + 1)

*JS(O) = - ~ eso'kJijk (0),

(22)

*K' =

(23)

*Pik = 2 Vk * K i - 2~ik ~s *Ks ,

(24)

with *Pik being the contracted dual tensor of Pijkl obeying

P[ ik ] = 2 eik ll p[jl ] • =

(25)

-

= V.R-~K

V.RvK = ~¢]~vu -

(18)

Using the field equations (see (19) below)ff~v~ is thus seen to be expressible in terms of covariant derivatives L of and r = Moreover,L.. satisfies = 0 while the 0-part of the current is assumed to be completely antisymmetric in all three indices obeying thus J(Kvu}(O) = 3JKvu(O). With (16) eqs. (1) reduce to Einstein's equations of general relativity (with/~ denoting the curvature scalar in a V 4) /~,v - ~guuR ' - = KTuv ,

(19)

together with an expression defining Tu~,(O ) in terms o f g u v P (with P = P~rxg°?') and Puv ; the latter possessing both a symmetric and an antisymmetric part in/~ and u (see eqs. (14) and (15)). This relation between Tu~,(O ) and Pu~, will be considered as an identity defining T,v(0 ). The effects which matter distributed in quantum mechanical or wave function form exerts on the underlying geometry is expressed through the second set of equations (which follows from eqs. (2)) involving the source current JKvu(O) and the torsion tensor K~v u. Taking the ~8 That part of matter which is described in a quantum mechanical (wave function) form givesrise to the contributions T~v(O) and J~vu(0) in eqs. (16) and (17) below. For a detailed discussion concerning the wave function description of matter in this framework see ref. [ 1 ].

Expressed in terms of the axial vectors *K s and *Js(O), eqs. (20) and (21) take, finally, the form: *

I

*

S

~Oi( K s K ) -

--

vJ vj % - as

*

--S*

KiV

Ks,

(26)

*Kj)

- esi/k *K i Vj *K k = -~K3 - *Js (0) .

(27)

Eqs. (27) are four-coupled nonlinear wave equations for an axial vector torsion field, *Ks, induced in the U 4 geometry by an axial current associated with the spin degrees of freedom of matter described in wave function form. Eqs. (26) represent four nonlinear constraints which could also be written as

½a i P = *K i *P,

(28)

with

P = - 6 *K s *K s = Kij k K i/k

(29)

being the torsion contribution to the curvature scalar in a U 4 (R =/~ + P), and with

*P = - 6 ~s *Ks

(30)

being the contracted dual tensor, i.e., *P = ~i/*Pij" Note that P is quadratic in the *K s while *P is a differential expression linear in the *K s . We, furthermore, mention that from eqs. (27) one obtains the following divergence equation for *Js(0):

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Volume 107B, number 6

31 December 1981

[] K s - (1/Ro) esqk

=

X (esi]k(Vs*Ki)(V]*Kk) + Vi(RiS*Ks)}.

(31)

= - ~ KRo*Js(¢),

(36)

Eq. (31) reduces to an associated divergence equation for *Js(4)) which is analogous to the axial vector anomaly relation found in spinor electrodynamics [9], i.e.,

with *Ks*KS = - 1 . Thus 1/R 0 determines the coupling strength of the nonlinear term in the wave equation. It is now an interesting question to ask whether eqs. (36) can have solutions for vanishing *Js(¢), i.e., whether there exist radiation-type phenomena for the K-field which are disconnected from the sources. These solutions must clearly satisfy esi/k(~s*Ki)(a / *Kk) = 0. Despite serious efforts we were so far unsuccessful in the search for a general time-dependent solution of the homogeneous nonlinear wave equations obtained from (36) by putting the right-hand side equal to zero. The next question is whether there are static solutions of the sourcefree nonlinear equations (36). It is easy to show that eqs. (36) turn in this case into a system of three coupled nonlinear integrodifferential equations for */~ = (*/~1, */~2, K3)The zero component of the sourcefree eqs. (36) reads in the static case [ i} = (~1, 32, 03)]

aS*Js(CP ) = (2/3g) e s i ] k O s * K i ) O j * K k ) .

A*/~0 = (1/R0)*K • a × *K,

Here an explicit coupling between riemannian and torsion quantities appear in the form Vi (h i s *Ks) involving the Ricci tensor. Let us now study the dynamics of the K-field in a metrically fiat background space, i.e., let us assume that the metric, determined according to eqs. (19) by the classical energy momentum distribution of distant masses, is Minkowskian and the space-time thus degenerates to a T 4 described above. In this case one has, with 310] = [] denoting the d'Alembert operator,

a i (*K s *K s) = *K i a s *K s ,

(32)

[] *K s - a s (3/*K]) -

esi/k*Kia]*K k --

3-*Js(q 0 .

- ~ g

(33)

(34)

To proceed further we now demand an additional condition to be satisfied by the solutions of eqs. (32) and (33). Let us impose the condition

~S*Ks = 0 ,

(35)

which is analogous to the Lorentz condition in electrodynamics but has here nothing to do with gauge fixing since torsion is a tensor field. Group theoretically (35) would mean that the spin zero content of the K-field is eliminated. With eq. (35) the constraints (32) imply that the K-field has a constant absolute value *K s *K s = - P / 6 . It was shown in [1] that *Js($) can be constructed from a matter field ~ with the help of the Pauli-Lubanski operator of the Poincar6 group. This has the consequence that *Js($) is a spacelike vector. Hence the same is true for *K s. Eq. (35) thus means that P assumes a f i x e d positive value throughout spacetime. Since *K s has the dimension [L -1 ] (L = length), and P has the dimension [L-2], one can use the constant value o f P - or, what is the same thing, the norm I l g l l = ( - * K s *KS) 1/2 of the field *K s - to introduce a length scale R 0 = (6/P) 1/2 into the nonlinear equations and go over to dimensionless fields *Ks = Ro*Ks" Rewriting eqs. (33) for the fields */~s one has

418

(37)

which - if */~ • curl */~ were known and nonvanishing - would have the solution •K 0 -

4-R0f

_-

d3x '

(38)

with r = hx - x ' l and a / = a / a x ' / ; / = 1, 2, 3. Inserting this expression into the remaining three equations for •/( yields 47rR~

+ (*K X a )

r

d3x '

OX

r-

d3x

=0

(39)

which is a set of Laplace-Poisson-type equations with a complicated self-coupling. Solutions to eqs. (39) are not known: It was Einstein's original hope with regard to the theory of absolute parallelism involving an asymmetric affine connection, i.e. involving torsion, to relate the electromagnetic interaction to gravitation and thereby establish a common geometric origin for both interactions. In the geometric framework discussed in this paper torsion appears to be linked with another inter-

Volume 107B, number 6

PHYSICS LETTERS

action different f r o m e l e c t r o m a g n e t i s m as well as f r o m gravitation. This new interaction, however, shows up on the scene t o g e t h e r with gravitation to which it is only loosely related. Tile dynamics o f the n e w interaction is governed by a nonlinear wave e q u a t i o n for an axial vector field, *Ks, representing the torsion o f an underlying R i e m a n n - C a r t a n s p a c e - t i m e U 4 . It w o u l d be e x t r e m e l y interesting to see whether the solutions o f these equations show features k n o w n f r o m strong interaction physics.

31 December 1981

References [ 1 ] W. Drechsler, Poincar6 gauge field theory and gravitation, MPI-PAE/PTh 15/81, March 1981, [2] A. Einstein, Math. Ann. 102 (1930) 685. [3] E. Cartan, Math. Ann. 102 (1930) 698. [4] A. Trautman, Symposia Mathematica 12 (1973) 139. [5] F.W. Hehl, P. von der Heyde and G.D. Kerlick, Rev. Mod. Phys. 48 (1976) 393. [6] K. Hayashi and T. Shirafuji, Phys. Rev. D19 (1979) 3524. [7] E. Caftan and A, Einstein, Letters on Absolute parallelism 1929-1932, ed. R. Debever (Princeton University Press, 1979). [8] K. Hayashi and T. Nakano, Progr. Theor. Phys. 38 (1967) 491. [9] S.L. Adler, Phys. Rev. 117 (1969) 2426.

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