Electrostatic interaction of a pointlike charge with a wormhole S Krasnikov 9 0 The Central Astronomical Observatory of Russian Academy of Science, M-140, 0 Pulkovo,St. Petersburg, Russia 2 E-mail: [email protected] n a J Abstract. A pointlike static (or quasistatically moving) electric charge q is 7 considered in the spacetime which is a wormhole connecting two otherwise ] Minkowskianspaces. The electrostaticforce actingonthe chargeis foundto be a sum c of two terms. One of them is uniquely determined by the value of q and the geometry q - of the wormhole. The other has the Coulomb form and is proportional to a freely r g specifiable parameter (the “charge of the wormhole”). These terms are interpreted, [ respectively, as the self-force and the force exerted on the charge by the wormhole. 2 The self-force is found explicitly in the limit of vanishing throat length. The result v differs from that obtained recently by Khusnutdinov and Bakhmatov. 8 5 3 1 PACS numbers: 04.20.Gz, 03.50.De . 2 0 8 1. Introduction 0 : v What electric force (if any) acts on a pointlike charge at rest outside a wormhole, if i X there are no more charges in the space? This question is of interest by, at least, two r reasons. The first is its relation to the famous concept of “charge without charge” [1]. a Suppose, in a flat region of a spacetime we observe the electric field E = Qr/r3, r > r . (1) 0 From this we need not conclude that the field is generated by a charge (sitting, say, at r = 0). It may well happen that there is a wormhole mouth inside the sphere r = r 0 (so that the “coordinate” r does not, in fact, take the zero value) and the field force lines do not terminate at all, see figure 1. Which suggests that maybe there is no charge — as “substance” — in nature and the electromagnetic field is described by the source-free Maxwell equations, while all elementary “charges” are, in fact, mouths of wormholes . In developing such a theory it would be important to know how strong the ‡ resemblance is between a wormhole threaded by force lines and a pointlike charge Q. The flux conservation together with the spherical symmetry guarantees that (1) is valid in both cases. So, one might think that, as long as we restrict ourselves to the region Throughout the paper we regard the matter supporting the wormhole as electrically neutral. ‡ Electrostatic interaction of a pointlike charge with a wormhole 2 Figure 1. A section ϕ = const of M(3). The divergent gray lines are the force lines of a source-free field which “imitates” the Coulomb field for an observer outside the sphere r =d+a. r > r , the resemblance is perfect. As we shall see, however, this is not the case: the 0 force F experienced by a finite charge q put in a point p outside the wormhole would ∗ not be just the Coulomb force F = qQr(p )/r3(p ) (moreover, F may turn out to be C ∗ ∗ C a negligible part of F). Another reason of interest in finding F is the possible existence of macroscopic traversable wormholes. At the moment the only observational restriction on their abundance has been obtained on the basis of unusual lensing properties of negative mass [2] and is valid only for a very special type of wormholes. To improve the situation it would be desirable, of course, to know more about physical effects involving wormholes. The consideration of the electrostatic problem in the wormhole background can be viewed as a step in that direction. An interesting, in this sense, result of this paper is that self-interaction leads to appearance of the attraction infinitely growing (in the approximation of infinitely short throat) as the charge approaches a mouth of a wormhole. Tempted by the resemblance between the electrostatic and (Newton’s) gravitational forces — which differ in the sign, though — one might speculate therefore that wormholes are possible which are macroscopic and static, but nevertheless non- traversable for massive bodies. Our analysis will be confined to a simplest wormhole: ds2 = dt2 +dx2 +r2(x)(dθ2 +sin2θdϕ2) (2a) − x ’, r C∞, r( x) = r(x), r > 0, r = x+a. (2b) ∈ ∈ − x>d The wormhole is obviously static and spherically symmetric. Each its spacelike section t = const — we shall denote such sections by M(3) — is a pair of flat three-dimensional spaces M and M (they are defined by the inequalities x > d and x < d, respectively, + − − and either is just the Euclidean space minus a ball of radius d + a) connected with a ‘tunnel’, see figure 1. Electrostatic interaction of a pointlike charge with a wormhole 3 It is well known [3, problem 14.16] that in curved spacetime the Maxwell equations writtenforthevector-potentialAi have, inthegeneralcase, two non-equivalent versions. Fortunately, R0 = 0 for our metric and the difference does not lead to any ambiguity in i the equation on Φ = A0. It reads: Φ, ;a(p,p ) = 4πqδ(p p ), p M(3), (3a) a ∗ ∗ − − ∈ where a = 1,2,3 and the derivatives are by the coordinates of p (not p ). Equation (3a) ∗ canbesolved bystandardmethods, seethefollowing section, buttherearetwo problems in finding the force F from Φ: A. The solutions of (3a) diverge in p , where the force is to be found, and thus one ∗ needs a “renormalization” procedure to derive a meaningful and finite value for the force. The problem is quite hard in the general case, see, e. g., [4, 5], and references therein, but in the case under discussion p is restricted to M , where the procedure ∗ + is trivial due to flatness: to obtain the renormalized solution Φren one simply subtracts the Coulomb part from Φ, see [4, 6]. B. Themoreseriousproblemisthat(3a)hastoomanysolutions: ifsomeΦ solves (3a) 1 then so also does Φ (p,p )+f(p )Z(p), 1 ∗ ∗ where f is arbitrary andthe “source-free” potential Z is an arbitraryharmonic function. In the ordinary electrostatics the problem is solved by requiring the electric field to fall at infinity Φ, 0 at r r(p ) = const, r(p) , (3b) i ∗ ∗ → ≡ → ∞ which physically means that we are not interested in field configurations with infinite energies. We adopt the restriction (3b) too, but in our case this does not fix the problem, because in M(3) there are non-zero harmonic functions satisfying (3b). Thus, the (absolute value of) the force experienced by a pointlike charge near the wormhole is arbitrary and the question posed in the beginning of the paper has no meaningful answer. To overcome this problem I introduce “the charge” of the wormhole defined — up to the factor 4π — as the flux of E through the throat and prove (see the proposition in the following section) that for the wormhole of a given charge Q the solution of (3) is unique up to an additive constant. The solution depends on Q in quite a natural way: Φren(q,p,p ) = Q/r(p)+Φ (q,p,p )+const. ( ) ∗ sf ∗ ∗ [cf. (17)]. The desired force acting on the charge is found now by, first, differentiating this expression by the coordinates of p and then setting p = p : ∗ F(r) = qQ(r)r/r3 q Φ (r,r) (4a) sf − ∇ (as before, in this expression acts on the first argument). ∇ Electrostatic interaction of a pointlike charge with a wormhole 4 Formally, equation (4a) solves the problem in discussion [an explicit expression for Φ is given by (17)]. It is, however, of little practical use yet. Indeed, by sf F(p) one normally understands the dependence of the force on position of the charge when everything except the position is assumed to be fixed. But this latter (perhaps, somewhat vague) condition inno way enters the derivationof(4a) andthe functionQ(r) is therefore arbitrary. To fix it suppose that the charge is transported (quasistatically, so that the radiation can be neglected) from p to some other point p M . In ∗ ∗∗ + ∈ section 3, I argue that Q in such a case will remain unchanged Q(r ) = Q(r ), which ∗ ∗∗ conservation justifies the name “charge”. So, (4a) must be complemented with Q(r) = const, (4b) which accomplishes the task. The structure of (4) with Q independent of p and Φ independent of Q suggests ∗ sf interpretation of the first term in (4a) as the force exerted on the charge by the source- free field, or by the wormhole. And the second term is naturally interpreted as the self-force. Note. Recently, Khusnutdinov and Bakhmatov [5] have found special solutions — let us denote them by Φ(1) and Φ(2) — of equations (3) for r = √a2 +x2 and r = a+ x , KB KB | | (1) respectively (later Φ was refound by Linet [7], who used a different method). Neither KB ofthoser(x)satisfies (2b), butthemainproblemwithfindingtheself-force[which ishow toidentify theself-interaction potentialamongthe infinitely manysolutions of(3)]is the ren(i) same as in our case. Correspondingly, as explained above, the quantity q Φ (r,r) − ∇ KB ren(1) need not be the self-force. And, indeed, calculating the flux of Φ through the ∇ KB sphere x = const one finds that it depends on p , see, e. g., [7, (20)] and the sentence ∗ below it. Likewise, for the wormhole of the second type the comparison of our formula (20) with that for Gren in [5] gives the flux qa/(2r ). So, F q Φren(i)(p ,p ) is not − ∗ ≡ − ∇ KB ∗ ∗ the self-force, but rather another, much less meaningful, quantity — the force acting on the pointlike charge located in p in the presence of a wormhole with the charge Q(p ). ∗ ∗ 2. The multipole expansion In this section we establish the uniqueness of the solution of equation (3) up to the term Qρ/r+Φ , where ρ is a certain function of r (specified below), while Q and Φ do not 0 0 depend on r. We begin by rewriting equation (3a) in the coordinate form 2r′ 1 ∂2 + ∂ + (∂2 +cotθ∂ +sin−2θ∂2) Φ x r x r2 θ θ ϕ h i 4πq = δ(ϕ)δ(θ)δ(x x ) −r2sinθ − ∗ (we have set ϕ = θ = 0, which obviously does not lead to any loss of generality). ∗ ∗ Electrostatic interaction of a pointlike charge with a wormhole 5 Expanding ∞ l Φ = φ(m)(x)Ym(ϕ,θ), l l l=0 m=−l X X where Ym are spherical functions [8] l 2l+1 (l m )! Ym(ϕ,θ) −| | |m|(cosθ)eimϕ, l ≡ s 4π (l+ m )!Pl | | m dm Plm(µ) ≡ (1−µ2)2 dµm Pl(µ) ( are the Legendre polynomials) one gets l P ∞ l 2r′ l(l+1) ∂2 + ∂ φ(m)(x)Ym(ϕ,θ) x r x − r2 l l Xl=0 mX=−lh i 4πq = δ(ϕ)δ(θ)δ(x x ).(5) −r2sinθ − ∗ Multiply both sides of (5) by rYm′∗sinθ and integrate over ϕ and θ. The result (Ym′ l′ l′ are orthonormal on the sphere) is r′′ l(l+1) 4πq ∂2 + rφ(m)(x) = δ(x x )Ym∗(0,0), x − r r2 l − r − ∗ l h (cid:16) (cid:17)i It is convenient to treat the cases of zero and non-zero m separately, because 2l+1 Y0(0,0) = , Ym(0,0) = 0, m = 0 l 4π l 6 r So, we define v 2l+1rφ(0), v rφ(m), m = 0. l ≡ 4π l l,m ≡ l 6 q For v we have l r′′ l(l+1) 2l+1 ∂2 + v = qδ(x x ) (6) x − r r2 l − r − ∗ ∗ h (cid:16) (cid:17)i while v irrespective of m must solve the equation l,m r′′ l(l+1) ∂2 + z(l,x) = 0. (7) x − r r2 h (cid:16) (cid:17)i Thus, the solution of (3) is the function ∞ ∞ l 1 1 Φ = v (x) (cosθ)+ v (x)Ym(ϕ,θ), (8) r l Pl r l,m l l=0 l=1 |m|=1 X X X where v and v are the solutions, respectively, of (6) and (7) which [because of (3b)] l l,m grow at x not faster than x . | | → ∞ | | To proceed note that in the flat regions M the term with r′′ vanishes in (7) and ± the equation is easily solved: the solution is a superposition of r−l and rl+1. Electrostatic interaction of a pointlike charge with a wormhole 6 Notation By z and z we denote the solutions of (7) which are equal to r−l at, − + respectively, x < d and x > d. And z , z are the solutions of (7) defined by the initial e o − data z (0) = 1, z′(0) = 0, z (0) = 0, z′(0) = 1 e e o o Evidently z and z are even and odd, respectively, and any solution of (7) is their linear e o combination. Proposition. If z(l,x) is a solution of (7) with l > 0, the function r−1z grows unboundedly as x ( ) . → − ∞ Proof. We start with the observation that if a solution z of (7) satisfies the condition z(x ) > 0, W[z,r](x ) 0, (9a) 0 0 ≥ where W is the Wronskian W[f ,f ] = f′f f f′, then 1 2 1 2 − 1 2 z′/z > 0 and z/r grows at x > x . (9b) 0 Indeed, rewrite (7) as l(l+1) W′[z,r] = z. (10) r Integrating this equation one gets z′ r′ 1 l(l+1) x zdx = W[z,r](x )+ . (11) 0 z − r rz rz r Zx0 Due to (9a) the r. h. s. is positive at least up to x , where x is , if z(x) has no zeroes, 1 1 ∞ and the first zero of z otherwise. Thus, z′/z > 0 and z/r grows at x (x ,x ). The 0 1 ∈ latter means, in particular, that x cannot be finite (because if it were, r(x ) would have 1 1 been less than z(x ) = 0), which proves (9b). 1 Nownotethatbothz = z andz = z satisfy(9a)withx equaltozerointheformer e o 0 case and to some (sufficiently small) positive number in the latter. So, z /r grows at e(o) all x > x and hence, z cannot be proportional to r−l at large x. Consequently, 0 e(o) z (x) c rl+1, x , c = 0. e(o) 2 2 ∼ → ∞ 6 The same is true for x , since z is even (odd). And, finally, it is true, when e(o) → −∞ x or x , for every z because any of them is a superposition of z = z and e → ∞ → −∞ z = z . o Corollary 1. If l > 0, the solutions z and z are linearly independent. + − Corollary 2. The second term in the r. h. s. of (8) is zero. Electrostatic interaction of a pointlike charge with a wormhole 7 Corollary 3. Denote by ϑ the Heaviside step function. Then at l > 0 the function (2l+1)q ϑ(x x )z (l,x)z (l,x )+ϑ(x x)z (l,x)z (l,x ) ∗ + − ∗ ∗ − + ∗ v = − − (12a) l − r W[z ,z ] ∗ + − is the unique solution of (6) that grows slower than rl+1 at the infinities. Now let us turn to the case l = 0. Equation (7) [as seen from (10)] transforms into W′[z,r] = 0. This gives (z/r)′ = C/r2, where C is an arbitrary constant. Thus, z(0,x) is a linear combination of r and ρ r(x) x dx ρ(x) r(x) . ≡ r(d) − r2(x) Zd Inthiscase z isproportionaltoz andthe formula(12a)doesnot define v . The latter, + − 0 however, can be easily found by using — as independent solutions of the homogeneous equation (7) — the functions ρ and r instead of z and z (note that W[r,ρ] = 1): + − q v = ϑ(x x )ρ(x)r +ϑ(x x)rρ +Qρ+Φ r (12b) 0 ∗ ∗ ∗ ∗ 0 r − − ∗ (cid:16) (cid:17) Here ρ = ρ(x ) and Q, Φ are arbitrary, but do not depend on r. ∗ ∗ 0 Summing up, ∞ 1 Φ = v (x) (cosθ), (13) l l r P l=0 X where v (x) are given by formulae (12). l 3. Self-interaction Equation (13) gives, in principle, the electrostatic field of a pointlike charge in the wormhole background. However, as mentioned in the Introduction, to find the force acting on the charge it remains to cope with the fact that the field diverges in the point p where the charge is located. To this end we take advantage of the fact that p is in ∗ ∗ a flat part of the wormhole (let it be M , for definiteness). In this region we define the + potential (for the second equality see, e. g., [8, (II 2.13)]) q Φ (p,p ) = Eucl ∗ ≡ p,p ∗ | | ∞ 1 q[ϑ(x x )(r /r)l +ϑ(x x)(r/r )l+1] (cosθ), (14) ∗ ∗ ∗ ∗ l r − − P l=0 X where p,p is the distance between p and p in the space …3, obtained by gluing a ∗ ∗ | | usual Euclidean ball of radius d to M . From the usual electrostatics we know that the + field Φ exerts no force on the charge. So, in finding the self-force we are only Eucl −∇ interested in the difference Φren Φ Φ (15) Eucl ≡ − Electrostatic interaction of a pointlike charge with a wormhole 8 (which is defined, of course, only in M ). It is Φren that plays the rˆole of the “external + field”, i. e. the force acting on the charge is F = q Φren(p ,p ). ∗ ∗ − ∇ To rewrite the expression (13) for Φ in a more convenient form let us substitute the equalities (in fact, the second one is a definition of α ) l z (x,l) = r−l, z (x,l) = C(rl+1 +α r−l), at l > 0, x > d, (16) + − l into (12a): α v = q[ϑ(x x )(r /r)l +ϑ(x x)(r/r )l+1 + l(rr )−l], l ∗ ∗ ∗ ∗ ∗ − − r ∗ at l > 0, x,x > d ∗ Substituting this together with an obvious (notice that ρ(x) = 1 at x > d) equality v = q[ϑ(x x )+(r/r )ϑ(x x)]+Q+rΦ , at x,x > d. 0 ∗ ∗ ∗ 0 ∗ − − into (13) and, then, the result — combined with (14) — into (15), we finally obtain Φren(p,p ) = Φ (p,p )+Φ (p,p ), p,p M , (17a) ∗ sf ∗ wh ∗ ∗ + ∈ where ∞ Φ q α (rr )−l−1 (cosθ), Φ Q/r+Φ . (17b) sf l ∗ l wh 0 ≡ P ≡ l=1 X Note In the region under consideration (i. e., at x,x > d) Φ is smooth. ∗ sf Proof. By definition [see, (16)] (z r−l−1)′r2(l+1) − at x = d α = = l − (2l+1)C (z r−l−1)′r2(l+1) (2l+1)r2l l/r z′ /z +1/r − = − − − r2l+1(d) (18) − (2l+1) (z rl)′ l/r+z′ /z − − − On the other hand, z satisfies the condition (9a) with x = d. Hence, by (9b), z′ /z − 0 − − − is positive at x = d. It follows then from (18) that at l → ∞ α = A r2l+1(d), A = O(1) l l l and ∞ A r(d) r(d) l+1 l Φ (x,x ) = q (cosθ). sf ∗ l d+a r(x) r(x ) P ∗ Xl=1 h i Obviously for any x > d the series converges uniformly on [x , ) and so do all the 1 1 ∞ series obtained from this one by termwise differentiation in r. We interpret Φ and Φ as the parts of Φren generated by the charge and by the sf wh wormhole, respectively. To justify this interpretation note that 1) Φ , for a given p, sf depends only on q and p and 2) Φ , in contrast, does not depend on p in the following ∗ wh ∗ sense. Suppose the charge is at rest up to some moment t and is then quasistatically 0 moved from p (t ) to some p (t ) M , where — at the moment t — is again put to ∗ 0 ∗ 1 + 1 ∈ Electrostatic interaction of a pointlike charge with a wormhole 9 rest. Let t be a moment when at small x the disturbance in the potential caused by 2 | | the motion of the charge has already settled down and the potential became constant (in time) . Then at times t > t in the vicinity of the wormhole the equations (17) 2 § remain valid with r (t ) replaced by r (t ) and with the same Q. ∗ 0 ∗ 1 Proof. Indeed, at t > t the flux of Φthrough thesphere x = d is F(d,t ) = 4πQ(t ), 2 2 2 ∇ − because neither Φ , nor Φ give any contribution to it. At the same time, there is a Eucl sf sphere x = D > x such that F(D,t ) = 4π[Q(t )+q], because if D > x +c(t t ), ∗ 2 0 ∗ 0 − − the field is not disturbed there yet. Thus the flux F (t ) through the boundary of the B 2 layer d < x < D,t = t is 4π[Q(t ) Q(t ) q]. On the other hand, the total charge 2 2 0 { } − − inside the layer has not changed and hence F (t ) = F (t ) = 4πq by the Gauss B 2 B 0 − theorem. So, Q(t ) = Q(t ). 2 0 4. Short wormhole It is seen from formulae (17) that the force acting on a charge depends on the form of the wormhole — the information about the form being encoded in the coefficients α . l But today we have no reason to consider any particular form as more realistic than any other. So, it would be interesting to find a form-independent effect. To this end we consider in this section the limit d 0 for the wormhole (2) with a > 0. In doing so → we allow the the throat to be arbitrary, the only additional requirement on r(x) being r′ < c d (19) r ∀ (c is a constant), which, among other things, guarantees that r(0) a. r → Let us, first, present z as the solution of the following differential equation (which − does not contain the large quantity r′′) ∂2 +2r′r−1∂ l(l+1)r−2 y = 0, x x − hy( d) = (d+a)−l−1. i − Here the first line is simply (7) in terms of y z /r, while the second follows from the − ≡ definition of z . The coefficients of the equation by (19) are uniformly (by d) bounded, − so at a fixed l and d 0 → ln′y( d) ln′y(d). − → Hence, ln′z( d) ln′r( d) ln′z(d) ln′r(d) and thus − − − → − ln′z (d) ln′z ( d)+2/a (l+2)/a − − → − → (recall that z = (a x)−l at x = d). On the other hand, by (16) − − − (l+1)rl lα r−l−1 (l+1)al lα a−l−1 ln′z (d) = − l = − l , − rl+1 +αlr−l r=a al+1 +αla−l The existence of such a moment is an assumption, even though a very plausible one. If a wormhole § is such that in its vicinity the electro-magnetic waves caused by stirring the charge do not dissipate with time, one probably cannot develop electrostatics in that spacetime at all. Electrostatic interaction of a pointlike charge with a wormhole 10 combining which with the equation above we find in the limit d 0 → a2l+1 α = . l −2(l+1) Thus, asymptotically, ∞ a2l+1 Q Φren q (rr )−l−1 (cosθ)+ +Φ ∗ l 0 ∼ − 2(l+1) P r l=1 X ∞ q 1 Q = (a2/rr )l (cosθ)+ +Φ . ∗ l−1 0 −2a l P r l=2 X In particular, at θ = 0 ∞ q 1 Q Φren (a2/rr )l + +Φ ∗ 0 ∼ −2a l r l=2 X q a2 a2 Q = ln(1 )+ + +Φ (20) 0 2a − rr r r r (cid:20) ∗ ∗ (cid:21) and the electric field on the axis is r r qa3 Q Φ, = (Φren +Φ ), q − ∗ + + . − x − Eucl x∼ r r 3 2r r2(a2 rr ) r2 ∗ ∗ ∗ | − | − Thus, asymptotically, inthepresenceofaninfinitelyshortwormholewiththeradius a a pointlike charge q experiences the (radial) force q2a3 qQ F(r ) = qΦren = + . ∗ − ,x −2r3(r2 a2) r2 ∗ ∗ − ∗ Its first term — the self-force F — can be presented, if desired, in the form s q2 F (r ) = U(r ), U(r ) = ln[1 (a/r )2]+(a/r )2. s ∗ ∗ ∗ ∗ ∗ −∇ 4a − (cid:16) (cid:17) Acknowledgements I am grateful to N. R. Khusnutdinov and R. R. 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