Some Spherically Symmetric Exact Solutions of the Metric-Affine ...

CHINESE JOURNAL OF PHYSICS

DECEMBER 1997

VOL. 35, NO. 6-I

Some Spherically Symmetric Exact Solutions of the Metric-Affine Gravity Theory J. K. Ho, De-Ching Chern, and James M. Nester’ a r t m e n t o f Physics, National Central University, Chungli, Taiwan 320, R.O.C. (Received December 2, 1996; revised manuscript received August 20, 1997) D

e

p

We present four new spherically symmetric exact solutions of the Metric-Affine Gravity theory. In each case, just as in the other known spherically symmetric solutions, the metric has the Reissner-Nordstrom (anti) de-Sitter form. The nonmetricity is of the Weyl type with a Coulomb potential form due to a dilatation charge. The torsion exhibits a variety of asymptotic behaviors including asymptotically constant. For three solutions the torsion depends on an arbitrary function. Each of the solutions is rather general, requiring only 7 or 8 conditions on the 23 parameters of the theory. PACS 04.20.Jb - Solutions to equations.

PACS 04.50.Sh - Unified field theories and other theories of gravitation. I. Introduction The success of gauge theories in other branches of physics inspired the development of gauge theories of space time symmetries. At low energies the spacetime group associated with the matter fields is the Poincure’ group. This fact, in addition to geometric considerations, encouraged the development of the Poincare’ Gauge theory of gravity (PGT) in which the matter energy-momentum and spin densities are +,he source of the curvature and t o r sion of a Riemann-Curtan geometry. (Einstein’s General Relativity (GR) is a very special

case where the torsion vanishes leaving a Riemannian geometry.) However there are indications of more general symmetries in higher energy domains. Recently an extension of the Poincare gauge theory to the most general spacetime symmetry gauge theory, the MetricAffine Gauge Theory of Gravity (MAG), has been developed by Hehl and collaborators [l]. This theory has the most general type of covariant derivative: in addition to curvature and torsion, the MAG also has nonmetricity, i.e., a nonmetric compatible connection. Hence parallel transport no longer preserves length and angle. Although the theoretical structure of this theory has been developed, we do not yet have much understanding of what new physics is allowed by the MAG theory. One source of improved understanding is exact solutions. However, due to the highly nonlinear nature of the theory, exact solutions are not easily found unless they have a great deal of symmetry. While highly symmetric solutions are very special, they are actually of considerable interest. In GR it was found that, concerning the physical effects which are practically observable, most are actually a consequence of the exact spherically symmetric solution. Physically this is not surprising, for observable relativistic effects will only show up in strongly 640

@ 1997 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

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gravitating systems which are thus very massive and hence nearly spherical. We recall that for GR the Birkhoff theorem assures us that there is a uniqzle spherically symmetric solution for the metric-Schwarzschild, for the asymptotically flat case, and its asymptotically constant curvature generalization: the Schwarzschild (anti) de-Sitter metric. However for the PGT this is no longer the case, the metric has the same form but a fair number of distinct (presumably they correspond to different spinning sources) exact spherically symmetric solutions for the PGT with non-trivial torsion are now known [2]. Now it is timely to find analogous solutions for the MAG. Two spherically symmetric MAG exact (vacuum) solutions with nontrivial nonmetricity were found by Tresguerres [3]. T h e metric is of the Reissner-Nordstrom (anti) de-Sitter type-like the metric of an electrically charged massive particle in an asymptotically constant (negative) curvature space. However in the MAG there is no need for a source or any electromagnetic fields present to generate this part of the metric. Instead it is a non-linear consequence of the Weyl one-form which resembles a Coulomb potential that has originated from a “dilatation” charge. In order to gain more insights into the MAG theory, we looked for more solutions under similar conditions. With the aid of the symbolic computer program EXCALC, we have found four new solutions [4]. Like the solutions of Tresguerres, our solutions also have a metric of the Reissner-Nordstrom (anti) de-Sitter form and the nonmetricity is of the Weyl type with a Coulomb potential form. The torsion, however, exhibits a variety of fall off behaviors including asymptotically constant. Each of the new solutions is rather general, requiring only 7 or 8 conditions on the 23 parameters of the theory. For three of the solutions the torsion, like certain PGT solutions, depends on an arbitrary function. The next section gives a short review of the MAG theory of Gravity. In section III. the special case of Weyl-Cartan geometry to which our new solutions belong is described. Section IV. presents the restrictions due to the assumption of spherical symmetry. In section V. we present the MAG exact solutions of Tresguerres, as well as our new solutions. The concluding section briefly discusses our results and compares them with several other new MAG solutions. An appendix is included giving the irreducible decomposition of torsion and curvature in the MAG geometry. II. The Metric-Afflne gravity theory

Here we briefly outline the essential features of the Metric-Affine gauge theory of gravity, referring the interested reader to a comprehensive review [l] for further details. The potentials for the MAG gauge theory are the metric gap, the (co)frame one-form 19” (not assumed to be orthonormal) and the completely independent and unrestricted connection one-form Ia p. The corresponding field strengths are the nonmetricity one-form

the torsion 2-form

T” := 019” := dd” + I’$ and the curvature 2-form

A @,

(2)

SOME SPHERICALLY SYMMETRIC EXACT SOLUTIONS

642

I’$

R/ := dI’,/ +

I’@?.

A

VOL. 35

(3)

The (vacuum) gravitational Lagrangian L := VEC + VR + VT +

VQ + VQT,

(4)

includes all (parity preserving) scalar valued 4-forms constructed from the gauge field strengths up to quadratic order: VEC is the Einstein-Car&r Lagrangian, VR, VT, and VQ are quadratic curvature, torsion and nonmetricity terms and VQT is bilinear in nonmetricity and torsion. Specifically,

(5) (6) (7) VQ := -

Rap,

;Qea A

(8)

4 VQT := c c(L) L=2

tL’Qmp

A

19~ A* T”,

(9)

where 7 is the unit volume 4-form, ~~0 := gorgPb*(tiY A 19’) and, in terms of the irreducible parts of the curvature, torsion and nonmetricity defined in the appendix,

(11)

qM)hr 4 c d(h-)g

K=l

cry 9 ,m(W

(12)

From the Lagrangian one gets the gauge field momenta:

JpP :=

av = Ma” + 2c,& /j*(l) TP)

__2_

8QcYp

+2C@ A*(2)To) +

Ha := -g = p, +*

f: ctLj (L)Q,p L=2

A

;(c4 _ Q)~‘-$Q

116) ,

(13)

,,+y,

(14)

- ____..__

J.K.HO,DE-CHING CHERN,AND JAMES M.NESTER

VOL.35 dV

Hap := -dR," -

643

(15)

The variation of the action with respect to the independent gauge potentials leads to the vacuum field equations: DMQP - map = 0,

(0th)

(16)

DH,-E,=O,

W)

(17)

DHap -Eap = 0,

(2nd)

(18)

where

Ea = g =~cz~V Emp =

+(e,]T’) AHp

-$$ = -6” A

+(e,JRpY)/\HP7 + k(e,]Qp-l)M"',

(1%

HP-g&F,

l3V map=2-= r9aAEp+QP7~iM~7-TQAHp-R7QAHN-rp+RprAHQy. dsc!

(21)

Using the Noether identities, the zeroth field equation can be shown to be redundant [l], provided the matter equations hold. III. Weyl-Cartan geometry An interesting special case of the general metric-affine geometry is Weyl-Cartan geometry. In this case the nonmetricity one-form is restricted to be purely a trace:

Qclp := -Dgap = gapQ,

(22)

with Q := aQ,y being the Weyl one-form. In this type of geometry angles are preserved under parallel transport but lengths are not. As a consequence the number of irreducible decomposition pieces of nonmetricity reduces from four to one (only the Weyl one-form survives), the number of irreducible decomposition pieces of the curvature reduces from eleven to seven (there is only a trace term in the symmetric part of curvature). But the torsion still has three irreducible decomposition pieces. The Lagrangian and the vacuum field equations retain the same forms. Naturally this large reduction of the number of nonvanishing quantities makes things much easier. IV. Spherical symmetry We now consider spherical symmetry. One can apply Birkhoff’s theorem to the underlying Riemannian geometry and deduce the necessary form of the metric. For convenience we work in terms of orthonormal frames, then the metric coefficients are constant: gap = diag(- + tt), and our spherical symmetric orthonormal coframe has the form

SOME SPHERICALLY SYMMETRIC EXACT SOLUTIONS

644

tit = f(T)di,

19’ =

tie = rde, LdT, f(T)

@ = rsinfJd4,

VOL. 35

(23)

where f ( T )

:=

1-

hoN12 T + $&T2 + ~ 2tcaor2

(24)

.

Thus our metric is of the Reissner-Nordstrom (anti) de-Sitter type just as that used by Tresguerres [3] for his MAG exact solutions. Such a metric allows for both asymptotically constant curvature and long range connection terms acting like an effective Einstein source. For SO(3) spherical symmetry the torsion depends on 8 functions of r in general:

(25)

and the nonmetricity depends on 12 functions of

T

in general:

Qtt = '?I(+~ t 4.2(T)dT,

Qv = Q3(T)dt t Q(T)‘f?

Qee = q+')@

Q+4 = 4%(T)tit + q6(T)fir, Qte = Qs(T)fi’ + @o(T)@,

t Qt7. = 47(T)fit t

q6(T)flr, q8(T)%

= -'?m(T)~e t ‘3(T)@, QT4 = -Qn(T)fi' t Q&‘)@,

Qt+

(26)

= Qn(T)fi’ + qn(T)@, Qeqt = 0.

Q,-e

For the special case of Weyl-Cartan geometry, things are much simpler, then 41 = -q3 = -45,

q2 = -q4 = -46,

(27)

with all the other q; vanishing. V. Exact solutions

A symbolic algebra computer routine using REDUCE [5] and EXCALC [S] was written to compute the MAG field equations. The program was tested on the known spherically symmetric exact solutions, those of the PGT type (a special case of MAG) [2] and two solutions of the MAG type [3]. We now briefly describe these MAG solutions of Tresguerres. His two solutions have certain common parameter conditions: a2 = -2c4 = %d4,

c3 = 0,

J.

VOL. 35

K.HO,DE-CHINGCHERN,AND JAMES M.NESTER

645

and properties. In particular they both have the torsion trace connected to the Weyl oneform:

t2)TQ =

;Q A ti”,

(29)

which has a Coulomb potential like structure Q =

-%I’, Tf(T)

consequently, NO 2t2 = 2t3 = q1 = -q3 = -45 = -.

(31)

Tf (4

One solution is of the Weyl-Cartan type with the additional specific restrictions: all other t; = qj = 0,

br = -64 = bs = 0.

(32)

The other has a more general nonmetricity with the additional specific geometric restrictions:

NI

2t4 = q10 = ___

N2

2te = q12 = -

T2 f(T) ’

T2f(T) ’

(33)

where Nr and N2 are arbitrary constants with all other ti and qj vanishing, and the additional parameter restrictions 1 ae = -2aa = -Sal - 3C2

4

2

= id2 f 3C2.

(34)

Briefly, his solutions impose 6 conditions on the parameters of the theory, they depend on O(~/T~) fall off for the nonmetrieity and a total of 3 or 5 constants, and have O(l/T and ) torsion, respectively. For our new exact solutions we have confined our considerations to those with WeylCartan geometry. This imposes the vanishing of various of the irreducible pieces. First, The whole nonmetricity is proportional to the Weyl l-form: Qclp = gapQ, for which we, like Tresguerres, use the Maxwell potential-like form (35) For the torsion, we assumed that (‘IT* vanishes, and that t2)Tcr is, as above, a function of the Weyl l-form: t2)T” I ;Q A 19”.

(36)

SOME SPHERICALLY SYMMETRIC EXACT SOLUTIONS

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VOL. 35

This again means

No

2t2 = 2t3 = q1 = -q3 = -qs = __

Tf (T) ’

with all other qK vanishing. For the torsion we further assume that t1 = ts = 0,

t7 = t4 = C(T),

t8 =

-t6 =

(38)

p ( T ) ,

where a(r), and p(T )are unknown functions. We now require that the vacuum field equations are satisfied. We found four different cases. They impose certain conditions on the parameters ae,a;,cj,dk and on the undetermined functions CI(T),~(T). There are some common parameter conditions (one more in addition to those of Tresguerres) an = -2~3,

8

a2 = -2c4 = id4,

c3 =

0 .

(39)

The remaining specifics of the four special cases representing new solutions are given in Table I. Three of them contain an arbitrary constant, N7 or Ns. Note that @(T) is a free function in case (ii), and Q(T) is a free function in case (iii), while in case (iv) one function can also be chosen freely. TABLE I. Parameter conditions and functional forms for new MAG solutions. conditions on bj

conditions on

a(~)

and

6)

bl = -b4 = b5 = be,

a(T) = 0,

(ii)

bl = -b2 = -b4 = b5 = be,

N7 Q ( T ) = T;?(T) ’

(iii)

bl = -b2 = b3 = -b4 = b6,

m> = fol

(iv>

bl = -b2 = b3 = -b4 = be,

Q2(T) - p2(T) = &.

P(T)

P(T) = &f(T),

VI. Discussion

We have found four new solutions of the Metric-Affine theory with a Weyl-Cartan geometry. Each of our solutions is rather general, requiring only 7 or 8 conditions oil the 23 parameters of the theory. Our new solutions have the same 3 parallleter ReissnerNordstrom (anti) de-Sitter type metric as the two solutions first found by Tresguerres. Moreover, for our solutions, just as for the solutions of Tresguerres, the scalar curvature is asymptotically equal to the cosmological constant while the curvature trace two form

VOL.35

J.K.HO,DE-CHING CHERN, AND JAMES M. NESTER

t R : = R,7 = 2dQ =2No --+I A TJ’,

647

(40)

reveals a new long range coulomb-like gravitational effect presumably due to a “dilatation charge ”. For our solutions, unlike the two found previously, the torsion part can have a variety of fall offs and in three cases contains a free function. In view of our past experience in the PGT, where free functions have been also found, these results are not surprising. In the PGT there was an attempt to identify this phenomena with some kind of gauge freedom [7]. Although we now know that is not correct, the physical significance of such free functions has not yet been identified. Presumably the various behaviors are related to the detailed spin and hypermomentum of the source. One consequence is that in the MAG, just as in the PGT, black holes have lots of hair. Judging from the PGT experience, we expected that these 6 exact solutions would represent only a small portion of the possible spherically symmetric exact MAG solutions. Indeed since our investigation was completed it has come to our attention that several more exact MAG solutions have been found by Tresguerres and his coworkers [8-lo] (additional related investigations have also appeared [11,12]). We have used our program to verify these additional solutions (and thereby have further tested our program). The additional spherically symmetric solutions all have the same form for the metric as we have used. One solution [8] depends on a total of 5 constants, has l/r, l/r2 and l/r3 torsion fall off in the asymptotically flat case and requires 16 constraints on the parameters of the theory leaving only 7 free parameters. Another [9] depends only on the 3 constants which appear in the metric, has l/r torsion and requires only 7 constraints on the parameters leaving 16 free. This latter solution has been generalized to an axisymmetric solution [lo] depending on an additional Kerr type angular momentum parameter but now allowing only 11 of the parameters of the theory free. None of these solutions allow free functions as appear in several of our solutions. Comparing these latest solutions with the four we have found suggests several lines of investigation for generalizations which we intend to pursue. Concerning the physical significance of all of these solutions, we are not yet in a position to say much. Clearly it depends largely upon the various parameter combinations which appear, but the meaning of these combinations cannot really be ascertained until a Hamiltonian analysis as well as a linearized theory analysis of the MAG is available. In any case, one of the things we wish to report here is that we now have a reliable symbolic program which can be used with more general ansatz to find additional exact solutions for the MAG as well as explore their significance. Acknowledgments We thank Dr. R. S. Tung for his generous advice and assistance. We also wish to thank H. J. Yo for double checking our computer program and verifying several solutions. We appreciate the continued computer support from the department of Physics, National Central University. This work was supported by the National Science Colmcil of the R.O.C. under the grants NSC84-2112-M-008-004, NSC85- 2112-M-008-003.

SOME SPHERICALLY SYMMETRIC EXACT SOLUTIONS .

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VOL. 35

Appendix: Irreducible decomposition of curvature, torsion and nonmetricity

The irreducible decomposition of the nonmetricity is:

(3)Qcrp =

;[29(& - 1 Sap Al 7 4

(A.2)

(4)Qap = gaaQ,

(A4

(l)Qap = Qa/d2) Qap- (3) Qap -t4) Qap.

(A4

Q := fQT7

(A.5)

where

11, :=

ePJ

(A.6)

!Zap,

(A.7)

A := R,P,

0, :=* (i&p A 0 := 19” A

@),

64.8)

o,,

(A*9)

R, := 0, - +ea]O.

(A.lO)

The irreducible decomposition of the torsion is: (2)Ta = $290 A (ep]@),

(AX)

(‘)Tcr = -5 *[,, A* (?+ A tip)],

(A.12)

(r)Ta = Ta _t2) T” _c3) T”.

(A.13)

The irreducible decomposition of the czlrvat~re is obtained by first separating the curvature two form into its antisyrmmetric and symmetric parts, &p = Kp + &PC,

w h e r e Wap := R[,pl, Z,p := Rt,p).

(A.14)

Then the irreducible decomposition of Wap is: ‘2’w,,

= -*(‘L~L, A ql,j),

(A.15)

J.K.HO,DE-CHINGCHERN,AND JAMES M.NESTER

VOL. 35

649

(A.16) (A.17) (A.18) (A.19)

(A.20)

where W” : = ep] W”“ ,

(A.21)

W : = eaJWQ,

(A.22)

X” : =*

(A.23)

(WPa A tip),

(A.24)

X : = ea]Xo,

(A.25) (A.26) And the irreducible decomposition of Zap is: P)z,+ : = _ f

(3)Zap : =

*[fi(,

A

TP)]

7

$319(, A (ep)] A) - 2w.41,

wzap : = (5)Zap : =

(A.27) (A.28) (A.29)

A zp), li7 2 (a

(‘),q_& : zz zap -(2) z,p -C3) zap -t4) zap -t5) z,p.

(A.30) (A.31)

(A.32)

SOME SPHERICALLY SYMMETRIC EXACT SOLUTIONS

650

VOL. 35

(A.33) (A.34) (A.35) (A.36) (A.37)

(A.38) Finally we take (‘)R =(‘I W for i 2 6 and (‘I& =(i-6) Z for i 2 7. References t email:

[email protected]

[ 1 ] F. W. Hehl, J. D. McCrea, E. Mielke and Y. Ne’eman, Phys. Rep. 258, l-171 (1994). [ 2 ] C. M. Chen, D. C. Chern, J. M. Nester, P. K. Yang and V. V. Zhytnikov, Chin. J. Phys. 32, 29 (1994). [ 31 R. Tresguerres, Zeit. Phys. C65, 347 (1995). [ 4 ] J. K. H O , SpheticaIly Symmetric Exact Solutions for Metric-Affine Gauge Theory, MSc. thesis, (National Central Univ., Chungli, Taiwan 1996), unpublished. [ 5 ] A. C. Hearn, REDUCE User ’s Manual, Rand Publication CP78 (The Rand Corporation, Santa Monica, CA, 1985). [ 6 ] E. Schriifer, F. W. Hehl, J. D. McCrea, Gen. Rel. Grav. 19, 171 (1987). [ 71 P. Baekler and E. W. Mielke, Fortscr. Phys. 36, 549 (1988). [ 8 ] R. Tresguerres, Phys. Lett. A200, 405 (1995). [ 9 ] Yu. N. Obukhov, E. J. Vlachynsky, W. Esser, R. Tresguerres, and F. W. Hehl, Phys. Lett. A220, 1 (1996). [lo] E. 3. Vlachynsky, R. Tresguerres, Yu. N. Obukhov and F. W. Hehl, Class. Quantum Grav. 13, 3253 (1996). [ll] R. W. Tu ck er and C. Wang, Class. Quantum Grav. 12, 2587 (1995). [12] T. Dereli, M. &der, J. Schray, R. W. Tucker and C. W. Wang, Class. Quantum Grav. 13, L103 (1996).