on neutrino fields in einstein-cartan theory

Volume 132, number 2,3 PHYSICS LETTERS A 26 September 1988

ON NEUTRINO FIELDS IN EINSTEIN-CARTAN THEORY

P. SINGH and J.B. GRIFFITHS Department of Mathematics, University of Technology, Loughborough. Leicestershire LEl I 3TU, UK

Received I8 May 1988; accepted for publication 7 July 1988 Communicated by J.P. Vigier

An approach to classical neutrino fields in Einstein-Cartan theory is described using a generalised method of spin-coefficients, and some corrections to a previous paper are pointed out. It is also pointed out that this approach is particularly suitable for the generation of new exact solutions from known solutions in general relativity, and an example is given.

1. Introduction and field equations

In a previous paper [ 11, classical two-component neutrino fields were considered in the Einstein- Cartan theory [2,3]. In this theory the dynamical energy-momentum tensor is defined in terms of the lagrangian density of the matter field by

However, in ref. [ 11, the negative sign was omitted in the derivation of a,,, and so the results of this paper should be ammended accordingly.

It is convenient to take the lagrangian density in the form

Ze= - 2iJ-g bp A*(@AV/#-$~VII$A). (2)

This then leads to the field equations

R,, - tg,,R= -kz;, , (3)

&v = - k+KJK , (4)

@,,#XV,oA =0 , (5)

where the canonical energy-momentum tensor is given by

c py = Zicr,x( @“V# -$‘V,@” ) (6)

and Jp- -cT~~,&~~P is the neutrino current vector.

2. The spin-coeffkient formalism

As in ref. [ 11, we now introduce the generalised Newman-Penrose formalism of Jogia and Griffiths [4 ] and align one of the basis spinors with the neutrino spinor. Putting

gA=(boA, (7)

the neutrino-Weyl equation (5) can be written as two complex equations

D@= (p-t)+, 8@= (r-8)@. (8)

The only nonzero components of the contortion tensor (4) are

pl =ik& y1 = - jik@- . (9)

From (3) and (6) andusing (7) and (8), thecom- ponents of the symmetric part of the Ricci tensor are given by

@Jo =O, @or =-fik&%c,

@cl2 = -ik&%r, c9, I = fik$&i-p) ,

Q, 12=&ik[@8$+@&c?-2r)],

@22 =ik[@Aq?-$A++@&y-y)] ,

A=0 (10)

and the components of the antisymmetric part are

@=-iik$@c, @,=-ik@$>,

@ = - fik(@++@cr) . (11)

88 037%9601/88/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )


The terms ( 11) do not occur in the space-times Vq. It can also be seen that the only component in ( 10) which differs in a U4 is given by

022 =e2 -k2(&Q2. (12)

The other components of the curvature tensor in a U4 which are generated by the torsion are now given

by

i&0 ~0, i&, = - fik&&c ,

i@,, = -ik&%, i6,, = - fik@(p+p) ,

i@,,=-tik[@66+&@+27)],

ie,, = -ik[@A6+@@+@(F+r) 1 ,

z=o. (13)

It can also be shown using the identities in the ap- pendix of ref. [ 41, that the components of the Weyl tensor in the associated V4 are unaltered by the pres- ence of the torsion given by (9),

!&=lu”, for A=l,2,3,4,5. (14)

3. General properties

With the correction in the sign of the torsion components, it can be seen that the theorems stated in ref. [ 1 ] are incorrect. However the exact solutions given only require one or two minor corrections.

It may be observed that, in a Vq, the neutrino equation is identical to (8), and the Ricci tensor is given by the V4 components of ( 10). It follows that any solution of the Einstein-Weyl equations in general relativity has a corresponding solution in Ein- stein-Cartan theory [ 51 in which the metric is the same, the additional contortion components are given by (4) and the additional components of the curvature tensor are given by ( 11) and ( 13 ). This en- ables solutions in Einstein-Cartan theory to be generated easily from known solutions in general relativity. Moreover the Weyl tensor has identical components ( 14). There is, therefore, no difference in the algebraic classification of the U, and V, parts. Also, with the choice of basis spinor (7), the only change in the symmetric components of the Ricci tensor is given by ( 12 ) .

The solutions given in ref. [ 1 ] correspond to the V4 “pure radiation” solutions of algebraic type III, N and 0 obtained by Collinson and Morris [ 61. The type N solution includes the only possible “ghost” solution [ 7 ] in which the canonical energy-momentum tensor ( 6 ) vanishes, though in view of ( 12 ) , the stronger conditions for ghostness suggested by Letelier [ 83 cannot be satisfied.

To illustrate the method, consider the type D solution in ref. [ 61. When corrected, this has the line element

&2=2dudr-2mOdu2- $ (dx2+dy2) I

(15)

and the null tetrad may be taken as

1,&L, n,=- y s: +s;,

m,=- 5 (6:+i$), (16)

where x(u) is an arbitrary function. The neutrino field is given by

a(u) A rpG.0 ) (17)

where a ( u ) is arbitrary and the metric function m ( u ) must satisfy riz= -ku2j. The only nonzero components of the curvature tensor are

@2= -ti/r2, !Pj= -ml?, (18)

where the degree signs are used to indicate V4 components.

The corresponding solution in Einstein-Cartan theoryhasidenticalexpressions(15), (16)and(17), but introduces the contortion components

p, =ika2/r2, y1 = - fika2/r2 . (19)

The nonzero components of the curvature tensor are now

@22 = -r)l/r2-k2a4/r4, Y, = -m/r’,

@, =ika2/r3, ie,, =ika2/r3,

ie,, = -2iti/r2+ika2m/r4. (20)

89


It may be observed that, in a V4, a ghost solution occurs when 9~0. However, no ghost solution occurs in the askciated U,. _

Other solutions may be extended in the same way.

References

[ I] J.B. Griftiths, Gen. Rel. Grav. 13 ( 198 1) 227. [2] F.W. Hehl, Gen. Rel. Grav. 4 (1973) 333; 5 (1974) 491. [ 3 ] F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester.

Rev. Mod. Phys. 48 (1976) 393.

Acknowledgement [4] S. Jogia and J.B. Griftiths, Gen. Rei. Grav. 12 (1980) 597. [ 51 M. Seitz, Class. Quantum Grav. 2 (1985) 919. [ 61 C.D. Collinson and P.B. Morris, Int. J. Theor. Phvs. 5 ( 1972)

One of us (P.S. ) wishes to acknowledge the award of a research scholarship by the Science and Engi- neering Research Council while this work was car- ried out.

293. [ 7 J J.B. Griffiths, Phys. Lett. A 75 ( 1980) 441. [8] P.S. Letelier, Phys. Lett. A 79 (1980) 283.

90

on neutrino fields in einstein-cartan theory

Documents