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A charged Taub-NUT metric with torsion: A new axially symmetric solution of the poincare gauge field theory
Volume
IOOA, number
PHYSICS
8
A CHARGED TAUB-NUT A NEW AXIALLY
LETTERS
20 February
1984
METRIC WITH TORSION:
SYMMETRIC SOLUTION OF THE POINCARE GAUGE FIELD THEORY *
Peter BAEKLER and Friedrich W. HEHL Instrtute for Theoretical Physrcs, University of Cologne, D-5000 Cologne 41, W Germany Recerved
18 October
1983
Within the framework of the Poutcare gauge field theory, McCrea has recently drscovered d Taub-NUT-like metric with torsion. The metric is axially symmetric, whereas the torsion turns out to be S0(3)-symmetric. We find the corresponding solution with an additional electric charge.
I. Lorentz gauge bosom and PG. If one assumes that Lorentz gauge bosom exist m nature, then the Poincare gauge field theory (PC) appears to be the most natural starting point for the description of gravity plus Lorentz gauge bosons (cf. refs. [l-4] In order to allow the Lorentz gauge bosons to propagate, the gauge field lagrangian T’ should be polynomial at least to second order in the curvature FQprd (= -FpaY6) and the torsion FLY,7 (= -FPcuY):
*r.
I/= l//J4
+d2F70a
+(1/12)[(1/2~)F+$F,BY(d1FQ;
- (1/4M,/376
(F@Ys
+ fiFar@
t
td36,PF,,‘l)]
f2F'+@ t f@F@Y
t f4@Frfl)
.
(1)
The term -l//J4 represents the cosmological constant, 1 the Planck length, K the couphng constant of the Lorentz gauge bosons, and x, dA, and fA are 8 further dimensionless coupling constants. Moreover, the Rrcci tensor and the curvature scalar read, respectively, Fap := FYapY, F := T@F,~; 77a@is the local Mmkowskr metric. This theoretical framework of a general relativistic structure (cf. ref. [l] for details and the conventions) will be referred to as PG. 2. The purely quadratic gauge lagrangian. In analogy to GR (general relativity), one would hke to study exact solutions of the field equations of the PC and, for simplicity, vacuum solutions at first. A very efficient strategy turned out to be the following. The quadratic model lagrangran [6] Vo = (1/4Z2)(-Fi,“F”,
+ 2F”~F,,‘)
- (1/4~)F,p~s
Fcu’ra
(2)
is much simpler to handle than V of eq. (1). (Anholonomrc or Lorentz indices are denoted by Q, /3, y . .. = 0 .. . 3, holonomic or coordinate indices by i, j, k . .. = 0 . . . 3.) As soon as one finds a solution belonging to (2), there 1s a systematic way, pointed out in refs. [7-91, of generalizing this solution to the case of the lagrangian of eq. (I), provided the 10 constants of (1) fulfill certain weak algebraic constraints. The PG with the purely quadratrc lagrangian (2) will be called the QPG in the future. Hence the QPG is a special caseofthePGwrthA=l/x=fA=O,dl=-l,d2=0,d3=2. 3. The McCrea-NUT solution. In the past few years spherically symmetric * Supported by a grant of the “Minister fur Wissenschaft und Forschung des Landes *’ For attempts to quantize such theories, see ref. [5] and references given there.
392
and cylindrically symmetric
vacuum
Nordrhein-Westfalen”.
0375-9601/84/$03.00 0 Elsevier Science Publishers B.V (North-Holland Physics Publishing Division)
PHYSICS LETTERS
Volume lOOA, number 8
20 February 1984
solutions of the PG and of the QPG have been found by different authors. About a year ago, we gave a catalog of these solutions [lo] *2. The first electrically charged solution was derived recently in a highly interesting paper by Lee [ 151. The goal of these investigations is, of course, to find a Kerr-Newman-like solution with torsion. In the PG spin and other quantities of 3rd rank play an essential role. Accordingly, a Kerr-like solution with torsion is expected to represent some sort of generic solution of the PG. On this way McCrea took a very important step: He discovered a solution of the QPG which isaxially symmetric and NUT-like in the metric and carries an additional dynamic torsion field [16]. As will be shown below, the torsion of McCrea’s solution is SO(3)symmetric. McCrea’s procedure for finding his solution was similar to the one he used in “deriving” cylindrically symmetric solutions [17]. Because of the presumed weak double duality of the solution (cf. ref. [IS]), he started with the Taub-NUT-solution of CR with “cosmological” constant and made a clever ansatz for the torsion components. The metric and the torsion were then substituted into the weak double duality relation. The emerging equations can be specialized and thereby simplified by requiring the torsion to occur only linearly. Then they can be resolved with respect to the torsion. We will generalize the McCrea-NUT solution of the QPG to the case of electrical charge. 4. The field equations of the QPG with a Maxwell field as source. We add to (2) a Maxwell lagrangian and find the field equations (L$ = covariant exterior derivative, e r det e{(Y)modified torsion Til” := F$ t 2e [I.“F.Ilk k): D,(@l,)
Di(eFiiolp) t (eK/12)Ti [OrPI=0 ’
_ l2(tr &,i + rot &,i) = 12 Max T&i ,
aiw
aIiFjk1 = 0,
=0,
(374) (5,6)
The electromagnetic field strength is Fii, its contravariant density reads FFik := eg’lgklFjr. Observe that the Maxwell equations, in the form (5), (6), are valid in a U4. The translational, rotational, and electromagnetic energy tensors, entering (3) are defined as follows: tr cO’ := (e/12)(F,,UT’v, *Ot&oli:= (e/K)(F,,,7F’uu, Max T&1 := y”F,,
- ~ei(yFp,,u TpV,) , - ~ei~Fp,,rF,~uu~)
- ;e’,?k’Fk,
(7) ,
.
(8) (9)
The system of field equations (3)-(6) is much more complicated than the Einstein-Maxwell system. 5. The new solution. The procedure for finding the solution was described above. We start with the charged Taub-NUT metric of GR, cf. ref. [19]. For the computations we made heavy use of the computer algebra system REDUCE [20] and of some programs of Dautcourt et al. [21]. We work in Eddington-Finkelstein coordinates (u,r,9,cp)=(O,1,2,3)andassume~
ds2 = -2(du + 2(1/&l cos 9 dlp)[@(r)(du + 2(1/T)K cos 0 dq) t dr] t (r2 t 12/r7)(da2 + sin26 dq2) .
(10)
The function a(r) is defined according to (Q = electric charge): cqr)
r
_Mr -
Q2 t (r7/812)(r2 +12/k) .
r2 + 12/E
(11)
As the basis one-forms we take the half-null tetrad *’ Additionally plane torsion waves [ 11,121 and certain so-called “cosmological” solutions with matter [ 13,141 have been found in the framework of the PG.
393
Volume lOOA, number 8
eG = du + 2(1/d)
PHYSICS LETTERS
t? = a(r)&
cos 19dq ,
20 Februdry 1984
+ dr ,
e? = (r2 + 12/t?)1/2 sin 9 dp .
es = (r2 + 12/K)‘i2 dS ,
(12)
The local metric is
(13)
If we put the (anholonomrc) tornon in matrix form (UT = 1, US = 2, uq = 3, i?q = 4, cpr = 5, rt9 = 6), cf. ref. [lo], then for the S0(3)-symmetric case, in the notation of McCrea with U, = Ui(r), we have the following pattern.
r-u,3 -u,
1
.
Ul
@‘alp -9 =
-u8
-u,
-u3
u3
Ul
.
.
-u9
U, = FGaa = FGzG =
U3 +J6
(14)
u9
-u,o
We find 8 independent ception of
\
-U10
/ cf. refs. [22,23]. As in the uncharged case, the Ui vamsh with the ex-
torsion components,
r(Mr -
Q2)
(r2 + zqq2
=FE_a=F3;rz=
U, = FGTL =
r(Mr - 2Q2) - III~~/K
’
(r2 + 12/K)2
f Fgza = -W&Wr
(15, 16)
’
- Q2)
(r2 t 12/iZj2
(17)
*
(Note that by putting 12/K = 0, we find the torsion of the Baekler-Lee solution 1151.) The reflection symmetry IS destroyed by the existence of U3 and U6 [for O(3)-symmetry in the torsion we have U3 = U6 = U8 = UIo = 0 in (14)]. The factor 2 in front of the Q2-term prevents the application of the substitution Mr + Mr - Q2 for finding the charged solution. For the (anholonomic components of the) electromagnetic field, we find F ;r~=2Q
r2 - 12/K (r2 t 12/~)2 ’
Fs? = -4Q
(lIG)r
(18)
(r2 t 12/i2)2.
The charged McCrea-NUT solution is completely specified by the eqs. (1 l)-(13) (18). For Q = 0, we recover the uncharged version [16].
together with (lS)-(
17) and
6. Concluding remarks. We will compute some further quantities related to our solution, which may help in its future physical interpretation: For the U4curvature tensor we find the remarkably simple expression
394
PHYSICS LETTERS
Volume lOOA, number 8
Incidentally, Baekler-Lee tr&ii?_i-
20 February 1984
by dropping 12/ii in the parentheses, one arrives at the curvature of the O(3)-symmetric charged solution [15] in Eddington-Finkelstein coordinates. The gravitational gauge field energies read 2eQ2(Mr - Q2) 12(r2 t 12/i)3
and the electromagnetic MaxT__F = MaxT!s
rot& iiii = Ce(Mr - Q2) 214(r2 + P/i)
’
’
energy turns out to be
= M”T-.z
=
2Q2
(21)
12(r2 + 12/i)2 ’ The gravitational
spin carries, like the torsion, no totally antisymmetric e ii~~ %G
&&s,Z = 2”;aa
=2&a;;
piece:
-e(M12/ii t Q2r) =- 212(r2 + 12/i)2 ’
- Q2) = +(ll$)(Mr 12(r2 + 12/r7)2 ’
(22)
Eq. (19) shows that the solution embodies a microscopic constant curvature background, similar to the de Sitter “microuniverses” of Salam and Strathdee [24]. This background is presumably produced by attractive Lorentz gauge bosons. According to (20)-(22), on top of this microuniverse, there is gauge energy and gauge spin distributed mainly in the domain of small values of r; also the torsion decays for big values of r. Certainly the possible application of the charged McCrea-NUT solution has to be found in the microcosmos: Besides the weak gravitational potential -l/r, there enters a rising strong gravitational potential -r2, as is evident from (11). The question is whether this strong gravitational potential can be linked to microscopic properties of matter. We thank Professors C.H. Lee (Seoul) and J.D. McCrea (Dublin) for sending us their solutions prior to publication. Furthermore we are grateful to H.J. Lenzen for help in the computations and to B. Mashhoon and J.D. McCrea for useful remarks. References [II F.W. Hehl, Four lectures in Poincare gauge field theory, in: Cosmology and gravrtation. Spin, torsion, rotatron, and supergravtty, Proc. Erice-School, May 1979, eds. PG. Bergmann and V. de Sabbata (Plenum, New York, 1980) p. 5.
VI K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 66 (1981) 2258, and references therein. [31 E.W. Mielke, Habilitation theses, Kiel (1982). 141 A.A. Tseytlin, Phys. Rev. D26 (1982) 3327. [51 S. Hamamoto, Z. Phys. Cl9 (1983) 353. 161 F.W. Hehl, Y. Ne’eman, J. Nitsch and P. von der Heyde, Phys. Lett. 78B (1978) 102. [71 E.W. Mrelke, J. Math. Phys. (1983), to be published. 181 P. Baekler, Phys. Lett. 96A (1983) 279; prepnnt, UNV. Cologne (1983). [91 H.J. Lenzen, preprint, Univ. Cologne (1983). [lOI P. Baekler, F.W. Hehl and H.J. Lenzen, in. Proc. 3rd M. Grossmann Meeting on Recent developments
in general relativity
(Shanghai, 1982), ed. Hu Ning, to be published. 1111 W. Adamowicz, Gen. Rel. Grav. 12 (1980) 677.
395
Volume
[ 121 [13] [ 141 [15] [16] [17] [18]
[ 191 [ 201 [21]
[ 221 [23] (241
396
ZOOA, number
8
PHYSICS
LETTERS
M.-Q. Chen, D.-C. Chern, R.-R Hsu and W.-B. Yeung, Phys. Rev. D28 (1983) 2094. A.V. Minkevrch, Phys. Lett. 80A (1980) 232,95A (1983) 422. F. Muller-Horssen, Phys. Lett. 92A (1982) 433. C.H. Lee, Phys Lett. 130B (1983) 257 J.D. McCrea, Phys. Lett. 1OOA (1984) 397 J.D. McCrea, J. Phys. Al6 (1983) 997. P. Baekler, F.W. Hehl and E.W. Mrelke Preprmt IC/80/140, Trreste (1980), m Proc. 2nd M. Grossmann relatrvrty, ed. R. Ruffim (North-Holland, Amsterdam, 1982). J.G. Miller, J. Math. Phys. 14 (1973) 486. A.C. Hearn, REDUCE 2 User’s Manual, 2nd Ed., Umv. of Utah (1973). D. Dautcourt, K.P. Jann, E. Rremer and M. Rremer, Astronom. Nachr. 302 (1981) 1. P.D. Yasskin, thesis, Univ. Maryland (1979). R. Rauch, J.C. Shaw and H.T. Nieh, Gen. Rel. Grav. 14 (1982) 331. A. Salam and J. Strathdee, Phys. Rev. D18 (1978) 4596, and references therem.
20 1 ebruary
1984
Meetmg on General
IOOA, number
PHYSICS
8
A CHARGED TAUB-NUT A NEW AXIALLY
LETTERS
20 February
1984
METRIC WITH TORSION:
SYMMETRIC SOLUTION OF THE POINCARE GAUGE FIELD THEORY *
Peter BAEKLER and Friedrich W. HEHL Instrtute for Theoretical Physrcs, University of Cologne, D-5000 Cologne 41, W Germany Recerved
18 October
1983
Within the framework of the Poutcare gauge field theory, McCrea has recently drscovered d Taub-NUT-like metric with torsion. The metric is axially symmetric, whereas the torsion turns out to be S0(3)-symmetric. We find the corresponding solution with an additional electric charge.
I. Lorentz gauge bosom and PG. If one assumes that Lorentz gauge bosom exist m nature, then the Poincare gauge field theory (PC) appears to be the most natural starting point for the description of gravity plus Lorentz gauge bosons (cf. refs. [l-4] In order to allow the Lorentz gauge bosons to propagate, the gauge field lagrangian T’ should be polynomial at least to second order in the curvature FQprd (= -FpaY6) and the torsion FLY,7 (= -FPcuY):
*r.
I/= l//J4
+d2F70a
+(1/12)[(1/2~)F+$F,BY(d1FQ;
- (1/4M,/376
(F@Ys
+ fiFar@
t
td36,PF,,‘l)]
f2F'+@ t f@F@Y
t f4@Frfl)
.
(1)
The term -l//J4 represents the cosmological constant, 1 the Planck length, K the couphng constant of the Lorentz gauge bosons, and x, dA, and fA are 8 further dimensionless coupling constants. Moreover, the Rrcci tensor and the curvature scalar read, respectively, Fap := FYapY, F := T@F,~; 77a@is the local Mmkowskr metric. This theoretical framework of a general relativistic structure (cf. ref. [l] for details and the conventions) will be referred to as PG. 2. The purely quadratic gauge lagrangian. In analogy to GR (general relativity), one would hke to study exact solutions of the field equations of the PC and, for simplicity, vacuum solutions at first. A very efficient strategy turned out to be the following. The quadratic model lagrangran [6] Vo = (1/4Z2)(-Fi,“F”,
+ 2F”~F,,‘)
- (1/4~)F,p~s
Fcu’ra
(2)
is much simpler to handle than V of eq. (1). (Anholonomrc or Lorentz indices are denoted by Q, /3, y . .. = 0 .. . 3, holonomic or coordinate indices by i, j, k . .. = 0 . . . 3.) As soon as one finds a solution belonging to (2), there 1s a systematic way, pointed out in refs. [7-91, of generalizing this solution to the case of the lagrangian of eq. (I), provided the 10 constants of (1) fulfill certain weak algebraic constraints. The PG with the purely quadratrc lagrangian (2) will be called the QPG in the future. Hence the QPG is a special caseofthePGwrthA=l/x=fA=O,dl=-l,d2=0,d3=2. 3. The McCrea-NUT solution. In the past few years spherically symmetric * Supported by a grant of the “Minister fur Wissenschaft und Forschung des Landes *’ For attempts to quantize such theories, see ref. [5] and references given there.
392
and cylindrically symmetric
vacuum
Nordrhein-Westfalen”.
0375-9601/84/$03.00 0 Elsevier Science Publishers B.V (North-Holland Physics Publishing Division)
PHYSICS LETTERS
Volume lOOA, number 8
20 February 1984
solutions of the PG and of the QPG have been found by different authors. About a year ago, we gave a catalog of these solutions [lo] *2. The first electrically charged solution was derived recently in a highly interesting paper by Lee [ 151. The goal of these investigations is, of course, to find a Kerr-Newman-like solution with torsion. In the PG spin and other quantities of 3rd rank play an essential role. Accordingly, a Kerr-like solution with torsion is expected to represent some sort of generic solution of the PG. On this way McCrea took a very important step: He discovered a solution of the QPG which isaxially symmetric and NUT-like in the metric and carries an additional dynamic torsion field [16]. As will be shown below, the torsion of McCrea’s solution is SO(3)symmetric. McCrea’s procedure for finding his solution was similar to the one he used in “deriving” cylindrically symmetric solutions [17]. Because of the presumed weak double duality of the solution (cf. ref. [IS]), he started with the Taub-NUT-solution of CR with “cosmological” constant and made a clever ansatz for the torsion components. The metric and the torsion were then substituted into the weak double duality relation. The emerging equations can be specialized and thereby simplified by requiring the torsion to occur only linearly. Then they can be resolved with respect to the torsion. We will generalize the McCrea-NUT solution of the QPG to the case of electrical charge. 4. The field equations of the QPG with a Maxwell field as source. We add to (2) a Maxwell lagrangian and find the field equations (L$ = covariant exterior derivative, e r det e{(Y)modified torsion Til” := F$ t 2e [I.“F.Ilk k): D,(@l,)
Di(eFiiolp) t (eK/12)Ti [OrPI=0 ’
_ l2(tr &,i + rot &,i) = 12 Max T&i ,
aiw
aIiFjk1 = 0,
=0,
(374) (5,6)
The electromagnetic field strength is Fii, its contravariant density reads FFik := eg’lgklFjr. Observe that the Maxwell equations, in the form (5), (6), are valid in a U4. The translational, rotational, and electromagnetic energy tensors, entering (3) are defined as follows: tr cO’ := (e/12)(F,,UT’v, *Ot&oli:= (e/K)(F,,,7F’uu, Max T&1 := y”F,,
- ~ei(yFp,,u TpV,) , - ~ei~Fp,,rF,~uu~)
- ;e’,?k’Fk,
(7) ,
.
(8) (9)
The system of field equations (3)-(6) is much more complicated than the Einstein-Maxwell system. 5. The new solution. The procedure for finding the solution was described above. We start with the charged Taub-NUT metric of GR, cf. ref. [19]. For the computations we made heavy use of the computer algebra system REDUCE [20] and of some programs of Dautcourt et al. [21]. We work in Eddington-Finkelstein coordinates (u,r,9,cp)=(O,1,2,3)andassume~
ds2 = -2(du + 2(1/&l cos 9 dlp)[@(r)(du + 2(1/T)K cos 0 dq) t dr] t (r2 t 12/r7)(da2 + sin26 dq2) .
(10)
The function a(r) is defined according to (Q = electric charge): cqr)
r
_Mr -
Q2 t (r7/812)(r2 +12/k) .
r2 + 12/E
(11)
As the basis one-forms we take the half-null tetrad *’ Additionally plane torsion waves [ 11,121 and certain so-called “cosmological” solutions with matter [ 13,141 have been found in the framework of the PG.
393
Volume lOOA, number 8
eG = du + 2(1/d)
PHYSICS LETTERS
t? = a(r)&
cos 19dq ,
20 Februdry 1984
+ dr ,
e? = (r2 + 12/t?)1/2 sin 9 dp .
es = (r2 + 12/K)‘i2 dS ,
(12)
The local metric is
(13)
If we put the (anholonomrc) tornon in matrix form (UT = 1, US = 2, uq = 3, i?q = 4, cpr = 5, rt9 = 6), cf. ref. [lo], then for the S0(3)-symmetric case, in the notation of McCrea with U, = Ui(r), we have the following pattern.
r-u,3 -u,
1
.
Ul
@‘alp -9 =
-u8
-u,
-u3
u3
Ul
.
.
-u9
U, = FGaa = FGzG =
U3 +J6
(14)
u9
-u,o
We find 8 independent ception of
\
-U10
/ cf. refs. [22,23]. As in the uncharged case, the Ui vamsh with the ex-
torsion components,
r(Mr -
Q2)
(r2 + zqq2
=FE_a=F3;rz=
U, = FGTL =
r(Mr - 2Q2) - III~~/K
’
(r2 + 12/K)2
f Fgza = -W&Wr
(15, 16)
’
- Q2)
(r2 t 12/iZj2
(17)
*
(Note that by putting 12/K = 0, we find the torsion of the Baekler-Lee solution 1151.) The reflection symmetry IS destroyed by the existence of U3 and U6 [for O(3)-symmetry in the torsion we have U3 = U6 = U8 = UIo = 0 in (14)]. The factor 2 in front of the Q2-term prevents the application of the substitution Mr + Mr - Q2 for finding the charged solution. For the (anholonomic components of the) electromagnetic field, we find F ;r~=2Q
r2 - 12/K (r2 t 12/~)2 ’
Fs? = -4Q
(lIG)r
(18)
(r2 t 12/i2)2.
The charged McCrea-NUT solution is completely specified by the eqs. (1 l)-(13) (18). For Q = 0, we recover the uncharged version [16].
together with (lS)-(
17) and
6. Concluding remarks. We will compute some further quantities related to our solution, which may help in its future physical interpretation: For the U4curvature tensor we find the remarkably simple expression
394
PHYSICS LETTERS
Volume lOOA, number 8
Incidentally, Baekler-Lee tr&ii?_i-
20 February 1984
by dropping 12/ii in the parentheses, one arrives at the curvature of the O(3)-symmetric charged solution [15] in Eddington-Finkelstein coordinates. The gravitational gauge field energies read 2eQ2(Mr - Q2) 12(r2 t 12/i)3
and the electromagnetic MaxT__F = MaxT!s
rot& iiii = Ce(Mr - Q2) 214(r2 + P/i)
’
’
energy turns out to be
= M”T-.z
=
2Q2
(21)
12(r2 + 12/i)2 ’ The gravitational
spin carries, like the torsion, no totally antisymmetric e ii~~ %G
&&s,Z = 2”;aa
=2&a;;
piece:
-e(M12/ii t Q2r) =- 212(r2 + 12/i)2 ’
- Q2) = +(ll$)(Mr 12(r2 + 12/r7)2 ’
(22)
Eq. (19) shows that the solution embodies a microscopic constant curvature background, similar to the de Sitter “microuniverses” of Salam and Strathdee [24]. This background is presumably produced by attractive Lorentz gauge bosons. According to (20)-(22), on top of this microuniverse, there is gauge energy and gauge spin distributed mainly in the domain of small values of r; also the torsion decays for big values of r. Certainly the possible application of the charged McCrea-NUT solution has to be found in the microcosmos: Besides the weak gravitational potential -l/r, there enters a rising strong gravitational potential -r2, as is evident from (11). The question is whether this strong gravitational potential can be linked to microscopic properties of matter. We thank Professors C.H. Lee (Seoul) and J.D. McCrea (Dublin) for sending us their solutions prior to publication. Furthermore we are grateful to H.J. Lenzen for help in the computations and to B. Mashhoon and J.D. McCrea for useful remarks. References [II F.W. Hehl, Four lectures in Poincare gauge field theory, in: Cosmology and gravrtation. Spin, torsion, rotatron, and supergravtty, Proc. Erice-School, May 1979, eds. PG. Bergmann and V. de Sabbata (Plenum, New York, 1980) p. 5.
VI K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 66 (1981) 2258, and references therein. [31 E.W. Mielke, Habilitation theses, Kiel (1982). 141 A.A. Tseytlin, Phys. Rev. D26 (1982) 3327. [51 S. Hamamoto, Z. Phys. Cl9 (1983) 353. 161 F.W. Hehl, Y. Ne’eman, J. Nitsch and P. von der Heyde, Phys. Lett. 78B (1978) 102. [71 E.W. Mrelke, J. Math. Phys. (1983), to be published. 181 P. Baekler, Phys. Lett. 96A (1983) 279; prepnnt, UNV. Cologne (1983). [91 H.J. Lenzen, preprint, Univ. Cologne (1983). [lOI P. Baekler, F.W. Hehl and H.J. Lenzen, in. Proc. 3rd M. Grossmann Meeting on Recent developments
in general relativity
(Shanghai, 1982), ed. Hu Ning, to be published. 1111 W. Adamowicz, Gen. Rel. Grav. 12 (1980) 677.
395
Volume
[ 121 [13] [ 141 [15] [16] [17] [18]
[ 191 [ 201 [21]
[ 221 [23] (241
396
ZOOA, number
8
PHYSICS
LETTERS
M.-Q. Chen, D.-C. Chern, R.-R Hsu and W.-B. Yeung, Phys. Rev. D28 (1983) 2094. A.V. Minkevrch, Phys. Lett. 80A (1980) 232,95A (1983) 422. F. Muller-Horssen, Phys. Lett. 92A (1982) 433. C.H. Lee, Phys Lett. 130B (1983) 257 J.D. McCrea, Phys. Lett. 1OOA (1984) 397 J.D. McCrea, J. Phys. Al6 (1983) 997. P. Baekler, F.W. Hehl and E.W. Mrelke Preprmt IC/80/140, Trreste (1980), m Proc. 2nd M. Grossmann relatrvrty, ed. R. Ruffim (North-Holland, Amsterdam, 1982). J.G. Miller, J. Math. Phys. 14 (1973) 486. A.C. Hearn, REDUCE 2 User’s Manual, 2nd Ed., Umv. of Utah (1973). D. Dautcourt, K.P. Jann, E. Rremer and M. Rremer, Astronom. Nachr. 302 (1981) 1. P.D. Yasskin, thesis, Univ. Maryland (1979). R. Rauch, J.C. Shaw and H.T. Nieh, Gen. Rel. Grav. 14 (1982) 331. A. Salam and J. Strathdee, Phys. Rev. D18 (1978) 4596, and references therem.
20 1 ebruary
1984
Meetmg on General