We determine in closed form the electrostatic potential generated by a point
charge at rest in a simple model of static spherically symmetric wormhole. From
this, we deduce the electrostatic self-energy of this point charge.Comment: latex, 5 pages, a slight correctio

Linet, Bernard

English

arXiv

5 pagesWe determine in closed form the electrostatic potential generated by a point charge at rest in a simple model of static spherically symmetric wormhole. From this, we deduce the electrostatic self-force of this point charge

HAL Université de Tours

Electrostatics in a wormhole geometryBernard LinetTo cite this version:Bernard Linet. Electrostatics in a wormhole geometry. 5 pages. 2008. <hal-00203389>HAL Id: hal-00203389https://hal.archives-ouvertes.fr/hal-00203389Submitted on 9 Jan 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00203389, version 1 - 9 Jan 2008Electrostatics in a wormhole geometryBernard Linet ∗Laboratoire de Mathe´matiques et Physique The´oriqueCNRS/UMR 6083, Fe´de´ration Denis PoissonUniversite´ Franc¸ois Rabelais, 37200 TOURS, FranceAbstractWe determine in closed form the electrostatic potential generated by a pointcharge at rest in a simple model of static spherically symmetric wormhole. Fromthis, we deduce the electrostatic self-energy of this point charge.The physical effects in wormholes has been a subject of considerable investigation. Thepresent work is devoted to solving the electrostatic equation in closed form in the back-ground metric of Morris and Thorne [1] which describes a simple wormhole. We take upagain the question of the electrostatic self-energy analysed by Khusnutdinov and Bakhma-tov [2] in the framework of the multipole formalism. We emphasize that we considerelectrostatics in a static spherically symmetric wormhole connecting two different asymp-totically flat spacetimes. Of course, electromagnetism in a static spherically symmetricwormhole connecting two distant regions in the same spacetime is radically different [3].In the paper of Morris and Thorne [1], a simple geometry of wormhole is described bythe metricds2 = −dt2 + dl2 +(l2 + w2) (dθ2 + sin2 θdϕ2)(1)in the coordinate system (t, l, θ, ϕ) with −∞ < l <∞. The throat is located at l = 0. Thetwo different asymptotically flat spacetimes are defined for l → ±∞.To solve in the next the electrostatic equation, it is convenient to write metric (1)in isotropic coordinates (t, r, θ, ϕ). The radial cordinate r is related to the proper radialcoordinate l byr =12(l +√l2 + w2)(2)with the range 0 < r <∞. Now the throat is located at r = w/2. We note thatdrdl=l +√l2 + w22√l2 + w2satisfyingdrdl> 0∗E-mail: linet@lmpt.univ-tours.frand thus coordinate transformation (2) is valid eveywhere. The inverse transformation isgiven byl = r − w24r. (3)Taking into account (2), metric (1) is written asds2 = −dt2 +(1 +w24r2)2 (dr2 + r2(dθ2 + sin2 θdϕ2))(4)in the coordinate system (t, r, θ, ϕ) with r > 0.We set the Cartesian coordinates (xi), i = 1, 2, 3, associated to the spherical coordinates(r, θ, ϕ). Metric (4) takes then the formds2 = −dt2 +(1 +w24r2)2 ((dx1)2 + (dx2)2 + (dx3)2)(5)with r =√(x1)2 + (x2)2 + (x3)2.In the background metric (5), we consider a test electromagnetic field generated by apoint charge e at rest. The charge density ρ of this point charge located at xi0 has theexpressionρ(xi) = eδ(3) (xi − xi0)√−g (6)where g is the determinant of the metric (5),√−g =(1 +w24r2)3.We note (r0, θ0, ϕ0) the position of the point charge with r0 > 0.The Maxwell equations in metric (5) for the electric component A of the electromagneticpotential reduce with source (6) to the electrostatic equation△A+ h(r)xir∂iA = − 4pie1 + w2/4r2δ(3)(xi − xi0) (7)where △ is the Laplacian operator and the function h has the expressionh(r) =11 +w24r2ddr(1 +w24r2)= − w22r (r2 + w2/4). (8)The aim of this work is to determine in closed form the solution to equation (7) which issmooth everywhere except of course at the point xi0.In a recent paper [4], we have shown that if h obeys certain differential relations thenthere exists a solution to equation (7) which can be written in the formA(xi) = eNg(r)g(r0)F (s(xi, xi0)) (9)2withs(xi, xi0) =Γ(xi, xi0)k(r)k(r0)(10)where Γ(xi, xi0) = (x1 − x10)2 + (x2 − xi0)2 + (x3 − x30)2. The functions g, k and F are to bedetermined and N is a numerical factor.The first step is to determine the elementary function in the Hadamard sense Ae toequation (7). This solution is defined by requiring in the vicinity of the point xi0 thefollowing development:Ae(xi) =1√Γ(xi, xi0)(U0(r, r0) + U1(r, r0)Γ(xi, xi0) + · · ·). (11)In the case of metric (5), we have from (7) the first coefficientU0(r, r0) =e√1 + w2/4r2√1 + w2/4r20.By inserting expression (8) into the differential relation of our previous paper [4], weobtainh′ +12h2 +2rh =w22(r2 + w2/4)2.In consequence, we know to be able to find a solution in form (9) by considering the caseII of our paper. By comparing, we put a = w/2, b = w/2 and A = −4. The functions gand k are then given byk(r) =r2 + w2/4w/2, g(r) =rr2 + w2/4, (12)and the differential equation for F iss(1− s)F ′′ + (−3s + 3/2)F ′ − F = 0. (13)According to (10) with (12), we have 0 ≤ s ≤ 1.We are now in a position to determine in closed form the elementary solution in theHadamard sense Ae to equation (13) by taking as solution Fe to equation (13)Fe(s) =1√s(1− s)(14)sinceFe(s) ∼ 1√s+12√s as s→ 0. (15)Using (9), we get finallyAe(xi) =12ewg(r)g(r0)Fe(Γ(xi, xi0)k(r)k(r0))(16)3with N = w/2 to satisfy (11). We can directly verify that Ae given by expression (16)obeys the partial differential equation (7).However, we are in a case where solution (16) presents another singularity. Indeed,it defines a new point charge e diametrically opposed to xi0 on the sphere r = r1 withr1 = w2/4r0. This corresponds to s = 1 in expression (14) sinceFe(s) ∼ 1√1− s as s→ 1.We must look for another solution Fv to equation (13) which leads only a point chargelocated at xi0. We take the solutionFv(s) =12√s(1− s)+arcsin(1− 2s)pi√s(1− s). (17)The regularity at s = 1 of expression (17) results from the fact that arcsin(−1) = −pi/2.We note the expansion of function (17)Fv(s) ∼ 1√s− 2pi+12√s− 43pis as s→ 0. (18)This choice of function Fv yields the electrostatic potential V generated by a point chargee located at xi0V (xi) =12ewg(r)g(r0)Fv(Γ(xi, xi0)k(r)k(r0)). (19)We point out that the electric flux through the sphere at the infinity r → ∞ of theelectrostatic potential (19) is4pie[12+1piarcsin(r20 − w2/4r20 + w2/4)]. (20)We have the opportunity to add to potential (19) the electrostatic monopole solution toequation (7) which is smooth everywhere but we have no prescription to fix this homoge-neous solution.We turn now to calculate the electrostatic self-energy. We develop expression (19) inthe vicinity of the point xi0 by using expansion (18). We getV (xi) ∼ U0(r, r0)√Γ(xi, xi0)− ewpig(r)g(r0) as xi → xi0. (21)Thus, the electrostatic self-energyW of the point charge e at rest in the wormhole geometry(5) can be immediately deduces from development (21)W (r0) = − 12pie2wg2(r0) = − e2wr202pi (r20 + w2/4)2 . (22)4The electrostatic self-energy (22) can be expressed with the proper radial coordinate l inthe formW (l0) = − e2w2pi (l20 + w2)(23)Result (23) coincides with the one of Khusnutdinov and Bakhmatov [2].References[1] M. S. Morris and K. S. Thorne Am. J. Phys. 56 395 (1988).[2] N. R. Khusnutdinov and I. V. Bakhmatov Phys. Rev. D 76 124015 (2007).[3] V. P. Frolov and I. D. Novikov Phys. Rev. D 42 1057 (1990).[4] B. Linet Gen. Rel. Grav. 37 2145 (2005).5

Electrostatics in a wormhole geometry

Abstract

Similar works

Full text

Available Versions

HAL Université de Tours