Massless particles in three-dimensional Lorentzian warped products

Massless particles in three-dimensional Lorentzian warped products Jose L. Cabrerizo, Manuel Fernandez, and Miguel Ortega Citation: Journal of Mathematical Physics 48, 012901 (2007); doi: 10.1063/1.2409522 View online: http://dx.doi.org/10.1063/1.2409522 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/48/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Classical gauge theory in Riem J. Math. Phys. 52, 082501 (2011); 10.1063/1.3603990 Engineering warp drives AIP Conf. Proc. 504, 1079 (2000); 10.1063/1.1290909 The relativistic theory of gravity and Mach’s principle Phys. Part. Nucl. 29, 1 (1998); 10.1134/1.953058 Three-dimensional Lorentzian manifolds with constant principal Ricci curvatures ρ1=ρ2 ≠ρ3 J. Math. Phys. 38, 1000 (1997); 10.1063/1.531880 Gravitational energy of conical defects J. Math. Phys. 38, 458 (1997); 10.1063/1.531827 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 Massless particles in three-dimensional Lorentzian warped products Jose L. Cabrerizoa兲and Manuel Fernandezb兲 Departamento de Geometría y Topología, Facultad de Matematicas, Universidad de Sevilla, 41012-Sevilla, Spain Miguel Ortegac兲 Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain 共Received 24 July 2006; accepted 16 November 2006; published online 18 January 2007兲 The model of a massless relativistic particle with curvature-dependent Lagrangian is well known in 共d+1兲-dimensional Minkowski space. For other gravitational fields less rigid than those with constant 共zero兲curvature only a few results are known. In this paper, we give a geometric approach in order to solve the field equations associated with that Lagrangian in the setting of an interesting threedimensional background, namely, a three-dimensional warped product with Lorentzian fibers. When some rigidity conditions are imposed to the fiber 共constant Gauss curvature兲, the trajectories can be totally described. Several examples help us clarify this. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2409522兴 I. INTRODUCTION A natural variational problem can be defined on a space of suitable curves ⌫in a semi- Riemannian manifold 共M ¯ ,g ¯ 兲, associated with the Lagrangian L:⌫→R,L共 ␥ ¯ 兲= 冕 ␬ ¯ ds,共1兲 where ␬ ¯ is the Frenet first curvature of ␥ ¯ , that measures the total curvature of these curves in 共M ¯ ,g ¯ 兲. Concerning this, we found the following in the literature. 1. In the Euclidean plane R2, the total curvature action of any closed curve is not arbitrary, but an integer multiple of 2 ␲ ,L共 ␥ ¯ 兲=2 ␲ i共 ␥ ¯ 兲, where the integer number i共 ␥ ¯ 兲is the rotation number of ␥ ¯ . Moreover, a classical result due to Whitney and Grauenstein ensures that the rotation number characterizes the regular homotopy class of closed curves. Hence, the functional Lis constant on any regular homotopy class of closed curves. This fact also holds true for curves connecting to fixed points of the Euclidean plane and with the same direction in these boundary points. 2. Another classical result, due to Fenchel,9states that the total curvature action of any closed curve in the Euclidean space satisfies the inequality L共 ␥ ¯ 兲艌2 ␲ , and the equality holds if, and only if, ␥ ¯ is a convex plane curve. In particular, every minimum of the total curvature functional, on closed curves, in the Euclidean three space is a plane curve, and so, a trivial one. This holds not only for minima but also for critical points. In fact, the critical points of L:⌫→R共where ⌫is the space of closed curves, or curves connecting two points with given a兲Author to whom correspondence should be addressed; electronic mail: [email protected] b兲Electronic mail: [email protected] c兲Electronic mail: [email protected] JOURNAL OF MATHEMATICAL PHYSICS 48, 012901 共2007兲 48, 012901-10022-2488/2007/48共1兲/012901/12/$23.00 © 2007 American Institute of Physics Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 tangent vectors at these two points兲in the Euclidean n-space are just the plane curves and so they are again trivial.1 From a physical point of view, Lagrangians are a very powerful tool to introduce mathematical models. The following nonexhaustive, but illustrative, list displays some of the uses of Lagrangians in physics. 1. Models of n-dimensional relativistic objects with rigidity attract considerable attention. Some of the most remarkable examples are point particles 共n=0兲,15,17 strings 共n=1兲,8,18 membranes 共n=2兲,16 etc. The action functionals of these models depend on the external curvature of the 共n+1兲-dimensional world surfaces swept by the n-dimensional objects in the process of evolution. The starting point was the model of a relativistic string with rigidity proposed by Polyakov.18 In a d-dimensional Lorentz space 共Pd,g兲we have the bosonic conformal string theory associated to the combined Nambu-Goto-Polyakov 共NGP兲action,3 S=W+␭ 冕 dA= 冕 共储H储2+R+␭兲dA, where Hand Rare the mean and the extrinsic curvature of the surface, respectively, and W=兰共储H储2+R兲dAis the Willmore action on surfaces. This Willmore action is invariant under conformal changes of the metric gin Pd,25 and it was generalized to m-dimensional submanifolds.7The critical points of Ware called the Willmore surfaces in 共Pd,g兲. When 共Pd,g兲is flat, the action Sis given by S=兰共储H储2+␭兲, and it coincides with the classical NGP action.11 2. A 共2+1兲-dimensional space-time offers new possibilities which are not present in any higher-dimensional case. In fact, thanks to the Abelian nature of the spatial rotation group SO共2兲, and the topology of many-particle configuration space, the spin of relativistic particles can be an arbitrary real number, and a generalized statistics, intermediate between Bose and Fermi statistics, is also possible.26 The considerable interest to the field models of such particles, called anyons, is due to their application to different planar physical phenomena: the fractional quantum Hall effect, high-Tcsuperconductivity, and description of the physical processes in the presence of cosmic strings.27 In particular, 共2+1兲-dimensional models of relativistic particles described by the action Adepending linearly on the world-trajectory curvature ␬ ¯ and torsion ␶ ¯ have been studied,28 A=− 冕 共m+␭ ␬ ¯ + ␮␶ ¯ 兲ds, where mis a parameter with the dimension of mass, whereas ␭and ␮ are dimensionless parameters. This model contains only spin internal degrees of freedom, its classical and quantum spectra contain tachyonic states, and depending on the relation between the parameters ␭and ␮ , may also have massive and massless states. 3. Plyushchay21 proved the particularity of the case of the linear dependence on the first curvature, A1=− 冕 共m+␭ ␬ ¯ 兲ds.共2兲 In fact, only spin internal degrees of freedom are present on the system, whereas any other dependence also implies the existence of radial-like degrees of freedom. The model of a particle described by the action A1with m=0,4,22,29 which is essentially given by L,20 corresponds to a massless particle moving classically at a super-relativistic velocity. Nevertheless, this model has been shown not to contradict relativity.22 4. On the other hand, in 共2+1兲dimensions, the relativistic particle with torsion is described by the action 012901-2 Cabrerizo, Fernandez, and Ortega J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 A2=− 冕 共m+ ␮␶ ¯ 兲ds, and the spectrum of the model contains massive, massless, and tachyonic states.23 The interest is the appearance of the action of this model under consideration of the problem describing high-Tcsuperconductivity,19 which allows us to use it as the basis for constructing models for the relativistic anyon 共in the massive sector the spin of the quantum states can take fractional values兲. 5. The action depending quadratically on the curvature of the particle path was considered not only in several papers,2,16,24 but also from a geometric point of view concerning the problem of the elastic rod.5,6,12,13 Awarped product M ¯ =B⫻fFof the semi-Riemannian manifolds 共B,gB兲共the base兲and 共F,gF兲 共the fiber兲with f:B→Ra given positive smooth function 共the warping function兲is an interesting geometric tool to easily model some spaces. The warped metric is defined as gM=gB+f2gF.14 For example, a surface of revolution M ¯ in R3can be obtained by rotation around the OZ axis of an arclength parametrized curve in the OXZ plane, ␣ 共t兲=共x共t兲,z共t兲兲,x共t兲⬎0,t苸I=共−␧,␧兲.If ␪ is the rotation angle, then the induced metric on M ¯ is ds2=dt2+x2共t兲d ␪ 2. But dt2共d ␪ 2兲is the standard metric on I共the unit circle S1兲, and hence M ¯ can be viewed as the warped product M ¯ =I⫻x共t兲S1, with f共t兲=x共t兲as the warping function. The standard space-time models of the universe are warped products 共the Robertson-Walker space-time is M ¯ =I⫻fS, where Sis a three-dimensional connected manifold of constant curvature兲, as are the simplest models of neighborhoods of stars and black holes. Inspired by all this information, we have considered the problem of finding critical points of the total curvature Lagrangian Lon M ¯ =I⫻fM, that is, trajectories of massless particles in the setting of three-dimensional warped products with Lorentzian fiber M, for curves contained in the fibers. The content of this paper is as follows. In Sec. II, we obtain the expression of the Euler- Lagrange equation corresponding to the action Lfor clamped 共or closed兲spacelike curves ␥ ¯ in any three-dimensional semi-Riemannian manifold, obtaining the motion equation 共5兲. Then, in Sec. III, we adapt this equation to our case, bearing in mind that we are handling a manifold which is a warped product, which means that we can use some very well-known formulas that can be found in Ref. 14. In this way, the Euler-Lagrange equation can be reduced to a pair of ordinary differential equations whose unknown is the curvature function ␬ of the original curve ␥ in M. Section IV is devoted to study the case when the Gaussian curvature of the given Lorentz surface is a constant along the original curve, since this condition allows us to completely solve the motion equations 共14兲and 共15兲. In particular, we remark the important case when the given Lorentz surface 共M,g兲has constant Gaussian curvature G, because we can completely describe all critical points of the action L. In Sec. V, general methods to construct explicit examples are explained, following some ideas that can be found in Ref. 10. Finally, in Sec. VI, we consider the situation if timelike curves are considered instead. II. THE EULER-LAGRANGE EQUATION The problem of finding the critical points of the Lagrangian Lcan be considered in a more general setting, namely, by considering a family of Frenet curves in a three-dimensional semi- Riemannian manifold. Let 共M ¯ ,g ¯ 兲be a three-dimensional semi-Riemannian manifold. Let ␥ ¯ be an arclength parametrized spacelike curve in M ¯ admitting a Frenet system 兵T ¯ = ␥ ¯ ⬘,N ¯ ,B ¯ 其with curvature and torsion functions ␬ ¯ and ␶ ¯ , respectively 共the timelike case is considered in the last section兲. We have the Frenet equations ⵜ ¯ T ¯ T ¯ =␧2 ␬ ¯ N ¯ , 012901-3 Particles in 3D Lorentzian warped products J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 ⵜ ¯ T ¯ N ¯ =−␧1 ␬ ¯ T ¯ +␧3 ␶ ¯ B ¯ , ⵜ ¯ T ¯ B ¯ =−␧2 ␶ ¯ N ¯ ,共3兲 where ⵜ ¯ is the Levi-Cività connection of g ¯ ,␧1=g ¯ 共T ¯ ,T ¯ 兲=1, ␧2=g ¯ 共N ¯ ,N ¯ 兲, and ␧3=g ¯ 共B ¯ ,B ¯ 兲,␧i =±1, i=1,2,3. Given a suitable space of curves ⌫, say, clamped or closed, we are interested in studying the critical points of the total curvature Lagrangian, L:⌫→R, ␥ ¯ 哫L共 ␥ ¯ 兲= 冕 ␥ ¯ ␬ ¯ .共4兲 For this end, we take a tangent vector at ␥ ¯ 苸⌫, which is nothing but a vector field Walong the curve ␥ ¯ . Also, it defines in M ¯ a variation of ␥ ¯ in ⌫. Next, if ⌽=⌽共t,r兲:关a,b兴⫻共−␧,␧兲→M ¯ is the corresponding variation of ␥ ¯ ,⌽共t,0兲= ␥ ¯ 共t兲, with variational field W=W共t兲=共 ⳵ ⌽/ ⳵ r兲共t,0兲, the critical points of the variational problem are those curves ␥ ¯ 苸⌫such that ␦ L共 ␥ ¯ 兲关W兴=0, ∀W苸T ␥ ¯ ⌫. Some standard arguments, involving integration by parts, let us compute the first derivative of L, ␦ L共 ␥ ¯ 兲关W兴= 冕 ␥ ¯ g ¯ 共⍀共 ␥ ¯ 兲,W兲ds+关B共 ␥ ¯ ,W兲兴a b, where ⍀共 ␥ ¯ 兲and B共 ␥ ¯ ,W兲denote the Euler-Lagrange and the boundary operators, respectively, which, in this case, are given by ⍀共 ␥ ¯ 兲=ⵜ ¯ T ¯ 2N ¯ +␧1 ␬ ¯ ⬘T ¯ +␧1 ␬ ¯ ⵜ ¯ T ¯ T+R ¯ 共N ¯ ,T ¯ 兲T ¯ , B共 ␥ ¯ ,W兲=g ¯ 共ⵜ ¯ T ¯ W,N ¯ 兲−g ¯ 共W,ⵜ ¯ T ¯ N ¯ 兲−␧1 ␬ ¯ g ¯ 共W,T ¯ 兲, where R ¯ stands for the curvature operator of g ¯ and ␬ ¯ ⬘is the derivative of the function ␬ ¯ with respect to the arclength parameter. Now, if we set V= ⳵ ⌽/ ⳵ t=vT ¯ , we have ⵜ ¯ T ¯ W= ⳵ r共log v兲T ¯ +ⵜWT ¯ . This formula shows that if the curves in variation ⌽are in ⌫, then W共a兲=W共b兲=0, ⵜ ¯ WT ¯ 共a兲=ⵜ ¯ WT ¯ 共b兲=0, and consequently 关B共 ␥ ¯ ,W兲兴a b=0. By using Eq. 共3兲, the Euler-Lagrange operator becomes ⍀共 ␥ ¯ 兲=−␧2␧3 ␶ ¯ 2N ¯ +␧3 ␶ ¯ ⬘B ¯ +R ¯ 共N ¯ ,T ¯ 兲T ¯ , where ␶ ¯ ⬘is the derivative of ␶ ¯ with respect to the arclength. Finally a curve ␥ ¯ in ⌫is a critical point of Lif, and only if, it satisfies the Euler-Lagrange equation R ¯ 共N ¯ ,T ¯ 兲T ¯ =␧2␧3 ␶ ¯ 2N ¯ −␧3 ␶ ¯ ⬘B ¯ .共5兲 III. SOLVING THE EULER-LAGRANGE EQUATION Now we adapt the general case of Sec. II to a warped product M ¯ =I⫻fM, where Iis an interval endowed with the metric ␧dt2共␧is1or−1兲,共M,g兲is an oriented Lorentzian surface 共the fiber兲, and f:I→Ra positive smooth function. We set the warping metric g ¯ =␧dt2+f2g. Thus, 共M ¯ ,g ¯ 兲is a three-dimensional manifold of index 1 or 2, according to ␧be 1 or −1, respectively. For a given spacelike unit speed curve ␥ 共s兲in the surface M, let us denote by 兵T,N其its Frenet frame and by 012901-4 Cabrerizo, Fernandez, and Ortega J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 ␬ its geodesic curvature. A positive orthonormal Frenet frame along ␥ t共s兲=共t, ␥ 共s兲兲 in 共M ¯ ,g ¯ 兲is 兵T ¯ = ␥ t⬘=共1/f兲T, ␰ =共1/f兲N, ⳵ t其. Note that ␧1=g ¯ 共T ¯ ,T ¯ 兲=1, g ¯ 共 ␰ , ␰ 兲=−1, g ¯ 共 ⳵ t, ⳵ t兲=␧, and it is easy to see that the gradient of fis given by ⵜf=␧f ˙ ⳵ t.共6兲 Next, we compute the Frenet system of ␥ t. Let ⵜ ¯ be the Levi-Cività connection of 共M ¯ ,g ¯ 兲.We have ⵜ ¯ T ¯ T ¯ =␧2 ␬ ¯ N ¯ , ⵜ ¯ T ¯ N ¯ =− ␬ ¯ T ¯ +␧3 ␶ ¯ B ¯ , ⵜ ¯ T ¯ B ¯ =−␧2 ␶ ¯ N ¯ , where ␬ ¯ and ␶ ¯ are the curvature and torsion of ␥ tin 共M ¯ ,g ¯ 兲, respectively. In our case, some well-known formulas in warped spaces allow us to express the geometry of 共M ¯ ,g ¯ 兲in terms of the the warping function fand the geometries of 共I,␧dt2兲and 共M,g兲共Ref. 14,p.206兲. Thus, after some computations, we obtain ⵜ ¯ T ¯ T ¯ =− ␬ f ␰ −␧f ˙ f ⳵ t.共7兲 Note that the vector ⵜ ¯ T ¯ T ¯ is lightlike if, and only if, ␧f ˙2共t兲− ␬ 2共s兲=0. We will always assume that the curve ␥ tis a spacelike Frenet curve, and hence we have ␧f ˙2共t兲− ␬ 2共s兲⫽0 for any t苸Iand any s.共8兲 First, let us suppose that ␥ tis a geodesic. From Eq. 共7兲, we have ␬ =0 and f ˙共t兲=0, that is, ␥ is a geodesic and the slice Mt=兵t其⫻Mis critical. In that case, the Euler-Lagrange equation reduces trivially and ␥ tis a critical point of L. Now, we assume that f共t兲=0 for a certain t苸I.If ␥ is a geodesic, we already know that ␥ tis a geodesic on M ¯ . Next, if f ˙共t兲=0 and ␬ ⫽0, simple computations let us obtain ⵜf=0, ⵜ ¯ T ¯ T ¯ =− ␬ f ␰ , ␬ ¯ = ␬ f, N ¯ = ␰ ,B ¯ = ⳵ t,ⵜ ¯ T ¯ ␰ =− ␬ fT ¯ , ␶ ¯ =0, ⵜ ¯ T ¯ ⳵ t=f ˙ fT ¯ =0. In this case, the Euler-Lagrange equation reduces to R ¯ 共N ¯ ,T ¯ 兲T ¯ =0, or equivalently 共Ref. 14,p.210兲, 0=RM共N,T兲T=GN, where RMis the curvature operator of 共M,g兲and Gis the Gauss curvature of M共recall that G=−g共RM共T,N兲N,T兲兲. As a consequence, we have the following. Theorem 3.1: Assume that t苸I is such that f ˙共t兲=0. Let ␥ 共s兲be a unit spacelike curve in M such that either ␥ is a geodesic or G共 ␥ 共s兲兲=0 for any s 共that is, ␥ is a curve of parabolic points兲. Then, ␥ tis a critical point of L. 012901-5 Particles in 3D Lorentzian warped products J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 From now on, we assume f ˙⫽0. After some computations, recalling O’Neill’s book 共Ref. 14, p. 206兲, we obtain N ¯ =−␧2 ␬ ␰ −␧␧2f ˙ ⳵ t 冑␧2共␧f ˙2− ␬ 2兲 ,B ¯ =f ˙ ␰ + ␬ ⳵ t 冑␧2共␧f ˙2− ␬ 2兲 ,共9兲 ␬ ¯ =冑␧2共␧f ˙2− ␬ 2兲 f, ␶ ¯ =f ˙ ␬ ⬘ f共␧f ˙2− ␬ 2兲,␧2␧3=−␧.共10兲 In order to solve the Euler-Lagrange equation 共5兲, we have to distinguish several cases. Case 1: Assume that ␥ is a geodesic in the fiber M, i.e., ␬ =0. According to Eq. 共8兲,f ˙⫽0. If we set ␴ =sign共f ˙兲, Eqs. 共5兲,共7兲,共9兲, and 共10兲become R ¯ 共 ⳵ t,T ¯ 兲T ¯ =0, T ¯ =1 fT,␧2=␧,N ¯ =− ␴⳵ t,B ¯ = ␴␰ , ␬ ¯ =兩f ˙兩 f, ␶ ¯ =0. Once again, the well-known formulas in Ref. 14, p. 210, gives 0=R ¯ 共 ⳵ t,T ¯ 兲T ¯ =␧ff ¨ ⳵ t. Then, we have obtained the following. Theorem 3.2: Let ␥ be a unit spacelike geodesic in M.Let t苸I such that f ˙共t兲⫽0and f ¨共t兲 =0. Then, ␥ tis a nongeodesic critical point of L. Case 2: Now, we suppose that ␬ ⬎0 is a constant. Once again, from Eqs. 共9兲and 共10兲we obtain the following information: N ¯ =− ␬ ␰ −␧f ˙ ⳵ t 冑␧2共␧f ˙2− ␬ 2兲 , ␬ ¯ =冑␧2共␧f ˙2− ␬ 2兲 f, ␶ ¯ =0. Thus, the Euler-Lagrange equation reduces to 0=R ¯ 共N ¯ ,T ¯ 兲T ¯ , and by using the formulas for warped products, 0= 1 冑␧2共␧f ˙2− ␬ 2兲共 ␬ 共␧f ˙2−G共 ␥ 共s兲兲兲 ␰ −f ˙f ¨ ⳵ t兲. Therefore, f ¨共t兲=0 and G共共 ␥ 共s兲兲兲−␧f ˙2共t兲=0. Therefore, we can state the following result. Theorem 3.3: Let us assume that the curvature function ␬ of ␥ is a positive constant.Let t 苸I such that f ˙共t兲⫽0and f ¨共t兲=0. If in addition G共 ␥ 共s兲兲=␧f ˙2共t兲for any s,then ␥ tis a critical point of L. Case 3: Finally, we assume that the curvature ␬ is not a constant and the current slice is not critical, that is, f ˙共t兲⫽0 for t苸I. By using some formulas in O’Neill’s book14, we obtain R ¯ 共N ¯ ,T ¯ 兲T ¯ =␧2 ␬ 共␧f ˙2−Gⴰ ␥ 兲 f2冑␧2共␧f ˙2− ␬ 2兲 ␰ +␧2f ˙f ¨ 冑f␧2共␧f ˙2− ␬ 2兲 ⳵ t,共11兲 where Gⴰ ␥ means the Gauss curvature of Malong ␥ . From Eqs. 共9兲–共11兲we obtain the equations ␧2 ␬ 共␧f ˙2−Gⴰ ␥ 兲 f2=−␧3 ␬␶ ¯ 2−␧3f ˙ ␶ ¯ ⬘,␧2f ˙f ¨ f=−␧␧3f ˙ ␶ ¯ 2−␧3 ␬␶ ¯ ⬘. Since −␧2␧3=␧, we easily obtain the following equations: 012901-6 Cabrerizo, Fernandez, and Ortega J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 ␶ ¯ 2= ␬ 2共f ˙2−␧Gⴰ ␥ 兲−␧ff ˙2f ¨ f2共 ␬ 2−␧f ˙2兲,共12兲 ␶ ¯ ⬘=␧ ␬ f ˙共ff ¨−f ˙2+␧Gⴰ ␥ 兲 f2共 ␬ 2−␧f ˙2兲.共13兲 By recalling the expression of ␶ ¯ , bearing in mind Eq. 共10兲, we have 共 ␬ ⬘兲2= 冉 f ˙2−␧Gⴰ ␥ f ˙2 ␬ 2−␧ff ¨ 冊 共 ␬ 2−␧f ˙2兲.共14兲 Now, we compute the derivative of ␶ ¯ , and by Eqs. 共13兲and 共14兲, we have k⬙=2f ˙2−␧Gⴰ ␥ f ˙2 ␬ 3+ 冉 ␧f ˙2−Gⴰ ␥ f−␧f ¨−2␧ff ¨ 冊 ␬ .共15兲 IV. RIGIDITY CONDITIONS We first assume that the Gaussian curvature of Malong ␥ isaconstant. Then, Eq. 共15兲can be integrated by multiplying by 2 ␬ ⬘, 共k⬘兲2=f ˙2−␧Gⴰ ␥ f ˙2 ␬ 4+ 冉 ␧f ˙2−Gⴰ ␥ f−␧f ¨−2␧ff ¨ 冊 ␬ 2+A,共16兲 where Ais an integration constant. Now, Eq. 共14兲compared with Eq. 共16兲gives A=ff ˙2f ¨,共17兲 ␧Gⴰ ␥ =f ˙2−ff ¨.共18兲 Next, we insert Eq. 共18兲into Eq. 共14兲, obtaining 共k⬘兲2=ff ¨ f ˙2共 ␬ 2−␧f ˙2兲2.共19兲 Since we are assuming that the curvature function ␬ of ␥ is not constant, then f ¨⬎0. Next, we solve Eq. 共19兲by discussing the value of ␧. ␧=1, ␬ 共s兲=f ˙tanh共s冑ff ¨+f ˙c兲,共20兲 ␧=−1, ␬ 共s兲=f ˙tan共s冑ff ¨+f ˙c兲,共21兲 where cis an integration constant. Therefore, we have proved the following. Theorem 4.1: Fix a point t苸I.We assume 1. f ˙共t兲⫽0,f ¨共t兲⬎0; 2. Gⴰ ␥ is a constant function and ␧Gⴰ ␥ =f ˙2共t兲−f共t兲f ¨共t兲; and 3. the curvature of ␥ is given by either Eq. (20) or Eq. (21). Then, the curve ␥ tis a critical point of L. 012901-7 Particles in 3D Lorentzian warped products J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55 Next, we assume that the Gaussian curvature Gof the whole surface Mis constant. There is no loss of generality if we assume G=0,1,−1. We recall that dS2共1兲is the two-dimensional de Sitter space of Gaussian curvature G=1, and AdS2共−1兲is the two-dimensional anti de Sitter space of Gaussian curvature G=−1. We denote by M ␦ the space dS2共1兲or AdS2共−1兲according to ␦ = ±1, respectively. Now, let M ¯ ␦ be the warped product of an open interval and M ␦ . From Theorems 3.1, 3.2, 3.3, and 4.1 we have the following. Corollary 4.1: A unit spacelike curve ␥ t苸⌫in M ¯ ␦ is a critical point of Lif, and only if, one of the following statements holds: 1. f ˙共t兲=0 and ␥ is a geodesic; 2. f ˙共t兲⫽0,f ¨共t兲=0, and ␥ is a geodesic; 3. f ˙2共t兲=␧ ␦ ⬎0,f ¨共t兲=0, and ␬ ⫽0is constant; and 4. f ˙共t兲⫽0,f ¨共t兲⬎0,␧ ␦ =f ˙2共t兲−f共t兲f ¨共t兲, and the curvature function is given by Eq. (20) or Eq. (21), according to ␧=1 or ␧=−1, respectively. Next, let L2be the Lorentz-Minkowski 2-plane. Corollary 4.2: A unit spacelike curve ␥ t苸⌫in M ¯ =I⫻fL2is a critical point of Lif, and only if, one of the following statements holds: 1. f ˙共t兲=0 and ␬ is any smooth function; 2. f ˙共t兲⫽0,f ¨共t兲=0, and ␥ is a geodesic; 3. 0⫽f ˙2共t兲=f共t兲f ¨共t兲,f ¨共t兲⬎0, and the curvature function is given by Eq. (20) or Eq. (21), according to ␧=1 or ␧=−1, respectively. V. EXAMPLES Some of the ideas of Examples 5.1, 5.2, and 5.3 can be found in Ref. 10. Example 5.1: In L3with usual metric ds2=dx2+dy2−dz2, let ␣ :共a,b兲傺R→L3, ␣ 共u兲 =共x共u兲,0,z共u兲兲 be a timelike unit curve with x共u兲⬎0 for any u苸共a,b兲. Let Mbe the revolution surface around the zaxis whose profile curve is ␣ . A local parametrization of Mis given by X:共a,b兲⫻S1→L3,X共u, ␪ 兲=共x共u兲cos ␪ ,x共u兲sin ␪ ,z共u兲兲, for any 共u,ei ␪ 兲苸共a,b兲⫻S1. A unit spacelike normal vector field to Mis N共u, ␪ 兲=共z⬘共u兲cos ␪ ,z⬘共u兲sin ␪ ,x⬘共u兲兲 for any 共u,ei ␪ 兲苸共a,b兲⫻S1. Since ␣ is timelike, we have a Lorentzian surface whose Gaussian curvature is not a constant, in general. In fact, it is easy to obtain the following expression for the Gaussian curvature of M: G共u, ␪ 兲=z⬘共u兲共z⬘共u兲x⬙共u兲−x⬘共u兲z⬙共u兲兲 x共u兲for any 共u,ei ␪ 兲苸共a,b兲⫻S1. Now, we pick a point u苸共a,b兲. The curve ␥ given by ei ␪ 苸S1哫 ␥ 共 ␪ 兲=X共u, ␪ 兲is nothing but the orbit of point ␣ 共u兲by a subgroup of orthocronal isometries of L3. Therefore, both the Gaussian curvature of Malong ␥ and the geodesic curvature ␬ of ␥ are constant functions. Now, we choose f:I→Ra positive smooth function such that there is a point t苸Iand suitable ␧= ±1 satisfying f ˙共t兲⫽0, f ¨共t兲=0, f ˙2共t兲=␧z⬘共u兲共z⬘共u兲x⬙共u兲−x⬘共u兲z⬙共u兲兲 x共u兲. According to Theorem 3.3, the curve ␥ tis a critical point of the action Ldefined on M ¯ =I⫻fM. As a remark, the surface dS2共1兲can be obtained in this way by choosing the curve ␣ :R →L3defined by ␣ 共u兲=共cosh共u兲,0,sinh共u兲兲. 012901-8 Cabrerizo, Fernandez, and Ortega J. Math. Phys. 48, 012901 共2007兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:08:55