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On the skyrmion number of Schwarzschild black holes
Volume 126, number 5,6
PHYSICS LETTERS A
11 January 1988
ON THE SKYRMION NUMBER OF SCHWARZSCHILD BLACK HOLES D. LOHIYA and A. MUKHERJEE Department ofPhysics andAstrophysics, Universityof Delhi, Delhi 110007, India Received 5 May 1987; revised manuscript received 7 September 1987; accepted for publication 17 November 1987 Communicated by J.P. Vigier
It is shown that the coupled Skyrme—Einstein equations of motion do not admit any Schwarzschild-like solution with nonzero topological number. A Schwarzschild black hole thus has no “skyrmion hair”. More general static, spherically symmetric solutions with a horizon are discussed. Solutions with nonzero skyrmion number exist, but are unlikely to be stable.
In recent years there has been a lot of interest in the model first proposed by Skyrme [1] nearly a quarter of a century ago. The model describes the breakdown of the chiral symmetry SU(2)LXSU(2)R to the isospin SU(2). It has the advantage of being only second order in time derivatives, and is thus amenable to the usual hamiltonian techniques. Further, it possesses classical soliton solutions skyrmions which can be interpreted as baryons. This Letter reports on a search fora skyrmion black hole, i.e., a solution to the classical equations of motion for the Skyrme model coupled to gravity, with a nonzero skyrmion number as well as an event horizon. A natural candidatewould be a Schwarzschildlike solution with nonzero skyrmion number, which would describe a spherically symmetric black hole with “skyrmion hair”. Such an object could result, conceivably, from the collapse ofa star made ofskyrmions as well as other matter. An earlier analysis [2] solved the Skyrme equation of motion in a background Schwarzschild spacetime, ingnoring the back
Finally, we argue that such solutions are likely to be unstable against time-dependent perturbations. The model of our interest is described bythe action 1=1,
J
(la)
,
where j~=
—
~ ~
Tr(D,~UlYU-’)
+(l/32a2) Tr[UtD~U, U—’D~U]2}
—
reaction of the skyrmion on the metric. We, on the other hand, consider the solutions to the coupled Einstein—Skyrme equations of motion. We prove that a whole class of static, spherically symmetric metrics (including the Schwarzschild and Reissner— Nordstrom metrics) cannot support a nonzero skyrmion number. Nontrivial solutions outside this class do exist. We study the analytic structureof solutions with an event horizon and relate the value of the skyrmion field at the horizon to that of the metric.
+12
J
X ( —g)”2 d4x, 1= (—g) “2R d4x. —
(ib) (ic)
—
Here U is a matrix-valued function of spacetime, related to the pion fields r1(x) by U=exp(if;1a.~) where a’ are the Pauli matrices andf, (—~190 MeV) is the pion decay constant. The covariant derivative D,~,the determinant g and the curvature scalar R are defined as usual in terms of the metric tensor g,~ 3,.The classical solutions are the extrema of the action (I). We look for solutions where Usatisfies the Skyrme ansatz U(x) = exp [ih ( r)~.~/2]
(2)
and g,~,,is static and spherically symmetric: 2 = —f( r) dt2 +j( r) dr2 ds +r2(d02+sin2O dq2)
(3)
,
0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
.
307
Volume 126,number5,6
PHYSICSLETTERSA
In terms of the ansatz functionsf(r), h(r) and j(r), the matter part of the action becomes Ii =
—
~-~-
x [r2(l
J
dt dr (fl)”2
—cos h)(l —cos h+4k2r2)
+(l—cosh+k2r2)j’ (h’)2]
11 January 1988
Since k must be real for the solution to make any physical sense, we obtain h = const. The value of this constant must be 2rnt (n an integer) if we require the energy density T 0=p= (l/8a2r2)[(1/j)(h’)2(l —cos h+k2r2) 0 +(l/r2)(l—cosh)(l—cosh+4k2r2)] (10)
(4)
,
where k=f~,a/2,and prime denotes derivative with respect to r. Similarly the gravitational part of the action yields
and the pressure density Tp
—
—
(l/8a2r2)[( l/j)(h’)2(l —cos h+k2r2) (11)
(l/r2)(l —cos h)(l —cos h+4k2r2)]
to vanish as r—+~.The skyrmion number of a solution is given by
12—4~ Jdtdrr2(f~2f’i~2
B= (l/2it )[h(oo) —h(r +
~f_l/2
—
~
—
~f
11—1/2 —
f— 3/2 (f~)
2J—I/2
+
~fI/2j~j_3/2 —
If—
~fI/2j_I/2
1,2f J 3/2J, —
0)] where r0 is the origin for a spacetime without horizon, and the (outer) horizon for a possible black hole solution. The constancy of h for f=j rules out all such metrics as candidate solutions with a nonzero skyrmion number, irrespective of whether they have a horizon or not. Thus Schwarzschild, Reissner— Nordstrom and de Sitter-like solutions, in particular, have no “skyrmion hair”. In general, f=j~ -‘
(5)
I/2J1 /2).
Requiring I to be stationary under variations of f~ h andj yields the field equations
T 0° ~[fI/2jl/2(l_cosh+k2r2)2h~]
2J”2/r2)[sinh(l—cosh+4k2r2) =(f” +(l—cosh) sinh]+f”2j”2 sinh (h’)2,
(6)
rf’+f—fj_——(fj/16a2)
rj’
(1/f)
_f+j2
(1 —cos h+k2r2)(h’)2]
(7)
,
imply that
x [(h’)~(l 2(1
—cos h+k2r2)]
(12)
,
1/f) + ~-ln(j/J) I dr =(l/8a2r2)(l —cos h)(l —cosh+4k2r2).
= (12/16a2)
x [r~(l —cos h)(l —cos h+4k2r2) +(l/f)(1—cosh+k2r2)(h’)2J
This is clearly not satisfied in our case un-
~—ln(f/)=(l/8a2r)
x [r~(l —cos h)(l —cos h+4k2r2) —
Tr’.
less h=const. We now consider possible solutions horizon, withoutrequiringf=j’. Eqs.with (7) an andevent (8)
(8)
.
Eqs. (6).-(8) should be solved numerically in order to study the gravitational properties of a spherically symmetric skyrmion distribution. We are interested in the possible existence of a Schwarzschild-like solution. The Schwarzschild metnc belongs to a class of metrics withf=j For this class eqs. (7) and (8) imply that
—
(13)
The null generators of the horizon are given by J( ±) ...f_ 1/2
ô/ôt+j—”2
a/ar.
(14)
At the horizon, we require that the components satisfy TaP ‘a1fl = T°°f+T”J= 0.
‘a
—‘.
(l/j)(1 —cos h+k2r2)(h’)2 =0.
Thus we obtain the boundary condition on the horizon
(9) (h’)2j’ =0.
308
(15)
Volume 126, number 5,6
PHYSICS LETTERS A
We discuss below the consequences of the above on the structure of any nontrivial solution with a horizon. First, let j have a zero at r= r0. Eq. (12) implies that, if h’ is to be finite at r0, f must diverge at r0 as 3— Next, let f diverge at some radius r0 as (1 rc,/r)—a, where a>0. This will be the case when there is a horizon at r0, and in principle h’ could diverge at r= r0. Now the r.h.s. of eq. (12) is always nonnegative. Thus f must have a zero at 2must r0, of the diform at (l—r0/r)~,with /J~a.If fl>a, (h’) verge r 0 as (1 rr,/r) Then eq. (15) implies that a> 1. On the other hand, requiring that eq. (6) balance near r= r0 we get the condition a = 1. thus the only way to satisfy eqs. (6), (12) and (15) simultaneously is to have h’ bounded everywhere, and fi = a = 1. We conclude that for every zero and pole of j there is, respectively, a pole and zero off and futhermore that all are simple poles and zeros. Consider a possible (typical)andf, solution with—ra horizon forwhichj=(1—r,,/r)’ =f(r)(l 0/r), withf (r) regular and nonvanishing at r= r 0. On substitution in eq. (8), the l.h.s. vanishes identically. The r.h.s. is a sum of nonnegative terms and it vanishes only if 1 —cos h vanishes. Thus we must have h = const = 2nx everywhere. In general j= (1 r~/r) T°= 0 and this is not satisfied in our case unless h = 2nx. A more general solution with a horizon for which 1= [i~ (r) (1 ro/r)1’ and 1 f, (r)( 1 r0/r) (with f,,&j, ever3lwhere andf,(r) and j1(r) positive for all r>~r0) is not ruled out. For such solutions, the value of h( r) at the horizon, h0, is related toj,(r0) (from eq. (8)): 2r~)(1—cosho) 1—j,(r0)= (l/16a x (1 cos h 2r~) (16) 0 + 4k The arbitrariness of h 0 suggests that such solutions in general have a fractional skyrmion number. However, such a conclusion needs to be qualified. Observe first that the l.h.s. ofeq. (12) is bounded. Thus, at large distance, h’ must approach zero sufficiently 2r2 rapidly to cancel the contribution of the term in so theasr.h.s. of q. (12). Now consider the kcase of large black holes, defined by r 0>> k Then, in the of validity the solution (r>be r0),small the 2r2 domain term is always large.ofThus (h’ )2 must k everywhere. Similarly, from eq. (16), h( r) must ap—
—
‘.
—
-‘
~
—
—
—
.
-‘.
11 January 1988
proach 2rnt close to the horizon. On the other hand, h (r) must be monotonically decreasing if we are talking about a skyrmion distribution that is localised near the horizon. Thus in the limit kr0>’ 1, the only surviving solution is the trivial one, viz., h = 2nx everywhere. In the opposite limit ofvery small black holes (kr0 ~ 1), the thermal properties (Hawking effect) become important, and the classical analysis breaks down. Further, asfandj can have only simple zeros, the possibility of having “cold” (extreme Reissner—Nordstrom type) solutions also ruled out. This still allows the existence of is intermediate-size black holes. We argue below that such nontrivial solutions are unlikely to be stable. To investigate the instability of any nontrivial solution we have to consider its behaviour under arbitrary time dependent perturbations. For the system considered here, it suffices to consider perturbations of the form 2 = —j( r, t) dt2 +f( r, t) dr2 cit 2 do2 +r2 sin2O d9’2). (17) +k(r, t)(r (It has been shown by Vishveshwara [3] that taking k(r, t) = 1 leads to trivial perturbations.) The usual linear stability analysis [4] consists in solving a Schrödinger-like equation for the normal modes in an effective potential derived by a small perturbation around the classical solution. We have made a preliminary analysis on solutions of the type discussed above. We find that the effective potential in this case can support one or more points ofinflexion. Typically, systems with no point of inflexion are stable, while unstable modes tend to arise in systems with more than one point. This suggests (though it does not prove) that nontrivial solutions with a horizon are unstable: stable black holes have no skyrmion hair. A more detailed investigation of stability is now in progress. To conclude, we have proved that a whole class of spherically symmetric solutions of the Einstein— Skyrme equations, j, with or without horizon, cannot have awithf=f nonzero skyrmion number. aEqs. (6)—(8) can of course be solved numerically to get the gravitatinal properties of a skyrmion. For large a2MIr solution expected to ofhave where Misis the total energy the skyrmion. For reasons given above, it appears un-
r, such f=j 1 = 1
—
309
Volume 126, number 5,6
PHYSICS LETTERS A
likely that solutions with a horizon are stable. Thus the typical solution to eqs. (6—8) should have f and j regular and nonvanishing everywhere. Finiteness of the energy density at the origin then requires h(0)=2rnx, and a skyrmion number n—rn. The numerical analysis of ref. [2] ignores the back reaction of the skyrmion on a background Schwarzschild solution. We therefore feel that the numerical result [21 reporting black holes with a fractional skyrmion number should not be taken seriously. Note added. After the first version of this paper had been submitted for publication, we received a prepnnt by Luckock [5] in which eqs. (6)— (8) have been set up and solved numerically. The remarks in —‘
310
11 January 1988
this preprint regarding stability are misleading (only time-independent perturbations are mentioned which, as expected, are trivial) and are not supported by any systematic analysis, which must incorporate eq. (17).
References [I] [2] [3] [4]
T.H.R. Skyrme, Proc. R. Soc. A 260 (1961) 127. H. Luckock and I. Moss, Phys. Lett. B 176 (1986) 341. C.V. Vishveshwara, Phys. Rev. D 1(1970) 2870. T. Regge and J.A. Wheeler, Phys. Rev. 108 (1957) 1063.
[5] H. Luckock, Newcastle preprint, to be published in the proceedings ofthe 1986 Paris—Mendon Colloquium.
PHYSICS LETTERS A
11 January 1988
ON THE SKYRMION NUMBER OF SCHWARZSCHILD BLACK HOLES D. LOHIYA and A. MUKHERJEE Department ofPhysics andAstrophysics, Universityof Delhi, Delhi 110007, India Received 5 May 1987; revised manuscript received 7 September 1987; accepted for publication 17 November 1987 Communicated by J.P. Vigier
It is shown that the coupled Skyrme—Einstein equations of motion do not admit any Schwarzschild-like solution with nonzero topological number. A Schwarzschild black hole thus has no “skyrmion hair”. More general static, spherically symmetric solutions with a horizon are discussed. Solutions with nonzero skyrmion number exist, but are unlikely to be stable.
In recent years there has been a lot of interest in the model first proposed by Skyrme [1] nearly a quarter of a century ago. The model describes the breakdown of the chiral symmetry SU(2)LXSU(2)R to the isospin SU(2). It has the advantage of being only second order in time derivatives, and is thus amenable to the usual hamiltonian techniques. Further, it possesses classical soliton solutions skyrmions which can be interpreted as baryons. This Letter reports on a search fora skyrmion black hole, i.e., a solution to the classical equations of motion for the Skyrme model coupled to gravity, with a nonzero skyrmion number as well as an event horizon. A natural candidatewould be a Schwarzschildlike solution with nonzero skyrmion number, which would describe a spherically symmetric black hole with “skyrmion hair”. Such an object could result, conceivably, from the collapse ofa star made ofskyrmions as well as other matter. An earlier analysis [2] solved the Skyrme equation of motion in a background Schwarzschild spacetime, ingnoring the back
Finally, we argue that such solutions are likely to be unstable against time-dependent perturbations. The model of our interest is described bythe action 1=1,
J
(la)
,
where j~=
—
~ ~
Tr(D,~UlYU-’)
+(l/32a2) Tr[UtD~U, U—’D~U]2}
—
reaction of the skyrmion on the metric. We, on the other hand, consider the solutions to the coupled Einstein—Skyrme equations of motion. We prove that a whole class of static, spherically symmetric metrics (including the Schwarzschild and Reissner— Nordstrom metrics) cannot support a nonzero skyrmion number. Nontrivial solutions outside this class do exist. We study the analytic structureof solutions with an event horizon and relate the value of the skyrmion field at the horizon to that of the metric.
+12
J
X ( —g)”2 d4x, 1= (—g) “2R d4x. —
(ib) (ic)
—
Here U is a matrix-valued function of spacetime, related to the pion fields r1(x) by U=exp(if;1a.~) where a’ are the Pauli matrices andf, (—~190 MeV) is the pion decay constant. The covariant derivative D,~,the determinant g and the curvature scalar R are defined as usual in terms of the metric tensor g,~ 3,.The classical solutions are the extrema of the action (I). We look for solutions where Usatisfies the Skyrme ansatz U(x) = exp [ih ( r)~.~/2]
(2)
and g,~,,is static and spherically symmetric: 2 = —f( r) dt2 +j( r) dr2 ds +r2(d02+sin2O dq2)
(3)
,
0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
.
307
Volume 126,number5,6
PHYSICSLETTERSA
In terms of the ansatz functionsf(r), h(r) and j(r), the matter part of the action becomes Ii =
—
~-~-
x [r2(l
J
dt dr (fl)”2
—cos h)(l —cos h+4k2r2)
+(l—cosh+k2r2)j’ (h’)2]
11 January 1988
Since k must be real for the solution to make any physical sense, we obtain h = const. The value of this constant must be 2rnt (n an integer) if we require the energy density T 0=p= (l/8a2r2)[(1/j)(h’)2(l —cos h+k2r2) 0 +(l/r2)(l—cosh)(l—cosh+4k2r2)] (10)
(4)
,
where k=f~,a/2,and prime denotes derivative with respect to r. Similarly the gravitational part of the action yields
and the pressure density Tp
—
—
(l/8a2r2)[( l/j)(h’)2(l —cos h+k2r2) (11)
(l/r2)(l —cos h)(l —cos h+4k2r2)]
to vanish as r—+~.The skyrmion number of a solution is given by
12—4~ Jdtdrr2(f~2f’i~2
B= (l/2it )[h(oo) —h(r +
~f_l/2
—
~
—
~f
11—1/2 —
f— 3/2 (f~)
2J—I/2
+
~fI/2j~j_3/2 —
If—
~fI/2j_I/2
1,2f J 3/2J, —
0)] where r0 is the origin for a spacetime without horizon, and the (outer) horizon for a possible black hole solution. The constancy of h for f=j rules out all such metrics as candidate solutions with a nonzero skyrmion number, irrespective of whether they have a horizon or not. Thus Schwarzschild, Reissner— Nordstrom and de Sitter-like solutions, in particular, have no “skyrmion hair”. In general, f=j~ -‘
(5)
I/2J1 /2).
Requiring I to be stationary under variations of f~ h andj yields the field equations
T 0° ~[fI/2jl/2(l_cosh+k2r2)2h~]
2J”2/r2)[sinh(l—cosh+4k2r2) =(f” +(l—cosh) sinh]+f”2j”2 sinh (h’)2,
(6)
rf’+f—fj_——(fj/16a2)
rj’
(1/f)
_f+j2
(1 —cos h+k2r2)(h’)2]
(7)
,
imply that
x [(h’)~(l 2(1
—cos h+k2r2)]
(12)
,
1/f) + ~-ln(j/J) I dr =(l/8a2r2)(l —cos h)(l —cosh+4k2r2).
= (12/16a2)
x [r~(l —cos h)(l —cos h+4k2r2) +(l/f)(1—cosh+k2r2)(h’)2J
This is clearly not satisfied in our case un-
~—ln(f/)=(l/8a2r)
x [r~(l —cos h)(l —cos h+4k2r2) —
Tr’.
less h=const. We now consider possible solutions horizon, withoutrequiringf=j’. Eqs.with (7) an andevent (8)
(8)
.
Eqs. (6).-(8) should be solved numerically in order to study the gravitational properties of a spherically symmetric skyrmion distribution. We are interested in the possible existence of a Schwarzschild-like solution. The Schwarzschild metnc belongs to a class of metrics withf=j For this class eqs. (7) and (8) imply that
—
(13)
The null generators of the horizon are given by J( ±) ...f_ 1/2
ô/ôt+j—”2
a/ar.
(14)
At the horizon, we require that the components satisfy TaP ‘a1fl = T°°f+T”J= 0.
‘a
—‘.
(l/j)(1 —cos h+k2r2)(h’)2 =0.
Thus we obtain the boundary condition on the horizon
(9) (h’)2j’ =0.
308
(15)
Volume 126, number 5,6
PHYSICS LETTERS A
We discuss below the consequences of the above on the structure of any nontrivial solution with a horizon. First, let j have a zero at r= r0. Eq. (12) implies that, if h’ is to be finite at r0, f must diverge at r0 as 3— Next, let f diverge at some radius r0 as (1 rc,/r)—a, where a>0. This will be the case when there is a horizon at r0, and in principle h’ could diverge at r= r0. Now the r.h.s. of eq. (12) is always nonnegative. Thus f must have a zero at 2must r0, of the diform at (l—r0/r)~,with /J~a.If fl>a, (h’) verge r 0 as (1 rr,/r) Then eq. (15) implies that a> 1. On the other hand, requiring that eq. (6) balance near r= r0 we get the condition a = 1. thus the only way to satisfy eqs. (6), (12) and (15) simultaneously is to have h’ bounded everywhere, and fi = a = 1. We conclude that for every zero and pole of j there is, respectively, a pole and zero off and futhermore that all are simple poles and zeros. Consider a possible (typical)andf, solution with—ra horizon forwhichj=(1—r,,/r)’ =f(r)(l 0/r), withf (r) regular and nonvanishing at r= r 0. On substitution in eq. (8), the l.h.s. vanishes identically. The r.h.s. is a sum of nonnegative terms and it vanishes only if 1 —cos h vanishes. Thus we must have h = const = 2nx everywhere. In general j= (1 r~/r) T°= 0 and this is not satisfied in our case unless h = 2nx. A more general solution with a horizon for which 1= [i~ (r) (1 ro/r)1’ and 1 f, (r)( 1 r0/r) (with f,,&j, ever3lwhere andf,(r) and j1(r) positive for all r>~r0) is not ruled out. For such solutions, the value of h( r) at the horizon, h0, is related toj,(r0) (from eq. (8)): 2r~)(1—cosho) 1—j,(r0)= (l/16a x (1 cos h 2r~) (16) 0 + 4k The arbitrariness of h 0 suggests that such solutions in general have a fractional skyrmion number. However, such a conclusion needs to be qualified. Observe first that the l.h.s. ofeq. (12) is bounded. Thus, at large distance, h’ must approach zero sufficiently 2r2 rapidly to cancel the contribution of the term in so theasr.h.s. of q. (12). Now consider the kcase of large black holes, defined by r 0>> k Then, in the of validity the solution (r>be r0),small the 2r2 domain term is always large.ofThus (h’ )2 must k everywhere. Similarly, from eq. (16), h( r) must ap—
—
‘.
—
-‘
~
—
—
—
.
-‘.
11 January 1988
proach 2rnt close to the horizon. On the other hand, h (r) must be monotonically decreasing if we are talking about a skyrmion distribution that is localised near the horizon. Thus in the limit kr0>’ 1, the only surviving solution is the trivial one, viz., h = 2nx everywhere. In the opposite limit ofvery small black holes (kr0 ~ 1), the thermal properties (Hawking effect) become important, and the classical analysis breaks down. Further, asfandj can have only simple zeros, the possibility of having “cold” (extreme Reissner—Nordstrom type) solutions also ruled out. This still allows the existence of is intermediate-size black holes. We argue below that such nontrivial solutions are unlikely to be stable. To investigate the instability of any nontrivial solution we have to consider its behaviour under arbitrary time dependent perturbations. For the system considered here, it suffices to consider perturbations of the form 2 = —j( r, t) dt2 +f( r, t) dr2 cit 2 do2 +r2 sin2O d9’2). (17) +k(r, t)(r (It has been shown by Vishveshwara [3] that taking k(r, t) = 1 leads to trivial perturbations.) The usual linear stability analysis [4] consists in solving a Schrödinger-like equation for the normal modes in an effective potential derived by a small perturbation around the classical solution. We have made a preliminary analysis on solutions of the type discussed above. We find that the effective potential in this case can support one or more points ofinflexion. Typically, systems with no point of inflexion are stable, while unstable modes tend to arise in systems with more than one point. This suggests (though it does not prove) that nontrivial solutions with a horizon are unstable: stable black holes have no skyrmion hair. A more detailed investigation of stability is now in progress. To conclude, we have proved that a whole class of spherically symmetric solutions of the Einstein— Skyrme equations, j, with or without horizon, cannot have awithf=f nonzero skyrmion number. aEqs. (6)—(8) can of course be solved numerically to get the gravitatinal properties of a skyrmion. For large a2MIr solution expected to ofhave where Misis the total energy the skyrmion. For reasons given above, it appears un-
r, such f=j 1 = 1
—
309
Volume 126, number 5,6
PHYSICS LETTERS A
likely that solutions with a horizon are stable. Thus the typical solution to eqs. (6—8) should have f and j regular and nonvanishing everywhere. Finiteness of the energy density at the origin then requires h(0)=2rnx, and a skyrmion number n—rn. The numerical analysis of ref. [2] ignores the back reaction of the skyrmion on a background Schwarzschild solution. We therefore feel that the numerical result [21 reporting black holes with a fractional skyrmion number should not be taken seriously. Note added. After the first version of this paper had been submitted for publication, we received a prepnnt by Luckock [5] in which eqs. (6)— (8) have been set up and solved numerically. The remarks in —‘
310
11 January 1988
this preprint regarding stability are misleading (only time-independent perturbations are mentioned which, as expected, are trivial) and are not supported by any systematic analysis, which must incorporate eq. (17).
References [I] [2] [3] [4]
T.H.R. Skyrme, Proc. R. Soc. A 260 (1961) 127. H. Luckock and I. Moss, Phys. Lett. B 176 (1986) 341. C.V. Vishveshwara, Phys. Rev. D 1(1970) 2870. T. Regge and J.A. Wheeler, Phys. Rev. 108 (1957) 1063.
[5] H. Luckock, Newcastle preprint, to be published in the proceedings ofthe 1986 Paris—Mendon Colloquium.