Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds

a r X i v :h e p -t h /9806171v 1 19 J u n 1998UCB-PTH-98/35,LBNL-41948,NSF-ITP-98-070hep-th/9806171Glueballs and Their Kaluza-Klein CousinsHirosi Ooguri,Harlan Robins and Jonathan Tannenhauser Department of Physics,University of California at Berkeley,Berkeley,CA 94720Theoretical Physics Group,Mail Stop 50A-5101,Lawrence Berkeley National Laboratory,Berkeley,CA 94720Institute for Theoretical Physics,University of California,Santa Barbara,CA 93106Abstract Spectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been computed using the conjectured duality between superstring theory and large N gauge theory.The Kaluza-Klein states of supergravity do not correspond to any states in the Yang-Mills theory and therefore should decouple in the continuum limit.On the otherhand,in the supergravity limit g 2Y M N →∞,we find that the masses of theKaluza-Klein states are comparable to those of the glueballs.We also showthat the leading (g 2Y M N )−1corrections do not make these states heavier than the glueballs.Therefore,the decoupling of the Kaluza-Klein states is not evident to this order.1IntroductionSpectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been calculated[1]using the conjectured duality between string theory and large N gauge theory[2–5].The results are apparently in good numerical agreement with available lattice gauge theory data,although a direct comparison may be somewhat subtle,since the supergravity computation is expected to be valid for large ultraviolet couplingλ=g2Y M N,whereas we expect that QCD in the continuum limit is realized forλ→0[5,6].As explained in[6,1],the supergravity computation atλ≫1 gives the glueball masses in units of thefixed ultraviolet cutoffΛUV.Forfiniteλ,the glueball mass M is expected to be a function of the formM2=F(λ)Λ2UV.(1.1)In the continuum limitΛUV→∞,M should remainfinite and of orderΛQCD.This would require F(λ)→0asλ→0.In[1],the leading string theory corrections to the masses were computed and shown to be negative and of orderλ−3/2,in accordance with expectation.Witten has proposed[5]that three-dimensional pure QCD is dual to type IIB string theory on the product of an AdS5black hole and S5.This proposal requires that certain states in string theory decouple in the continuum limitλ→0.One class of such states are Kaluza-Klein excitations on S5.The supergravityfields on the AdS5black hole×S5 can be classified by decomposing them into spherical harmonics(the Kaluza-Klein modes) on S5[7,8].They fall into irreducible representations of the isometry group SO(6)of S5, which is the R-symmetry of the four-dimensional N=4supersymmetric gauge theory from which QCD3is obtained by compactification on a circle.Consequently,only SO(6) singlet states should correspond to physical states in QCD3in the continuum limit.These are the glueball states studied in[1].However,wefind that,in the supergravity limit, masses of the SO(6)non-singlet states are of the same order as the SO(6)singlet states. Since these states should decouple in the limitλ→0,it was speculated in[1]that the string theory corrections should make the non-singlet states heavier than the singlet states.The purpose of this paper is to test this idea.We compute the masses of the SO(6)non-singlet states coming from the Kaluza-Klein excitations of the dilaton in ten dimensions. Wefind the masses in the supergravity limit to be of the same order as those of the SO(6) singlet states.We then calculate the leading string theory corrections to the masses.We find that the leading corrections do not make the Kaluza-Klein states heavier than theglueballs.Therefore,the decoupling of the Kaluza-Klein states is not evident to this order.This suggests that the quantitative agreement between the glueball masses from supergravity and the lattice gauge theory data should be taken with a grain of salt.2The Supergravity LimitWe calculate the masses of the Kaluza-Klein states following the analysis of[1].Ac-cording to[5],QCD3is dual to type IIB superstring theory on the AdS5black hole×S5 geometry given bydx24πg2Y M N =dρ2ρ2 +ρ2−b4dρ (ρ4−1)ρd f0this shooting method can be used to compute k 20and the wavefunction f 0(ρ)to arbitrarilyhigh precision.The results of the numerical work are listed in Table 1.As expected,the masses are all of the order of the ultraviolet cutoffΛUV =b .l135711.5929.2654.9388.6034.5363.60100.6145.668.98109.5157.9214.3l 2sρ2−b 4ρ2 dτ2+ρ23 i =1dx 2i +d Ω25,(3.1)where δ1=+15γ 5b 4ρ8−19b 12ρ4+5b 8ρ12 ,(3.2)and γ=18γ b 42ρ8+b 1216πG 10 d 10x √2g µν∂µΦ∂νΦ+γe −3periodicity2πR ofτis also modified toR= 1−152b.(3.5) It is the inverse radius R−1that serves as the ultraviolet cutoffof QCD3.To solve the dilaton wave equation in theα′-corrected geometry(3.1),we writeΦ=Φ0+f(ρ)e ikx Y l(Ω5),(3.6) whereΦ0is the dilaton background given by(3.3),and expand f(ρ)and k2inγasf(ρ)=f0(ρ)+γh(ρ),k2=k20+γδk2.(3.7) Here f0(ρ)obeys the lowest order equation(2.3)and is a numerically given function,and k20is likewise determined from(2.3).The second-order differential equation obtained from the action(3.4)in the background metric(3.1)and dilatonfield(3.3)is,in units in which b=1,ρ−1ddρ −(k20+l(l+4)ρ2)h==(75−240ρ−8+165ρ−12)d2f0dρ+(δk2−120(k20+l(l+4)ρ2)ρ−12−405ρ−14)f0(ρ).(3.8) With f0(ρ)and k20given,one may regard this as an inhomogeneous version of the equation (2.3).We solve this equation for h(ρ)andδk2.We are now ready to present our results.Let us denote the lowest mass of the l-th Kaluza-Klein state by M l.In units of the ultraviolet cutoffΛUV=(2R)−1,with R given by(3.5),wefindM20=11.59×(1−2.78ζ(3)α′3+···)Λ2UVM21=19.43×(1−2.66ζ(3)α′3+···)Λ2UVM22=29.26×(1−2.62ζ(3)α′3+···)Λ2UVM23=41.10×(1−2.61ζ(3)α′3+···)Λ2UVM24=54.93×(1−2.63ζ(3)α′3+···)Λ2UVM25=70.76×(1−2.66ζ(3)α′3+···)Λ2UVM26=88.60×(1−2.69ζ(3)α′3+···)Λ2UVM27=108.4×(1−2.72ζ(3)α′3+···)Λ2UV.(3.9)Similar behavior is observed for the excited levels of each Kaluza-Klein state.Thus the corrections do not make the Kaluza-Klein states heavier than the glueballs, and the decoupling of the Kaluza-Klein states is not evident to this order.According to Maldacena’s duality,theλ−1/2expansion of the gauge theory corresponds to theα′-expansion of the two-dimensional sigma model with the AdS5black hole×S5as its target space.It is possible that the decoupling of the Kaluza-Klein states takes place only non-perturbatively in the sigma model.AcknowledgmentsWe thank Csaba Cs´a ki,Aki Hashimoto,Yaron Oz,John Terning,and especially David Gross for useful discussions.We thank the Institute for Theoretical Physics at Santa Barbara for its hospitality.This work was supported in part by the NSF grant PHY-95-14797and the DOE grant DE-AC03-76SF00098,and in part by the NSF grant PHY-94-07194through ITP.H.R. and J.T.gratefully acknowledge the support of the A.Carl Helmholz Fellowship in the Department of Physics at the University of California,Berkeley.Appendix:The Boundary Condition at the HorizonIn this appendix,we show that the boundary condition at the horizonρ=b used in the shooting method[1]is consistent,and that the eigenvalue k2and the wavefunction f(ρ)can be evaluated to an arbitrarily high precision using this method.In the neighborhood ofρ=b,the dilaton wave equation takes the form∂ρ(ρ−b)∂ρf(ρ)+···=0.(3.10) Its general solution is of the formf(ρ)=c1[1+α(ρ−b)+···]+c2[log(ρ−b)+···](3.11)with arbitrary coefficients c1,2(the constantαis determined by the wave equation and is in general non-zero).The regularity of the dilatonfield requires c2=0.In the shooting method,we numerically integrate the differential equation starting from a sufficiently large value ofρdown to the horizon.For generic k2,the function thus obtained,when expanded as in(3.11),would have c2=0.The task is to adjust k2so that c2=0.Since f(ρ)is divergent atρ=b for generic k2,it is numerically difficult to impose the boundary condition directly atρ=b.Instead,in[1]and in this paper,we required f′=0 atρ=b+ǫfor a given smallǫ(for example,ǫ=0.0000001b in this paper).By(3.11), this condition impliesc2=−c1αǫ+···.(3.12)Therefore,c2can be made arbitrarily small by adjustingǫ.This justifies the numerical method used in[1]and in this paper.We thank Aki Hashimoto for discussions on the numerical method.References[1]C.Cs´a ki,H.Ooguri,Y.Oz and J.Terning,“Glueball Mass Spectrum from Supergravity,”hep-th/9806021.[2]J.M.Maldacena,“The Large N Limit of Superconformal Field Theories and Supergravity,”hep-th/9711200.[3]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,“Gauge Theory Correlators from Non-Critical String Theory,”hep-th/9802109.[4]E.Witten,“Anti-de Sitter Space and Holography,”hep-th/9802150.[5]E.Witten,“Anti-de Sitter Space,Thermal Phase Transition,And Confinement in GaugeTheories,”hep-th/9803131.[6]D.J.Gross and H.Ooguri,“Aspects of Large N Gauge Theory Dynamics as Seen by StringTheory,”hep-th/9805129.[7]H.J.Kim,L.J.Romans and P.Van Nieuwenhuizen,“Mass Spectrum of Chiral Ten-Dimensional N=2Supergravity on S5,”Phys.Rev.D32(1985)389.[8]M.G¨u naydin and N.Marcus,“The Spectrum of The S5Compactification of Chiral N=2,D=10Supergravity and The Unitary Supermultiplets of U(2,2/4),”Class.Quant.Grav.2(1985)L11.[9]S.S.Gubser,I.R.Klebanov and A.A.Tseytlin,“Coupling Constant Dependence in theThermodynamics of N=4Supersymmetric Yang-Mills Theory,”hep-th/9805156.[10]M.T.Grisaru,A.E.M.van de Ven and D.Zanon,“Four-Loop Beta Function for theN=1and N=2Supersymmetric Nonlinear Sigma-Model in Two Dimensions,”Phys.Lett.B173(1986)423;M.T.Grisaru and D.Zanon,“Sigma-Model Superstring Corrections to the Einstein-Hilbert Action,”Phys.Lett.B177(1986)347.[11] D.J.Gross and E.Witten,“Superstring Modifications of Einstein Equation,”Nucl.Phys.B277(1986)1.。

a rXiv:solv-int/9593v312Se p1995Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz Craig A.Tracy Department of Mathematics and Institute of Theoretical Dynamics University of California,Davis,CA 95616,USA E-mail address:tracy@ Harold Widom Department of Mathematics University of California,Santa Cruz,CA 95064,USA E-mail address:widom@ Abstract We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel e −(u (θ)+u (θ′))2.I.Introduction Thermodynamic Bethe Ansatz techniques were introduced in the pioneering analysis of Yang and Yang [11]of the thermodyamics of a nonrelativistic,one-dimensional Bose gas with delta function ter this method was extended to a relativistic system with a factorizable S -matrix to give an exact expression for the ground state energy of this systemon a cylindrical space of circumference R [5,12].This was done by relating the ground state energy to the free energy of the same system on an infinite line at temperature T =1/R .In all cases one expresses the various quantities of interest in terms of “excitation energies”εa (θ)which are solutions of nonlinear integral equations of the formεa (θ)=u a (θ)− b φab (θ−θ′)log1+z b e −εb (θ′) dθ′matter.The methods used are either numerical or perturbative and there are,as far as the authors are aware,no known explicit solutions to the TBA equations.It thus came as a surprisewhen Cecottiet al.[2](see also [3]),in their analysis of certain N =2supersymmetric theories [1],discovered that a certain quantity (the “supersymmetric index”),expressible in terms of the solution of the pair of “TBA-like”integral equationsε(θ)=t cosh θ−1cosh(θ−θ′)dθ′η(θ)=2λ ∞−∞e −ε(θ′)2(cosh θ+cosh θ′)2.Zamolodchikov [13]then conjectured that the system of nonlinear equations could actually be solved in terms of this resolvent kernel.More precisely,if we denote the operator by K ,the kernel of the operator K (I −λ2K 2)−1by R +(θ,θ′),and set R +(θ):=R +(θ,θ)then the system should be satisfied ife −ε(θ)=R +(θ)and ηis defined by the second equation.In fact he conjectured that this should hold for operators with kernels of the more general forme −(u (θ)+u (θ′))2(1.1)if the first equation is replaced byε(θ)=2u (θ)−1cosh(θ−θ′)dθ′.In addition he conjectured that,with the same function η,R −(θ)=1cosh 2(θ−θ′)dθ′,where R −(θ,θ′)is the kernel of K 2(I −λ2K 2)−1and R −(θ):=R −(θ,θ).(We state everything in terms of kernels here;in the cited work the functions R ±were given by infinite series.2That they are the same follows from the Neumann series representation for the kernel of (I−λ2K2)−1.)We prove these conjectures here.One of the main ingredients is the fact that the equa-tions are in a sense equivalent to relations among the analytic continuations of the functions R±(θ)into a strip.(See formula(6.8)of[13].)Another is a particularly convenient represen-tations for these functions in terms of other functions,which we call Q(θ)and P(θ).(That these latter functions are fundamental is known from earlier work[4,8,9,10].)These rep-resentations are stated in Lemma1,and in Lemmas2and3we state precise versions of the equivalence alluded to before.In Lemmas4and5we derive general properties of functions in the range of the operator K in order to derive,as Lemma6,some basic properties of the functions Q(θ)and P(θ).If we try to prove the desired relation among the analytic continuations of R±(θ)wefind that we have to prove a certain crucial identity involving Q(θ)and P(θ)which is by no means obvious.But once conjectured it is not hard to prove,given the previous preparatory work, and that is stated as the Proposition which follows the lemmas.We show that a certain combination of these functions,which is clearly analytic in the strip,extends by periodicity to an entire bining this fact with the use of Liouville’s theorem,we deduce the identity.It should be mentioned that the main part of the argument may only be used if u belongs to a restricted class,but the result for general u follows by an approximation argument.This will be presented in the Appendix,as will proofs of some of the lemmas and some facts about Fourier transforms we shall use.II.PreliminariesWe shall assume throughout that u is continuous and bounded from below and that0<λ<e2min u/2π.(2.1) This assures that the series defining R±(θ)converge uniformly and that the operatorλK, acting on any of the usual function spaces,has norm less than1,so that I−λ2K2is invertible. (This follows from(4.5)below.)Since the parameterλmay be incorporated into u we may assume that in factλ=1.If we set√E(θ):=.(2.3)eθ+eθ′Our functions Q and P are defined byQ:=(I−K2)−1E,P:=(I−K2)−1KE.3Lemma1.We have the representationsR+(θ)=Q(θ)2−P(θ)2eθ.(2.4)Proof.We use the notations[A,B]:=AB−BA,{A,B}:=AB+BA and write X⊗Y for the operator with kernel X(θ)Y(θ′)and M for multiplication by eθ.Then we have immediately{M,K}=E⊗Efrom which it follows that also that[M,K2]=E⊗KE−KE⊗Eand then that[M,(I−K2)−1]=Q⊗P−P⊗Q.SinceK2(I−K2)−1=(I−K2)−1−I,(2.5) we deduce that the kernel of this operator is given by the formulaR−(θ,θ′)=Q(θ)P(θ′)−P(θ)Q(θ′)eθ+eθ′,and thefirst part of(2.4)follows.4Recall that a function f defined on R is said to belong to the Wiener space W if itsFourier transformˆf belongs to L1.Such a function is necessarily continuous and vanishes at ±∞.A sufficient condition that f∈W is that f and f′belong to L2.(See the Appendix.) We use the notation S a to denote the strip|ℑθ|<a in the complexθ-plane,and A(S a)to denote those functions g which are bounded and analytic in the strip,continuous on its closure,and for which g(θ+iy)→0asθ→+∞through real values when y∈R isfixed and satisfies|y|<a.The proofs of the next three lemmas will be found in the Appendix.Lemma2.Assume f∈W.Ifg(θ)=1cosh(θ−θ′)dθ′(2.6)then g∈A(Sπ/2)and its boundary functions satisfyg(θ+iπ/2)+g(θ−iπ/2)=f(θ)(2.7) for realθ.Conversely,if g∈A(Sπ/2)and if(2.7)holds then so does(2.6).Lemma3.Assume f,f′∈W.Ifg(θ)=1cosh2(θ−θ′)dθ′then g∈A(Sπ/2)and its boundary functions satisfyg(θ+iπ/2)−g(θ−iπ/2)=if′(θ)for realθ.Conversely,if g∈A(Sπ/2)and the second relation holds then so does thefirst. Lemma4.Assume f∈L1and set h(x)equal to f(log x)/x for x>0and equal to0for x≤0.If h∈W andg(θ)= ∞−∞f(θ′)this that if f∈L2then Kf and its derivative are exponentially small at±∞.In particular,any function in the range of K satisfies the hypothesis of the next lemma.Lemma5.If f is a bounded function on R with bounded derivative then Kf(θ)/E(θ) extends to a function in A(Sπ).The boundary functions of Kf satisfyKf(θ+iπ)+Kf(θ−iπ)=4πv(θ)f(θ)for realθ,wherev(θ):=e−(u(θ)+u(θ+iπ)).Proof.If we look at the the expression(2.3)for the kernel of K we see that Kf(θ)/E(θ) is of the form of the function g of Lemma4if f(θ)there is replaced by our E(θ)f(θ).It iseasy to see that if our f satisfies the stated conditions then the function h in the statement of Lemma4belongs to L2and has an L2derivative,so h∈W and the conclusion ofthe Lemma holds.Thus Kf/E∈A(Sπ).For the boundary function identity we use theexpression(1.1)for the kernel of K.If we make the substitutionsθ→2θ,θ′→2θ′we see that e u(2θ)(Kf)(2θ)=2eθ(Kf)(2θ)/E(2θ)is exactly of the form of the function g in thestatement of Lemma2if the function f(θ)there is replaced by our present4πe−u(2θ)f(2θ). Applying the identity stated there and using the periodicity of u give the identity stated here.We apply this to the functions Q(θ)and P(θ).Lemma6.The functionsQ(θ)(2.9)E(θ)belong to A(Sπ)and the boundary functions of Q and P satisfy the identities Q(θ+iπ)+Q(θ−iπ)=4πv(θ)P(θ),P(θ+iπ)+P(θ−iπ)=4πv(θ)Q(θ)(2.10) for realθ.Proof.Because of the relations Q=E+K2(I−K2)−1E,P=KQ all statements of thelemma except for thefirst part of(2.10)follow from Lemma5and the remark preceding it. Since Q−E=KP Lemma5gives the identity(Q−E)(θ+iπ)+(Q−E)(θ−iπ)=4πv(θ)P(θ).But it follows from the definition(2.2)of E and the fact that u has period2πi thatE(θ+iπ)+E(θ−iπ)=0.Thus we obtain the desired identity for Q.Using Lemma6and the fact that v is an entire function we can conclude that Q and P have analytic continuations to entire functions ofθ;each use of the pair of identities allows us6to widen byπthe strip of analyticity.Here is the crucial identity relating these continuations from which our results will follow.Proposition.We haveQ(θ+iπ/2)Q(θ−iπ/2)−P(θ+iπ/2)P(θ−iπ/2)=E(θ+iπ/2)E(θ−iπ/2).Proof.SetS(θ):=Q(θ+iπ/2)Q(θ−iπ/2)−P(θ+iπ/2)P(θ−iπ/2).ThenS(θ+iπ/2)=Q(θ+iπ)Q(θ)−P(θ+iπ)P(θ),S(θ−iπ/2)=Q(θ−iπ)Q(θ)−P(θ−iπ)P(θ),and so by(2.10),S(θ+iπ/2)+S(θ−iπ/2)=4πv(θ)[P(θ)Q(θ)−Q(θ)P(θ)]=0.It follows that S(θ)extends to an entire function of period2πi whose values atθ±iπ/2are negatives of each other.Therefore1−S(θ)/E(θ+iπ/2)E(θ−iπ/2)extends to an entire function of periodπi whose values atθ±iπ/2are equal.(We used again the fact that E(θ−iπ)=−E(θ+iπ).)To show that this is0(this is equivalent to the claimed identity)it suffices,by Liouville’s theorem,to show that it is bounded and that it tends to0asθ→+∞through real values.For aπi-periodic function it suffices to show that these properties hold in the strip Sπ/2.They do hold there because forθin this stripθ±iπ/2lie in the strip Sπ, for which we have the conclusions of Lemma6.III.Proof of the ConjecturesWe have to show that ifε:=−log R+and ifηis defined byη(θ)=2 ∞−∞e−ε(θ′)2π ∞−∞log(1+η2(θ′))πR+(θ) ∞−∞arctanη(θ′)We shall assume that we have a functionηsatisfying(3.1)–(3.3),and formally apply the first parts of Lemmas2and3to obtain the three identities(3.4)–(3.6)below.From these we shall see thatηmust have a certain representation in terms of Q and P.Then,using the identities of the Proposition,we shall show that(3.4)–(3.6)in fact hold ifηis defined this way.Finally,using the second parts of Lemmas2and3,we show that(3.1)–(3.3)hold.(Of course thefirst step is unnecessary for the proof,but it provides motivation for the eventual definition ofη.)Applying Lemmas2and3to(3.1)–(3.3)give4πR+(θ)=η(θ+iπ/2)+η(θ−iπ/2),(3.4) log(1+η2(θ))=2u(θ+iπ/2)−ε(θ+iπ/2)+2u(θ−iπ/2)−ε(θ−iπ/2),(3.5)2i η′(θ)R+(θ+iπ/2)−R−(θ−iπ/2)E(θ+iπ/2)2E(θ−iπ/2)2.(3.8)Lemma1shows that(3.6)may be writteni η′(θ)Q2−P2(θ+iπ/2)−Q′P−P′Q1+η2(θ)=Q′Q−P′PQ2−P2(θ−iπ/2)−E′E(θ−iπ/2).We have a choice now of either adding or subtracting the last two displayed formulas. Choosing the former,we obtainη′(θ)Q−P (θ+iπ/2)+Q′+P′E(θ+iπ/2)−E′η(θ)−i =ddθlog(Q+P)(θ−iπ/2)−dIntegrating and exponentiating gives the desired formula(Q−P)(θ+iπ/2)·(Q+P)(θ−iπ/2)η(θ)−i=−i.(3.10)E(θ+iπ/2)E(θ−iπ/2)We shall now show that ifηis defined by(3.9)then(3.10)also holds,as do relations (3.4)–(3.6).First,the statement that the right side of(3.10)minus the right side of(3.9)is equal to 2i follows from the Proposition.Thus(3.9)and(3.10)are completely equivalent.Second,taking the product of(3.9)and(3.10)gives(3.8)and hence(3.5).Third,reversing the argument that showed(3.8)and(3.6)imply(3.9)we see that(3.9) and(3.8),which we now have,imply(3.6).Finally,to obtain(3.4)we use(3.10)to expressη(θ+iπ/2)and(3.9)to express η(θ−iπ/2)andfind that their sum equals(Q−P)(θ)[(Q+P)(θ+iπ)+(Q+P)(θ−iπ)]i[Q(θ)2−P(θ)2]E(θ)E(θ+iπ)and by(2.4)and the definitions of E(θ)and v(θ)this equals4πR+(θ).So(3.4)–(3.6)are established.Now we show that they imply(3.1)–(3.3).By Lemmas2 and3this will be true if the functionsR+,log(1+η2),arctanη,η′/(1+η2)(3.11) belong to W and the functionsη,2u−ε,R−/R+(3.12) belong to A(Sπ/2).Our assumption has been thatλ=1satisfies inequality(2.1).This identity is still satisfied if u is increased or,equivalently,if E is decreased.It follows that we could,in our representation(2.3)of the kernel of K,have replaced E byδE for anyδ∈[0,1].It is clear that all quantites in(3.1)–(3.3)would then be real-analytic functions ofδforδ∈[0,1].So9if the relations hold for sufficiently smallδthey hold for all,includingδ=1.If we retrace the steps leading to bounds for the functions(2.9)wefind that they tend to0asδ→0. (In fact,they are O(δ2).)Hence we may assume that the bounds for these functions are as small as we like.In fact we assume that|Q(θ)4and|P(θ)4(θ∈Sπ).(3.13) We take in succession the items we have to verify.Recall that a function belongs to W if it and its derivative belong to L2.This is what we shall show for the functions(3.11).First, from(2.2)and(2.4)we see haveR+(θ)=Q(θ)2−P(θ)2E (θ+iπ/2)−Kfe2θ+e2θ′dθ′=O(sechθ)if,say,f∈L∞.This holds for|ℑθ|strictly less thanπ/4.Applying this to f=Q and f=P we deduce thatPE (θ−iπ/2)=O(sechθ),QE(θ−iπ/2)=O(sechθ)as long as|ℑθ|<π/8,for example.Now adding(3.9)and(3.10)gives the representation η(θ)=iQ(θ−iπ/2)P(θ+iπ/2)−Q(θ+iπ/2)P(θ−iπ/2)E(θ+iπ/2)+O(sechθ) P E(θ+iπ/2) PNow we show that the functions(3.11)all belong to A(Sπ/2).First,by(2.4),R−Q2−P2=2Q′P−P′Q1−(P/Q)2.By(3.13)P/Q is bounded in the larger strip Sπby1/3and so it follows that the last factor above is bounded.It also follows that P/Q∈A(Sπ)from which it follows that (P/Q)′∈A(Sπ/2).Hence R−/R+∈A(Sπ/2).Next,we use(2.4)again to write2u−ε=2u+log Q2−P2E2.By(3.13)we have|1−(Q±P)/E|<1/2in Sπ/2and we deduce as above that2u−ε∈A(Sπ/2).Finally since1−Q/E and P/E belong to A(Sπ),the functions1−Q(θ±iπ/2)E(θ±iπ/2)belong to A(Sπ/2).It follows from this and the representation(3.14)thatη∈A(Sπ/2).Thus we have proved the conjectures in the case where u(θ)has the special form(2.8). For a general u,more precisely for any u which is continuous and bounded below,we canfind a sequence of u n of the special type such that e−u n converge boundedly and locally uniformly to e−u.(This will be demonstrated in the Appendix.)This is enough to deduce the result for u from the results for the u n.IV.AppendixWe give details here of certain matters postponed from the previous sections.First we recall some facts about the Fourier transform,which we denote,as usual,by a circumflex:ˆf(ξ)= ∞−∞e−iξθf(θ)dθ.Ifˆf∈L1,in other words,if f∈W,we have the Fourier inversion formulaf(θ)=12π||ˆf||1.(4.1)Parseval’s identity reads||ˆf||2=12π||f||2,11and we have the general formula f′(ξ)=iξˆf(ξ).Proof that f,f′∈L2implies f∈W.It suffices to show that if we writeˆf(ξ)as [ˆf(ξ)(ξ+i)](ξ+i)−1then both factors on the right belong to L2.The second factor surely does,and the square of the absolute value of thefirst factor equals|ˆf(ξ)|2(ξ2+1)=| f′(ξ)|2+|ˆf(ξ)|2.The right side belongs to L1by Parseval’s identity and the assumption f,f′∈L2.Proof of Lemma2.It is clear that g,when defined by(2.6),is analytic in the strip.If we write g y(θ):=g(θ+iy)for realθtheng y(ξ)=ˆf(ξ)e−yξ2sinhπξ/2=ˆf(ξ)(ξ+i)ξe−yξand so the difference of the limiting values of g has Fourier transform−ˆf(ξ)ξ= if′.This establishes thefirst part of the lemma and the second follows just as before.Proof of Lemma4.Forfixedθ′the factor(eθ+eθ′)−1in the integral defining g(θ)is analytic inθ.Forθin any subset of Sπof the formℜθ≥θ0,|ℑθ|≤π−δ(δ>0)this factor is bounded uniformly inθ′and tends to0pointwise asℜθ→+∞.Since f∈L1 this is enough to conclude that g is analytic in Sπand tends to0in this strip asℜθ→+∞andℑθisfixed.It remains to prove boundedness of g and continuity near the boundary of Sπ,and for this we use the function h.In the lower part of the strip,−π<y<0,we set z=−eθ+iy withθreal,so thatℑz>0,and we can writeg(θ+iy)= ∞0h(x)x−z dx.Using the Fourier inversion formula and interchanging the order of integration,wefind the representationg(θ+iy)=i ∞0e izξˆh(ξ)dξ.Sinceˆh∈L1the integral is bounded uniformly forℑz>0and we deduce as in the Lemmas 2and3that it extends continuously toℑz=0.Thus g(θ+iy)is bounded for0≤y<πand extends continuously to y=π.A similar argument holds for the upper half of Sπ,and so g∈A(Sπ).Extension to general u.We begin with an approximation fact,reminiscent of the Weier-strass approximation theorem.Recall the notation C0(R)for the space of continuous func-tions f on R satisfying f(±∞)=0,which is a Banach space under the norm||f||:=sup{|f(x)|:x∈R}.The fact is that for eachδ>0thefinite linear combinations of the functionssinh kθe−δsinh2θ(k=0,1,···)are dense in C0(R);in other words,for any f∈C0(R)and anyδ′>0there exists afinitelinear combinationp(θ)=Nk=0a k sinh kθsuch that|p(θ)e−δsinh2θ−f(θ)|<δ′(4.3)13for allθ.This is true because the change of variable t=sinhθconverts it to the statementthat thefinite linear combinations of the functions t k e−δt2are dense in C0(R).And this in turn is true because if it weren’t then the Hahn-Banach theorem and Riesz representationtheorem([7],Thms.5.19and6.19)would imply that there there is a function of boundedvariation(=signed measure)µon R,not identically zero,such that ∞−∞t k e−δt2dµ(t)=0for all k≥0.But then the entire function F(z):= ∞−∞e izt e−δt2dµ(t)would satisfy F(k)(0)=0 for k≥0and so F≡0.This in turn implies e−δt2dµ(t)≡0and soµ(t)≡0,a contradiction.Here is how to construct the sequance u n described at the end of Section III.We may clearly assume that u is uniformly positive,i.e.that for someα>0we have u(θ)≥αfor allθ.Let n be given and definew:=min(u,n).Thenfind p(θ),a linear combination of the powers of sinhθ,such that(4.3)holds withf(θ)=w(θ)|<n−1e n−1sinh2θ.It is an easy exercise to deduce from this that for sufficiently large n|p(θ)2−u(θ)|<6n−1/2if u(θ)<n and sinh2θ<n.(We use here the facts that u is uniformly positive and that e<3.)The function p(θ)2is our u n(θ).We now deduce the identities(3.1)–(3.3)for general u.Denote by R n±the R±functionsassociated with the functions u n.If we can show that R n±(θ)→R±(θ)boundedly and pointwise then(3.1)–(3.3)for u will follow from the corresponding identities for the u n,since by the dominated convergence theorem we could take the limits as n→∞under the integral signs.The function R+(θ)is given by the series∞λ2m ∞−∞··· ∞−∞K(θ,θ1)···K(θ2m,θ)dθ1···dθ2m(4.4) m=0where K(θ,θ′)is given by(1.1).It follows from the fact∞dθ−∞u n≥0fo all n,and thatλ<1/2π.Denote by K n(θ,θ′)the kernel corresponding to u n so that R n+(θ)is given by the series∞λ2m ∞−∞··· ∞−∞K n(θ,θ1)···K n(θ2m,θ)dθ1···dθ2m.(4.6) m=0It follows from(4.5)and the inequality e−u n≤1that the integral in the m th term of(4.6) is at most(2π)2m for all n and so,sinceλ<1/2π,the series converges uniformly in n. Thus we may take the limit as n→∞under the summation sign.Next,each integrand K n(θ,θ1)···K n(θ2m,θ)is uniformly bounded by K(θ,θ1)···K(θ2m,θ),which hasfinite integral over R2m,and so we may take each limit as n→∞under the integral sign(again by the dominated convergence theorem).The result is the series(4.4),and this giveslimR n+(θ)=R+(θ).n→∞Since0<R n+(θ)≤R+(θ)we have established that R n+(θ)→R+(θ)boundedly and pointwise.A similar argument applies to R n−(θ),and the proof is complete.AcknowledgementsThe authors wish to thank Paul Fendley for elucidating the conjectural status of the identities we prove here.This work was supported in part by the National Science Foundation through grants DMS–9303413and DMS–9424292.15References[1]Cecotti,S.,Vafa,C.:Topological–anti-topological fusion,Nucl.Phys.B367,359–461(1991);Ising model and N=2supersymmetric theories,Commun.Math.Phys.157, 139–178(1993)[2]Cecotti,S.,Fendley,P.,Intriligator,K.,Vafa,C.:A new supersymmetric index,Nucl.Phys.B386,405–452(1992)[3]Fendley,P.,Saleur,H.:N=2supersymmetry,Painlev´e III and exact scaling functionsin2D polymers,Nucl.Phys.B388,609–626(1992)[4]Its,A.R.,Izergin,A.G.,Korepin,V.E.,Slavnov,N.A.:Differential equations forquantum correlation functions,Int.J.Mod.Physics B4,1003–1037(1990)[5]Klassen,T.R.,Melzer,E.:The thermodynamics of purely elastic scattering theories andconformal perturbation theory,Nucl.Phys.B350,635–689(1991)[6]McCoy,B.M.,Tracy,C.A.,Wu,T.T.:Painlev´e functions of the third kind,J.Math.Phys.18,1058–1092(1977)[7]Rudin,W.:Real and Complex Analysis,3rd ed.New York:McGraw-Hill,1987[8]Tracy,C.A.,Widom,H.:Fredholm determinants,differential equations and matrixmun.Math.Phys.163,33–72(1994)[9]Tracy,C.A.,Widom,H.:Systems of partial differential equations for a class of operatordeterminants.Operator Theory:Adv.and Appls.78,381–388(1995)[10]Tracy,C.A.,Widom,H.:Fredholm determinants and the mKdV/sinh-Gordon hierar-chies,to appear in Commun.Math.Phys.,solv-int/9506006[11]Yang,C.N.,Yang,C.P.:Thermodynamics of a one-dimensional system of bosons withrepulsive delta-function interaction,J.Math.Phys.10,1115–1122(1969)[12]Zamolodchikov,Al.B.:Thermodynamic Bethe Ansatz in relativistic models:Scaling3-state Potts and Lee-Yang models,Nucl.Phys.B342,695–720(1990)[13]Zamolodchikov,Al.B.:Painlev´e III and2D polymers.Nucl.Phys.B432[FS],427–456(1994)16。

Integral Equation Formulation for ScatterDensity ProblemJ.H¨a m¨a l¨a inen,S.Savolainen,R.Wichman,K.Ruotsalainen and J.YlitaloAbstractIntegral equation formulation for the problem offinding a circularly symmetric scatter density(SD)around the mobile is deduced,and the SD is computed assuming that the distribution of angle-of-arrival(AoA)in base station is known.The corresponding integral equation is solved by using the spline collocation method.Thus,instead offitting a priori selected SD to estimated AoA distribution,we solve the SD based on AoA characteristics.Introduction:Recently,geometrical-based single bounce channel models have been proposed by different authors [1],[2],[3],where the distribution of the scatterers defines the model through the selected simple geometry.In[1], it was assumed that scatterers are uniformly spread over a disc with radius R and vanishes outside of this region. Similar analysis was carried out in[2]where the scatterers are uniformly distributed inside the ellipse with foci at Base Station(BS)and Mobile Station(MS).In[3],it was shown that angle-of-arrival(AoA)in BS,computed assuming Gaussian scatter density(SD),provides a betterfit to measurements of[4]when compared to model applying uniform distribution of scatterers.In[1],[2,[3],the form of SD isfirst selected,after that the distribution of AoA is computed,andfinally the result is compared to existing measurements.However,in order to maximise thefit between the AoA provided by the model and measurements the direction in the selection process concerning to SD should be reversed.We propose a novel approach where AoA in BS isfirst selected based on measurements and SD is then computed from the integral equation that defines the relation between SD and AoA seen by BS.The resulting SD can be used when building up a channel simulator modelling a certain environment.System Model and Notations:The notation used in the sequel is introduced in Fig.1,where r refers to the distance between MS and scatterer,d is the distance between scatterer and BS,D is the distance between MS and BS,andαandβdefine the departing(arriving)and arriving(departing)angles of the signal path(dashed lines)respectively.We assume that SD in polar coordinates(fα,r)depends only on r.Hence,AoA seen by MS is uniformly distributed. Furthermore,we assume that D≥R,i.e.,there are no local scatterers around the BS.This assumption is valid in macro-cell environments where BS antenna is placed well over the rooftops and angular spread(AS)is small.Due to the symmetry about the line joining MS and BS we can concentrate on the positive values ofβ.Integral Equation Formulation:We compute AoA in BS(fβ)by using the joint density function fβ,r that is obtained from fα,r after a standard change of variables.For that purpose we need the derivative ofβwith respect toα.From Fig.1β=arcsin(r/d·sinα),D+r cosα=d cosβ,(1) and the derivative ofβis computed from thefirst equation in(1),∂αβ=r(d2cosα+rD sin2α)d2d2−r2sin2α=d−D cosβd,(2)where the latter equality is obtained with the help of equations in(1).Furthermore,from Fig.1wefind thatd=D cosβ±r2−D2sin2β.(3)Here the sign changes when r=D sinβ.By(2)and(3)the joint distribution ofβand r attains the formfβ,r(β,r)=D cosβ±r2−D2sin2βr2−D2sin2βf r(r)2π,(4)where f r/2π=fα,r.The pdf of AoA in BS is obtained by integrating(4)over r.By taking into account the change of the sign in(4)we obtainfβ(β)=DπRD|sinβ|cosβf r(r)drr2−D2sin2β,|β|≤sin−1RD.(5)Numerical Solution of the Integral Equation:The numerical solution of(5)is obtained by using spline collocation method that is known to be fast and simple[5].For that purpose we define collocation pointsβn whereβ0=0,βn<βn+1andβN+1=sin−1(R/D).Furthermore,we define the mesh points r n by r n=D sinβn.In spline collocation method,f r in(5)is expressed in spline space,spanned by standard spline base functionsψn.Here we apply piecewise linear splines,i.e.ψn is a normalised hat function centred at r n and vanishing outside the interval (r n−1,r n+1).The coordinates of f r in spline space are denoted by u n.Since SD vanishes outside the circle with radius R,we have f r(r N+1)=0,and thusfβ(βn)=Nk=nu kπRr nD2−r2nr2−r2nψk(r)dr.(6)If fβis known,the coordinates u n can be solved from(6)in a recursive manner and they provide an approximation to values f r(r n).On the other hand,if f r is known,the point values of fβcan be computed directly from(6).We note that with piecewise linear splines,integral in(6)admits a closed-form expression in terms of elementary transcendental functions.The numerical accuracy of the method was studied in the case where the closed-form expression for fβwas deduced from(5)assuming that f r is Rayleigh,and the corresponding spline approximation˜f r was computed from(5)using the collocation method.Results show that the rms error between original f r and approximation˜f r decays proportionally to N−2.Absolute rms error was of the order of4·10−4when N=256and AS22=12◦.Fig.2depicts one-dimensional SDs for two example AoAs:Gaussian AoA[4]and the AoA following Student’s t-distribution[6].In both cases we take R=D=1and truncate distributions such that AoA vanishes if|β|>90◦and AS=12◦.Results show that the form of SD greatly depends on the selected distribution of AoA.The difference between SDs in Fig.2reflects the fact that tails of the Gaussian AoA decay rapidly while Student’s t-distributed AoA has a larger peak atβ=0.Furthermore,although the AS is the same for both AoAs,it is noticed that the expected scatterer distance(¯r)may vary depending on the AoA distribution.Finally,we note that relative scale in Fig.2can be easily replaced by absolute values having a physical meaning,Conclusions:Integral equation formulation was deduced tofind the SD provided that AoA in BS is known.The formulation provides effective means to compute SD numerically when AoA in BS is estimated from measurements. Resulting SD can then be used when building channel simulators for various types of environments.References[1]Petrus,P.,Reed J.H.,and Rappaport,T.S.:‘Geometrical-Based Statistical Macrocell Channel Model for Mobile Environments’,IEEE Trans.on Comm.,V ol.50,No.3,March2002.[2]Ertel,R.B.,and Reed,J.H.:‘Angle and Time Arrival Statistics for Circular and Elliptical Scattering Models’, IEEE J.on Selected Areas in Comm.,V ol.17,No.11,November1999.[3]Janaswamy,R.:‘Angle and Time of Arrival Statistics for the Gaussian Scatter Density Model’,IEEE Trans.on Wireless Comm.,V ol.1,No.3,July2002.[4]Pedersen,K.I.,Mogensen,P.M.,and B.H.Fleury,B.H.:‘A Stochastic Model of the Temporal and Azimuthal Dispersion Seen at the Base Station in Outdoor Propagation Environments’,IEEE Trans.Veh.Technol.,V ol.49,March2000.[5]Saranen,J.,and Vainikko,G.:‘Periodic Integral and Pseudodifferential Equations with Numerical Approximation’, Springer Verlag,Berlin,2002.[6]Andersen,J.B.,and Pedersen,K.I.:‘Angle-of-Arrival Statistics for Low Resolution Antennas’,IEEE Trans.on Antennas and Propagation,V ol.50,No.3,March2002.Author’s affiliations:J.H¨a m¨a l¨a inen and R.Wichman(Helsinki University of Technology,P.O.Box3000,FIN–02015HUT,Finland)S.Savolainen(Department of Mathematical Sciences,University of Oulu,FIN–90401Oulu,Finland)K.Ruotsalainen(Mathematics Division,University of Oulu,FIN–90401,Finland)J.Ylitalo(4G Lab/CWC,University of Oulu,P.O.Box4500,FIN–90014,Finland)Corresponding Author:R.Wichman,wichman@wooster.hut.fiFigure captions:Fig.1Problem geometry and notationsFig.2One-dimensional SD as a function of r when R=D=1and AoA follow Gaussian distribution(o)and Student’s t-distribution(x)with AS=12◦.Dashed lines refer to the expected scatterer distances。

Maxwell’s Equations in Medium *Zhang TaoInstitute of Low Energy Nuclear Physics, Beijing Normal University, Beijing Radiation Center, Beijing100875, Chinataozhang@AbstractReactions (magnetization, polarization, induced magnetization) of medium toelectromagnetic wave and propagation of electromagnetic wave in medium wereinvestigated. Faraday’s law in medium was presented, which is correct not only invacuum but also in medium. The Maxwell’s Equations in medium was modified.Keywords: Maxwell’s Equations, Faraday’s Law of induction, medium, electromagneticwave, total electric field, total magnetic induction1. IntroductionInteraction between electromagnetic wave and medium has been a very active research field. Maxwell’s Equations in vacuum have been verified by countless experiments. This theory resulted in the discovery of electromagnetic wave and has greatly changed our understanding of the world. The interaction between medium and electromagnetic wave is a very important subject and about it many problems have yet to be studied deeply. In this paper the behavior of electromagnetic wave in medium is analyzed and the Maxwell’s Equations in medium are reconsidered. For simplicity, the medium in this paper is an infinite, homogeneous and isolating medium, and there is no combining induced current in the medium [1].2. Problem with the Maxwell’s Equations in the mediumAs is well known, the existing theory of the Maxwell’s equations in the medium is0=⋅∇E , (1)t∂∂−=×∇B E , (2) 0=⋅∇B , (3)* Project supported by Beijing Science Technology New Star Program (Grant No. 952870400), the Beijing Municipal Commission of Education, Key Lab of Beam Technology and Material Modification of Ministry of Education in Beijing Normal University, and the Excellent Young Teachers Program of Ministry of Education, P. R. China.tµµt µ∂∂ε+×∇+∂∂=×∇E M P B 0000, (4) where E is electric field intensity, B magnetic induction, P polarization and M magnetization in the medium. Equation (2) is Faraday’s Law of induction and can be expressed as∫∫∫⋅∂∂−=⋅S B l E d d t. (5) Suppose that there is a varying magnetic field B through a toroid of a kind of the medium in vacuum (B is in vacuum). The symmetry axis of the magnetic field coincides with that of the toroid, as shown in Fig.1. Thus the induced electric field caused by ∫∫⋅∂∂−S B d t coincides with the center line l of the toroid, and the absolute value of E is the same one everywhere on the center line l . The induced electric field caused by ∫∫⋅∂∂−S B d t makes the medium (the toroid) polarized. Suppose the polarization is Pon the center line l . P is in the same direction as the induced electric field. Let t∂∂B keep unchanged during a period of time, then ∫∫⋅∂∂−S B d tin the area circled by the center line l keeps unchanged also. So the induced electric field keeps unchanged, and so does P . Therefore polarization current in the medium is 0 during this period of time, i.e., polarization of the medium does not influence the magnetic induction B through the toroid. P is different for different medium, and the macro electric field E and ∫⋅l l E d on the center line l are different for different medium, too. This means that ∫⋅ll E d is not equal to ∫∫⋅∂∂−S B d t . Hence, Eqs.(2) and (5) are not correct in the medium. Equations (2) and (5)are correct only in vacuum.3. Modified Maxwell’s Equations in MediumWhen propagating in the medium, electric field intensity E and magnetic induction B of electromagnetic wave cause polarization P and magnetization M in the medium [2]. We only consider refracted part and do not consider absorbed and scattered parts of electromagnetic wave in the medium. E , P , B and M are in the same phase. As E varies from its amplitude to 0, P varies synchronously. We separate this process into two steps to investigate the effects of E and P , respectively. Step 1: E varies from its amplitude to 0. Step 2: P varies from its amplitude to 0. Step 1 certainly causes a magnetic induction. Step 2 results in a varying electric field that causes another magnetic induction. So both E and P contribute to the magnetic induction of electromagnetic wave. Therefore P correlates an electric field, which should be E P =P /ε0. Since E P has the same phase and contributes to the magnetic induction of electromagnetic wave as E does, E P is part of the electric field of electromagnetic wave as E is. The same relation exists between B and M . The magnetic induction correlated to M is B M =−µ0M . B M has the same phase and contributes to the magnetic induction of electromagnetic wave as B does. B M is part of the magnetic induction of electromagnetic wave as B is. Maxwell’s theory believes that varying magnetic field produces electric field, and varying electric field produces magnetic field. Hence, propagation of electromagnetic wave in the medium should be the process that E +E P and B +B M translate into each other. That E , P , B and M are in the same phase meets the requirement of this process.For easy narrating, we call E T =E +E P =E +P /ε0 the total electric field, and B T =B +B M =B −µ0M the total magnetic induction. So Maxwell’s equations in the medium can be obtained by substituting electric field E and magnetic induction B of Maxwell’s equations in vacuum with total electric field E T and total magnetic induction B T , respectively, i.e.0T /ε=⋅∇ρE , (6)t∂∂−=×∇ΤB E T ， (7) 0T =⋅∇B , (8)tεµ∂∂=×∇T 00T E B . (9) Inserting E T =E +E P =E + P /ε0 and B T =B +B M = B −µ0M into Eqs. (6)-(9) yields00/)(ε=+⋅∇ρεP/E , (10)tε∂−∂−=+×∇)()(00M B P/E µ ， (11) 0)(0=−⋅∇M B µ, (12)tεεµµ∂+∂=−×∇)/()(0000P E M B . (13) Since E , B and M are curl vectors and there is no net charge in the medium, ∇⋅E =∇⋅P =∇⋅B =∇⋅M =0. Thus Eqs.(10)-(13) become0=⋅∇E , (14)00εtt P/M B E ×∇−∂∂+∂∂−=×∇µ, (15) 0=⋅∇B , (16)M P E B ×∇+∂∂+∂∂ε=×∇0000µtµt µ . (17) Except Eq.(15) which is derived from Eq.(11), Eqs. (14), (16) and (17) are consistent with Eqs (1), (3) and (4) of the existing theory, respectively. Equation (11) means that the change rate of B and the change rate of −µ0M in medium jointly induce an electric field, and this electric field becomes E and P /ε0 in the medium. Equations (11) and (15) are correct not only in vacuum, but also in the medium. They should be more universal forms of Faraday’s Law. The integral form of Eq. (11) is∫∫∫⋅∂−∂−=⋅ε+S M B l P/E d )(d )(00tμ (18) Equations (11) and (18) do not contradict existing experiments. For example, in the imagined experiment of Fig.1, since E +P /ε0 on the center line l is equal to the electric field E V in vacuum, i.e. the electric field without the toroid, and B −µ0M is equal to the magnetic induction B V in vacuum ( M =0 in vacuum), Eq. (18) becomes ∫∫∫⋅∂∂−=⋅S B l E d d V V tl . Obviously, this equation is correct. The reasonableness of Eq. (11) can be proved further by the symmetry property of electric field and magnetic field of electromagnetic wave. Equation (4) or (17) of the existing theory can be changed into Eq. (13). Equation (13) expresses the relation between curl of B +B M =B −µ0M and change rate of E +E P =E +P /ε0. According to the symmetry property, Faraday’s Law should be the relationship about curl of E +E P =E +P /ε0 and change rate of B +B M =B −µ0M , i.e., Eq. (11).Maxwell’s equations in the medium, i.e. Eqs. (6)-(9), have the same form as that in vacuum. This indicates that electromagnetic wave has the same propagation speed in the medium as in vacuum when there is no combining induced current. The refractive index (speed ratio) of electromagnetic wave is not directly related to polarization P and magnetization M of the medium.4. Conclusions Faraday’s law of induction of the Maxwell’s equations of the existing theory, i.e. ∫∫∫⋅∂∂−=⋅S B l E d d t, is correct only in vacuum. It is not correct in medium. When there is no combining induced current, the Faraday’s law of induction in medium should be expressed with total electric field and total magnetic induction, i.e. ∫∫∫⋅∂−∂−=⋅ε+S M B l P/E d )(d )(00tμ. Propagation speed of electromagnetic wave is not directly related to polarization P and magnetization M of medium, or relative dielectric constant and relative permeability.References[1] Zhang, T. (2004) Effect of magnetic field of light on refractive index, Chinese Physics 13 1358[2] Cai, S. S., Zhu, Y . and Xu, J. J. (2002) Electrodynamics , Beijing, Higher Education Press.注：本文中文（介质中的麦克斯韦方程组）已在中国科技论文在线发表，编号200603-519。

a r X i v :g r -q c /0607020v 2 7 D e c 2006Time (in)dependence in general relativityS.Deser ∗Department of Physics,Brandeis University,Waltham,Massachusetts 02454andLauritsen Laboratory,California Institute of Technology,Pasadena,California 91125J.Franklin †Department of Physics,Reed College,Portland,Oregon 97202Abstract We clarify the conditions for Birkhoff’s theorem,that is,time-independence in general relativity.We work primarily at the linearized level where guidance from electrodynamics is particularly useful.As a bonus,we also derive the equivalence principle.The basic time-independent solutions due to Schwarzschild and Kerr provide concrete illustrations of the theorem.Only familiarity with Maxwell’s equations and tensor analysis is required.I.INTRODUCTIONA major obstacle to teaching general relativity is the initially confusing mathematics underlying useful,physical simplifications.We focus in this paper on the conditions that lead to the simplest regime,time-independence.Because general relativity is coordinate-invariant,what does it mean to speak of a particular coordinate’s independence?The answer is illuminating.Loosely,we expect that there exists a choice of coordinate frame in which the gravitationalfield does not depend on t.But is this a meaningful,that is,invariant criterion?The answer is yes:it means that the spacetime geometry allows the existence of a Killing vectorfield fµ(x)that obeys the tensor equationDνfµ+Dµfν≡∂νfµ+∂µfν−gσρ(∂νgµρ+∂µgνρ−∂ρgµν)fσ=0,(1)where gµνis the metric and Dµis the covariant derivative with respect to it,as defined in Eq.(1).We use the signature(−+++)and units such that c=1.If fµis also timelike (f2<0),then the solution in the frame where fµ=g0µ(more manifestly,the contravariant form fµof the vector is fµ=δµ0)implies that∂0gµν=0,(2)and there is no time dependence.(A special property of time-independent geometries is that in(and only in)them,matter systems such as particles retain a conserved energy,just as in flat space.)Our main point is that we have re-expressed the issue of when a given geometry is time-independent,that is,when there exists a frame where Eq.(2)holds,as a covariant (coordinate-independent)criterion:the existence of solutions to Eq.(1).All this transcrip-tion makes no reference tofield equations.There exist many frames where t-dependence is present,but that is not the point.It is not true false that every geometry has a static frame –the Killing equation is a strong requirement.II.MAXWELLWe begin with electrodynamics whosefield equations outside sources,unlike general relativity,can be written entirely in terms of gauge invariantfield strengths,∇·E=0(3a)∇·B=0(3b)˙E=−∇×B(3c)˙B=∇×E.(3d)The˙E equation’s longitudinal part(see the following)implies that˙E L=0,which exhibits the fact that the“Coulomb”part of E is always time-independent,whatever the behavior of the interior charges.The remaining,dynamical transverse part E T and its partner B (transverse by definition)cannot depend on time if they vanish identically,which is the case for spherically symmetric configurations:any E(r)is necessarily of the form∇S(r)and is purely longitudinal.There is no monopole radiation;it is also the only guaranteed static case,as dipole and higher configurations define transverse vectors.Equation(3)does not therefore require time-dependence,or electro/magneto-statics would not exist.For future use we recall that the transverse/longitudinal division of any vectorfield V is a decomposition of unity,V i= (δij−ˆk iˆk j)+ˆk iˆk j V j,(4) along some arbitrary unit vector directionˆk.Its more familiar Fourier transform isV=V T+V L,(5)where∇·V T=∇×V L=0.Our discussion has been couched in terms of the gauge invariantfield strengths E and B, whose time(in-)dependence is unaffected by the choice of gauge.The underlying potentials (A0,A)are another story:even if(E,B)are static,there exist gauge choices for which the potentials do depend on t by adding gauge terms∂µΛ(r,t)that do not affect Fµν=∂µAν−∂νAµ.In any case the transverse vector potentials are unaffected,being gauge invariant.Only(A0,A L)can be altered,keeping E L unchanged.It is instructive to analyze the equations in terms of the Aµin parallel with the general relativity discussion in Sec.III where potentials are unavoidable.III.GENERAL RELATIVITYFor our purposes the gravitationalfield is a glorified tensor version of the vector Maxwell field Aµ,and we expect similar properties of the results there to apply.At the linearized level,the Einstein equations outside sources are2Gµν≡ hµν− ∂µ∂αhαν+∂ν∂αhαµ +∂µ∂νh−ηµν h−∂α∂βhαβ =0(6) for thefield hµνwith h≡hαα;all indices are moved by the Minkowski metricηµν.As for Maxwell’s equations,we decompose Eq.(6)into space and time components,with the simplifying notation h0i≡N i and h00≡N.The theory is invariant under linearized gauge/coordinate transformations hµν→hµν+∂µξν+∂νξµ,that is,Gµν(∂µξν+∂νξµ)=0, an invariance that is useful to exploit.The component form(the linearized version of a decomposition used long ago to analyze the full theory1)of Eq.(6)is2G00=∇2˜h−∂i∂j h ij(7a)2G0i=∇2N i−∂j˙h ji−∂i∂j N j+∂i˙˜h(7b)2G ij= h ij+∂i˙N j+∂j˙N i−(∂i∂k h kj+∂j∂k h ki)+ δij∇2−∂i∂j (N−˜h)+¨˜hδij+δij ∂m∂n h mn−2∂k˙N k ,(7c) with˜h≡h i i the trace of the spatial part of thefield.This slightly complicated set of equations simplifies when we decompose the spatial tensors h ij and the vectors N i,the latter into transverse/longitudinal parts via Eq.(4),the former by the following partition of unity:h ij=h T T ij+h T ij+∂i h j+∂j h i,(8a)∂i h T T ij=∂i h T ij=0=h T T ii(8b)1h T ij=be set to zero,leaving the gauge invariant set(h T T ij,h T,N T i,N)once we use the available gauge invariance.Now Eq.(7)reduces to2G00=∇2h T=0(9a)2G0i=∇2N T i+∂i˙h T=0(9b)2G ij= h T T ij+ ∂i˙N T j+∂j˙N T i + δij∇2−∂i∂j N−1(δij+∇−2∂i∂j)¨h T=0.(9c)2The time-independence of h T follows from the longitudinal part of Eq.(9b),and the relation N=1Unlike Maxwell,there is another category offields lacking a TT part,namely those with dipole character.As we saw there,dipoles permit a transverse vector,but their single direction is not generic enough to construct a TT tensor.Axial symmetry does permit TT, for example via the tensor harmonic P2(cosθ).To summarize at this point,both Maxwell and linearized general relativity gaugefields only allow time-dependence of their true dynamical excitations,and only when those modes can be present,which always excludes spherical symmetry and also dipole symmetry for the general relativity case.IV.KERR AND SCHW ARZSCHILDIt is instructive,at the linearized level,to relate the exterior solution properties to explicit matter sources.In electrodynamics the current consists of two parts:the charge densityρand the longitudinal current j L,which obey the continuity equation˙ρ+∇·j L=0,and the transverse current j T.The(ρ,j L)subset couples only to the longitudinal electricfield, which is equivalent to it,and as we saw,is time-independent away from sources.The transverse electric and magneticfields are generated by the transverse current and can be time dependent if j T is.Similar reasoning applies to general relativity:the source here is the tensor Tµν,whose(T00,T L0i)components are like(ρ,j L).They obey the same continuity equation and excite only the metric component h T,which is also t-independent outside of source distributions.Because general relativity is a tensor theory,there is another“charge”associated with momentum like T00was with energy,namely(T0i,T L ij),which also obeys continuity and is coupled to N T i.The remaining source part,T T T ij,which may,but need not, depend on time,excites the dynamical h T T ijfields.An important example of time-independence is furnished by the Kerr solution3,4of full general relativity,which we will reproduce in the following.In our linearized context,the static metric is generated by a time-independent spinning point mass withT00=mδ3(r),T0i=amǫijk s j∂kδ3(r),(11)where s j denotes the(constant)unit spin vector.As explained in Ref.5the space integral of T00is the total mass m,and that of T0i vanishes because there is no momentum.Itsfirst moment,the angular momentum J,is given by J=am s.The notation choice that expresses J∼am is historical,but has the virtue that m=0is actually justflat space(also in fullgeneral relativity)and the parameter a reduces to that defining ellipsoidal coordinates in ordinary euclidean 3-space.The opposite limit,a =0,defines the spherically symmetric static Schwarzschild solution.We will not discuss in detail the full general relativity extensions of our linear results.Consider,without deriving it (there is no simple way to do so)the full Kerr intervalds 2=−g tt dt 2+g rr dr 2+g θθdθ2+g φφdφ2+2g tφdtdφ.(12)There are five functions of (r,θ)which are (in units of c =1=16πG ),g tt =−(1−2Mr/ρ2)(13a)g rr =ρ2/∆(13b)g θθ=ρ2(13c)g φφ=sin 2θ r 2+a 2)+2a 2Mr sin 2θ/ρ2 (13d)g tφ=−2aMr sin 2θ/ρ2,(13e)with ρ2≡r 2+a 2cos 2θand ∆≡a 2−2Mr +r 2.The linearized limit of Eqs.(12)and (13),or equivalently its asymptotic form,is a superposition of the (linearized)Schwarzschild solution and a spin term h 0φcorresponding to the source (11)h 00=2mr(14b)h ij =2m r 2.(14c)We emphasize that the time-independence here is derivable directly from the exterior equa-tions,apart from details of the interior source,as we would expect for a spinning spherical ball of charge in E&M,its natural analogue.6V.CONCLUSIONSBy working primarily in the linearized limit,we have provided,using the Maxwell tem-plate,a framework for understanding the basis of time-independence in general relativity in terms of the underlying physics and source geometry.Our main conclusion is that thetime-dependence of solutions of gauge theories such as Maxwell’s or general relativity is a property of their radiation modes.If these are forbidden due to spherical(dipole)symmetry, then time-independence is guaranteed.In particular,the Kerr and Schwarzschild solutions illustrate the absence of dipole and monopole excitations.Although the full general relativ-ity is unavoidably more complicated(and involves global issues we have bypassed here),our results capture at least its long distance properties.AcknowledgmentsWe are grateful to Prof.J.Hartle for stimulating criticism that led to this(we hope) improved version of our earlier paper.This work was supported by NSF grant PHY-04-01667.∗Electronic address:deser@†Electronic address:jfrankli@1R.Arnowitt,S.Deser,and C.W.Misner,“The dynamics of general relativity,”in Gravitation: An Introduction to Current Research,edited by L.Witten(John Wiley&Sons,New York, 1962).Reprinted as gr-qc/0405109.2For the history and a modern derivation,see S.Deser and J.Franklin,“Schwarzschild and Birkhoffa la Weyl,”Am.J.Phys.73(3),261–264(2005),gr-qc/0408067.3Roy P.Kerr,“Gravitationalfield of a spinning mass as an example of algebraically special metrics,”Phys.Rev.Lett.11(5),237–238(1963).4Robert H.Boyer and Richard W.Lindquist,“Maximal analytic extension of the Kerr metric,”J.Math.Phys.8(2),265–281(1967).5Charles W.Misner,Kip S.Thorne,and John Archibald Wheeler,Gravitation(W.H.Freeman, New York,1973).6J.Franklin and P.T.Baker,“Linearized Kerr and spinning massive bodies:An electrodynamics analogy,”Am.J.Phys.,to appear.。