Can the torsion always prevent singularities?

Volume 57A, number 3

PHYSICS LErFERS

14 June 1976

CAN THE TORSION ALWAYS PREVENT SINGULARITIES? P.S. LETELIER Department ofPhysics, Boston University, Boston, Mass. 02215, USA Received 20 April 1976 A curvature invariant of spacetimes that are solutions to the Einstein—Cartan—Weyl equations is studied. It is found that in the static plane symmetric case this invariant is singular.

Speculations about the possibility that the torsion may prevent the occurrence of singularities have been made [1, 2]. In particular, several solutions to the Einstein—Cartan equations with nonsingular curvature have been found [3—5].This property is achieved by using particular models of spinning fluids. This can be done because the EC equations do not make severe restrictions on the particular model used. We have a different situation when we deal with neutrinos. The EC—Weyl equations form a complete set of equations, the spin and the torsion are completely determinated by the theory. The aim of this paper is to study a curvature invariant of spacetimes solutions to the EC—Weyl equations and in particular the static plane symmetric case. The “basis” of the EC—Weyl theory is the affine connection defined in an holonomic system of coordinates by [6] bC

=

+ k\/—g4~.J’1,

(1)

where k is a constant and ja the neutrino current. From (1) we find that the Riemann—Christoffel tensor can be split as follows [6] a

r

a

—

+ Sa

+

a

~ th~thn~ Sg~and Ug~({})have the symmetries — —S (3\ aba! [ab]a! ab[aI] cdab “ / Uaba!({}) = U[~,]a!({}) = Uab[a!] ({}) (4a) —

—

Uaba!({})

—

=

—

—

(4b)

Ua!~,({}).

Relation (4b) follows from j~~ja = 0 implies

=

0. We have that

Sa~~lSab& = 0.

(5)

A direct computation shows that

{} {}

2VaJbVaJb

Ua~~lUab~ = 8k

R~~({ })S

=

4k2R ({ })JaJb

abcd ~

/

‘~

(7)

ab

We want to study the curvature invariant ~2 = R~!~’~i(r\D ‘T~ ‘~

(6)

1’

abcd’

‘8

P

From (2)—(8) and the symmetries of R~b~({ }) we find ~~ ~ (9) 6~2(r)= ~2({ }) + 8k2 [JajbRab({ }) + V~J~ VaJb ]. In get~eral~2({ }) will be singular because the EC— Weyl field equations are equivalent to the Einstein— Weyl equations [6]. But we might have that the expression in the parenthesis cancels out the singular term and end up with a nonsingular 6~2(r). Let us take the particular case of a static plane symmetric spacetime, in this case the Einstein—Weyl equations give us [7] R —0 10 ab — ds2 = (1 + ax)1/2(dx2

—

dt2) + (1 +ax)(dy2 + dz2)

= 1bl2 (1 +ax)’I2(~5’j.f~) (11) where a and IbI are real constants. (11) is the Taub metric [8]. Note that in ref. [7], J” is given in a nonholonomic system of coordinates. Replacing (l0)—(l2) in (9) we find

ja

~2(r) = a2(l +axY3 (*a2 +/c2 lb ~).

(13)

Therefore, we have that ~R2(r)is singular for all Ibi and a * 0, i.e., the essential singularity of the Taub 211

Volume 57A, number 3

PHYSICS LETTERS

14 June 1976

spacetime can not be “cancelled” by the torsion. The caseaO was study in ref. [6]. In conclusion we have that the torsion does not prevent the singular behavior of ~R2(T)in the static plane symmetric case (a *0).

[2] F.W. Hehl, P. Von der Heyde and G.D. Kerlik, Phys. Rev. D10 (1974) 1066. [3] W. CopczyiSski,Cosmology Phys. Letters 39A (1972) 219. (?olish [4] B. Kuchowicz, spin and torsion Academy of Sciences, Inst. with of Astr. 1975), and references contained therein. [5] J. Tafel, Acta Phys. Polon. B6 (1975) 537. [6] P.S. Letelier, Phys. Lett. 54A (1975) 351.

References

[7] A.H. T.M. Taub, Davis and J.R. Ray, Rev. D9 [81 Phys. Rev. 103Phys. (1956) 454.

[1] A. Trautman, Inst. Naz. Alta Mat., Symp. Math. 12 (1973) 139.

212

(1974) 331.