- Email: [email protected]
Solitons and black holes in non-Abelian Einstein–Born–Infeld theory
3 August 2000
Physics Letters B 486 Ž2000. 431–442 www.elsevier.nlrlocaternpe
Solitons and black holes in non-Abelian Einstein–Born–Infeld theory V.V. Dyadichev, D.V. Gal’tsov Department of Theoretical Physics, Moscow State UniÕersity, 119899, Moscow, Russia Received 23 May 2000; accepted 16 June 2000 Editor: P.V. Landshoff
Abstract Recently it was shown that the Born–Infeld modification of the quadratic Yang–Mills action gives rise to classical particle-like solutions in the flat space which have a striking similarity with the Bartnik–McKinnon solutions obtained within the gravity coupled Yang–Mills theory. We show that both families of solutions are continuously related within the framework of the Einstein–Born–Infeld theory via interpolating sequences of parameters. We also investigate an internal structure of the associated black holes and find that the Born–Infeld non-linearity changes drastically the black hole interior typical for the usual quadratic Yang–Mills theory. In the latter case a generic solution exhibits violent metric oscillations near the singularity. In the Born–Infeld case the generic interior solution is smooth, the metric tends to the standard Schwarzschild type singularity, and we did not observe internal horizons. Smoothing of the ‘ violent’ EYM singularity may be interpreted as a result of non-gravitational quantum effects. q 2000 Elsevier Science B.V. All rights reserved. PACS: 04.20.Jb; 04.50.q h; 46.70.Hg
Physical significance of the particle-like solutions to the Einstein–Yang–Mills ŽEYM. field equations found by Bartnik and McKinnon ŽBK. w1x Žfor more details see a review paper w2x. as well as their possible role in the string-inspired models remains rather obscure. These ‘particles’ where found to have sphaleron nature w3x, and they could be responsible for fermion number violating processes. However, it is not quite clear how these purely classical solutions can be embedded into some quantum framework. Another related puzzle is the singularity structure of the associated black holes w4–6x. It was shown that the metric inside the classical EYM black holes exhibits violent oscillations near the singularity, which go far beyond the classical bounds w7–9x. Presumably these oscillations must be regularized in the quantum theory, but no relevant explanation was suggested so far. To probe the effect of quantum corrections to the Yang–Mills lagrangian within the stringrM theory framework it is tempting to utilize the Born–Infeld modification of the Yang–Mills action describing the low
E-mail addresses: [email protected] ŽV.V. Dyadichev., [email protected] ŽD.V. Gal’tsov.. 0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 1 3 - 9
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
432
energy dynamics of D-branes w10x. Classical solutions to both the Abelian and non-Abelian Born–Infeld theories received recently much attention. Although there are certain complications in the definition of the trace over the gauge group generators in the Born–Infeld action w11–13x, it is believed that the simplified ‘square root’ form of the lagrangian gives correct Žat least qualitatively. predictions concerning solitons. Magnetic monopoles were shown to persist in such a theory w14–16x. A new type of particle-like solutions in the non-Abelian Born–Infeld model was discovered by Gal’tsov and Kerner ŽGK. w17x. It was shown that this theory gives rise to flat space classical glueballs which have a striking similarity with the BK solutions. More close relationship between these two particle-like configurations becomes clear within the Einstein– Born–Infeld ŽEBI. theory. As was shown recently by Wirschins, Sood and Kunz w18x, the GK solutions survive when gravity is added, in which case the corresponding black holes also come into play. There is a substantial difference between the above two families which has to be clarified, however. Both the BK and GK particles form the same kind of discrete sequences resulting from the ‘quantization’ of the parameter entering the boundary conditions near the origin. But contrary to the BK case for which this parameter sequence is convergent to a limiting value, in the GK case the corresponding sequence diverges. So it is necessary to check carefully whether the both families are continuously related indeed. Here we investigate non-Abelian Einstein–Born–Infeld solitons and black holes in more detail. We show that within the gravity coupled BIYM theory one has a unique family of the regular particle-like solutions smoothly interpolating between the GK and BK solutions when the gravitational coupling constant increases in units of the Born–Infeld ‘critical field’. We also investigate the interior structure of the non-Abelian EBIYM black holes and find that the problem of violent metric oscillations is removed. Recall that the static spherically symmetric EYM field equations admit three smooth branches of local solutions near the singularity possessing the Schwarzschild, Reissner–Nordstrom ¨ and the imaginary-charge Reissner–Nordstrom ¨ type behavior w7x. Neither of these three, however, has a sufficient number of free parameters to be generic. Therefore when one moves from the event horizon into the black hole interior one can meet smooth local solutions near the singularity at best for discrete values of the black hole mass w7x. A generic EYM black hole interior looks very differently from other explicitly known examples. When the singularity is approached the metric exhibits oscillations with an infinitely growing amplitude and an infinitely decreasing period. Clearly, the classical bounds are exceeded after a few oscillation cycles. In the non-Abelian EBI theory we also find several local solution branches near the singularity, but the situation is drastically different. The local solution which has a sufficient number of parameters to be generic has a smooth behavior of the Schwarzschild type. As in w14,17,18x we assume the ‘square root’ form of the non-Abelian Born–Infeld action which, as we believe, describes well enough particle-like solutions being at the same time much simpler than the favored by strings symmetrized trace w11x action. Thus the EBIYM action is chosen as Ss
1 16p G
H ŽyR q 4Gb
2
Ž 1 y R . . 'y g d 4 x ,
Ž 1.
where Rs
(
1q
1 2b
2
Fmna Famn y
1 16 b
4
a mn
žF
F˜amn
2
/
.
Ž 2.
Without loss of generality the dimensionless gauge coupling constant Žin the units " s c s 1. will be set to unity, so we are left with two dimensional parameters: the BI ‘critical field’ b of dimension Ly2 , and the Newton constant G of dimension L2 ŽPlanck’s length.. from these one can form a dimensionless constant g s Gb ,
Ž 3.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
™
433
™
which is the only substantial parameter of the theory w18x. The decoupling of gravity corresponds to the limit g 0, while the limit g ` after rescaling gives the EYM theory. We consider the case of the SUŽ2. gauge group assuming for the YM field a usual spherically symmetric static purely magnetic ansatz A s Ž 1 y w Ž r . . Ž Tu sin u d w y Tw du . ,
Ž 4.
where a rotated basis for the gauge group generators is used. The spacetime metric is parameterized as follows: ds 2 s Ns 2 dt 2 y
dr 2 N
y r 2 Ž du 2 q sin2u d w 2 . .
Ž 5.
The equations of motion after rescaling of the radial coordinate Ž b .1r2 r form of two coupled equations for N and w: X
NwX
ž /
s
w Ž w 2 y 1. r2R
R
2 gwX 3 N y
R2
™ r Žmaking r dimensionless. take the
,
Ž 6.
X
Ž Nr . s 1 q 2 gr 2 Ž 1 y R . ,
Ž 7.
where now
Rs
(
1q2
NwX 2 r2
q
Ž1yw2 . r4
2
,
Ž 8.
and primes denote the derivatives with respect to r. The third is a decoupled equation for s , X
Ž ln s . s
2 gwX 2 rR
,
Ž 9.
which can be easily solved once the YM function w is known. We are interested in the asymptotically flat configurations such that the local mass function mŽ r ., defined through Ns1y
2 mŽ r . r
™
,
Ž 10 .
™
™
has a finite limit m M as r `, while s 1. Like in the EYM case, it can be easily derived from the Eq. Ž7. that finiteness of M implies the following asymptotic behavior of the YM function: w s "1 q O Ž ry1 .. In this limit R 1 so the BI non-linearity is negligible. Let us first discuss the globally regular solutions which start at the origin with the following series expansion Žits convergence in a non-zero domain can be proved by the standard methods w19x.:
™
w s 1 y br 2 q
Ns1q
2r2 3
br 4 Ž 3b Ž 44 b 2 q 3 . q 4 g 28 b 2 q 3 y Ž 4 b 2 Ž 48 b 2 q 13 . q 3 . R0y1
g Ž 1 y R0 . y
2
30 Ž 4 b q 1 . 16 gb 2 r 4 15 Ž 4 b 2 q 1 .
Ž g Ž R0 y 1. 2 q 3b R0 . q O Ž r 6 . ,
. qO
Ž r6. ,
Ž 11 . Ž 12 .
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
434
where R0 is a limiting value of the square root at the origin: R
™ R s '1 q 12 b . 2
Ž 13 .
0
This local solution has a unique free parameter b. In the limiting case g s 0 it coincides with that found in the flat space w17x, while to make contact with the corresponding expansion of the EYM theory one has to take the limit g ` with a simultaneous rescaling b brg. Near the origin the spacetime is flat. Like in the EYM case, matching of the solution which departs from the origin as Ž11., Ž12. and meets the conditions of the asymptotic flatness at infinity can be achieved for a discrete sequence of the free parameter values b s bnŽ g . labeled by the number n of zeroes of w Ž r .. The proof of existence may be given along the lines of w17,19x, here we rather concentrate on the qualitative physical picture. We have investigated numerical solutions in a large range of g. Typical behavior for small g is shown on Figs. 1, 2. The YM curves for g s 0.001 are practically indistinguishable from those found previously by Kerner and one of the authors w17x in the flat space BIYM model. The region of oscillations corresponds to an unscreened Coulomb charge inside the particles. One can see that with growing n this region expands both in the directions of small and large r. The metric deviates from the flat one rather weakly for small g Žweak gravity.. The amplitude of w-oscillations in the intermediate region decreases, so in this region the solution can be regarded as approaching the embedded Abelian solution w ' 0. As was observed by Wirschins, Sood and Kunz w18x, the metric for small g also approaches the corresponding Abelian Bion metric w20x Žwith zero ‘seed’ mass in the singularity.. To avoid confusion it is worth noting that for odd n the Yang–Mills topology of the EBIYM regular solutions Žkink. is essentially different form that of the embedded Abelian solution w ' 0 Žtrivial.. Thus
™
™
Fig. 1. First five solutions wnŽ r . for weak gravity Ž g s 0.001.. The oscillation region, which corresponds to the localization of an unscreened magnetic charge inside the BIYM particle, expands with growing n. The amplitude of the first and the last oscillation cycles remains roughly the same, while the amplitude of the intermediate cycles tends to zero with increasing n. All these curves practically coincide with those found in the flat space BIYM theory by Gal’tsov and Kerner.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
435
Fig. 2. Metric function N Ž r . for regular solutions with g s 0.001. For all solutions the deviation of the metric from a flat one is within 2%. For higher node numbers n one observes a stabilization of N near the minimal value, this corresponds to the oscillation region of w Ž r ..
it would be misleading to say that the sequence of non-Abelian EBIYM solutions converges to the Abelian solution in the global sense. For large g the regular EBIYM solutions tend to the BK solutions of the EYM system after the coordinate rescaling r g r, see Fig. 3 for g s 10. Comparing this with the weak gravity behavior ŽFig. 1. we observe that gravity reduces the region of the unscreened charge. The metric deviates substantially form the flat one. The function N Ž r . for large n exhibits a deep well Žan ‘almost’ horizon., but remains always strictly positive. ŽSee Fig. 4.. The sequences bn of discrete values of the parameter in the expansion Ž11. behave very differently for small and large g. As was found in w17x, bn in the flat space are quite large with respect to the corresponding BK values, and the sequence does not converge with growing n. Contrary to this, the BK sequence bn rapidly converges to a limiting value b` . In this case there is an additional limiting solution with different space-time structure w19x. We have obtained numerically the interpolating functions bnŽ g . for several n clearly demonstrating that these two extrema are continuously related indeed ŽFig. 5.. For small values of g these functions tend to the GK values w17x
™'
bn Ž 0 . s bnGK , while in the opposite limit g
™ ` one recovers the convergent sequence of rescaled BK values w1,2x
™`
lim bn Ž g . rg s bnBK .
g
Ž 14 . Ž 15 .
The corresponding solutions of the EYM system with the standard YM lagrangian are recovered in terms of the rescaled coordinate r˜ s rr g . In the intermediate region all parameter functions bnŽ g . were found to be monotonously varying between the above extrema.
'
436
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
Fig. 3. Regular solutions wnŽ r . for strong gravity, g s 10. These curves are close to BK solutions in terms of a rescaled coordinate r˜ s rrg 1r 2 .
Fig. 4. Metric functions NnŽ r . for regular solutions with g s 10. For higher n the curves come very close to but never reach zero: N3min f 0.003, N4min f 0.0003.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
437
Fig. 5. Dependence of the parameters bn for regular solutions on the effective gravity coupling constant g Žin logarithmic coordinates.. The left side of the picture correspond to the GK limit. With growing g higher-n curves merge, this is reminiscent of the rapid convergence of the sequence bnBK s lim g ` bnŽ g .rg.
™
Masses of the regular solutions as functions of the effective gravitational coupling are shown on Fig. 6. For vanishing g one recovers the GK masses after a rescaling
™0
lim Mn Ž g . rg s MnGK .
g
Ž 16 .
We recall w17x that the sequence MnGK converges to the mass of an embedded Abelian solution. The reason is that the main contribution to the mass comes from the region of oscillations where w approaches an Abelian value w ' 0 with growing n. But for any n the limiting values w Ž0.,w Ž`. are equal to "1, so, as we have already noted, the global topology of non-Abelian solutions is entirely different. In the limit of strong gravity one recovers the BK masses after a rescaling
™`
lim Mn Ž g . r g s MnBK .
g
'
Ž 17 .
Now discuss the black holes. These are parameterized by the horizon radius r h and the value of the YM function wh s w Ž r h . at the horizon. The series expansions near the regular horizon reads: w s wh q NhX s
1 rh
wh Ž wh2 y 1 . r h NhX
2 2 Ž r y rh . q O Ž Ž r y rh . . , N s NhX Ž r y rh . q O Ž Ž r y rh . . ,
ž (
2
1 q 2 g 2 r h2 1 q 1 q Ž w h2 y 1 . rrh4
/.
Ž 18 .
Asymptotically flat solutions satisfying these boundary conditions are likely to exist for any horizon radii r h . Exterior black hole solutions are very similar to regular solutions, especially for small r h . So our main interest is in the interior solutions. We start by listing various series solutions that can be obtained near the singularity.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
438
Fig. 6. Dependence of masses of regular solutions on the gravity coupling g.
Let us first explore the series expansion for an embedded Abelian solution w20,21x. For the unit magnetic charge Žwhat corresponds to w ' 0. the local mass function satisfies the equation mX s g Ž 'r 4 q 1 y r 2 . .
Ž 19 .
Expanding the square root at small r and integrating one obtains: 2 m0 Nsy q 1 y 2 g q 23 gr 2 y 15 gr 4 q O Ž r 8 . . Ž 20 . r The ‘seed’ mass parameter m 0 may be positive, negative or zero. Positive m 0 corresponds to a timelike singularity of the Schwarzschild type. Negative m 0 corresponds to a spacelike singularity, in which case an internal horizon also exist Žthough contrary to a more familiar example of the Reissner–Nordstrom ¨ metric, the ‘local charge’ term Q 2rr 2 now is absent.. The case of vanishing m 0 is particularly interesting. The metric near the origin is then locally flat unless g s 1r2, in which case the limiting value N Ž0. shrinks to zero. For other values of g there is a conical singularity w20x, the critical value g s 1r2 corresponding to an extremal deficit angle. Within the present framework the embedded Abelian black holes correspond to an identically vanishing function w. It can be shown that the local series solution starting at r s 0 with zero initial value w 0 s 0 and an arbitrary m 0 generates this global embedded Abelian solution. For non-Abelian solutions the value of the YM function at the singularity w Ž0. s w 0 should therefore be non-zero. We have found the generalization of the series expansion Ž20. with w 0 / 0. It is valid for the non-zero seed mass parameter m 0 :
ž
w s w0 1 q Nsy
2 m0 r
er 2
er q 2 m0
16 m20
q 1 y 2 ge q
3 Ž 2 e y 1 . y 4 ge 3 gew 02 r 2 m0
q gr 2
ž
/
2 q 3
qOŽ r3. , ew 02 12 m20
3 Ž 5e y 2 . y 8 ge
/
qOŽ r 3. ,
Ž 21 .
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
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Fig. 7. YM function w Ž r . for n s 1, 2 EBIYM black holes with rh s 1.0, g s 1.0. At the singularity these curves tend to constant values non-equal to unity.
where e s 1 y w 02 . This local solution has two free parameters w 0 ,m 0 . For w 0 s 0 it coincides with Ž20.. If w 0 / 0 it becomes singular in the limit m 0 s 0. The search for local solutions with m 0 s 0 shows that then either w 0 s 0 in which case we come back to the Abelian embedded solution w ' 0, or w 0 s "1 and we recover the regular solution Ž11., Ž12.. It is also worth noting that, contrary to the EYM case, in the EBIYM theory there are no local solutions with N ; ry2 behavior at the singularity Žthe Reissner–Nordstrom ¨ type.. Since the asymptotic solution Ž21. fails to contain a sufficient number of free parameters to be a generic solution, Žthis number is equal to three for the system of equations Ž6., Ž7.., the question is how a generic solution looks like near the singularity, in particular, whether it admits any series expansion. In the case of the ordinary Yang–Mills lagrangian such an expandable solution does not exist at all; one finds that a generic solution has a non-analytic oscillating behavior w7x. Here the situation is different, although the generic local solution around the singularity still exhibits non-analyticity in terms of the variable r. It turns out to be series expandable but in terms of the 'r : w s w 0 q a'r y y
a2 w 0 r
q
a Ž 24 g 2 a 2 y 32 g 2 a 2 e y 3c 2 . r 3r2 32 g 2 e 2
e
a2 Ž Ž 16 g 2 a 2 y 32 g 2 a 2 e y 15c 2 . w 0 y 8 g 2 cae . r 2
Ns1y2
32 g 2 e 3 e2 a2 r
a Ž 3w 0 c q ag 2 e . r ' qc r y q O Ž r 3r2 . . e
q O Ž r 5r2 . ,
Ž 22 .
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
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This local solution contains three free parameters w 0 , a, c. The singularity is of the Schwarzschild type, the seed mass m 0 s e 2ra2 is strictly positive. Two leading terms in the expansion of ‘kinetic’ Žnegative for N - 0. and ‘potential’ Žpositive. terms in R mutually cancel, so the leading behavior in the singularity is 3c Rf
4 gr 3r2
.
Ž 23 .
We have tested for various r h , g, n that a continuation of the exterior black hole solutions under the horizon meets this generic asymptotic solution indeed. In all numerical experiments the function N remained negative under the horizon and no internal Cauchy horizons were met. Typical global black hole solutions are shown on the Figs. 7, 8, 9 for r h s 1, g s 1, n s 1, 2. The YM function w outside the horizon has qualitatively the same behavior as in the regular case. Inside the horizon it remains smooth and tends to a finite limit w 0 / 0 at the singularity. Recall that for the ordinary quadratic YM lagrangian the function w inside the horizon of the EYM black holes has a rather sophisticated behavior: while w itself tends to a finite limit w 0 , its derivative exhibits a sequence of growing absolute values at tiny intervals whose length tends to zero w7,8x. Such a behavior causes oscillations of the local mass mŽ r . with an infinitely increasing amplitude. At the beginning of each oscillation cycle the metric function N Ž r . takes values very close to zero Ž‘almost’ Cauchy horizons., then an exponential growth of mŽ r . starts. After a few oscillation cycles the maximal values of m attained in the subsequent cycles become of the googolplex order, obviously lying beyond any classical bounds. Contrary to this, in the EBIYM case we observe a smooth mŽ r . inside the horizon up to the singularity where mŽ r . has a finite positive value ŽFig. 8.. Similarly, the second metric function s tends smoothly to a finite value at the singularity ŽFig. 9.. We conclude with the following remarks. The BK solutions were found in the gravity coupled Yang–Mills theory with the usual quadratic lagrangian. Now they were shown to emerge in the strong gravity limit of the Einstein–BIYM theory which can be interpreted as an effective EYM theory including corrections due to non-local nature of the underlying strings. The EBIYM theory has a limit of decoupled gravity in which
Fig. 8. The mass function mŽ r . for the EBIYM black holes with rh s 1.0, g s 1.0, n s 1, 2. Inside the horizon mŽ r . remains a smooth function tending to a constant limit at the singularity. There are no internal horizons, the singularity is timelike.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
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Fig. 9. The metric function s Ž r . for the EBIYM black holes with rh s 1.0, g s 1.0, n s 1, 2.
qualitatively similar solutions continue to exist. This indicates the way how the BK solutions can be incorporated into the Žquantum. string theory. One implication of this reasoning is the resolution of the problem of ‘ violent’ oscillations near the singularity of the EYM black holes. Born–Infeld non-linearity removes huge metric oscillations in the singularity but does not change its timelike nature. The local mass function tends to the finite limit in the singularity and no ‘mass-inflation’ phenomenon takes place. In numerical experiments we did not see any internal horizons. In principle, the even number of horizons is compatible with the general form of the local solution near the singularity, this possibility is worth to be investigated in more detail. In the gravity coupled Abelian Born–Infeld theory the mass function mŽ r . also tends to the finite limit Žcontrary to the Reissner–Nordstrom ¨ case in which mŽ r . diverges as 1rr .. However, the limiting value mŽ0. Ž may be positive timelike singularity., negative Žspacelike singularity. or zero Žlocally flat space with a conical defect.. Two latter possibilities are not realized in the non-Abelian theory. This work was supported in part by the RFBR grant 00-02-16306.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x
R. Bartnik, J. McKinnon, Phys. Rev. Lett. 61 Ž1988. 141. M.S. Volkov, D.V. Gal’tsov, Phys. Rep. C 319 Ž1999. 1, hep-thr9810070. D.V. Gal’tsov, M.S. Volkov, Phys. Lett. B 273 Ž1991. 255. M.S. Volkov, D.V. Galt’sov, JETP Lett. 50 Ž1989. 346; Sov. J. Nucl. Phys. 51 Ž1990. 747. P. Bizon, Phys. Rev. Lett. 64 Ž1990. 2844. H.P. Kunzle, A.K.M. Masood-ul-Alam, J. Math. Phys. 31 Ž1990. 928. E.E. Donets, D.V. Gal’tsov, M.Yu. Zotov, Phys. Rev. D 56 Ž1997. 3459. D.V. Gal’tsov, E.E. Donets, M.Yu. Zotov, JETP Lett. 65 Ž1997. 895. P. Breitenlohner, G. Lavrelashvili, D. Maison, Nucl. Phys. B 524 Ž1998. 427.
442 w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442 J. Polchinski, String Theory, Vols. I, II, Cambridge University Press, Cambridge, 1998. A. Tseytlin, Nucl. Phys. B 501 Ž1997. 41. See also the recent review hep-thr9908105. J.P. Gauntlett, J. Gomis, P.K. Townsend, JHEP 01 Ž1998. 003. D. Brecher, M.J. Perry, Nucl. Phys. B 527 Ž1998. 121. N. Grandi, E.F. Moreno, F.A. Shaposnik, hep-thr9901073. J.H. Park, Phys. Lett. B 458 Ž1999. 471. N. Grandi, R.L. Pakman, F.A. Schaposnik, G. Silva, Phys. Rev. D 60 Ž1999. 125002. D. Gal’tsov, R. Kerner, Phys. Rev. Lett., to appear; hep-thr9910171. M. Wirschins, A. Sood, J. Kunz, hep-thr0004130. P. Breitenlohner, P. Forgacs, D. Maison, Comm. Math. Phys. 163 Ž1994. 141. G.W. Gibbons, D.A. Rasheed, Nucl. Phys. B 454 Ž1995. 185. D.A. Rasheed, hep-thr9702087.
Physics Letters B 486 Ž2000. 431–442 www.elsevier.nlrlocaternpe
Solitons and black holes in non-Abelian Einstein–Born–Infeld theory V.V. Dyadichev, D.V. Gal’tsov Department of Theoretical Physics, Moscow State UniÕersity, 119899, Moscow, Russia Received 23 May 2000; accepted 16 June 2000 Editor: P.V. Landshoff
Abstract Recently it was shown that the Born–Infeld modification of the quadratic Yang–Mills action gives rise to classical particle-like solutions in the flat space which have a striking similarity with the Bartnik–McKinnon solutions obtained within the gravity coupled Yang–Mills theory. We show that both families of solutions are continuously related within the framework of the Einstein–Born–Infeld theory via interpolating sequences of parameters. We also investigate an internal structure of the associated black holes and find that the Born–Infeld non-linearity changes drastically the black hole interior typical for the usual quadratic Yang–Mills theory. In the latter case a generic solution exhibits violent metric oscillations near the singularity. In the Born–Infeld case the generic interior solution is smooth, the metric tends to the standard Schwarzschild type singularity, and we did not observe internal horizons. Smoothing of the ‘ violent’ EYM singularity may be interpreted as a result of non-gravitational quantum effects. q 2000 Elsevier Science B.V. All rights reserved. PACS: 04.20.Jb; 04.50.q h; 46.70.Hg
Physical significance of the particle-like solutions to the Einstein–Yang–Mills ŽEYM. field equations found by Bartnik and McKinnon ŽBK. w1x Žfor more details see a review paper w2x. as well as their possible role in the string-inspired models remains rather obscure. These ‘particles’ where found to have sphaleron nature w3x, and they could be responsible for fermion number violating processes. However, it is not quite clear how these purely classical solutions can be embedded into some quantum framework. Another related puzzle is the singularity structure of the associated black holes w4–6x. It was shown that the metric inside the classical EYM black holes exhibits violent oscillations near the singularity, which go far beyond the classical bounds w7–9x. Presumably these oscillations must be regularized in the quantum theory, but no relevant explanation was suggested so far. To probe the effect of quantum corrections to the Yang–Mills lagrangian within the stringrM theory framework it is tempting to utilize the Born–Infeld modification of the Yang–Mills action describing the low
E-mail addresses: [email protected] ŽV.V. Dyadichev., [email protected] ŽD.V. Gal’tsov.. 0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 1 3 - 9
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
432
energy dynamics of D-branes w10x. Classical solutions to both the Abelian and non-Abelian Born–Infeld theories received recently much attention. Although there are certain complications in the definition of the trace over the gauge group generators in the Born–Infeld action w11–13x, it is believed that the simplified ‘square root’ form of the lagrangian gives correct Žat least qualitatively. predictions concerning solitons. Magnetic monopoles were shown to persist in such a theory w14–16x. A new type of particle-like solutions in the non-Abelian Born–Infeld model was discovered by Gal’tsov and Kerner ŽGK. w17x. It was shown that this theory gives rise to flat space classical glueballs which have a striking similarity with the BK solutions. More close relationship between these two particle-like configurations becomes clear within the Einstein– Born–Infeld ŽEBI. theory. As was shown recently by Wirschins, Sood and Kunz w18x, the GK solutions survive when gravity is added, in which case the corresponding black holes also come into play. There is a substantial difference between the above two families which has to be clarified, however. Both the BK and GK particles form the same kind of discrete sequences resulting from the ‘quantization’ of the parameter entering the boundary conditions near the origin. But contrary to the BK case for which this parameter sequence is convergent to a limiting value, in the GK case the corresponding sequence diverges. So it is necessary to check carefully whether the both families are continuously related indeed. Here we investigate non-Abelian Einstein–Born–Infeld solitons and black holes in more detail. We show that within the gravity coupled BIYM theory one has a unique family of the regular particle-like solutions smoothly interpolating between the GK and BK solutions when the gravitational coupling constant increases in units of the Born–Infeld ‘critical field’. We also investigate the interior structure of the non-Abelian EBIYM black holes and find that the problem of violent metric oscillations is removed. Recall that the static spherically symmetric EYM field equations admit three smooth branches of local solutions near the singularity possessing the Schwarzschild, Reissner–Nordstrom ¨ and the imaginary-charge Reissner–Nordstrom ¨ type behavior w7x. Neither of these three, however, has a sufficient number of free parameters to be generic. Therefore when one moves from the event horizon into the black hole interior one can meet smooth local solutions near the singularity at best for discrete values of the black hole mass w7x. A generic EYM black hole interior looks very differently from other explicitly known examples. When the singularity is approached the metric exhibits oscillations with an infinitely growing amplitude and an infinitely decreasing period. Clearly, the classical bounds are exceeded after a few oscillation cycles. In the non-Abelian EBI theory we also find several local solution branches near the singularity, but the situation is drastically different. The local solution which has a sufficient number of parameters to be generic has a smooth behavior of the Schwarzschild type. As in w14,17,18x we assume the ‘square root’ form of the non-Abelian Born–Infeld action which, as we believe, describes well enough particle-like solutions being at the same time much simpler than the favored by strings symmetrized trace w11x action. Thus the EBIYM action is chosen as Ss
1 16p G
H ŽyR q 4Gb
2
Ž 1 y R . . 'y g d 4 x ,
Ž 1.
where Rs
(
1q
1 2b
2
Fmna Famn y
1 16 b
4
a mn
žF
F˜amn
2
/
.
Ž 2.
Without loss of generality the dimensionless gauge coupling constant Žin the units " s c s 1. will be set to unity, so we are left with two dimensional parameters: the BI ‘critical field’ b of dimension Ly2 , and the Newton constant G of dimension L2 ŽPlanck’s length.. from these one can form a dimensionless constant g s Gb ,
Ž 3.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
™
433
™
which is the only substantial parameter of the theory w18x. The decoupling of gravity corresponds to the limit g 0, while the limit g ` after rescaling gives the EYM theory. We consider the case of the SUŽ2. gauge group assuming for the YM field a usual spherically symmetric static purely magnetic ansatz A s Ž 1 y w Ž r . . Ž Tu sin u d w y Tw du . ,
Ž 4.
where a rotated basis for the gauge group generators is used. The spacetime metric is parameterized as follows: ds 2 s Ns 2 dt 2 y
dr 2 N
y r 2 Ž du 2 q sin2u d w 2 . .
Ž 5.
The equations of motion after rescaling of the radial coordinate Ž b .1r2 r form of two coupled equations for N and w: X
NwX
ž /
s
w Ž w 2 y 1. r2R
R
2 gwX 3 N y
R2
™ r Žmaking r dimensionless. take the
,
Ž 6.
X
Ž Nr . s 1 q 2 gr 2 Ž 1 y R . ,
Ž 7.
where now
Rs
(
1q2
NwX 2 r2
q
Ž1yw2 . r4
2
,
Ž 8.
and primes denote the derivatives with respect to r. The third is a decoupled equation for s , X
Ž ln s . s
2 gwX 2 rR
,
Ž 9.
which can be easily solved once the YM function w is known. We are interested in the asymptotically flat configurations such that the local mass function mŽ r ., defined through Ns1y
2 mŽ r . r
™
,
Ž 10 .
™
™
has a finite limit m M as r `, while s 1. Like in the EYM case, it can be easily derived from the Eq. Ž7. that finiteness of M implies the following asymptotic behavior of the YM function: w s "1 q O Ž ry1 .. In this limit R 1 so the BI non-linearity is negligible. Let us first discuss the globally regular solutions which start at the origin with the following series expansion Žits convergence in a non-zero domain can be proved by the standard methods w19x.:
™
w s 1 y br 2 q
Ns1q
2r2 3
br 4 Ž 3b Ž 44 b 2 q 3 . q 4 g 28 b 2 q 3 y Ž 4 b 2 Ž 48 b 2 q 13 . q 3 . R0y1
g Ž 1 y R0 . y
2
30 Ž 4 b q 1 . 16 gb 2 r 4 15 Ž 4 b 2 q 1 .
Ž g Ž R0 y 1. 2 q 3b R0 . q O Ž r 6 . ,
. qO
Ž r6. ,
Ž 11 . Ž 12 .
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434
where R0 is a limiting value of the square root at the origin: R
™ R s '1 q 12 b . 2
Ž 13 .
0
This local solution has a unique free parameter b. In the limiting case g s 0 it coincides with that found in the flat space w17x, while to make contact with the corresponding expansion of the EYM theory one has to take the limit g ` with a simultaneous rescaling b brg. Near the origin the spacetime is flat. Like in the EYM case, matching of the solution which departs from the origin as Ž11., Ž12. and meets the conditions of the asymptotic flatness at infinity can be achieved for a discrete sequence of the free parameter values b s bnŽ g . labeled by the number n of zeroes of w Ž r .. The proof of existence may be given along the lines of w17,19x, here we rather concentrate on the qualitative physical picture. We have investigated numerical solutions in a large range of g. Typical behavior for small g is shown on Figs. 1, 2. The YM curves for g s 0.001 are practically indistinguishable from those found previously by Kerner and one of the authors w17x in the flat space BIYM model. The region of oscillations corresponds to an unscreened Coulomb charge inside the particles. One can see that with growing n this region expands both in the directions of small and large r. The metric deviates from the flat one rather weakly for small g Žweak gravity.. The amplitude of w-oscillations in the intermediate region decreases, so in this region the solution can be regarded as approaching the embedded Abelian solution w ' 0. As was observed by Wirschins, Sood and Kunz w18x, the metric for small g also approaches the corresponding Abelian Bion metric w20x Žwith zero ‘seed’ mass in the singularity.. To avoid confusion it is worth noting that for odd n the Yang–Mills topology of the EBIYM regular solutions Žkink. is essentially different form that of the embedded Abelian solution w ' 0 Žtrivial.. Thus
™
™
Fig. 1. First five solutions wnŽ r . for weak gravity Ž g s 0.001.. The oscillation region, which corresponds to the localization of an unscreened magnetic charge inside the BIYM particle, expands with growing n. The amplitude of the first and the last oscillation cycles remains roughly the same, while the amplitude of the intermediate cycles tends to zero with increasing n. All these curves practically coincide with those found in the flat space BIYM theory by Gal’tsov and Kerner.
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Fig. 2. Metric function N Ž r . for regular solutions with g s 0.001. For all solutions the deviation of the metric from a flat one is within 2%. For higher node numbers n one observes a stabilization of N near the minimal value, this corresponds to the oscillation region of w Ž r ..
it would be misleading to say that the sequence of non-Abelian EBIYM solutions converges to the Abelian solution in the global sense. For large g the regular EBIYM solutions tend to the BK solutions of the EYM system after the coordinate rescaling r g r, see Fig. 3 for g s 10. Comparing this with the weak gravity behavior ŽFig. 1. we observe that gravity reduces the region of the unscreened charge. The metric deviates substantially form the flat one. The function N Ž r . for large n exhibits a deep well Žan ‘almost’ horizon., but remains always strictly positive. ŽSee Fig. 4.. The sequences bn of discrete values of the parameter in the expansion Ž11. behave very differently for small and large g. As was found in w17x, bn in the flat space are quite large with respect to the corresponding BK values, and the sequence does not converge with growing n. Contrary to this, the BK sequence bn rapidly converges to a limiting value b` . In this case there is an additional limiting solution with different space-time structure w19x. We have obtained numerically the interpolating functions bnŽ g . for several n clearly demonstrating that these two extrema are continuously related indeed ŽFig. 5.. For small values of g these functions tend to the GK values w17x
™'
bn Ž 0 . s bnGK , while in the opposite limit g
™ ` one recovers the convergent sequence of rescaled BK values w1,2x
™`
lim bn Ž g . rg s bnBK .
g
Ž 14 . Ž 15 .
The corresponding solutions of the EYM system with the standard YM lagrangian are recovered in terms of the rescaled coordinate r˜ s rr g . In the intermediate region all parameter functions bnŽ g . were found to be monotonously varying between the above extrema.
'
436
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
Fig. 3. Regular solutions wnŽ r . for strong gravity, g s 10. These curves are close to BK solutions in terms of a rescaled coordinate r˜ s rrg 1r 2 .
Fig. 4. Metric functions NnŽ r . for regular solutions with g s 10. For higher n the curves come very close to but never reach zero: N3min f 0.003, N4min f 0.0003.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
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Fig. 5. Dependence of the parameters bn for regular solutions on the effective gravity coupling constant g Žin logarithmic coordinates.. The left side of the picture correspond to the GK limit. With growing g higher-n curves merge, this is reminiscent of the rapid convergence of the sequence bnBK s lim g ` bnŽ g .rg.
™
Masses of the regular solutions as functions of the effective gravitational coupling are shown on Fig. 6. For vanishing g one recovers the GK masses after a rescaling
™0
lim Mn Ž g . rg s MnGK .
g
Ž 16 .
We recall w17x that the sequence MnGK converges to the mass of an embedded Abelian solution. The reason is that the main contribution to the mass comes from the region of oscillations where w approaches an Abelian value w ' 0 with growing n. But for any n the limiting values w Ž0.,w Ž`. are equal to "1, so, as we have already noted, the global topology of non-Abelian solutions is entirely different. In the limit of strong gravity one recovers the BK masses after a rescaling
™`
lim Mn Ž g . r g s MnBK .
g
'
Ž 17 .
Now discuss the black holes. These are parameterized by the horizon radius r h and the value of the YM function wh s w Ž r h . at the horizon. The series expansions near the regular horizon reads: w s wh q NhX s
1 rh
wh Ž wh2 y 1 . r h NhX
2 2 Ž r y rh . q O Ž Ž r y rh . . , N s NhX Ž r y rh . q O Ž Ž r y rh . . ,
ž (
2
1 q 2 g 2 r h2 1 q 1 q Ž w h2 y 1 . rrh4
/.
Ž 18 .
Asymptotically flat solutions satisfying these boundary conditions are likely to exist for any horizon radii r h . Exterior black hole solutions are very similar to regular solutions, especially for small r h . So our main interest is in the interior solutions. We start by listing various series solutions that can be obtained near the singularity.
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
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Fig. 6. Dependence of masses of regular solutions on the gravity coupling g.
Let us first explore the series expansion for an embedded Abelian solution w20,21x. For the unit magnetic charge Žwhat corresponds to w ' 0. the local mass function satisfies the equation mX s g Ž 'r 4 q 1 y r 2 . .
Ž 19 .
Expanding the square root at small r and integrating one obtains: 2 m0 Nsy q 1 y 2 g q 23 gr 2 y 15 gr 4 q O Ž r 8 . . Ž 20 . r The ‘seed’ mass parameter m 0 may be positive, negative or zero. Positive m 0 corresponds to a timelike singularity of the Schwarzschild type. Negative m 0 corresponds to a spacelike singularity, in which case an internal horizon also exist Žthough contrary to a more familiar example of the Reissner–Nordstrom ¨ metric, the ‘local charge’ term Q 2rr 2 now is absent.. The case of vanishing m 0 is particularly interesting. The metric near the origin is then locally flat unless g s 1r2, in which case the limiting value N Ž0. shrinks to zero. For other values of g there is a conical singularity w20x, the critical value g s 1r2 corresponding to an extremal deficit angle. Within the present framework the embedded Abelian black holes correspond to an identically vanishing function w. It can be shown that the local series solution starting at r s 0 with zero initial value w 0 s 0 and an arbitrary m 0 generates this global embedded Abelian solution. For non-Abelian solutions the value of the YM function at the singularity w Ž0. s w 0 should therefore be non-zero. We have found the generalization of the series expansion Ž20. with w 0 / 0. It is valid for the non-zero seed mass parameter m 0 :
ž
w s w0 1 q Nsy
2 m0 r
er 2
er q 2 m0
16 m20
q 1 y 2 ge q
3 Ž 2 e y 1 . y 4 ge 3 gew 02 r 2 m0
q gr 2
ž
/
2 q 3
qOŽ r3. , ew 02 12 m20
3 Ž 5e y 2 . y 8 ge
/
qOŽ r 3. ,
Ž 21 .
V.V. DyadicheÕ, D.V. Gal’tsoÕr Physics Letters B 486 (2000) 431–442
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Fig. 7. YM function w Ž r . for n s 1, 2 EBIYM black holes with rh s 1.0, g s 1.0. At the singularity these curves tend to constant values non-equal to unity.
where e s 1 y w 02 . This local solution has two free parameters w 0 ,m 0 . For w 0 s 0 it coincides with Ž20.. If w 0 / 0 it becomes singular in the limit m 0 s 0. The search for local solutions with m 0 s 0 shows that then either w 0 s 0 in which case we come back to the Abelian embedded solution w ' 0, or w 0 s "1 and we recover the regular solution Ž11., Ž12.. It is also worth noting that, contrary to the EYM case, in the EBIYM theory there are no local solutions with N ; ry2 behavior at the singularity Žthe Reissner–Nordstrom ¨ type.. Since the asymptotic solution Ž21. fails to contain a sufficient number of free parameters to be a generic solution, Žthis number is equal to three for the system of equations Ž6., Ž7.., the question is how a generic solution looks like near the singularity, in particular, whether it admits any series expansion. In the case of the ordinary Yang–Mills lagrangian such an expandable solution does not exist at all; one finds that a generic solution has a non-analytic oscillating behavior w7x. Here the situation is different, although the generic local solution around the singularity still exhibits non-analyticity in terms of the variable r. It turns out to be series expandable but in terms of the 'r : w s w 0 q a'r y y
a2 w 0 r
q
a Ž 24 g 2 a 2 y 32 g 2 a 2 e y 3c 2 . r 3r2 32 g 2 e 2
e
a2 Ž Ž 16 g 2 a 2 y 32 g 2 a 2 e y 15c 2 . w 0 y 8 g 2 cae . r 2
Ns1y2
32 g 2 e 3 e2 a2 r
a Ž 3w 0 c q ag 2 e . r ' qc r y q O Ž r 3r2 . . e
q O Ž r 5r2 . ,
Ž 22 .
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This local solution contains three free parameters w 0 , a, c. The singularity is of the Schwarzschild type, the seed mass m 0 s e 2ra2 is strictly positive. Two leading terms in the expansion of ‘kinetic’ Žnegative for N - 0. and ‘potential’ Žpositive. terms in R mutually cancel, so the leading behavior in the singularity is 3c Rf
4 gr 3r2
.
Ž 23 .
We have tested for various r h , g, n that a continuation of the exterior black hole solutions under the horizon meets this generic asymptotic solution indeed. In all numerical experiments the function N remained negative under the horizon and no internal Cauchy horizons were met. Typical global black hole solutions are shown on the Figs. 7, 8, 9 for r h s 1, g s 1, n s 1, 2. The YM function w outside the horizon has qualitatively the same behavior as in the regular case. Inside the horizon it remains smooth and tends to a finite limit w 0 / 0 at the singularity. Recall that for the ordinary quadratic YM lagrangian the function w inside the horizon of the EYM black holes has a rather sophisticated behavior: while w itself tends to a finite limit w 0 , its derivative exhibits a sequence of growing absolute values at tiny intervals whose length tends to zero w7,8x. Such a behavior causes oscillations of the local mass mŽ r . with an infinitely increasing amplitude. At the beginning of each oscillation cycle the metric function N Ž r . takes values very close to zero Ž‘almost’ Cauchy horizons., then an exponential growth of mŽ r . starts. After a few oscillation cycles the maximal values of m attained in the subsequent cycles become of the googolplex order, obviously lying beyond any classical bounds. Contrary to this, in the EBIYM case we observe a smooth mŽ r . inside the horizon up to the singularity where mŽ r . has a finite positive value ŽFig. 8.. Similarly, the second metric function s tends smoothly to a finite value at the singularity ŽFig. 9.. We conclude with the following remarks. The BK solutions were found in the gravity coupled Yang–Mills theory with the usual quadratic lagrangian. Now they were shown to emerge in the strong gravity limit of the Einstein–BIYM theory which can be interpreted as an effective EYM theory including corrections due to non-local nature of the underlying strings. The EBIYM theory has a limit of decoupled gravity in which
Fig. 8. The mass function mŽ r . for the EBIYM black holes with rh s 1.0, g s 1.0, n s 1, 2. Inside the horizon mŽ r . remains a smooth function tending to a constant limit at the singularity. There are no internal horizons, the singularity is timelike.
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Fig. 9. The metric function s Ž r . for the EBIYM black holes with rh s 1.0, g s 1.0, n s 1, 2.
qualitatively similar solutions continue to exist. This indicates the way how the BK solutions can be incorporated into the Žquantum. string theory. One implication of this reasoning is the resolution of the problem of ‘ violent’ oscillations near the singularity of the EYM black holes. Born–Infeld non-linearity removes huge metric oscillations in the singularity but does not change its timelike nature. The local mass function tends to the finite limit in the singularity and no ‘mass-inflation’ phenomenon takes place. In numerical experiments we did not see any internal horizons. In principle, the even number of horizons is compatible with the general form of the local solution near the singularity, this possibility is worth to be investigated in more detail. In the gravity coupled Abelian Born–Infeld theory the mass function mŽ r . also tends to the finite limit Žcontrary to the Reissner–Nordstrom ¨ case in which mŽ r . diverges as 1rr .. However, the limiting value mŽ0. Ž may be positive timelike singularity., negative Žspacelike singularity. or zero Žlocally flat space with a conical defect.. Two latter possibilities are not realized in the non-Abelian theory. This work was supported in part by the RFBR grant 00-02-16306.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x
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