EXPLICIT SOLUTIONS OF THE WAVE EQUATION ON THREE DIMENSIONAL SPACE-TIMES: TWO EXAMPLES WITH DIRICHLET BOUNDARY CONDITIONS ON A DISK ABSTRACT

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1 EXPLICIT SOLUTIONS OF THE WAVE EQUATION ON THREE DIMENSIONAL SPACE-TIMES: TWO EXAMPLES WITH DIRICHLET BOUNDARY CONDITIONS ON A DISK DANIIL BOYKIS, PATRICK MOYLAN Physics Department, The Pennsylvania State University, Abington College, Abington, PA 19001, USA ABSTRACT We give the general solutions to the wave equation with certain Dirichlet boundary conditions on two different three dimensional space times. The results of this paper should be useful for the calculation of the Casmir energy for a circular boundary in the plane, and to problems dealing with cylindrical wave guides. 1. Introduction Solutions of the wave equation with appropriate circular or cylindrical boundary conditions have many practical applications in engineering and physics. The paradim of such texbook problems is that of describing vibrations of a circular membrane (the shape of a drum) requiring solutions of the wave equation in a dimensional Minkowski space which vanish on the circular boundary of the membrane, and thus leading to zeros of Bessel functions.[4] Similarly, cylindrical wave guides in three space provide another important example. A theoretical application of much current interest requiring such solutions is the computation of Casimir energies for cylindrical and spherical boundary conditions. We show here that it is just as easy to set up such problems in a certain curved space of which the flat Minkowski space is a limiting case. The curved space which we consider in section 3 is a Lorentzian variant of the three sphere. In general relativity it is just the Einstein universe, only in one lower spatial dimension, since here we deal with dimensions instead of the (or 4) dimensional space-time of physical reality. The relation between Minkowski space and the Einstein universe is analogous to that of stereographic projection of 1

2 2 the three sphere onto the tangent plane of one of its points. Owing to the indefiniteness of the scalar product in Minkowski space the formulae of projection, eqns. (3.4), are more complicated than the usual formulae of stereographic projection, since it is a projection onto Minkowski space instead of onto Euclidean space. It is clear that as the radius R (c.f. section 3) of the curved space goes to infinity, solutions of the wave equation in the curved space must (at least locally) go over into corresponding solutions of the wave equation in flat space. Our main reasons for looking at such problems in curved space is: 1) the greater regularity and convergence properties afforded by working with fields over a compact space[1]; and 2) contrary to what one might expect, some problems formulated in the curved space may have considerably simpler solutions than those in flat space. Indeed, the second author has shown this to be the case for the calculation of the Casimir energy at least for the lowest dimensional case[2]. 2. Wave Equation for a Massless Scalar Field on a three dimensional Minkowski space Let M 0 denote three dimensional Minkowski space, which is IR 3 as a vector space, but whose metrical structure is that of a Lorentzian manifold with infinitesmal arc length in an inertial coordinate system given by dx 2 = dx 2 0 dx 2 1 dx 2 2 (2.1) We will be concerned with solution of the conformally invariant wave equation, which in Cartesian coordinates on M 0 is written as ( 1 c 2 2 x 02 2 x 12 2 x 22 ) ψ(x) = 0, (2.2) where ψ(x) (= ψ(x 0, x 1, x 2 )) is a suitably differentiable function on M 0 and c is the speed of light. Another set of coordinates on M 0 is (t, r, φ) with where x 0 = t, x 1 = r cosφ, x 2 = r sinφ. (2.3) < t <, 0 r π, 0 φ < 2π. (2.4) The wave equation (2.2) in these coordinates becomes ψ 1 c 2 2 ψ t 2 = 0. (2.5)

3 3 where = 2 r r r r 2 φ 2 (2.6) is the familiar Laplacian on IR 2. Solutions of equation (2.6) which are regular at the origin are: ψ ω, m (t, r, φ) = J m (kr) e imφ e i ω t (2.7) 0 < r < m = 0, ± 1, ± 2, ± 3... with ω = ± k c IR. A standard method of obtaining solutions (2.7) of eqn. (2.5) is that of separation of variables[3]. For completeness we present in the Appendix a detailed derivation of eqn. (2.7) using this method. Dirichlet boundary conditions on the circle of radius a, which we consider here, means that for each Bessel function of order n we require J n (k a) = 0 (2.8) This means k a is a root of this equation. Since the positive roots of eqn. (2.7) are countable in number we index them as k a = k n,j a (j = 1, 2, 3,....). (For each µ IR we have a countable number of real roots of the equation J µ (k a) = 0 [4]). It is easily verified that a general solution of eqn. (2.5) can be written as ψ(t, r, φ) = cos ( c t ) f(r, φ)+ ( c ) 1 sin ( c t ) g(r, φ), (2.9) where and f(r, φ) = ψ(0, r, φ) (2.10) g(r, φ) = ψ t (t, r, φ) t=0 (2.11) are the initial conditions which describe the field ψ(t, r, φ) at time t = 0. In fact, it is proved in ref. [4] (Vol. 2, p. 160) that eqn. (2.9) is the most general solution of eqn. (2.5). Since we have imposed the Dirichlet boundary conditions we must insist that f(a, φ) = 0 and g(a, φ) = 0. (2.12)

4 4 Due to the completeness of the basic solutions, eqns. (2.7), a we can expand f and g in terms of these basic solutions as: f(r, φ) = a m,j J m (k m,j r) e imφ, (2.13) and g(r, φ) = m,j=0 m,j=0 b m,j ω m,j J m (k m,j r) e imφ, (2.14) where k m,j a is a positive root of eqn. (2.7). Through the inversion formulae of these equations the coefficients a m,j and b m,j may explicitly be given in terms of the initial functions f(r, φ) and g(r, φ) and the J m (k m,j r) e imφ [8]. Very often for a particular problem it is only necessary to know the series, eqns. (2.13) ) and (2.14) to some finite number terms. Thus, in such cases, we are interested only in first N positive zeros of Bessel functions J n (x) of integer order n. Computational routines to to find these zeros using Maple are easy. We can do the job in Maple 9.5 with the following two simple lines of code: b g:=(i) fsolve(besselj(0,x)=0,x, (i-1)*2*pi..i*2*pi); seq(g(i),i=1..10); This routine has the advantage that it can easily be adapted to the Gegennbauer case in the next section. 3. Wave Equation for a Massless Scalar Field on a Lorentzian variant of the three sphere Next consider the space-time M, a three dimensional version of a compactification of the Einstein universe, namely: } M = {(u 0, u 1, u 2, u 3, u 4 ) u u2 4 = u2 1 + u2 2 + u2 3 = R2. (3.1) Clearly, M = S 1 S 2. We may introduce spherical coordinates on M as follows: a By completeness we mean completeness in a certain Hilbert space H of solutions of the zero mass wave equation, eqn. (2.5), c.f. [5]. This remark also makes the meaning of precise: it is a fact that has a self-adjoint closure in H[6], and thus we take to be the unique square root of the self-adjoint operator [7]. b Many such addtional routines can be found on e.g. Maple User Group. We give the relevant link to the site here: / We note that some of the routines given there, which are written for Maple 4 or 5, do not work for Maple 9.5 without modification.

5 5 u 4 = R cos τ, u 0 = R sin τ, u 1 = R sin θ cos φ, u 2 = R sin θ sin φ, u 3 = R cos θ (3.2) with ranges of the angular parameters being 0 < τ < 2 π, 0 < θ < π, 0 < φ < 2 π. An SO(2) SO(3) invariant metric on M is dσ 2 = R 2 (dτ 2 dρ 2 ) (3.3) where dτ and dρ denote infinitesmal arc length on the unit sphere S 1 in IR 2 and S 2 in IR 3, respectively. SO(2, 3) acts as the group of conformal transformations on M[9], and there is a conformal imbedding π 1 of the Minkowski space M 0 into M, given by π 1 (x) = u, with the inverse mapping given by a Lorentzian variant of stereographic projection. Specifically we have where i = 0, 1, 2 and u 1 = R p F, u i = R p x i, u 3 = R p D (3.4) p = {F 2 + x2 0 R 2 } 1 2, F = 1 x 2 4 R 2, D = 1 + x2 4 R 2 (3.5) where x 2 = x 2 0 x 2 1 x 2 2. Now π 1 is conformal, since Note that equations (3.2) and (3.4) give[9] x 0 = 2 R sin τ (cos τ + cos θ), x 1 dσ 2 = p 2 dx 2. (3.6) = 2 R sin θ cos φ (cos τ + cos θ), x 2 The conformally invariant wave equation on M is [10] = 2 R sin θ sin φ (cos τ + cos θ). (3.7) where S 2 2 τ 2 ψ S 2 ψ = 1 4 R 2 ψ, (3.8) is the Laplacian on S 2 which in spherical coordinates is S 2 = 1 ( sin θ θ sin θ θ ). (3.9) sin θ φ 2 Solutions of the wave equation are (compare the dimensional case in [11]): ψ l,±m (τ, θ, φ) = e ie τ sin m (θ) C m+ 1 ( ) 2 l+m 1 cos θ e ±i m φ (3.10)

6 6 with l = 0, 2,, 3,..., 0 m l, m Z +. The C m+ 1 ( ) 2 l+m 1 cos θ are Gegenbauer polynomials[12]. Substituting these solutions into the wave equation gives the following spectral equation: E l(l + 1) = R2 4 R 2. (3.11) Now for reasons made clear in the introduction we wish to consider the analog of eqn. (2.7) for the case at hand, i.e. Dirichlet boundary conditions on the region whose boundary is the great circle on S 2 which is contained in a plane parallel to the x 1 x 2 plane and is specified by the equation θ = π N. This means C m+ 1 ( π ) 2 l+m 1 cos( N ) = 0, (3.12) which leads us to the question of zeros of Gegenbauer polynomials. (If we let q = e 2πiθ then we seen the condition θ = π N means q2n = 1, i.e. q 1 2 is an Nth root of unity. We can easily adapt the Maple code of section 2 to write a program in Maple 9.5 for determining the zeros of Gegenbauer polynomials: h:=(i,n,a,b) fsolve(gegenbauerc(a,b,cos(x))=0,x,(i-1)*2*pi/n.. i*2*pi/n); S:=(N,a,b) seq(h(i,n,a,b),i=1..n); We can get all of the roots of a particular Ca b (cos x) by choosing N sufficiently large. If we take the case of C5 3 (cos θ) for example, we find that N = 12 is sufficient, and computing S(12, 5, 3) we get the following roots in the interval [0, 2π] (after cleaning up the output slightly): , , , , , , , Actually what we have just give is not quite what we need for our problem. However, we can use the just described routine to easily find out which Gegenbauer polynomials Ca b (cos x) have zeros at a particular value of x. Finally we easily write down in analogy to eqn (2.9) the general solution to eqn (3.8):[13], [14] ( ) ψ(τ, θ, φ) = cos 2 s + 1 4R 2 τ f(θ, φ) +

7 7 + ( 2 s + 1 4R 2 ) 1 sin ( ) 2 s + 1 4R 2 τ g(θ, φ) (3.13) where and f(θ, φ) = ψ(0, θ, φ) (3.14) g(θ, φ) = ψ t (t, θ, φ) t=0 (3.15) are the initial conditions which describe the field ψ(t, θ, φ) at time t = 0. The Dirichlet boundary conditions imply that f( π N, φ) = 0 and g( π, φ) = 0. (3.16) N As in the previous section we can expand f and g in terms of the basic solutions in eqn. (3.10) as: f(r, φ) = c l,m sin m (θ) C m+1/2 l m+1/2 (θ) eimφ, (3.17) and g(r, φ) = l, m l, m d l,m E l sin m (θ) C m+1/2 l m+1/2 (θ) eimφ, (3.18) where is the set of all admissible l, m which satisfy eqn. (3.12). As in the previous case the coefficients a m,j and b m,j may explicitly be given in terms of the initial functions f(r, φ) and g(r, φ) and the C m+1/2 l m+1/2 (θ) eimφ by the inversion formulae, which can be obtained by adapting methods in [14] to the case at hand. Appendix In this appendix we construct solutions of the wave equation in polar coordinates: 2 ψ t 2 c2 [ 2 ψ r 2 + ψ r ] ψ r 2 θ 2 = 0. (A.1) In this Appendix (only) we use, as is customary, θ to denote the angle of rotation in the plane. In the above we used φ for the azimuthal angle

8 8 instead of θ so as not to confuse it with the polar angle on S 2 in section 3. According to separation of variables we write:[3] ψ(r, θ, t) = R(r)Θ(θ)T (t). Substituting this into (A.1) we obtain: (A.2) R(r)Θ(θ)T (t) = c [R 2 (r)θ(θ)t (t) + 1 r R (r)θ(θ)t (t) + 1 ] r 2 R(r)Θ (θ)t (t) Dividing both sides of this equation by T (t) we get T [ (t) R T (t) = (r) c2 R(r) + 1 R (r) r R(r) + 1 Θ ] (θ) r 2 Θ(θ) Now in order for eqn. (A.4) to be always true we must have. (A.3) (A.4) T (t) T (t) = ω2, (A.5) with ω real. The most general complex solution of this equation having unit magnitude is: T (t) = e iω t = cos(wt) + i sin(ω t) Looking at the other side of eqn. (A.4) we get [ R ω 2 = c 2 (r) R(r) + 1 R (r) r R(r) + 1 Θ ] (θ) r 2 Θ(θ) or ω 2 r 2 c 2 We can rewrite this equation as r 2 R (r) + rr (r) R(r) = r2 R (r) + rr (r) R(r) + ω2 r 2 c 2 + Θ (θ) Θ(θ) = Θ (θ) Θ(θ), (A.6). (A.7) = λ c 2. (A.8) Just as for eqn. (A.4) we see that in order for this equation to be always true we must insist that r 2 R (r) + rr (r) R(r) + ω2 r 2 c 2 = Θ (θ) Θ(θ) = λ c 2 = n2. (A.9)

9 9 Since θ ranges over a compact set i.e. θ [0, 2π], it follows that n must be an integer. We thus have Θ (θ) + λ c 2 Θ(θ) = 0, (A.10) The solution of this differential equation is the same as eqn. (A.6) i.e. From the other side of eqn. (A.9) we obtain which we rewrite as r 2 R (r) + rr (r) R(r) Θ(θ) = e inθ. (A.11) = (n 2 ω2 r 2 c 2 ), (A.12) r 2 R (r) + rr (r) (n 2 ω2 r 2 )R(r) = 0. (A.13) c2 Now let us show that this can be transformed into Bessel s differential equation which is [4] x 2 y + xy + (x 2 n 2 )y = 0. (A.14) We let and dx dr = w c. Also d dr = dx dr and we have x = ωr c d dx = w c dr dr = w dr c dx ; r 2 ω 2 c 2 or, since x = rw c, we have x 2 d2 R(r) dx 2 d 2 R dx 2 + rω c Now we define R(x) = R(r) to get x 2 = ω2 r 2 d dx. Thus d 2 R dr 2 c 2 (A.15) = w2 c 2 d 2 R dx 2, (A.16) dr dx + (x2 n 2 )R(r) = 0 (A.17) + x dr(r) dx + (x2 n 2 )R(r) = 0. (A.18) x 2 R (x) + x R (x) + (x 2 n 2 ) R(x) = 0, (A.19) which by comparing with eqn. (A.14) we obtain R(x) = J n (x) or R(r) = J n ( ω r c ) where J n(x) is Bessel s function of integer order n.

10 10 Summarizing, we have shown that ψ(r, θ, t) = R(x)Θ(θ)T (t) = = R(x)e inθ e iωt = J n (x)e inθ e iωt = J n ( rω c )einθ e iωt (A.20) where n is any positive integer [9] and ω is any real number which, in order to satisfy the boundary condition that ψ(a, θ, t) = 0, is determined by J n ( a ω c ) = 0 i.e. a ω c is a root of J n (x) = 0. References 1. I.E. Segal, Z. Zhou, Ann. Phys. 218, (1992). 2. P. Moylan, The Casimir Effect in the Einstein Universe, in preparation. 3. Willard Miller, Jr., Symmetry and Separation of Variables, Academic Press (1975). 4. Richard Courant, David Hilbert, Methods of Mathematical Physics, Vol. 1, Wiley-Interscience (1989). 5. Jakobsen, Vergne, Jour. Funct. Anal. 47 (1982), Ji ř í Blank, Pavel Exner, Miloslav Havlíček, Hilbert Space Operators in Quantum Physics, AIP Press, New York (1994). 7. ibid, p N.Y. Vilenkin, Special Functions and the Theory of Group Representations, American Mathematical Society; Revised edition (June 1, 1983). 9. P. Moylan, Harmonic Analysis on Spannors, Jour. Math. Phys., 36 (6), (1995). 10. P. Moylan, The SO(4,4) Minimal Representation and the Rac Representation of SO(2,3), in Quantum Theory and Symmetries, eds. H.D. Doebner et. al., World Scientific, (2000), pp I.E. Segal, S.M. Paneitz, Jour. Funct. Anal. 47 (1982), I.C. Gradshteyn, I.M. Rhyshik, Tables of Integrals, Sums, Series and Derivatives, Academic Press, (2000). 13. Michael E. Taylor, Noncommutative Harmonic Analysis, American Mathematical Society, (1980), p Robert S. Strichartz, Jour. Funct. Anal. 12, 4, (1973),