UUNIVERSITY OF MIAMIGREEN FUNCTIONS, SOMMERFELD IMAGES, AND WORMHOLESByHASSAN ALSHALA THESISSubmitted to the Facultyof the University of Miamiin partial fulfillment of the requirements forthe degree of Master of ScienceCoral Gables, FloridaMay 2019 a r X i v : . [ phy s i c s . c l a ss - ph ] A p r bstract Thesis supervised by Professor Thomas L. Curtright.Electrostatic Green functions for grounded equipotential circular and elliptical rings,and grounded hyperspheres in n-dimension electrostatics, are constructed using Som-merfeld’s method. These electrostatic systems are treated geometrically as differentradial p-norm wormhole metrics that are deformed to be the Manhattan norm, namely“squashed wormholes”. Differential geometry techniques are discussed to show howRiemannian geometry plays a rule in Sommerfeld’s method. A comparison is madein terms of strength and position of the image charges for Sommerfeld’s method withthose for the more conventional Kelvin’s method. Both methods are shown to bemathematically equivalent in terms of the corresponding Green functions. However,the two methods provide different physics perspectives, especially when studying dif-ferent limits of those electrostatic systems. Further studies of ellipsoidal cases aresuggested.n tribute to Richard Feynman (1918-1988) and Arnold Sommerfeld (1868-1951). cknowledgements
I would like to show my deep gratitude to Prof. Dr. Thomas L. Curtright whohas never shown parsimony or hesitance to support and advise me through educationand research, either in this thesis or in any other curricular endeavors. I believe hisleading personality together with his wide knowledge of the topic were indispensablefactors for me to finish this thesis. I am forever indebted to him.HASSAN ALSHALUniversity of MiamiMay 2019 ontents
List of Figures iiPreface iv1 Introduction 12 2D Grounded Circular Ring 6 p -norm and Ellis wormholes . . . . . . . . . . . . . . . . . . . . . . . 122.4 Squashing p -norm wormholes into Manhattan wormhole . . . . . . . . 162.5 Relating Kelvin and Sommerfeld imagesthrough inversion transformation . . . . . . . . . . . . . . . . . . . . 19 n D Grounded Conducting Hyperspheres 21 n D curved space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 p -norm and Ellis wormholes in n D . . . . . . . . . . . . . . . . . . . . 273.3 Manhattan norm in n D . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Relating Kelvin and Sommerfeld imagesby inversion in n D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31ii ist of Figures
Figure 1.a: Frontal view of extended real coordinates geometry to Kelvin’s methodfor grounded ellipse (red) with U = 3 /
2. The “doorway” (blue) is in the middlebetween exterior charges (orange) and corresponding images (green) regions. No-tice the phase change in the image trajectories ∆ v = 2 v (cid:48) when they cross the“doorway” at u = 0. Grey curves are parameterized ellipses with fixed u and0 ≤ v ≤ π ...............................................................................................................38Figure 1.b: Lateral view of of extended real coordinates geometry to Kelvin’s methodwith for side view of grounded ellipse (red line) with U = 3 / v = 2 v (cid:48) when they cross the “doorway” at u = 0. Greycurves are parameterized ellipses with fixed u and 0 ≤ v ≤ π ...................................39Figure 2.a: Bird view of real coordinates geometry of Sommerfeld’s method forgrounded ellipse (red) at U = 0 with trajectories of exterior sources (orange) andtheir corresponding images (green). In contrary to Kelvin’s method, the “door-way” is the ring itself. Grey curves are parameterized ellipses with fixed u and0 ≤ v ≤ π ...............................................................................................................40iiivFigure 2.b: Frontal view of real coordinates geometry of Sommerfeld’s method forgrounded ellipse (red) at U = 0. Unlike Kelvin’s method, Sommerfeld’s method fixestrajectories of exterior sources (orange) and their corresponding images (green) atsame angle v . Grey curves are parameterized ellipses with fixed u and 0 ≤ v ≤ π .................................................................................................................................41 reface When George Green introduced his technique of Green functions, he didn’t realizehe had changed the face of both mathematics and physics forever. Being inspired bywork of Pierre-Simon Laplace and Sim´eon Poisson on electric potentials [4], Greenintroduced the concept of Green functions in his seminal work: “ An Essay on the Ap-plication of mathematical Analysis to the theories of Electricity and Magnetism. ”Ashe was aware he lacked an institutional scientific degree, Green was hesitant to submitit for publication in a journal of those managed by the only two scientific societiesin England at that time: British Royal Society and Cambridge Philosophical Soci-ety. Instead he printed it at his own expense and distributed it among his fellowsin Nottingham Subscription Library society. As it stayed unpublished in any jour-nal even after Green earned the Tripos degree from Cambridge and passed away twoyears later, his essay didn’t capture the attention of professional mathematicians untilWilliam Thomson [21], later Lord Kelvin, shared it with French mathematicians likeChasles, Liouville and Sturm, during his stay in Paris, and the French mathemati-cians were astounded by Green’s work. According to Kelvin, Sturm expressed hisadmiration: “Ah voil`a mon affaire!” [Oh that’s my business!] as he found what heindependently discovered was originally established by Green 16-17 years before him.After he returned home, Kelvin managed to republish it in Crelle’s journal (Berlin)in three separate parts in 1850, 1852, and 1854 . Indeed Green coined the term
Potential Theory [3]. It has been republished recently on [arXiv:0807.0088]. viIn fact the “revolutionary” Kelvin’s method of images is a direct applicationof Green functions in electrostatics problems. Also Green’s work inspired GeorgeStokes[3], a close Cambridgean friend of Kelvin, while he was persuing his study towave phenomena in hydrodynamical systems.Away for being known for introducing another set of conditions related to Green func-tions, Carl Neumann [5] focused on potential theory in 2D using Dirichlet boundaryconditions. He eventually came up with a solution he called
Logarithmischen Poten-tial , which will be discussed in chpater 2 when the potential problem with a grounded2D conducting ring is treated. Other German mathematicians [9] extensively studiedGreen function techniques, meanwhile Bernhard Riemann [3] was the one who coinedits name.By 1880s-90s, Green functions became a hot topic specially among theoretical andmathematical physicists who were interested in developing electromagnetic theorythat could fully adopt the concept of “Luminiferous Ether” [9, 10, 24]. Being alsointerested in studying diffraction patterns of electromagnetic and acoustic waves,Arnold Sommerfeld generalized Kelvin’s method of images to study Green functionsin 2D using the concept of complex Riemann surfaces [18, 20], which is the maintheme of this thesis. General consideration on how Sommerfeld built his method willbe presented in the introduction chapter of this thesis.Sommerfeld realized how powerful his technique is, so he wrote to Kelvin [10] that:“The number of boundary value problems solvable by means of my elab-orated Thomson’s method of images is very great.”As Whittaker [24] described Green: was introduced in Spring of Nations year; Thomson W., Geometrical Investigations with Ref-erence to the Distribution of Electricity on Spherical Conductors, Camb. Dublin Math. J. 3, 141(1848). ii“the founder of that Cambridge School of natural philosophers of whichKelvin, Stokes, Rayleigh, Clerk Maxwell, Lamb, Larmor and Love werethe most illustrious members in the later half of the nineteenth century”,perhaps it would not be inappropriate to describe him as “Desert father” of the propagators business in quantum electrodynamics that was introduced by RichardFeynman, Julian Schwinger and Sin-Itiro Tomonaga [4] and were awarded Nobelprize on it 1965. We can see that clearly in Schwinger’s lecture [19] as he reminiscedabout how he, and Feynman independently, developed the methodology of Green toconstruct their relativistic quantum theory of electrodynamics.This thesis is devoted to study Sommerfeld’s method to find Green functions of n D spherical and 2D elliptic “static”
Ellis wormholes [12], then to squash them intothe so-called Manhattan norm. It is based on the following papers:i. T Curtright, H Alshal, P Baral, S Huang, J Liu, K Tamang, X Zhang, andY Zhang. “The Conducting Ring Viewed as a Wormhole” arXiv: 1805.11147[physics.class-ph].ii. H Alshal and T Curtright, “Grounded Hyperspheres as Squashed Wormholes”arXiv: 1806.03762 [physics.class-ph].iii. H Alshal, T Curtright, S Subedi, “Image Charges Re-Imagined” arXiv: 1808.08300[physics.class-ph].By incorporating Riemannian geometry with Sommerfeld’s method, I hope this thesisconvinces the reader that Sommerfeld’s method, despite being not widely admired,could be considered as a precursor to studies of static wormholes [17, 15]. hapter 1Introduction
Method of Green functions is a robust technique to obtain solutions of many lin-ear, ordinary or partial, differential equations describing physical phenomena relatedto either classical or quantum field theory [4, 9]. Green functions can be seen as aresponse of a physical system to an impulse, e.g; a point like source of gravitationalor electric field. For such physical systems the equation of motion is usually on theform L [ y ( x )] = f ( x ), where L is the linear operator that the corresponding Greenfunction expresses the integral kernel of L − , i.e., G ( x, x (cid:48) ) = F − k { / F x [ L ] } . And f ( x ) = F − k (cid:20) F x [ y ( x )] ( k ) F x [ G ( x, x (cid:48) )]( k ) (cid:21) which can be expressed as a Fourier series, where F x and F − k are the direct and inverse Fourier transforms. So from integral equationspoint of view, a Green function is an integral kernel of the inverse of a given operatorsuch that Green function satisfies L [ G ( x, x (cid:48) )] = δ ( x − x (cid:48) ) where δ ( x − x (cid:48) ) is theDirac-delta function.For inhomogeneous partial differential equations, i.e., with λy ( x ) not being a so-lution for L [ y ( x ) ] = f ( x ) for some λ ∈ C , equations mainly come with three typesof boundary conditions (b.c) related to the surface S that bounds the volume V con-taining the impulse sources. Inhomogeneity comes either from b.c’s, i.e., for y ( x ) = α y (cid:48) ( x ) = β where α, β are either (cid:54) = 0 or vary from point to another in S , or fromthe nature of the equation of motion itself. And those conditions are:i. Dirichlet boundary conditions, where y ( x ) = α for some α ∈ C and x ∈ S = ∂V . α could be a function in x .ii. Neumann boundary conditions, where the flux, ∂ ˆn y ( x ) = ∂y ( x ) ∂ ˆn = ∇ y ( x ) . ˆn = α for some α ∈ C , is in the direction of the normal unit vector ˆn at x ∈ S = ∂V .iii. Robin boundary conditions, which is a combinition of both Dirichlet and Neu-mann b.c’s on the same boundary S = ∂V .For n -order linear operator L we integrate L [ G ( x, x (cid:48) )] = δ ( x − x (cid:48) ) over period x (cid:48) ∈ [ x (cid:48) − (cid:15), x (cid:48) + (cid:15) ] with (cid:15) →
0, we get d n − Gdx n − (cid:12)(cid:12)(cid:12) x = x (cid:48) = 0, while d n − Gdx n − (cid:12)(cid:12)(cid:12) x = x (cid:48) = c n , i.e.,the ( n − th -order of differentiation of a Green function has discontinuity ∀ x ∈ S ,and c n is the inverse of the coefficient of the n th term of the differential equation.Part of potential theory asserts that for an electric charge within the vicinity ofan idealized conductor, a geometric distribution of induced charge is produced to berestricted on the surface of the that conductor. In ideal electrostatic circumstancesthe interior of the conductor can not contain any induced charge as it’s a physicalequipotential volume. However, for the sake of finding the effect of the induced chargeon the surface, it would be expedient to imagine as if the surface is neutrally like amirror reflecting images of another charge distribution of opposite sign to the originalcharge . That image charges are not uniquely dictated although they are endowedwith electrostatic effects similar to those of the actual distribution of induced surfacecharge. The mutual effect of both charge and its image is determined only based on It’s different from the mixed b.c’s in that the mixed ones have Dirichlet and Neumann b.c’s onmutually exclusive boundaries, i.e., S ∩ S = φ and S ∪ S = ∂V . In general nothing precludes both charge and its image(s) to be on the same side of the conductorsurface depending on the geometry of the problem. the surface of the conductor, namely, the boundary . There are two geometrical tech-niques governing the mirror image methods: the well-known conventional Kelvin’smethod and Sommerfeld’s method. Those two methods render different geometricalimage regions despite they yield exactly the same results. This geometrical freedomis only restricted to the boundary determined by the surface separating the exteriorand the interior of the conductor. In general, Kelvin’s method is preferable when it ismanageable to extend Green functions from the charge domain to the image domaingeometries, while Sommerfeld’s method is recommended for conductor surfaces pro-ducing images at obvious conformational positions similar to the pattern of reflectionsof the real world on a flat mirror.Sommerfeld’s method imposes a split duplicate 2D space laying over the originalone [18]. Both spaces are endowed with Riemann surfaces properties. A Riemann sur-face is 2D real manifold (surface) with complex structure to adopt complex-analyticfunctions. The two spaces are separated through branch cuts which intersect thegrounded surface, where the impulse source of waves is forbidden to exist on, at itsboundaries. For the upper double space a point x up with θ ∈ [0 , π ) while for thelower one a point x down has θ ∈ [2 π, π ) such that each point on a space has corre-spondent coincident point upon squashing double-sheeted Riemann surface . Then themirror image of the source on the lower space would be away from the boundary atdistance exactly equal to that of the source on the upper space.Due to conformational symmetry corresponding to such boundary-value problem,Sommerfeld expected the behavior of the potential function of the source and the im-age separately to be identical up to a sign and angles difference in both double spaces.Meanwhile the difference in both functions that represent the actual potential, is not equal to zero, except for those points of the grounded surface with θ ∈ { , π } as Check figures [6-13] p.69-77 in [18]. branching locates there.Later Sommerfeld [20] used his generalization to Kelvin’s method to study the poten-tial of point charge near a grounded 2D disc by considering Green’s second identitywhich states that for harmonic and continuously differentiable functions u ( x ) and v ( x ) in volume V bounded by surface ∂V = S and the normal unit vector of thatsurface is ˆn we have: (cid:90) V (cid:2) u ( x ) ∇ v ( x ) − v ( x ) ∇ u ( x ) (cid:3) dV = (cid:90) S = ∂V [ u ( x ) ∇ v ( x ) − v ( x ) ∇ u ( x )] · ˆn dS . (1.1)So for any 2 nd order linear operator L , a Green function obeys b.c’s of y ( x ) togetherwith: G ( x, x (cid:48) ) (cid:12)(cid:12)(cid:12) x = x (cid:48) = 0 , and a discontinuity: ∂G ( x, x (cid:48) ) ∂x (cid:48) (cid:12)(cid:12)(cid:12) x = x (cid:48) = c. (1.2)Then by applying Green’s second identity to Poisson’s equation ∇ u ( (cid:126)r ) = (cid:37) ( (cid:126)r ) /(cid:15) and using suitable Dirichlet b.c’s: ∀ (cid:126)r ∈ S, u ( (cid:126)r ) = f ( (cid:126)r ) and G ( (cid:126)r, (cid:126)r (cid:48) ) = 0, Sommerfeldobtained: u ( (cid:126)r ) = (cid:90) V G ( (cid:126)r, (cid:126)r (cid:48) ) (cid:37) ( (cid:126)r (cid:48) ) dr (cid:48) + (cid:90) S f ( (cid:126)r (cid:48) ) ∂G ( (cid:126)r, (cid:126)r (cid:48) ) ∂n dr (cid:48) . (1.3)In Sommerfeld treatment, had b.c’s been homogeneous, the surface integral wouldhave vanished. Also G ( (cid:126)r, (cid:126)r (cid:48) ) vanishes at S together with ∇ [ G ( (cid:126)r, (cid:126)r (cid:48) )] = δ ( (cid:126)r − (cid:126)r (cid:48) ).Then Green function can be separated into fundamental solution G ( (cid:126)r, (cid:126)r (cid:48) ) and its“copies” G i ( (cid:126)r, (cid:126)r i (cid:48) ) that render mirror image solutions of the fundamental solutionoutside V in V i ’s for some i , That is: G o ( (cid:126)r, (cid:126)r (cid:48) ) = G ( (cid:126)r, (cid:126)r (cid:48) ) + N (cid:88) i =1 G i ( (cid:126)r, (cid:126)r i (cid:48) ) . (1.4)Both fundamental and mirror image solutions don’t necessarily satisfy the b.c’s. Whatactually matters is adjusting them such that:a) ∇ [ G ( (cid:126)r, (cid:126)r (cid:48) )] = δ ( (cid:126)r − (cid:126)r (cid:48) ) · b) ∇ G i ( (cid:126)r, (cid:126)r i (cid:48) ) = 0 ∀ i · c) G o ( (cid:126)r, (cid:126)r (cid:48) ) = 0 at (cid:126)r ∈ S · If we integrate condition (a) then consider the spherical symmetry G ( (cid:126)r, (cid:126)r (cid:48) ) = G ( | (cid:126)r − (cid:126)r (cid:48) | ) = G ( r ) with the fact that ˆn is parallel to (cid:126)r − (cid:126)r (cid:48) , then: (cid:90) S ∇ G ( (cid:126)r, (cid:126)r (cid:48) ) . ˆn dS = 2 π r ∂G ( r ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = R = 1 . (1.5)When G o ( r ) → | r | → ∞ , then Sommerfeld got the logarithmic potential , Neu-mann had studied before, for the upper surface as: G ( (cid:126)r, (cid:126)r (cid:48) up ) = 12 π ln | (cid:126)r − (cid:126)r (cid:48) up | + C , (1.6)where C is determined according to the boundary condition. If we want Green func-tion to vanish at (cid:126)r → ∞ then C = 0.In Sommerfeld treatment there is only one copy of double sheeted Riemann sur-face. Then we expect one mirror image function in the lower surface to be similar tothat of the upper surface, up to sign. And the general Green function is G o ( (cid:126)r, (cid:126)r (cid:48) ) = 12 π ln | (cid:126)r − (cid:126)r (cid:48) up | − π ln | (cid:126)r − (cid:126)r (cid:48)(cid:48) down | · (1.7)Finally by combining (1.7) and (1.3), the solution of potential function u ( (cid:126)r ) is ob-tained. hapter 22D Grounded Circular Ring Wormhole studies have captured interests since the introduction of the
Einstein-Rosen bridge [11]. The classic Ellis wormhole [12] can be seen in 2D, after beenfoliated and sliced, as circles with redii r ( w ) = √ R + w where w ∈ ( −∞ , + ∞ )is the parameter that relates the coordinates of vertical axis z , with circles alignedaround it, with coordinates of curves r that describes how to move from circle toanother. R is the fixed radius of the smallest circle at the equatorial near the bridgethat connects the double sheets of Riemann surface.Our approach is based on building Laplace-Beltrami operator of such manifold to getboth the fundamental solution and the image solution of Green functions after con-sidering the required Dirichlet boundary conditions together with the correspondingdiscontinuities.Next we strict the radii of those aligned circles to become r ( w ) = ( R p + | w | p ) /p for p ∈ [1 ,
2] and see how Green functions behave upon taking the limit p → r ( w ) = R + | w | is the Manhatten norm and equatorial circle with radius R serves as a “portal” ora “doorway”. Following Sommerfeld’s method, the interiors of these circles act asforbidden region of Sommerfeld screen; they are excluded from Riemann surfaces too.6 For a regular grounded conducting ring with radius R in Euclidean 2D space E ,the electrostatic potential function of a point-like charge at (cid:126)r (cid:48) is given by:Φ E ( (cid:126)r − (cid:126)r (cid:48) ) = − π ln (cid:18) | (cid:126)r − (cid:126)r (cid:48) | R (cid:19) , ∇ Φ E ( (cid:126)r − (cid:126)r (cid:48) ) = − δ ( (cid:126)r − (cid:126)r (cid:48) ) . (2.1)And the corresponding Green function obeying ∇ G ( (cid:126)r, (cid:126)r (cid:48) ) = − δ ( (cid:126)r − (cid:126)r (cid:48) ) is: G ( (cid:126)r, (cid:126)r (cid:48) ) = − π ln (cid:18) | (cid:126)r − (cid:126)r (cid:48) | R (cid:19) = − π (cid:20) ln (cid:16) r > R (cid:17) + ln (cid:18) r < r > − r < r > cos( θ ) (cid:19)(cid:21) , (2.2)where r ≷ = maxmin ( (cid:126)r, (cid:126)r (cid:48) ) and θ is the angle between (cid:126)r and (cid:126)r (cid:48) .For the image inside the ring at position (cid:126)r (cid:48)(cid:48) = R r (cid:48) (cid:126)r (cid:48) Green function is: G ( (cid:126)r, (cid:126)r (cid:48)(cid:48) ) = 12 π ln (cid:18) | (cid:126)r − (cid:126)r (cid:48)(cid:48) | R (cid:19) . (2.3)Then the general Green function is: G o ( (cid:126)r, (cid:126)r (cid:48) ) = − π ln (cid:18) | (cid:126)r − (cid:126)r (cid:48) | R (cid:19) + 12 π ln (cid:32) | (cid:126)r − R r (cid:48) (cid:126)r (cid:48) | R (cid:33) + 12 π ln (cid:18) | (cid:126)r (cid:48) | R (cid:19) . (2.4)The last term is some constant C we referred to before in (1.6) that’s to be determinedaccording to the b.c’s when G o ( (cid:126)r, (cid:126)r (cid:48) ) | r (cid:48) = R = 0. The last two terms are harmonic forall (cid:126)r, (cid:126)r (cid:48) , i.e., they obey the condition ∇ G i ( (cid:126)r, (cid:126)r i (cid:48) ) = 0 we stated before, and G ( (cid:126)r, (cid:126)r (cid:48) )is the only one contributes the Dirac delta since it’s the fundamental Green function.Those additional image functions break translational invariance on the plane but, ofcourse, they don’t change the symmetry of the arguments of the Green function uponchanging (cid:126)r ↔ (cid:126)r (cid:48) . A 2D Riemannian manifold a topological space that is endowed with a metric g µν , where g ≡ det g µν and 1 ≤ µ, ν ≤
2, with space intervals ds = g µν dx µ dx ν . Thenthe invariant Laplace-Beltrami operator ∇ = √ g ∂ µ ( √ gg µν ∂ ν ). For such manifold,the space interval is: ( ds ) = ( dw ) + r ( w ) ( dθ ) , (2.5)and its invariant Laplacian is: ∇ = 1 r ( w ) (cid:2) ∂ w ( r ( w ) ∂ w ) + ∂ θ (cid:3) . (2.6)Upon changing variable: u ( w ) = w (cid:90) d ˜ wr ( ˜ w ) , (2.7)the Laplacian can be read more clearly as: ∇ = 1 g (cid:18) ∂ ∂u + ∂ ∂θ (cid:19) . (2.8)Now harmonic functions h corresponding to Laplace equation ∇ h = 0 are periodicfor fixed w in θ ∈ [0 , π ]. Therefore they are: h ( u ) = a + b u and h ± m ( u, θ ) = c m e − mu e ± imθ for m ∈ Z \{ } . (2.9) A reader might wonder why to express it in partial derivatives rather than ∇ covariant ones. The Christoffel connections are implicitly expressed as Γ νµν = √ g ∂ µ ( √ g ) = g αβ ∂ µ ( g αβ ) upon applyingLeibntz rule to the Laplacian components. Remember the operator as whole is acting on a scalarfunction. After change of variables (2.9) can be written as: h ( w ) = a + b w (cid:90) d ˜ wr ( ˜ w ) and h ± m ( w, θ ) = c m e ± imθ exp − m w (cid:90) d ˜ wr ( ˜ w ) . (2.10)Notice that h is isotropic, so for w ≷ = maxmin ( w, w (cid:48) ) the radial part of Laplace equationfor h ( u ) can be read as: ddw (cid:20) r ( w ) ddw (cid:18)(cid:90) w > w < d ˜ wr ( ˜ w ) (cid:19)(cid:21) = ddw (cid:26) +1 , w>w (cid:48) − , w
12 1 r ( w ) ∂ w ( r ( w )) (cid:21) (cid:41) = − r ( w ) d r ( w ) dw = (1 − p ) R ( w ) ( p − / (cid:0) R p + ( w ) p/ (cid:1) · (2.30)The last result for the curvature assures that for p = 1 i.e., the Manhattan norm,the manifold is flat so the upper and the lower branches are perfectly superimposedover each other. Moreover, when we get close to the equatorial ring, i.e., w →
0, theradius corresponding to such curvature blows up ∀ p ∈ [1 , bridge ) becomes made of points that all are singular. More details on that arein [6], Appendix C.Final comment of the geometry before we return back to Green functions business.5We may notice: | w | = r (cid:18) − ( Rr ) p (cid:19) /p = r (cid:18) − p R p R p + | w | p + · · · (cid:19) . (2.31)So when w → ±∞ , the distance interval becomes:( ds ) ≈ w →±∞ ( dr ) + r ( dθ ) , (2.32)which guarantees such wormholes own flat curvature when w → ±∞ . It’s obviouswhen in the case of Manhattan norm for r (cid:29) R .Now for Green functions of different p -norms, there is a special case where p = 2(the Ellis wormhole ) as (2.17) becomes: ∇ G ( w, θ ; w (cid:48) , θ (cid:48) ) = 1 √ R + w ∂ w (cid:16) √ R + w ∂ w G (cid:17) + 1 R + w ∂ θ G = − √ g δ ( w − w (cid:48) ) δ ( θ − θ (cid:48) ) , (2.33)while (2.7) becomes: − πR w > (cid:90) w < d ˜ w (cid:113) ˜ w R = − πR ln (cid:34)(cid:114) ˜ w R + 1 + ˜ wR (cid:35) w > w < = 14 π ln (cid:34)(cid:32)(cid:114) w > R + 1 − w > R (cid:33) (cid:32)(cid:114) w < R + 1 + w < R (cid:33)(cid:35) . (2.34)We prefer expressing the solution of that integral as logarithmic function rather thanhyperbolic sine function as it’s related to the Green function of the logarithmic po-6tential. Substitute (2.34) in (2.18) to get: G ( w, θ ; w (cid:48) , θ (cid:48) ) = + 14 π ln (cid:34)(cid:32)(cid:114) w > R + 1 − w > R (cid:33) (cid:32)(cid:114) w < R + 1 + w < R (cid:33)(cid:35) − π ln (cid:32)(cid:114) w > R + 1 − w > R (cid:33) (cid:32)(cid:114) w < R + 1 + w < R (cid:33) − π (cid:34) − (cid:32)(cid:114) w > R + 1 − w > R (cid:33) (cid:32)(cid:114) w < R + 1 + w < R (cid:33) cos( θ − θ (cid:48) ) (cid:35) . (2.35)Then substitute it in (2.19) to get the desired general Green function G o ( w, θ ; w (cid:48) , θ (cid:48) ).Now we are ready to squash Ellis wormholes! p -norm wormholes into Manhattanwormhole Wormhole “flattening” is achieved as a continuous deformation by taking theradial function to be the p -norm of R and w . That is, a ± w symmetric p-normwormhole in 2D is defined taking p = 1 so the throat “height” becomes zero. Thatis: r ( w ) = R + | w | , (2.36)Then for the case where both w and w (cid:48) are on the same side from the branch cut ofthe double-sheeted space, i.e., the same wormhole branch, the integral (2.7) is: w > (cid:90) w < d ˜ wR + | ˜ w | = ln (cid:18) R + w > R + w < (cid:19) . (2.37) In the opposite extreme, where p → ∞ , the p -norm wormhole function becomes the right-circularcylinder of [15], p.488, eqn(3). w and w (cid:48) are on opposite side from the branchcut of the double-sheeted space, let’s say w > w (cid:48) <
0, the integral (2.7) is: w > (cid:90) w < d ˜ wR + | ˜ w | = ln (cid:18) R + wR (cid:19) + ln (cid:18) R − w (cid:48) R (cid:19) . (2.38)For the first scenario, (2.18) becomes: G ( w, θ ; w (cid:48) , θ (cid:48) ) | w> w (cid:48) > = − π ln (cid:34) (cid:20) R + w > R + w < (cid:21) (cid:34) (cid:20) R + w > R + w < (cid:21) − (cid:20) R + w > R + w < (cid:21) cos ( θ − θ (cid:48) ) (cid:35) (cid:35) = − π ln (cid:18) r + r (cid:48) − rr (cid:48) cos ( θ − θ (cid:48) ) rr (cid:48) (cid:19) . (2.39)While the second scenario makes (2.18)to be: G ( w, θ ; w (cid:48) , θ (cid:48) ) | w> w (cid:48) < = − π ln (cid:18) rr (cid:48) R + R rr (cid:48) − θ − θ (cid:48) ) (cid:19) . (2.40)where r = R + | w | and r (cid:48) = R + | w (cid:48) | . The last two equations (2.39) and (2.40) describethe general Green function for any w, w (cid:48) . We may check the consistency of these twoequations with the conventional Green functions got by Kelvin’s method for “the”traditional single Euclidean plane at identical points. After neglecting the azimuthaldependence, the difference for the first case is given by: G ( w, θ ; w (cid:48) , θ (cid:48) ) | w> w (cid:48) > − G E ( (cid:126)r, (cid:126)r (cid:48) ) = − π ln (cid:18) R rr (cid:48) (cid:19) , (2.41)while the difference in the second case is: G ( w, θ ; w (cid:48) , θ (cid:48) ) | w> w (cid:48) < − G E ( (cid:126)r, R ( (cid:126)r (cid:48) ) (cid:126)r (cid:48) ) = 14 π ln (cid:16) rr (cid:48) (cid:17) , (2.42)8where Kelvin’s method restricts the image point inside the ring (the equatorial circlein Sommerfeld’s method) at (cid:126) r (cid:48) = R ( (cid:126)r (cid:48) ) (cid:126)r (cid:48) . Since in Kelvin’s method r (cid:48) (cid:13) R , then (cid:126) r (cid:48) = R ( (cid:126)r (cid:48) ) (cid:126)r (cid:12) R (cid:48) . And obviously the RHS’s of (2.41) and (2.42) are harmonic on E where r (cid:54) = 0 (cid:54) = r (cid:48) .If we substitute (2.39) in (2.19), the more general Green function that describeselectrostatic potential function everywhere on both upper and lower branches forManhattan wormhole with grounded equator ( ring of radius R ) in the middle of themanifold bridge becomes: G o ( w, θ ; w (cid:48) , θ (cid:48) ) | w ≷ w (cid:48) ≷ = − π ln (cid:20) r + r (cid:48) − rr (cid:48) cos ( θ − θ (cid:48) ) rr (cid:48) (cid:21) + 14 π ln (cid:20) rr (cid:48) R + R rr (cid:48) − θ − θ (cid:48) ) (cid:21) = − π ln (cid:20) R r + r (cid:48) − rr (cid:48) cos ( θ − θ (cid:48) ) r r (cid:48) + R − R rr (cid:48) cos ( θ − θ (cid:48) ) (cid:21) . (2.43)If we substitute (2.40) in (2.19) then G o ( w, θ ; w (cid:48) , θ (cid:48) ) | w ≷ w (cid:48) ≶ = − G o ( w, θ ; w (cid:48) , θ (cid:48) ) | w ≷ w (cid:48) ≷ asit’s an odd function of w for fixed w (cid:48) .The corresponding general Green function, for both the source charge and its image,obtained by Kelvin’s method is given by: G o E ( (cid:126)r, (cid:126)r (cid:48) ) = G E ( (cid:126)r, (cid:126)r (cid:48) ) − G E ( (cid:126)r, R ( (cid:126)r (cid:48) ) (cid:126)r (cid:48) ) . (2.44)Then there is disparity shown in the difference between the general Green func-tion anywhere on the squashed wormhole and the general Green function of Kelvin’smethod: G o ( w, θ ; w (cid:48) , θ (cid:48) ) − G o E ( (cid:126)r, (cid:126)r (cid:48) ) = − π ln (cid:18) Rr (cid:48) (cid:19) . (2.45)We notice that such disparity comes from the fact that G o E is not symmetric under (cid:126)r ↔ (cid:126)r (cid:48) . However since the difference is harmonic on both the Euclidean plane and9the Riemannian wormhole for any r (cid:48) (cid:54) = 0, then both ∇ (cid:12)(cid:12)(cid:12)(cid:12) r,r (cid:48) G ( (cid:126)r, (cid:126)r (cid:48) ) = − δ ( (cid:126)r, (cid:126)r (cid:48) ) andDirichlet condition G ( (cid:126)r, (cid:126)r (cid:48) ) (cid:12)(cid:12) (cid:126)r (cid:48) = R ˆr = 0 are still satisfied for any points outside theforbidden region inside the Euclidean ring corresponding to the Riemannian bridge regardless whether Green function is considered to be G o or G o E . Part of complex structure that is endowed to Riemann surfaces is inversion map-ping: I : (cid:126)r (cid:55)→ (cid:126) r = R r (cid:126)r , (2.46)where R is the radius of Riemann’s sphere.Due to angular symmetry ˆ r = ˆr , we look to the traditional ring, with same radius R ,as an S bounding Riemann sphere. We can relate the mirror image got by Kelvin’smethod to that image got by Sommerfeld’s method through inversion transformation.Under such transformation Laplace-Beltrami operator is transformed as: I ( ∇ r ) = (cid:18) R r (cid:19) (cid:18) ∇ r + 2(2 − N ) r D r (cid:19) , D r = (cid:126) r · (cid:126) ∇ r , (2.47)such that, except for 2D, the Laplacian [1] is not an eigenvector of such transformation,it has extra piece ( scale operator ) to be added to it under inversion. In 2D case, Greenfunctions are invariant under inversion as: I ( G ( (cid:126)r, (cid:126)r (cid:48) ) ) = ( (cid:126) r ) N − ( (cid:126) r (cid:48) ) N − R N − G ( (cid:126) r ,(cid:126) r (cid:48) )= G ( (cid:126) r ,(cid:126) r (cid:48) ) , for N = 2 · (2.48)0And: I (cid:2) ∇ r G ( r, r (cid:48) ) (cid:3) = I (cid:2) − (cid:112) g ( r ) δ ( r − r (cid:48) ) δ ( θ − θ (cid:48) ) (cid:3) ↓∇ r G ( r , r (cid:48) ) = − (cid:112) g ( r ) δ ( r − r (cid:48) ) δ ( θ − θ (cid:48) ) , (2.49)leaving also the condition on Green functions unchanged under inversion.So these results lead us to map the lower branch of Manhattan norm wormhole tothe interior of the Euclidean ring, while it leaves the upper branch unchanged. Themore general case for N (cid:54) = 2 is to be discussed in next chapter. hapter 3 n D Grounded ConductingHyperspheres
Four years after Sommerfeld introduced his method, Hobson used it [14] to targetthe probleom of grounded conducting disk in 3D. Thirty-seven years later, Waldmann(a student of Sommerfeld) solved [22] for Green functions using Sommerfeld’s methodby mapping half-plane to a disk of finite radius. After another thirty-four years, Davisand Reitz [8] constructed Green functions for same problem using complex analysis.As how we did in the previous chapter, first we build Laplace-Beltrami operator in n Dto get Green functions after considering the required Dirichlet boundary conditionstogether with the corresponding discontinuities.Next we study the general characteristics of the harmonic functions for the p -normwormholes. We focus on the Ellis wormhole case for p = 2. Then we deform thewormhole such that the radii of those aligned hyperspheres become the Manhattannorm and see how Green functions behave after squashing the wormhole. Also wediscuss N = 4 case as N = 3 is in the previously mentioned references.Finally we emphasize on the inversion transformation to the and see how to mapKelvin’s method to Sommerfeld’s method.212 n D curved space
For a charged point-particle located at the origin of n D Euclidean space, E N , thepotential is governed by differential equation: ∇ S N Φ E N ( (cid:126)r ) = − δ N ( (cid:126)r ) , (3.1)with Laplace-Beltrami operator in S N sphere [13]: ∇ S N = 1 r N − ∂ r ( r N − ∂ r ) − r L E N , (3.2)where L (also − r ∇ S N − ) is the angular derivatives: L E N = − r ∇ S N − = ∂ θ + ( N −
2) cot θ ∂ θ − r ∇ S N − , (3.3)and θ represents the latitude from the S N hypersphere north pole. So for perceivable S sphere: ∇ S = 1sin θ (cid:20) sin θ ∂ θ (sin θ ∂ θ ) + ∂ φ (cid:21) . (3.4)Then for N ≥ E N ( (cid:126)r ) = 1( N −
2) Ω N r N − , (3.5)where the hyper solid angle Ω N is given by:Ω N = 2 π N/ Γ( N/ · (3.6)3The corresponding Green function is: G E N ( (cid:126)r, (cid:126)r (cid:48) ) = 1( N −
2) Ω N | (cid:126)r − (cid:126)r (cid:48) | N − · (3.7)The second fraction in last equation can be expanded in terms of Gegenbauer poly-nomials C N − l (cos θ ) as: 1 | (cid:126)r − (cid:126)r (cid:48) | N − = ∞ (cid:88) l =0 ( r < ) l ( r > ) l + N − C N − l (ˆ r · ˆ r (cid:48) ) , (3.8)where C N − l (cos θ ) = l/ (cid:88) k =0 ( − k ( N + 2 l − k − k !(2) l − k ( N − l − k )!! (2 cos θ ) l − k , with r > , r < , ˆ r , ˆ r (cid:48) named as before and ˆ r · ˆ r (cid:48) = cos θ . In N = 3 the Gegenbauer polynomials reduce toLegendre polynomials P lm (cos θ ).Now consider a generalized curved isotropic manifold with distance intervals:( ds ) = ( dw ) + r ( w )( d ˆ r ) , (3.9)where ˆ r represents the points on S N − , with w ∈ ( −∞ , + ∞ ), φ ∈ [0 , π ] and θ i ∈ [0 , π ] , ∀ i = 2 , · · · , N − d ˆ r = (sin θ ) N − (sin θ ) N − (sin θ ) N − · · · (sin θ N − ) dθ dθ · · · dθ N − dφ. The corresponding Laplace-Beltrami operator is: ∇ S N = 1 r ( w ) N − ∂ w ( r ( w ) N − ∂ w ) − r ( w ) L . (3.10)For 2D spherical harmonic eigenfunctions Y lm ( θ, φ ) = ( − m (cid:113) (2 l +1)( l − m )!4 π ( l + m )! P lm (cos θ ) e imφ of L E on S , the n D hyperspherical harmonic eigenfunctions Y lm m ··· m N − correspond-ing to L M N − are [23] :4 Y lm m ··· m N − (Ω) = (cid:40) N − (cid:89) i =1 A ( α i , m i )(sin θ i ) m i C α i / m i − − m i (cos θ i ) (cid:41) × (cid:40) Y m N − m N − ( θ N − , φ ) (cid:41) , (3.11)where A ( α i , m i ) = (cid:104) (2) α i − Γ ( α i /
2) Γ( m i − − m i + 1) ( α i + 2( m i − − m i )) π Γ( α i + m i − − m i ) (cid:105) / and α i = 2 m i + N − i + 1.Other properties of spherical harmonics:i. L Y lm m ··· m N − = l ( l + N − Y lm m ··· m N − .ii. (cid:90) Y lm m ··· m N − (ˆ r ) Y (cid:63)lm m ··· m N − (ˆ r (cid:48) ) d Ω = δ ll (cid:48) δ m m (cid:48) · · · δ m N − m (cid:48) N − .iii. (cid:88) lm m ··· m N − Y lm m ··· m N − (ˆ r ) Y (cid:63)lm m ··· m N − (ˆ r (cid:48) ) = δ N − (ˆ r − ˆ r (cid:48) )= 2 l + N − N − N C N − l (ˆ r · ˆ r ) · Using separation of variables we can write the harmonic functions on manifold withmetric (3.9) as: h lm m ··· m N − = h l ( w ) Y lm m ··· m N − (ˆ r ) . (3.12)Then the radial part of Poisson equation, which is homogeneous differential equation,reads: 1 r ( w ) N − ∂ w (cid:0) r ( w ) N − ∂ w h l ( w ) (cid:1) = 1 r ( w ) L h l ( w ) = l ( l + N − r ( w ) h l ( w ) ,∂ w h l + (cid:20) ( N − r (cid:48) ( w ) r ( w ) (cid:21) ∂ w h l − (cid:20) l ( l + N − r ( w ) (cid:21) h l ( w ) = 0 . (3.13)Then for such homogeneous differential equation with two linearly independent solu-5tions (1) h l and (2) h l , the Wronskian is: W (cid:2) (1) h l ( w ) , (2) h l ( w ) (cid:3) = (1) h l ( w ) ←→ ddw (2) h l ( w )= c l exp − w (cid:90) ( N − r (cid:48) ( ˜ w ) r ( ˜ w ) d ˜ w = c l r N − ( w ) , (3.14)where c l = r N − (0) × W (cid:2) (1) h l (0) , (2) h l (0) (cid:3) is a constant. For the examples of interest, r ( w ) = r ( − w ). Therefore if h l ( w ) is a solution, then so is h l ( − w ). Also when r ( w ) ≈ w →±∞ | w | , then the solution of (3.13) becomes: ( ± ) h l ( w ) ≈ (cid:40) | w | l + N − , w → ±∞ b l | w | l , w → ∓∞ , (3.15)where ( − ) h l ( w ) = (+) h l ( − w ). Notice in 3D space we have h l ∼ (cid:20) | w | l +1 + | w | l (cid:21) justlike the well-known potential function of grounded conducting S sphere.So the Wronskian becomes: W (cid:2) ( − ) h l ( w ) , (+) h l ( w ) (cid:3) ≈ w → + ∞ b l − N − lw N − , (3.16)as expected from (3.14) where c l = (2 − N − l ) b l · As expected, Green function must obey G ( ±∞ , ˆ r ; w (cid:48) , ˆ r (cid:48) ) = 0 , together with discon-tinuity conditions and: ∇ S N G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) = − r ( w ) N − δ ( w − w (cid:48) ) δ N − (ˆ r − ˆ r (cid:48) ) . (3.17)From hyperspherical harmonics property [iii] we discussed earlier, and for N > w ↔ w (cid:48) ) is: G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) = (cid:88) lm m ··· m N − c l (+) h l ( w > ) ( − ) h l ( w < ) Y lm m ··· m N − (ˆ r ) Y (cid:63)lm m ··· m N − (ˆ r (cid:48) )= 1( N − N ∞ (cid:88) l =0 b l (+) h l ( w > ) ( − ) h l ( w < ) C N − l (cos θ ) , (3.18)where c l is given by the radial discontinuity: c l = lim ε → (cid:20) (+) h l ( w (cid:48) ) 1 dw (cid:0) r ( w ) N − − ) h l ( w ) (cid:1) − ( − ) h l ( w (cid:48) ) 1 dw (cid:0) r ( w ) N − h l ( w ) (cid:1)(cid:21) w (cid:48) = w + ε (3.19)Then the more general Green function that describes electrostatic potential functioneverywhere on both upper and lower branches for curved manifold with groundedinnermost hypersphere with radius R is: G o ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) = G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) − G ( w, ˆ r ; − w (cid:48) , ˆ r (cid:48) ) , (3.20)with same comments on this function as those on G o at the end of both section (2.4)and section (2.2).The solution (3.13) is obtained after doing the usual change of variable as in (2.7)that renders ∂ w = r ∂ u . It guarantees r ( w ) ≈ w →±∞ | w | when u ( w ) ≈ w →±∞ ± ∞ . So (3.13)becomes: d du h l + ( N − r (cid:48) ( w ( u )) dh l du = l ( l + N − h l · (3.21)Instead of writing r (cid:48) ( w ( u )) as 1 r drdu we keep it as it is because eventually we describe r ( w ) radial function according to (2.20). But before we do that, (3.21) is solved forarbitrary r ( w ( u )) by considering another change of variable t = tanh u, for t ∈ [1 , − t ) ∂ t (cid:2) (1 − t ) ∂ t (cid:3) h l + ( N − r (cid:48) ( w (tanh − t ))(1 − t ) ∂ t h l = l ( l + N − − t ) h l , (3.22)with corresponding integrating factor: µ ( t ) = exp t (cid:90) ( N − r (cid:48) ( w (tanh − τ ))(1 − τ ) dτ . (3.23)Then (3.22) becomes: ddt (cid:18) µ ( t ) (1 − t ) ddt h l (cid:19) = l ( l + N − − t ) µ ( t ) h l , (3.24)which is determined based on the definition of r ( w (tanh − t )) in terms of variable t . p -norm and Ellis wormholes in n D Again for a class of p -norm radial functions defined as in (2.20), the asymptoticbehavior of harmonics derived from (3.13) is:1 | w | N − ddw (cid:18) | w | N − ddw h l (cid:19) ≈| w |(cid:29) R l ( l + N − | w | h l · (3.25)In light of Gauss hypergeometric functions F (cid:18) a , bc ; z (cid:19) = ∞ (cid:88) k =0 ( a ) k ( b ) k ( c ) k z k k ! where( a ) k = Γ( a + k )Γ( a ) etc, it’s more convenient to express the previously used change ofvariable trick in terms of hypergeometric functions: u ( w ) = w (cid:90) R p + | ( ˜ w ) | p/ ) d ˜ w = wR F (cid:18) /p , /p /p ) ; ( w R ) p/ (cid:19) . (3.26) Like how we chose the solution of the (2.7) to be logarithmic (2.34). u ( w ) = ln (cid:104) wR + (cid:114) w R (cid:105) and r (cid:48) ( w ( u )) = tanh u ,(3.21) has two linearly independent solutions, where independency is guaranteed fromthe Wronskian we discussed before in (3.14). (1) h l ( u ) = (1 + tanh u ) ( N + l − / (1 − tanh u ) l/ F (cid:18) − ( N/ , ( N/ − − ( N/ − l ; 1 − tanh u (cid:19) , (2) h l ( u ) = (1 − tanh u ) ( N + l − / (1 + tanh u ) l/ F (cid:18) − ( N/ , ( N/ − N/ l ; 1 − tanh u (cid:19) . (3.27)It is crucial to take into consideration a linear combination of (1) h l ( u ) and (2) h l ( u )as (2) h l ( u ) behaves right for N > u → + ∞ , meanwhile (1) h l ( u ) blows up forodd N upon u → −∞ .As we refered in the beginning of this chapter, harmonic potentials and theirGreen functions in 3D are discussed [14, 8, 22]. In 4D, harmonics are given by: (12) h l ( u ) = 1tanh u e ± ( l +1) u , (12) h l ( −∞ + ∞ ) = 0 , (3.28)such that (3.14) becomes: W (cid:2) (1) h l ( u ) , (2) h l ( u ) (cid:3) = − l + 1)cosh u · (3.29)Upon changing the variable t = tanh u , ( u = ∓∞ → t = ∓ (12) h l ( t = ∓
1) = 0, and (3.29) becomes W (cid:2) (1) h l ( t ) , (2) h l ( t ) (cid:3) = − l +1).So the linear independency of the two harmonics is still secured unless l = −
1, whichis non-physical from the definition of Gegenbauer polynomials (3.8) where l muststart from 0. Therefore, the Wronskian never vanishes. This also guarantees that anyother harmonic solution for same l would strictly comprise of (1) h l ( t ) and (2) h l ( t ).9Back to Green function, in 4D case, we use w > and w < as before so (3.18) renders: G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) == 14 π (cid:112) ( R + w > )( R + w < ) ∞ (cid:88) l =0 (cid:34) ( w > − (cid:112) R + w > )( w < + (cid:112) R + w < )( w > + (cid:112) R + w > )( w < − (cid:112) R + w < ) (cid:35) l +12 C (1) l (ˆ r. ˆ r (cid:48) )= 14 π (cid:112) ( R + w > )( R + w < ) × (cid:40) (cid:104) ( w > − (cid:112) R + w > )( w < + (cid:112) R + w < )( w > + (cid:112) R + w > )( w < − (cid:112) R + w < ) (cid:105) / + (cid:104) ( w > − (cid:112) R + w > )( w < + (cid:112) R + w < )( w > + (cid:112) R + w > )( w < − (cid:112) R + w < ) (cid:105) − / − r. ˆ r (cid:48) ) (cid:41) − (3.30)Asymptotically with w, w (cid:48) >
0, i.e., on the upper manifold branch, the last result is: G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) ≈ w,w (cid:48) (cid:29) R π | (cid:126)r − (cid:126)r (cid:48) | + O (cid:18) Rr , Rr (cid:48) (cid:19) . (3.31) n D Studying ( p =1)-norm of the radial function (2.20) reveals the two flattened sheetsof E N , where each sheet is indeed missing a N -ball of radius R , namely B N ( R ). Then,as p →
1, we get a single hypersphere S N − “creasing” the two manifolds E N − B N ( R ).Such hypersphere acts as “portal” or “doorway” between the identical copies of theRiemann double-space. As in footnote (6) of section (2.4) the opposite extreme, p → ∞ , for n D is endowed with an equatorial S N − slice of the wormhole that turnsout to be cylindrical tube of [15] p. 488, Eqn(3).0With Manhattan norm (2.36), Green function (3.18) becomes: G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) = N − N ∞ (cid:88) l =0 ( r ( w < )) l ( r ( w > )) l + N − C ( N − ) l (ˆ r. ˆ r (cid:48) ) , if both (cid:110) w > w < > N − N ∞ (cid:88) l =0 ( R ) l + N − ( r ( w < )( r ( w > )) l + N − C ( N − ) l (ˆ r. ˆ r (cid:48) ) , if both (cid:110) w > w < > < (3.32)It ensures the continuity of G as w < → r ( w ) is around R . Also G → w ≷ → ±∞ .Then for the case where both w and w (cid:48) are on the same side from the branch cut ofthe double-space E N − B N ( R ), i.e., the same wormhole hyperbranch, and in light of(3.8) the first line of the last equation gives: G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) | w> w (cid:48) > = 1( N − N r ( w ) + r ( w (cid:48) ) − r ( w ) r ( w (cid:48) ) ˆ r. ˆ r (cid:48) ] N − = 1( N − N | (cid:126)r − (cid:126)r (cid:48) | N − , (3.33)where if w and w (cid:48) are on opposite branches, say w > w (cid:48) <
0, then the secondline of (3.32) becomes: G ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) | w> w (cid:48) < = 1( N − N R N − [ r ( w ) + r ( w (cid:48) ) + R − R r ( w ) r ( w (cid:48) ) ˆ r. ˆ r (cid:48) ] N − = 1( N − N R N − [ r ( w ) r ( w (cid:48) ) + R − R (cid:126)r.(cid:126)r (cid:48) ] N − . (3.34)Combining the last two equations gives the more general Green function: G o ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) | w> w (cid:48) > = 1( N − N (cid:34) | (cid:126)r − (cid:126)r (cid:48) | N − − | (cid:126)r − (cid:126)r (cid:48) ( Rr (cid:48) ) | N − (cid:18) Rr ( w (cid:48) ) (cid:19) N − (cid:35) , (3.35)1where such Green function shows antisymmetric behavior for w (cid:48) in each branch, i.e., G o ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) | w> w (cid:48) > = − G o ( w, ˆ r ; w (cid:48) , ˆ r (cid:48) ) | w> w (cid:48) < .And for N = 4 the corresponding Green function is: G o ( (cid:126)r, (cid:126)r (cid:48) ) = G o ( (cid:126)r (cid:48) , (cid:126)r ) = 14 π (cid:20) r ( w ) + r ( w (cid:48) ) − r. ˆ r (cid:48) − R r ( w ) r ( w (cid:48) ) + R − R (cid:126)r.(cid:126)r (cid:48) (cid:21) , (3.36)where symmetry of Green function is maintained through G o ( (cid:126)r, (cid:126)r (cid:48) ) = G o ( (cid:126)r (cid:48) , (cid:126)r ), andcontour plots of G and G o are shown for the N = 4 in Figures of [1]. n D Inversion map we introduced in section 2.5 presents the position of the negativeimage, of the original charge at (cid:126)r (cid:48) , to be the reduced strength at (cid:126)r (cid:48)(cid:48) = (cid:0) Rr (cid:1) N − (cid:126)r (cid:48) .Also the Laplacian (2.47), which is not an eigenvector of such transformation expect for 2D, maintains the asymptotic behavior of harmonic functions as in (3.15). In 2Dharmonics become effectively of order r l or r − l , while for ( n (cid:54) = 2)D harmonics are oforder r l or r − N − l .According to (2.48) and for N (cid:54) = 2 Green functions are not invariant under inversion.However G is still symmetric under r ↔ r (cid:48) . Then (2.49) becomes: I (cid:2) ∇ r G ( r, r (cid:48) ) (cid:3) = I (cid:2) − (cid:112) g ( r ) δ N ( (cid:126)r − (cid:126)r (cid:48) ) (cid:3) ↓ (cid:20) r R (cid:21) (cid:20) ∇ r + 2(2 − N ) r D r (cid:21) (cid:20) ( (cid:126) r ) N − ( (cid:126) r (cid:48) ) N − R N − G ( (cid:126) r ,(cid:126) r (cid:48) ) (cid:21) = − r N +1 R N δ N ( r − r (cid:48) ) δ N − (ˆ r − ˆ r (cid:48) ) . (3.37)2Last results map the lower branch of wormhole with Manhattan norm to the interiorof innermost hypersphere while leaving the upper branched as it is.In Ref.[1] we find a discussion with more details about how Green functions andcorresponding Kelvin images of grounded conducting disk in 3D are constructed byHobson [14], by Waldmann [22] (one of Sommerfeld students), and by Davis andReitz [8]. Despite neither of these projects bring up Riemannian geometry to thestudies of the corresponding Green functions, it is easy to see that the inversion map(3.37), in N = 3 case, relates Sommerfeld’s method—we modified by incorporatingRiemannian geometry into game—to complex variables analysis that is used in thepreviously mentioned studies. hapter 42D Grounded Elliptic Ring The problem of conducting ellipse (ellipsenfl¨ache) has been extinsively studiedusing complex methods since the 19 th century [9]. However we still have freedom tochoose another choice of real variables rather than the conventional complex plane toget the same results. We will show that for such coordinate choice, and for the sakeof perceivable visual reasons, it is more advantageous to use Sommerfeld’s methodrather than Kelvin’s one. Elliptic coordinates in real xy -plane with two foci on the x -axis at ± a are given by: x = a cosh u cos v, y = a sinh u sin v, ≤ u ≤ ∞ , ≤ v ≤ π . (4.1)Then for a complex variable z = x + iy = (cid:107) r (cid:107) e iθ , z ∈ C with r = x + y it can bereparameterized such that: z = a cosh( u + iv ) . (4.2)Then Euler formula reads: u + iv = ± cosh − ( x + iya ) + 2 iπk, for some k ∈ Z . (4.3)334By choosing the positive solution with k = 0 we get: u = (cid:60) e (cid:18) cosh − (cid:18) x + iya (cid:19)(cid:19) , v = (cid:61) m (cid:18) cosh − (cid:18) x + iya (cid:19)(cid:19) . (4.4)Now we construct the metric and Laplacian to get the corresponding Green function. Building unique space intervals for 2D elliptic case will not change the fact thatthe Green function is for logarithmic potential. As Neumann emphasized on 2Delectrostatic problems [5], we expect the ellipse to be exactly like that of the 2D ring.However we can double check this fact in the following: dx = ( a sinh u cos v ) du − ( a cosh u sin v ) dv ,dy = ( a cosh u sin v ) du + ( a sinh u cos v ) dv , ( dx ) + ( dy ) = a (sinh u + sin v ) (cid:20) ( du ) + ( dv ) (cid:21) . (4.5)Then the corresponding Laplacian is: ∇ = 1 a (sinh u + sin v ) (cid:18) ∂ ∂u + ∂ ∂v (cid:19) , (4.6)with potential function equation: ∇ u,v Φ( u, v ) = − a (sinh u + sin v ) (cid:37) ( u, v ) (cid:15) · (4.7)Then, as expected, Green function obeys: (cid:18) ∂ ∂u + ∂ ∂v (cid:19) G ( u, v ; u (cid:48) , v (cid:48) ) = − δ ( u − u (cid:48) ) δ ( v − v (cid:48) ) , (4.8)5with, as usual, “a” symmetric solution similar to (2.18): G ( u, v ; u (cid:48) , v (cid:48) ) = − π | u − u (cid:48) | − π ln (cid:16) e − | u − u (cid:48) | − e −| u − u (cid:48) | cos( v − v (cid:48) ) (cid:17) . (4.9) For a grounded ellipse at the center and its points are at u = U with v ∈ [0 , π ],the general Green function is: G o ( u, v ; u (cid:48) , v (cid:48) ) = G ( u, v ; u (cid:48) , v (cid:48) ) − G ( u, v ; (2 U − u (cid:48) ) , v (cid:48) ) , (4.10)where the first G is the fundamental Green function, while the second G is the Greenfunction copy produced by the image located at u (cid:48)(cid:48) = 2 U − u (cid:48) .Before we discuss Green functions, it is worth accentuating we deal with mixed ho-mogeneous Dirichlet and Neumann boundary conditions as: G ( u, v ; u (cid:48) , v (cid:48) ) ≈ u →∞ π ( u (cid:48) − U ) , which means Green functions at infinity are not fixed. However ∂∂ u G ( u, v ; u (cid:48) , v (cid:48) ) ≈ u →∞ in particular is exceptedto yield solutions under false impression that it is governed by Dirichlet boundaryconditions alone, which is not true. The Neumann boundary conditions are stillconsidered even if they do not contribute to the solution.For u ≤ U and u (cid:48) ≤ U , i.e., both charge and field locations are inside the groundedellipse, the image is always outside the ellipse with U ≤ (2 U − u (cid:48) = u (cid:48)(cid:48) ) ≤ U ina contrast with the location of image in S N spheres that is produced by inversion mapping. Keeping in mind the ellipse is bona fide ellipse not a circle, a (cid:54) = 0 , U (cid:54) = ∞ ,the image is never at infinity, it is always located at u (cid:48) ∈ [ U, U ].6For u ≥ U and u (cid:48) ≥ U , i.e., both charge and field locations are outside the groundedellipse, with the charge outside the grounded ellipse that is not too far way fromcenter( U ≤ u (cid:48) ≤ U ), the image is inside the ellipse (0 ≤ (2 U − u (cid:48) = u (cid:48)(cid:48) ) ≤ U ). Theother case with u (cid:48) ≥ U we have “negative” position of the image (0 ≥ U − u (cid:48) ). Thisis not perceivable unless we consider that negative position to be in the second sheetof Riemann surface, i.e., the image moves through the part of the semi-major axis( portal or doorway ) connecting the elliptic foci to that opposite world. Also since(0 ≥ U − u (cid:48) ≥ U ) with respect to the original world, an observer in the secondsheet would see the image as if it is the original charge with a location far way fromthe ellipse in that world ( u (cid:48)(cid:48) | upper sheet (cid:55)→ u (cid:48) | lower sheet with u (cid:48) | lower sheet ≥ U | lower sheet ).So for Kelvin images, to solve (4.10) with real coordinates as (4.1), the real interior ofsuch 2D grounded elliptic conductor needs more interior extension for the case whena point-like charge is far from the conductor. Bird view figures visualizing differentlocations for sources and corresponding images are in [2] Appendix. A. To avoid being repetitive, we jump immediately to the squashed wormhole casewith Manhattan norm for the u parameter: u = U + | w | . (4.11)For a reader uncomfortable with du/dw discontinuity, we consider u = ( U p + w p ) / p for p ≥ / −∞ ≤ w ≤ ∞ and repeat again what we discussed before.Then for 0 ≤ w, w (cid:48) ≤ + ∞ , (4.9) becomes: G ( w, v ; w (cid:48) , v (cid:48) ) = − π | w − w (cid:48) | − π ln (cid:104) e − | w − w (cid:48) | − e −| w − w (cid:48) | cos( v − v (cid:48) ) (cid:105) . (4.12)7While for 0 ≤ w ≤ ∞ and −∞ ≤ w (cid:48) lower ≤ G ( w, v ; w (cid:48) lower , v (cid:48) ) = G ( w, v ; − w (cid:48) up , v (cid:48) )= − π | w + w (cid:48) up | − π ln (cid:104) e − | w + w (cid:48) up | − e −| w + w (cid:48) up | cos( v − v (cid:48) ) (cid:105) . (4.13)By suppressing lower,up and considering only w, w (cid:48) ∈ [0 , + ∞ ), the more generalGreen function becomes: G o ( w, v ; w (cid:48) , v (cid:48) ) = G ( w, v ; w (cid:48) , v (cid:48) ) − G ( w, v ; − w (cid:48) , v (cid:48) )= − π (cid:104) | w − w (cid:48) | + ( w + w (cid:48) ) (cid:105) − π ln (cid:16) e − | w − w (cid:48) | − e −| w − w (cid:48) | cos( v − v (cid:48) ) (cid:17) + 14 π ln (cid:16) e − w + w (cid:48) ) − e − ( w + w (cid:48) ) cos( v − v (cid:48) ) (cid:17) . (4.14)Also notice G ( w, v ; w (cid:48) , v (cid:48) ) = G o ( w (cid:48) , v ; w, v (cid:48) ) with G o ( − w, v ; w (cid:48) , v (cid:48) ) = − G o ( w, v ; w (cid:48) , v (cid:48) )for positive w and negative w (cid:48) . For the sake of comparison, both Kelvin and Som-merfeld methods can be visualized in figures (1.a), (1.b) and (2.a), (2.b) respectively.Notice how parameterized curves behave when they pass by the ring from the per-spective of each method.8Figure 1.a: Frontal view of extended real coordinates geometry to Kelvin’s methodfor grounded ellipse (red) with U = 3 /
2. The “doorway” (blue) is in the middlebetween exterior charges (orange) and corresponding images (green) regions. Noticethe phase change in the image trajectories ∆ v = 2 v (cid:48) when they cross the “doorway”at u = 0. Grey curves are parameterized ellipses with fixed u and 0 ≤ v ≤ π .9Figure 1.b: Lateral view of of extended real coordinates geometry to Kelvin’s methodwith for side view of grounded ellipse (red line) with U = 3 / v = 2 v (cid:48) when they cross the “doorway” at u = 0.Grey curves are parameterized ellipses with fixed u and 0 ≤ v ≤ π .0Figure 2.a: Bird view of real coordinates geometry of Sommerfeld’s method forgrounded ellipse (red) at U = 0 with trajectories of exterior sources (orange) andtheir corresponding images (green). In contrary to Kelvin’s method, the “doorway”is the ring itself. Grey curves are parameterized ellipses with fixed u and 0 ≤ v ≤ π .1Figure 2.b: Frontal view of real coordinates geometry of Sommerfeld’s method forgrounded ellipse (red) at U = 0. Unlike Kelvin’s method, Sommerfeld’s method fixestrajectories of exterior sources (orange) and their corresponding images (green) atsame angle v . Grey curves are parameterized ellipses with fixed u and 0 ≤ v ≤ π .2 As the focal distance a → ∞ , the elliptical coordinates ( u, v ) in the region inbetween the two foci near ( u, v ) ≈ (0 , π/
2) become more rectangular Cartesian ones.This is achieved by applying these limits to (4.1) so that lim a →∞ u → ( a sinh u ) = aya = y remains finite. Also when x = 0 , v = π/ U = Y /a , then y ( U/ , π/ → Y . Forsuch limits, the Green function (4.9) becomes: G ( u = ya , v = π u (cid:48) = y (cid:48) a , v (cid:48) = π ≈ a →∞ − π ln (cid:18) | y − y (cid:48) | a + O (cid:18) a (cid:19)(cid:19) . (4.15)while (4.10) becomes: G o ( u = ya , v = π u (cid:48) = y (cid:48) a , v (cid:48) = π ≈ a →∞ − π ln (cid:18) | y − y (cid:48) || y − Y + y (cid:48) | + O (cid:18) a (cid:19)(cid:19) . (4.16)If y, y (cid:48) ≥ Y then (4.16) becomes: G o ( u = ya , v = π u (cid:48) = y (cid:48) a , v (cid:48) = π ≈ a →∞ − π ln (cid:18) | y − y (cid:48) || y − Y + y (cid:48) | + O (cid:18) a (cid:19)(cid:19) . (4.17)This is exactly G o ( y ≥ Y, v = π ; y (cid:48) ≥ Y, v (cid:48) = π ) for a grounded charge line ( y = Y )parallel to the x -axis with Kelvin image at (0 , y (cid:48)(cid:48) ) = (0 , Y − y (cid:48) ).When x (cid:54) = x (cid:48) with the same a → ∞ limit, the corresponding Green function ofgrounded half-plane is: G half plane ( x, y ; x (cid:48) , y (cid:48) ) = − π ln (cid:32) (cid:112) ( x − x (cid:48) ) ( y − y (cid:48) ) (cid:112) ( x − x (cid:48) ) ( y + y (cid:48) − Y ) (cid:33) , (4.18)which is much more easy to derive using the complex variables rather that the realones.3 Since the density of the induced charge on the 2D ellipse is proportional tothe normal component of the electric field near the ellipse u ≈ U , it is known byconsidering the charge density induced on the 2D ellipse by an outsider point-likecharge u (cid:48) ≥ U to find the normal component of the electric field. This normal field isin Kelvin’s method given by: E n ( u ) (cid:12)(cid:12)(cid:12) u = U = − ∂G o ∂u (cid:12)(cid:12)(cid:12) u = U , (4.19)or in Sommerfeld’s method: E n ( w ) (cid:12)(cid:12)(cid:12) w =0 = − ∂G o ∂w (cid:12)(cid:12)(cid:12) w =0 . (4.20)So in Sommerfeld’s view the linear charge density for a point-like charge ( Q ) at w (cid:48) > U is: λ ( v ; w (cid:48) , v (cid:48) ) = E n ( w ) (cid:12)(cid:12)(cid:12) w =0 = − ∂G o ∂w (cid:12)(cid:12)(cid:12) w =0 = − π e − w (cid:48) − e − w (cid:48) − e − w (cid:48) cos( v − v (cid:48) ) + 1 Q · (4.21)In the infinite limit, the last equation shows constant induced charge density on theellipse: λ ( v ; w (cid:48) , v (cid:48) ) ≈ w (cid:48) →∞ − π Q · (4.22)Notice total induced charge on the ellipse is: (cid:90) π λ ( v ; w (cid:48) , v (cid:48) ) dv = − Q , (4.23)regardless wither u (cid:48) ≈ U or u (cid:48) → ∞ .4As our journey through wormholes is about to end, the last thing to be mentionedis that applying Sommerfeld’s method to n D grounded ellipsoids comes with a majordifficulty; there is no exact analytic function that can describe Green functions forsuch case. The known Green functions so far are in form of infinite summations of ellipsoidal harmonics . However, the principle stays the same, and therefore, uponfinding an exact solution to Green functions in ellipsoidal cases, we will be able toconstruct and compare between both Kelvin and Sommerfeld images in different p -norms wormhole backgrounds. More on Green functions for n D ellipsoidal cases arediscussed in [7]. hapter 5Conclusion
This thesis has focused on some mathematical aspects of Green functions usingSommerfeld’s method for p -norm wormholes.In chapter 1 we generally discussed Green functions and the necessary boundaryconditions involved in solving for them. We heuristically presented the concept ofRiemann surfaces and how to use it in combination with Sommerfeld’s method tosolve for Green functions of second order linear differential equations.In chapter 2 we discussed the logarithmic potential for Poisson’s equation and itscorresponding Green functions in 2D curved surfaces. Inspired by the example of Elliswormhole, we restricted the background to be a p -norm wormhole in preparation toobtain a Green function for a grounded ring. Then we squashed the wormhole intoManhattan norm and found the corresponding Green function for the grounded ring,along with all of its various properties. Later we related the Kelvin and Sommerfeldimages and Green functions using inversion mapping, and we found the two view-points give mathematically equivalent Green functions.456In chapter 3 we investigated the same problem for grounded conducting hyper-spheres in n D, and we compared and contrasted Green the functions of groundedconducting hyperspheres to that of the 2D grounded conducting ring.In chapter 4 we analyzed the same problem using real variables to compare andcontrast Kelvin and Sommerfeld methods for the 2D grounded conducting ellipse. Weillustrated different ways of extending the interior region of the ellipse and placingimages such that Dirichlet conditions are fulfilled. We also studied the straight linelimit and showed how it exactly reproduced the Green function of a grounded half-plane. Then, we calculated the density of induced charge and proved that althoughthe induced charge density itself behaves differently depending on where the originalcharge is, the total induced charge is the same as calculated using the conventionalKelvin’s method. Finally, we referred to difficulties and suggestions on expandingthis study to include ellipsoidal cases.I hope this thesis has shown in principle that Sommerfeld’s method simplifiesthe mathematics required to construct Green functions, in contrast with Kelvin’smethod, both conceptually and practically. I also hope that I have presented the notwidely embraced mixture of Riemann surfaces and Green functions, a.k.a Sommer-feld’s method, as a precursor toy model for the studies of wormholes. eferences [1] Alshal, H., and Curtright, T., “Grounded Hyperspheres as Squashed Wormholes”,J. Math. Phys 60, 032901 (2019). arXiv:1806.03762 [physics.class-ph].[2] Alshal, H., Curtright, T., Subedi, S., “Image Charges Re-Imagined”arXiv:1808.08300 [physics.class-ph].[3] Cannell, D. M., and Lord, N. J., “George Green, Mathematician and Physicist:1793-1841”, The Mathematical Gazette, Vol. 77, No. 478 (1993).[4] Challis, L., Sheard, F., “The Green of Green Functions”, Physics Today 56, 12,41 (2007).[5] Cheng, A., Cheng, D. T., “Heritage and early history of the boundary elementmethod”, Eng. Anal. Bound. Elem., 29, 268–302 (2005).[6] Curtright, T., Alshal, H., Baral, P., Huang, S., Liu, J., Tamang, K., Zhang, X.,and Zhang, Y., “The Conducting Ring Viewed as a Wormhole” Euro .J. Phys. 40(2019) 015206. arXiv:1805.11147 [physics.class-ph].[7] Dassios, G.,
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