A spherically symmetric vacuum solution of the quadratic Poincaré gauge field theory of gravitation with newtonian and confinement potentials

Volume 99B, number 4 PHYSICS LETTERS 26 February 1981

A SPHERICALLY SYMMETRIC VACUUM SOLUTION

OF THE QUADRATIC POINCARI~ GAUGE FIELD THEORY OF GRAVITATION

WITH NEWTONIAN AND CONFINEMENT POTENTIALS

Peter BAEKLER Institute for Theoretical Physics, University of Cologne, D-5000 KOln 41, West Germany

Received 13 November 1980

We derive a stationary spherically symmetric vacuum solution in the framework of the Poincar6 gauge field theory with a recently proposed quadratic lagrangian. We find a metric of the Schwarzschild-de Sitter type, both torsion and curvature are non vanishing, with torsion proportional to the mass and curvature proportional to the strong coupling constant K. The metric exhibits two pieces, a newtonian potential describing the gravitational behavior of macroscopic matter, and a confining potential ~ Kr 2 presumably related to the strong-interaction properties of hadrons. To our knowledge this is a new feature of a classical solution of a Yang-Mills type gauge theory.

A number of authors [ I - 8 ] considered the Poincar~ group as the underlying structure group of a general relativistic gauge field theory. The independent variables of this theory, representing the gauge potentials, are the tetrad coefficients e~ and the anholonomic components of a general metric compatible connection

l~ic~13 (= -Pit3c~)' The geometrical description of this theory takes place in a Riemann-Car tan spacetime U 4. Varying the action function independently with respect to ei c~ and F i ~ , yields the two general field equations of PG:

D / H j j - e~ i= e~,~ i , (1)

DjHc~ ij - ee~ i = ezc~ i , (2)

with the field momenta

Hai /= 2 a ~ / a F / i ~, H ~ i~ = a~V/aF/i~ .

The field lagrangian is denoted b y q ~ , ~c~ i and 7a[3 i are the canonical momentum and spin currents, respectively,

eu ~ : = e i C ~ - F~/~/H, ] i - F ~ j ~ H , ~ j i ,

ea~ i :=H[t~M i ,

gauge fields. Holonomic indices are denoted by i, j, k, .... anholonomic ones by a,/3, 7 ..... Our tetrads are always orthonormal, the metric is ~t~ = d i a g ( - 1 , +1, +1, +1), l is the Planck length, furthermore e := det (eia). The operator D] is the covariant exterior derivative. Torsion and curvature, the gauge field strengths are defined as usual:

Fi] ~ := 2(~[ie j l~ + F[il~ffe/.lt3), (3)

F i j ~ := 2(a[irjl~ + rEi, frjl ). (4)

Observe, that the field lagrangian c)y has not yet been specified.

The matter lagrangian is a U 4 with matter variables q~(x), which are representations of the Poincar6 group, is assumed to be

~?m = 6 ( '7~ , 7 ~, '~(x), ~i 'I '(x), ei~(x), r i ~ ( x ) ) .

The action function of matter W m = f d 4 x ./.2m (x) i s

required to be a Poincar6 scalar. Provided the matter equation 6~m/Oq~ is fulfilled, we find the following Noether identities:

e~,u i := 822m/Sei ~, e ~ c j := 622m/6Fi ~ ,

Di (eN i) -~ eFc~i~,[/ + eF~i~'YrB.r i , represent the momentum and the spin densities of the

Di(e ' r~ i ) - e 2 ; [ ~ l -- 0 .

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company

(st (6)

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In the special relativistic limit, i.e. FtT~ = 0 and F / / ~ = 0, we rediscover the momentum and angular momentum laws, respectively.

For scalar matter we have 7t3 i = 0, i.e., according to (6), 22[~t~ ] = 0. Hence we are only left with

D i (eF, ai ) = eF_,~iFai ~ . (7)

Eq. (7) can be rearranged into the two alternative forms [9,24]:

~i(eY.a i) = 2e~2od~Z~ i , (8)

V~ ) ~k i = 0 , (9)

where fZqa := ~ [iej] ~ is the object of anholonomity and Vf ) denotes the covariant derivative with respect to the Christoffel symbol.

Eq. (9) shows that scalar matter does not couple to the anholonomic c o n n e c t i o n l~i~ at all, but only to the tetrad coefficients ei '~. Hence scalar matter should only produce gravity mediated via the tetrads ei ~, i.e. "tetrad-gravity", cf. Nitsch [ 10]. Accordingly we have two possibilities to describe scalar matter in an appropriate geometrical framework. First (7), via (9), looks like the energy momentum law in a riemannian spacetime V 4. Secondly, (7) can be brought into the form (8) which displays this law in a Riemann-Cartan spacetime with vanishing curvature, a so-called T 4 or teleparallelism spacetime. In a T 4 we can globally transform the c o n n e c t i o n Fio~fl to zero. Hence ~2//.a is the sur- viving part of the torsion.

Up to this point we did not decide whether a V 4 or a T 4 is more appropriate for the description of macroscopic (scalar) matter. Only after the choice o f the field lagrangian Q3 can this question be settled.

There are two suggestive ways to choose a field lagrangian q3. A first choice is the (presumably unphysical) gauge lagrangian o f the ECSK-theory [11,1 ] :

cl~ = QPEC = ( e / 2 1 2 ) e [ i e / l ~ F l i ~ = eF/212 • (10)

Substitution of (10) into (1) and (2) yields the field equations:

FTa i3" - ½ eiaF,r6 ~ q' = l 2 ~-,a i , (1 1 )

½ Fa~ i + ei[c~F~]~, 3' = 12ra~ i . (12)

Because o f the algebraic relation between torsion and spin in (12) vanishing spin yields vanishing torsion, and we get a V 4. Consequently, scalar matter in the ECSK-

theory can only be described in the framework of a riemannian spacetime. In the spherical symmetric case we recover the interior and exterior Schwarzschild solution.

If we require the field equations to be quasi-linear in the second derivatives of the gauge potentials, up to quadratic terms in torsion and curvature are allowed to appear in the field lagrangian (7). In the following we will restrict ourselves to the quadratic lagrangian proposed by vonder Heyde et al.:

Q9 = ~ Q = (e/412) (-FijaFil '~ + 2 Fi'r. r FiB ~ )

+ (e/4K) ( - F i / ~ F i l ' o ~ ) , (13)

where K is a new dimensionless strong coupling constant. The corresponding coupled gauge field equations

read:

Dl(eFi]a+ ee[i,~F]l'~.r) - 12eo/= el2F, o/ , (14)

Dj(eFi/,~#) + K e/lZ(Fi[a~] + e t [~FM.r "r) = eKTc~ J . (15)

By construction macroscopic gravity is mediated by the /2-term, i.e. the motion of macroscopic matter can only be described in a T 4. Hence we expect the curvature to be proportional to K.

In the following we will make a spherically symmetric ansatz in order to find a simple vacuum solution. The most general spherical symmetric metric can be put into the form

ds 2 = - e 2 U d t 2 + e2Xdr 2 + r2(d0 2 + sin20 d~b 2) .

Imposing 0 (3 ) spherical symmetry on torsion and taking space reflections into account, we find 4 independent components of the torsion tensor. We denote the anholonomic torsion components as

R - h , ~ = - k , FT R T = f , F T R = F T ° o = FTc~

FRo 0 =FRo ~ = g ,

The connection components as

['TR T = V, I" R T R = X, I" 0 T O = I"¢p T c~ = Y ,

I-'oR 0 = ['OR ~ = -W, l~b0 q~ = r - 1 cot 0

and of the curvature components as

F T R R T = A , FTOoT=FTeaeaT = C ,

FROoT= F R c ~ T = - - D , FTOoR = FTo(oR = - - G ,

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I:ROOR = FR4e) R = H , Fo4~o = L .

Using the metric and these definitions, the (TT)-component of the vacuum field equation (14) has for spherical symmetry the following concrete form:

I-2 {2 r -2e -X(g r2 ) '+ 2 k r - 2 e - . - X ( r e X ) "

+ 2h~r - 2 e - ~ + k 2 - g2 + 2hk) (16)

+ t c - l ( - A 2 - 2C 2 - 2 D 2 + 2 G 2 + 2 H 2 + L 2) = 0 .

We take all components with respect to the frame which is naturally attached by the metric, the other components look similarly. For the (RTR)-component of (15) we find

K -1 [r-2e-U(r2A) " - 2WD + 2YH] + l - 2 k = 0 . (I 7)

Dots denote differentiation with respect to T, primes with respect to R. We used here the abbreviations of Yasskin and Ramaswamy [12].

It is relatively easy to integrate the coupled field eqs. (14) and (15)wi th the assumption - h = k = g = f and/~ + X = 0. We find for the metric:

ds 2 = - ( 1 - 2~/r + KrZ/4lZ)dt 2

+ ( 1 - - 2a/r+Kr2/412) -1 dr 2 +rZdg22 , (18)

for the torsion:

T FT R = FTR R = _ FTo o = _ FTc)¢) = FRo o = FR¢(a

= - -~ r -2 (1 -- 2~/r + Kr2/412) -1 /2 , (19)

and for the curvature [q5 := (a/r) e-Zu]:

Fa~v5 = (K/412)

1 0 0

0 1 + ~ 0

0 0 l+q5 X

0 0 0

0 0

0 _q5 0

0 0 0

0 0 q5

0 _q5 0

1 0 0

0 1 - ~ 0

0 0 1 -q~

(20)

The metric is of Schwarzschi ld-de Sitter type, i.e. not asymptotical ly flat, a is an integration constant. Further- more we represented the anholonomic components of the curvature in a 6 X 6 matrix, its rows and columns

are numbered in the following order (01 ,02 , 03, 23, 31 ,12) e.g. F0212 = qb. Observe that our solution for Fij a and F i j ~ does not exhibit t ime reflection symmetry.

It is remarkable that our solution carries both torsion as well as curvature. To our knowledge this is a new feature of a classical solution of a Yang-Mills type gauge theory. In the linearized form the potential

takes the form

~o = - a / r + g r 2/812 , (21 )

i.e., we find a newtonian plus a "conf inement" type of potential as foreseen and discussed in some detail in ref. [2].

As we saw above, in a quadratic PG the gravitational behaviour of macroscopic scalar matter can only be described in a teleparallel spacetime T 4 - and in no other way. Look at our solution: In the limit • -+ 0 (rotons suppressed [71) we find

ds 2 = - ( 1 - 2 a / r ) d t 2 + (1 - 2a/r) -1 dr 2 + r 2 d ~ 2 ,(22)

Fip ~ a r - 2 e - u , Fi/~O - O . (23)

This is the Schwarzschild solution in a teleparallelism theory.

As found earlier [ 3 ,13 -16 ] , the exterior Schwarz- schild solution is the unique spherical symmetric vacuum solution in a T 4 and is equivalent to the corresponding solution in General Relativity. Furthermore, in a T 4 the exterior Schwarzschild solution can be matched to an interior solution for a perfect fluid sphere [17]. Consequently, in this limit a = 12M, where M is the mass of the sphere. Accordingly, we recognize that our solution (18) to (20) has exactly the satisfac- tory macroscopic limit we had hoped for. Hence, in this limit, PG fulfills all the normal tests of GR [2,3,16] i.e. it is a viable gravitational theory.

There is another limit inherent in (18) to (20). I f we neglect the mass, then the torsion vanishes and we find a de Sitter metric in a V 4 :

ds 2 = - ( 1 +Kr2 /412)d t 2

+ (1 + ~/412) -1 d r 2 + r2dg22 , (24)

Fi/~ = 0, F~t3 "r~ = (~/4/2) 1 , (25)

where 1 is the six dimensional unit matrix. We remark that only in this fairly unphysical limit we have a riemannian spacetime.

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Spherical symmetric solutions of PG for vanishing torsion have been found earlier [ 18 -20 ] :

ds 2 = - ( 1 + c 1 / r + c 2r 2 ) d t 2

+ (1 + cl /r + c2r2) -1 dr 2 + r 2 d~22 , (26)

ds 2 = ~b2r -2 ( -d t2 + dr 2 + r2d02 +r2 sin20 d~2), (27)

where q~ obeys the fiat wave equation and c 1 and c 2 are integration constants. It is important to recognize that there is no direct relation botween (26) and (18)-(20) . Rather (26), see the discussion of Wallner [21] can be put into the form

f0.c~ = 0 ,

Fc~378 = diag (c 2 + 2~, c 2 - ~, c 2 - ~,

c2+2Lc 2 -~ ,c 2-~) , (28)

where ~ is c 1/2r 3. In comparing (26), (28) with (18), (19), (20) the distinctive features of our solution are clearly visible. The integration constant c 2 in (26), (28), in contrast to the claim of Wallner [21 ], is not determined in the unphysical limit of a V 4 spacetime. Incidentally, the V 4 limit (25) of our solution is a special case of (26), (28) with c 2 = K/412 and c 1 = 0.

To give a physical interpretation of our solution

(18) - (20) we will follow the suggestion of HeN et al. [2,3] that the long range newtonian gravitation is

mediated by the tetrad fields ei ~ and that the P-field mediates a short range interaction of the Yang-Mills type with "confinement" potential --(K/12)r 2. Hence the appropriate description of macroscopic matter takes place in a T4, indeed, we can read from (20) that the curvature of our solution is proportional to the strong coupling constant •. A metric of the Schwarzschild-de Sitter type was also found in the f -g- theory of gravity used for describing confinement properties of hadrons [22,23], an interpretation which we also would like to follow here. In analogy we could also try to regard a hadron as a de Sitter micro universe with radius R ~ h~ -1/2.

I would like to thank Professor F.W. Hehl and J. Nitsch for many helpful and illuminating discussions and for their continuous support.

References

[1] F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393.

[2] F.W. Hehl, Y. Ne'eman, J. Nitsch and P. vonder Heyde, Phys. Lett. 78B (1978) 102.

[3] F.W. Hehl, J. Nitsch and P. yon der Heyde, Gravitation and Poincar~ gauge field theory with quadratic lagrangian, in: General relativity and gravitation (the Einstein memorial volume), ed. A. Held (Plenum, New York, 1980).

[4] P. vonder Heyde, Phys. Lett. 58A (1976) 141. [5 ] Y. Ne'eman, Lecture notes in mathematics, Vol. 676

(Springer, 1978) p. 189. [6] Y. Ne'eman, and Dj. Sijacki, Ann. Phys. (NY) 120

(1979) 292. [7] F.W. Hehl, Four lectures on Poincar6 gauge field theory,

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[9] H. Meyer, Diploma Thesis (K~51n, 1979), unpublished. [10] J. Nitsch, The macroscopic limit of the Poincar6 gauge

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[14] J. Nitsch and F.W. Hehl, Phys. Lett. 90B (1980) 98. [15 ] L.L. Smalley, Phys. Rev. D, to be published. [16] M. Schweizer, N. Straumann and A. Wipf, preprint Univ.

of Zurich (1980). [17] P. Baekler, Phys. Lett. 94B (1980) 44. [18] R. Waller, Notes on recent U4 theories of gravitation,

preprint Universit~it Wien (1979), to be published in Gen. Rel. Gray.

[19] W. Kopczyfiski, private communication. [20] P.B. Yasskm and P. Baekler, unpublished. ]21 ] R. WaUner, The cosmological constant in the U4 theory

of gravity, preprint, Universit~it Wien (1979). [22] C. Sivaram and K.P. Sinha, Phys. Rep. 51 (1979) 111. [23] C.J. Isham and D. Storey, Phys. Rev. D18 (1978) 1047. [24] W.R. Stoeger and P.B. Yasskin, Gen. Rel. Grav., to be

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A spherically symmetric vacuum solution of the quadratic Poincaré gauge field theory of gravitation with newtonian and confinement potentials