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Classical and quantum analysis of repulsive singularities in four-dimensional extended supergravity
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Please note that terms and conditions apply. Class. Quantum Grav. 16 (1999) 2231–2246. Printed in the UK PII: S0264-9381(99)01984-X
Classical and quantum analysis of repulsive singularities in four-dimensional extended supergravity
I Gaida, H R Hollmann and J M Stewart Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, UK E-mail: [email protected] and [email protected]
Received 19 February 1999
Abstract. Non-minimal repulsive singularities (‘repulsons’) in extended supergravity theories are investigated. The short-distance antigravity properties of the repulsons are tested at the classical and the quantum level by a scalar test-particle. Using a partial wave expansion it is shown that the particle is totally reflected at the origin. A high-frequency incoming particle undergoes a phase shift of π/2. However, the phase shift for a low-frequency particle depends upon the physical data of the repulson. The curvature singularity at a finite distance rh turns out to be transparent for the scalar test-particle and the coordinate singularity at the origin serves as the repulsive barrier to bounce back the particles.
PACS numbers: 0470, 0465, 0367L
1. Introduction
In the last few years there has been a lot of progress in understanding black hole physics in supergravity and string theory in N>1 supersymmetric vacua (for recent reviews see [1]). A lot of these black hole solutions can be interpreted as certain (non-singular) p-brane solutions, too [2]. This opens the possiblity to obtain a microscopic understanding of the macroscopic Bekenstein–Hawking entropy [3]. Another interesting point in this context is the appearance of massless black holes at particular points in moduli space giving rise to gauge symmetry enhancement or supersymmetry enhancement [4, 5]. However, in [4, 6–9] it has also been shown that not only attractive singularities (black holes) are stable BPS solutions of these supersymmetric vacua, but that repulsive naked singularities (‘repulsons’) appear too. Thus, repulsons are non-perturbative manifestations of antigravity effects in supersymmetric vacua. With antigravity we mean in this context the property of these solutions to reflect particles with a given mass or angular momentum [6]. It has been known for a long time that antigravity effects occur at the perturbative level in extended supergravity [10]. It has been pointed out in [9] that repulsons, sometimes also called ‘white holes’, are as generic in moduli space as their ‘dual’ attractive singularities (black holes). In addition, a ‘minimal’ repulson background has been analysed in [6, 9] using a scalar test-particle and expanding the corresponding wavefunction in partial waves. It has been shown that at the classical level no massive scalar test-particle can reach the ‘outer’ curvature singularity at rh > 0. Moreover, at the quantum level the ‘inner’ singularity at the origin is reflecting, but
0264-9381/99/072231+16$19.50 © 1999 IOP Publishing Ltd 2231 2232 I Gaida et al the ‘outer’ singularity is transparent. In this context it has been assumed that the scalar test- particle can tunnel through the ‘outer’ curvature singularity. In addition, it has been suggested that for r>rh spacetime is Minkowskian and for r 2. Static solutions of N = 2 supergravity The repulsons we are investigating are solutions of the bosonic sector of N = 2 supergravity in four dimensions. In this section we briefly outline [4, 12–14], how these solutions are obtained. The action of the bosonic truncation of N = 2 supergravity in four dimensions includes a gravitational, vector and hypermultiplets. The hypermultiplets are assumed to be constant. The complex scalars of the vector multiplets form a sigma model, the target space of which is special Kahler.¨ That is, the real Kahler¨ potential K, which defines the sigma model metric, is entirely determined by a holomorphic and homogeneous function, the prepotential F. Repulsive singularities in 4D extended supergravity 2233 I All the fields of the theory can be expressed in terms of the vector = (X , FI ), where I = 0,...,Nv counts the number of the physical vectors and Nv the scalars. The components of are the holomorphic sections of a line bundle over the moduli space. With FI we denote I FI = ∂F(X)/∂X . In order to enforce the configuration to be supersymmetric the so-called stabilization equations [8] have to be satisfied I ¯ I I µ ¯ µ i(X − X ) = H (x ), i(FI − FI ) = HI (x ). I Because of the equations of motion and the Bianchi identities, the functions H and HI are harmonic. To derive explicit solutions in general and the repulson solution in particular, the prepotential and the harmonic functions have to be specified. The repulsons arise in the context of a model with prepotential F(S,T,U)=−ST U, with (S,T,U)=−iz1,2,3. The z1,2,3 are defined by so-called special coordinates X0(z) = 1,XA(z) = zA,A = 1,...,Nv. In string theory this prepotential corresponds to the classical heterotic ST U-model with constant hypermultiplets. The microscopic interpretation, the higher-order curvature corrections, the near extremal approximation and the effect of quantum corrections of this class of N = 2 models have been studied extensively in [5, 15, 16]. For simplicity we take all moduli to be axion-free, that is purely imaginary. Solving the stabilization equations with respect to these constraints yields H 1,2,3 p S,T,U = ,X0 = 1 −D/H , (1) 2X0 2 0 with D = H 1H 2H 3. The harmonic functions are given by the constants h, the electric and magnetic charges q and p, respectively, pI q H I (r) = hI ,H(r) = h I . + r I I + r The charges satisfy the Dirac quantization condition pq = 2πn,n ∈ Z. If we restrict ourselves to static spherically symmetric solutions only, the most general ansatz for the metric is 1 s2 =− t2 F(r)( r2 r2 2). d F(r) d + d + d 2 (2) The metric function F(r)is given by −K ¯ A A F = e = i(X FA − X FA), A = 1, 2, 3. Here F 2(r) reads X4 α F 2(r) =− H D = n . 4 0 rn n=0 It is possible to choose the parameters of the harmonic functions so that the solutions correspond to repulsive singular supersymmetric states [7]. It is straightforward to see whether these solutions are gravitationally attractive or repulsive. 1 2 3 q0 p p p (anti-)gravity # + + + + Attractive 1 − + + + Repulsive 4 −− + + Attractive 6 −− − + Repulsive 4 −− − − Attractive 1 2234 I Gaida et al Here we denote p, q > 0 (< 0) by + (−) and include possible permutations when denoting the number # of repulsive and attractive solutions. Note that any positive (negative) charge appearing in the solution can be interpreted as an (anti-)brane [8, 9]. Moreover, the solution with four antibranes, i.e. all charges are negative, yields a negative ADM mass. 3. Scalar particles in the background of a repulson In the following the repulson singularities are investigated. For that we follow the motion of a test-particle moving in the metric background produced by the repulson. The repulson is modelled by a family of potentials, defined by the metric function F 2(r). The parameters α1,...,α4 determine the shape of the potentials. In order to obtain flat spacetime at spatial infinity, α0 is chosen to be equal to 1. In addition, we shall consider charge configurations with positive ADM mass, i.e. α1 > 0. For a repulson configuration the coefficient α2 in the metric function is negative. The short-distance behaviour is dominated by the 1/r4-term in the metric function. Therefore the solution is gravitationally attractive at short distances if α4 > 0 and gravitationally repulsive if α4 < 0. We restrict ourselves to the case of α4 < 0 only. We put in a further simplification: α3 is defined by α3 α − α α = . 1 +8 3 4 1 2 0 (3) 2 With these restrictions on α1,...,α4 the function F (r) has the generic form illustrated in 2 figures 1(a) and (b). Figure 1(a) shows the shape of F (r) for α3 = α4 = 0. The parameters (α1,α2) are chosen to be (10.0, −10.0), (11.0, −9.0), (12.0, −8.0) and (13.0, −7.0) with lines of increasing thickness. In figure 1(b) the shape of the metric function is plotted for some values of αi , with α3,α4 6= 0. The triples of parameters (increasing thickness of the lines) are (α1,α2,α4) = (4.0, −1.0, −4.0), (3.0, −1.0, −3.0), (2.0, −1.0, −2.0) and (1.0, −1.0, −1.0). The coefficients which appear in the metric function are related to the physical parameters. 2 α1 = 4 m > 0, where M denotes the ADM mass, and α2 =−4Z < 0. Z is the central charge 2 2 of the configuration. With (3) we find α3 =−8M(M + Z )<0 and α4 is a free parameter, for a repulson configuration chosen to be less than zero. 2 F (r) has exactly one zero at a distance rh > 0, which is given by q 1 2 1 rh = α v − α , 16 1 + + 4 1 (4) with q v =−1 α − 1 α2 1 α − 1 α2 2 − α , + 2 2 4 1 + 4 2 4 1 4 if α3 and α4 are not simultaneously equal to zero. In terms of the physical data rh is given by p 2 rh = M + v+ − M, (5) with p 2 2 2 2 2 v+ = 2(Z + M ) + 4(Z + M ) − α4. If α3 and α4 both vanish, the position of the zero is given by q √ 1 2 1 rh = α − α − α = M2 Z2 − M . 4 1 2 2 1 2 + (6) Repulsive singularities in 4D extended supergravity 2235 2 Figure 1. (a) Shape of the metric function F (r) for α3 = α4 = 0; (b) shape of the metric function 2 F (r) for α3,α4 6= 0. At r = rh the metric changes its signature. For r>rh it is Lorentzian with the line element given by (2). As r −→ ∞ the spacetime is asymptotically Minkowskian and Schwarzschild. The Ricci scalar in this region is given by F 00 1 F 02 2 R =− + − F 0F 2. (7) F 2 2 F 3 r As r approaches rh from above a curvature singularity occurs. In the region 0 6 r 6 rh the line element becomes Euclidean, 1 s2 = t2 |F | r2 r2 2 . d |F | d + d + d (8) The Ricci scalar is again given by the expression (7). We find a curvature singularity when r approaches rh from below. Ar r = 0 all the curvature invariants remain finite. It is a coordinate singularity. 4. Classical analysis In the classical limit, that is in the limit of large r, the Newtonian potential 8(r) is given by M Z2 1 8(r) =− (gtt +1) =− + . (9) 2 r r2 The corresponding strength of the gravitational field 80 = M/r2 − 2Z2/r3 is gravitationally attractive at large distances (r>rc) and gravitationally repulsive for r 0 =−Et + Lφ + 0r (r), (12) with energy E and angular momentum L. This yields Z p 0r (r) = dr E2F 2 − L2/r2 − m2F. (13) Thus, from ∂0/∂E = 0 it follows that a test-particle takes the following time to move from r1 to r2: Z r 2 F 2 t = E dr p . (14) 2 2 2 2 2 r1 E F − L /r − m F For L = 0 a massive test-particle becomes reflected by the repulson at 2 √ m4 rmin = M2 + Z2 − M >rh,= 1 − . (15) E4 Moreover, for L 6= 0 a massless test-particle becomes reflected by the repulson at p rmin = 2 M2 + Z2 + L2/4E2 − M >rh. (16) The trajectory itself is determined by ∂0/∂L = 0, i.e. Z r 2 −L φ = dr p . (17) 2 2 2 2 2 2 r1 r E F − L /r − m F 5. Quantum mechanical analysis For the quantum mechanical analysis we consider a massless scalar test-particle with wavefunction ψ˜ satisfying the Klein–Gordon equation in the background of the repulson: √ µν ˜ ∂µ −gg ∂ν ψ = 0. (18) ˜ Writing ψ = 9(t,r)Ylm(θ, φ) we obtain the same equation with the metric in either the Lorentzian (2) or the Euclidean region (8) 2 19 − ν|F |9tt = 0, (19) where 1 is the flat space Laplace operator in spherical coordinates l(l ) 2 2 +1 1 = ∂ + ∂r − , (20) r r r2 Repulsive singularities in 4D extended supergravity 2237 and ν is 1 for r > rh and −1 for r 1r ψ + (E − V (r))ψ = 0, (22) with E = ω2, c c c c V(r)=− 1 + 2 + 3 + 4 , r r2 r3 r4 (23) 2 1 = ∂2 ∂ . r r + r r The coefficients in terms of the physical data are given by 2 2 2 c1 = 4 mω ,c2 =−4Z ω − l(l +1), 2 2 2 2 c3 =−8 mω (Z + M ), c4 = α4ω . For l = 0 the potential is, up to a factor −ω2, equal to the metric function F 2, so that the shape of the potential can be read off figures 1(a) and (b), respectively. Let us first study the special situation α3 = α4 = 0 [9]. In this case the differential equation (22) simplifies to 1r ψ + (E − V (r))ψ = 0, (24) with 2 E = ω c c V(r)=− 1 + 2 . (25) r r2 Here we have taken into account that at the border of the Euclidean and Minkowski region of the metric not only ν but also F 2(r) changes its sign. We now solve the differential equation. The transformation to a new coordinate ρ = 2iωr brings the differential equation into the form n s(s ) 2 2 1 +1 ∂ρ ψ + ∂ρ ψ + − + − ψ = 0, ρ 4 ρ ρ2 s −ρ/2 where n =−ic1/(2ω) and −s(s +1) = c2. The ansatz ψ(ρ) = (iρ) e f(ρ) yields the Kummer equation for f 2 ρ∂ρ f + (2s +2− ρ)∂ρ f + (n − s − 1)f = 0, so that the wavefunction turns out to be ψ(r) = ( ωr)s −iωr 2 e −(1+2s) × C1F(1+s − n, 2s +2, 2iωr) + C2(2iωr) F(−s − n, −2s,2iωr) , (26) s > − 1 F where we have chosen 2 . The functions are the confluent hypergeometric functions. The second term in the square brackets of equation (26) is singular at r = 0. If we require ψ(r) to be regular at the origin, C2 has to be zero. 2238 I Gaida et al Figure 2. (a) Wavefunctions for α3 = α4 = 0; (b) phase space diagram for α3 = α4 = 0. The solutions are illustrated in figures 2(a) and (b). In figure 2(a) the wavefunctions are plotted for l = 0 and ω = 1.0. The pairs of parameter values (α1,α2) = (11.0, −9.0), (12.0, −8.0), (13.0, −7.0) are indicated by lines of increasing thickness. In figure 2(b) we see the phase space diagram for the wavefunctions with parameter values as indicated above. If α3 and α4 are not zero we have to integrate the differential equation (22) numerically. The wavefunctions and the phase space diagrams are shown in figures 3(a) and (b). Again we have chosen l to be equal to zero and ω = 1.0. The lines of increasing thickness correspond to the following triples of parameter values (α1,α2,α4) = (4.0, −1.0, −4.0), (3.0, −1.0, −3.0), (2.0, −1.0, 2.0) (α3 is determined by equation (3)). In figures 4(a) and (b) and figures 5(a) and (b) we show how the shape of the wavefunction and the phase space diagrams change with increasing or decreasing value of ω. In these cases the parameters are α1 = 1.0,α2 =−1.0 and α4 =−1.0. Some information about the scattering can be obtained analytically in terms of a WKB approximation [18]. Applying the transformation ψ(r) = θ(r)/r to (22), we obtain a differential equation for θ(r) 2 ∂r θ − Q(r)θ = 0, (27) with l(l +1) Q(r) = ω2Q (r) + Q (r) =−ω2F 2(r) + . 0 2 r2 If we set = 1/ω and substitute the WKB ansatz X∞ θ(r) = 1 nS (r) exp n n=0 Repulsive singularities in 4D extended supergravity 2239 Figure 3. (a) Wavefunctions for α3,α4 6= 0; (b) phase space diagram for α3,α4 6= 0. into equation (27) differential equations for the Sn’s are obtained. If the series is truncated after S1 the result is called the physical optics approximation. The corresponding differential equations are S02 = Q , 0 0 (28a) S0 S0 S00 = , 2 0 1 + 0 0 (28b) S0 S0 S00 S02 = Q . 2 0 2 + 1 + 1 2 (28c) The solutions are Z r p S(r) = Q (t) t, 0 0 d (29a) rh S (r) =−1 ln |Q (r)|, (29b) 1 4Z 0 r Q00 Q02 S (r) = 0 − 5 0 t, l = . 2 3/2 5/2 d for 0 (29c) r Q 32 Q h 8 0 0 Therefore, the physical optics approximation is Z Z 1 1 r p 1 r p θ(r) = √ C1 exp Q0(t) dt + C2 exp − Q0(t) dt . 4 |Q (r)| 0 rh rh It should be noted that the angular momentum component of the potential does not contribute to the physical optics approximation. At r = rh the potential Q0(r) vanishes and the WKB approximation is not valid. Therefore, we need to split the interval [0, ∞) into three regions, region I near r = 0, region II near r = rh and region III far outside, and they need to be investigated separately. 2240 I Gaida et al Figure 4. (a) Wavefunction for ω = 10; (b) phase space diagram for ω = 10. Figure 5. (a) Wavefunction for ω = 0.5; (b) phase space diagram for ω = 0.5. For r As mentioned above the WKB approximation does not hold near r = rh. However, the potential near rh can be approximated by a linear function with negative slope so that equation (27) in region II becomes approximately 00 a 0 θ = (r − rh)θ(r), where a = Q (rh)<0. 2 0 The solution of this approximate equation is 1 √ √ ψ (r) = C −2/3 3 −a(r − r) C −2/3 3 −a(r − r) . II r a Ai h + b Bi h In region III Q0(r) is negative, so that S0(r) is purely imaginary and the wavefunction ψIII(r) can be written as Z Z 1 i r p i r p ψIII(r) = √ Co exp |Q0(t)| dt + Ci exp − |Q0(t)| dt . r 4 |Q (r)| 0 rh rh We now try to patch these approximations together to obtain a solution on [0, ∞). The matching procedure is done by estimating the range of validity of the solution in each region. For the WKB approximation to be valid on an interval it is necessary that S 0 S S ··· S (x) −→ . 1 2 n−1 n for 0 (30) We shall come back to this point in section 6. In the overlapping regions we take into account the asymptotic behaviour of the respective solutions. To match the functions ψI (r) and ψII(r) the asymptotic behaviour of the wavefunctions has to agree. For large positive arguments the Airy functions behave like 1 −1/4 − 2 t3/2 1 −1/4 2 t3/2 Ai(t) ∼ √ t e 3 , Bi(t) ∼ √ t e 3 . 2 π π √ 1/6 This shows that Cb = 0 and Ca = (2 π/(−a) )C1. We now fit the wavefunction in region II to region III. For large negative arguments the Airy function Ai behaves like √ / 3/2 1 √ 1 6 sin (2/3) −a(r − rh) + π −2/3 3 −a(r − r) ∼ √ 4 , Ai h / / π (−a)1 12(r − rh)1 4 that is √ / 3/2 1 1 6 sin (2/3) −a(r − rh) + π ψ (r) ∼ C √ 4 . II a / / πr (−a)1 12(r − rh)1 4 In the overlapping region ψIII(r) is approximated by 1 ψIII(r) ∼ √ √ r 4 −a 4 r − r h √ √ 2 3/2 2 3/2 × Co exp i −a(r − rh) + Ci exp − i −a(r − rh) . 3 3 The matching determines Co and Ci via C (−a)1/6 a 1 ix −ix √ sin x + π = Coe + Ci e , π 4 2242 I Gaida et al √ 3/2 with x = (2/3) −a(r − rh) . It follows that (−a)1/6 (−a)1/6 1 iπ − 1 iπ Co = Ca √ e 4 ,Ci =−Ca √ e 4 2i π 2i π and therefore Z (−a)1/6 r p 1 1 1 ψIII(r) = Ca √ √ sin |Q0(r)| dt + π . π r 4 |Q (r)| 4 0 rh Thus we have obtained the approximations ( = 1/ω): Z (−a)1/6 1 1 r p ψI (r) = Ca √ √ exp |Q0(t)| dt , (31a) 2 π r 4 |Q (r)| r √ 0 h −2/3 3 Ai −a(rh − r) ψ (r) = C , II a r (31b) Z (−a)1/6 r p 1 1 1 ψIII(r) = Ca √ √ sin |Q0(r)| dt + π . (31c) π r 4 |Q (r)| 4 0 rh 6. Discussion A comparison of the numerical solution and the physical optics approximation is given in figures 6(a) and (b). Figure 6(a) shows the numerical and the WKB solution for the parameter values l = 0.0,ω = 1.0,α1 = 1.0,α2 =−1.0 and α4 =−1.0. Figure 6(b) is a plot of the numerical solution and the WKB solution for a larger value, ω = 2.0. The normalizations of the solutions are such that the numerical and the approximate solution agree at a position r0, with 0 Figure 6. (a) Numerical (broken curve) and WKB solution (full curve) for ω = 1.0; (b) numerical (broken curve) and WKB solution (full curve) for ω = 2.0. Figure 7. (a) Comparison of ωS0,S1 and S2/ω for ω = 1.0; (b) comparison of ωS0,S1 and S2/ω for ω = 2.0. 2244 I Gaida et al term yields (−2iωr)n−s−1 (2iωr)−n−s−1 ψ(r) ∼ (2ωr)s e−iωr +eiωr 0(n + s +1) 0(−n + s +1) 1 ∼ sin ωr +2Mωln 2ωr − 1 πs +arg[0(1+s − 2iMω)] . r 2 Therefore, the moduli of the amplitudes of the ingoing and the outgoing mode are still equal. δ =−1 πs 0( s − Mω) . However, we find a phase shift of 2 +arg[ 1+ 2i ] In particular, the phase shift is no longer independent of the physical data of the repulson. In order to obtain a feeling for where the physical optics approximation ‘leaks’, we apply it to the repulson with α3 = α4 = 0. The potential in this case is given by 4M s(s +1) Q (r) =−1 − + . (32) 0 r r2 S+(r) ∼ r M r Substitution of an asymptotic expansion of 0 +2 ln into (31c) yields 1 1 ψIII(r) = sin ωr +2Mωln r + π . (33) r 4 The physical optics approximation predicts total reflection and a phase shift of π/2 independent of the repulson data. The phase shift is constant, i.e. it contains the zeroth-order contribution only. Let us expand the phase shift of the analytical solution in powers of ω. δ =−1 πs 0( s − Mω) ∼ 1 π 1 πω 0( s − Mω) 2 +arg 1+ 2i 4 + 2 +arg 1+ 2i ∼ 1 π ω( M ) ω2| − M |(− M − m M), 4 + 2 +1 + 1 2 i sin 4 2 cos s s ∼−1 ω where we have approximated by 2 + ; that is, the WKB approximation predicts the zeroth-order term in the phase shift. However, the exact solution also contains first- and second- order contributions. This shows that although the physical optics approximation represents the functional behaviour nicely it does not approximate the numerical value of the phase shift to high precision. This agrees with the observations we made in the case of α3 and α4 not equal to zero. To summarize: we studied the behaviour of a scalar test-particle in the metric background of a repulson. It has a curvature singularity at a finite distance rh and a coordinate singularity at r = 0. Although the metric changes at rh from a Lorentzian one to one with a Euclidean signature there is nothing peculiar about the position rh of the curvature singularity at the quantum level. In contrast, near the coordinate singularity at the origin the particle feels a potential barrier and becomes reflected. In terms of a semiclassical approximation, i.e. high frequency of the ingoing particle, there is an equal incoming and outgoing flux, i.e. the scattering matrix is unitary. The particle is phase shifted by π/2. The numerical data indicate— as expected—that the WKB approximation does not work very well for incoming particles with low frequency. For special values of the repulson data (α3 = α4 = 0) we succeeded in working out the asymptotic behaviour analytically. We again find total reflection but the phase shift for a low-frequency incoming particle depends on the physical data of the repulson. In particular, the scattering behaviour of the scalar test-particle at the repulson supports the conjecture that gravitational singularities might be smeared out quantum mechanically. There are other indications that these naked singularities deserve further attention. First of all—as they are repulsive in nature—they are not a complete disaster from the cosmic censorship point of view. The singularity is not shielded by a horizon but instead by an effective repulsive barrier at which scalar test-particles bounce off. In addition, they very often correspond to solutions of a higher-dimensional Kaluza–Klein-like theory, which usually has Repulsive singularities in 4D extended supergravity 2245 different properties. Singularities are resolved or naked singularities correspond to black holes in higher dimensions. It might be possible to establish some relations between these solutions and their properties. Unfortunately—although the repulsons can be associated to solutions of five-dimensional low-energy effective string theories—they do not correspond to black holes. At least not if we take the definition of a black hole seriously. They correspond to extremal dilatonic configurations, which represent spacetimes with timelike singularities [20–22]. We think it would be very interesting to study non-extremal repulson configurations to which black holes and subsequently entropy, temperature, etc, can be associated. Even more interesting would be to investigate the behaviour of a wavepacket travelling towards the repulsive barrier. The wavepacket would serve as a model of a test-string, which consists of an infinite number of modes. However, for that it is necessary to treat the partial differential equation (19) in an appropriate manner, which we leave for further investigations. 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