From Hopf fibrations to exotic causal replacements
Topological solitons are relevant in several areas of physics Manton (). Recently, these configurations have been investigated in contexts as diverse as hydrodynamics Klech (), Bose-Einstein condensates Hall (), ferromagnetism Coo (), knotted light Irv () and non-abelian gauge theories Fad (). In this paper we address the issue of wave propagation about a static Hopf soliton in the context of the Nicole model. Working within the geometrical optics limit we show that several nontrivial lensing effects emerge due to nonlinear interactions as long as the theory remains hyperbolic. We conclude that similar effects are very likely to occur in effective field theories characterized by a topological invariant such as the Skyrme model of pions.
Start with a rectangular rubber sheet and draw a collection of non-intersecting horizontal lines on it. Make sure you use a thin tip so as to fill the material with as many lines as possible. Glue the top edge of the rectangle to its bottom and give a full twist to the cylinder before you paste its left and right hand sides to form a torus. Now take the toroid structure and look at the original lines. Somehow they became linked! Repeat this recipe with slightly bigger (and also smaller) rectangles and try to assemble the resulting torii one inside the other, very much in the same way as matryoshka nested dolls are constructed. If you are patient enough to perform this continuously, you’ll end up with a fibered structure filling most of which captures the essence of the Hopf fibration Seven ().
In mathematics, the Hopf fibration Hopf () describes the 3-sphere in terms of a disjoint union of circles and an ordinary 2-sphere , with fiber structure
denoting the projection map. In one of its simpler forms, may be defined as follows. Identify with the subset , such that , and with the subset , such that . Then the projection map reads
It turns out that a point , with , will be mapped to the same point of if and only if , thus generating the structure of circular fibers embedded in . The remarkable structure of filled with nested torii made of linked circles described above appears when we compose with stereographic projection of onto . It can be shown projection () that such stereographic projection maps the Hopf fibers to linked circles in , the only exception being the Hopf circle containing the projection point which is mapped to a straight line in , a “circle through infinity”.
This paper deals with wave propagation in a nonlinear field theory –the so called Nicole model Nicole (); Radu ()– determining nontrivial fibrations of of the Hopf type induced by exact finite-energy solutions to the equations of motion. For the Nicole model, the projection map emerges as the critical point of a Lorentz invariant Lagrangian theory of maps from flat space-time into the 2-sphere, or equivalently into 111In order to distinguish between the former Hopf map and its spacetime version considered here, we shall denote the latter by and, following references Adam (); Gil (), call it a Hopf map as well.. The Lagrangian is a simple non-polynomial generalization of the -model Tataru (), and the associated Euler-Lagrange equations are given by a system of second order quasi-linear PDE’s for the map, see equation (12) below. Nicole’s motivation was to improve earlier attempts Enz () at constructing extended solutions of the ‘twisted ring’ type in three spatial dimensions. In order to achieve this, Nicole modified the action appearing in Tataru () in an exotic fashion so as to give a scaling neutral theory Derrick (), very much in the same way as was done before by Deser et al in Deser (). Nowadays, similar solutions are often discussed in the literature in various contexts ranging from the models of condensed matter physics Baba (); Met () to high energy physics and cosmology Bran (); Davis (); Vil (); AFZ ().
The main question we address here is the following: How do small perturbations about a given Hopf map propagate in space-time, according to Nicole’s equations? Working within the geometrical optics limit we shall show that several lensing effects emerge, yielding an exotic causal replacement governing the kinematics of perturbations. Using the effective metric technique (see, for instance, Viss ()), we work out numerically several interesting situations and discuss how the topological charge may affect the behaviour of rays. In this sense, the present work is a natural generalization of previous results sketched in Er1 ().
Ii The model
We write for the dimensional Minkowski spacetime with metric signature and for the unit 2-sphere with metric . Inspired by the classical paper Eells (), we are interested in surjective maps
see also the interesting review Bruhat (). If and denote local coordinates in the base and target spaces, respectively, the map reads
The differential of at is the best linear approximation of near
and we use it to pullback tensors living on to the spacetime In particular, we write
with and denoting the metric and area 2-form to the unit sphere, respectively. We shall call the strain for the map and the Hopf curvature.
In physical applications, it is natural to consider configurations such that the field tends to a single value as spatial infinity is approached in any direction. We choose this ‘constant state’ such that ‘infinity’ corresponds to the north pole on the 2-sphere. This procedure is effectively equivalent to a one-point compactification of , which we denote by . Since the compactified three-dimensional Euclidean space is topologically equivalent to a topological three-sphere we have, at any given time a function
Now, as the third homotopy group of the target space is , there is an associated integer topological charge called the Hopf charge. As a consequence, the possible mappings (3) fall into different homotopy classes, and a Hopf soliton is a configuration that is a critical point for an energy functional (see subsection II.B) within a fixed homotopy class.
Whitehead first showed in reference White () that it is possible to express the Hopf charge as an integral of the form
in which is the dual tensor to the Hopf curvature, is a normalized timelike vector field () orthogonal to the space slices and (using that is a closed 2-form) we define locally via . Roughly speaking, will remain invariant under arbitrary smooth deformations of the map and, in particular, under time evolution. Interestingly, for a given time , the preimage of a point will be, in general, a closed loop in . Heuristically, we can compute by counting the linking number between two such preimage curves. We define the position of the Hopf soliton as the preimage of the south pole , as it corresponds to the position in which is the most distant to the vacuum.
Nicole’s model Nicole () consists of maps which are stationary points of the action
where is the volume element determined by and is the first elementary symmetric polynomial constructed with the pulled-back metric . For the sake of conciseness we write henceforth for the Lagrangian density, and . With these conventions, the Euler-Lagrange equations become
where sum over repeated indices is understood, and denotes the Christoffel symbols of the Levi-Civita connection on .
where stands for semilinear terms in and is the principal part of the system. As it is well known, the highest-order terms in derivatives almost completely controls the qualitative behaviour of solutions of a partial differential equation Cour (). We obtain
where and .
Iii Regular Hyperbolicity
A key issue to elucidate about systems of the form (12) is whether they admit a well-posed Cauchy problem. In other words, we wish to determine the map –at least for some finite interval of time – if initial data , are given in a non-characteristic hyper-surface . In order to guarantee well-posedness for the Nicole model it is convenient to work in the framework of regular hyperbolicity introduced by Christodoulou Christ () (see also Wong () for a short review of Christodoulou’s work, and Kamran () for a geometric approach to classical hyperbolicity). Roughly speaking, hyperbolicity is an algebraic property of the principal part given by (14), entailing the existence of solutions for arbitrary smooth initial data, and uniqueness and continuous dependence on the initial data.
If we are given and at a spacetime point we can evaluate the principal part (14) corresponding to that point. Formally, we have the map between the fibre bundles
For the sake of conciseness we use the notation
for an arbitrary covector . This object is called the principal symbol and, according to Christodoulou Christ (), the system will be regular hyperbolic if:
There exists a scalar function such that is negative definite;
There exists such that is positive definite for all satisfying =0;
Functions are called time functions while vectors are called observer fields. The existence of these quantities is sufficient to construct algorithmically a compatible energy current in order to apply the energy estimates. The latter imply that solutions exist and depend continuously on the data given in a level set of M. Furthermore, one can also guarantee that the evolution of small disturbances (linearized waves) about some given smooth data is properly-posed for the corresponding linearized equations.
In order to check the existence of and for the symbol (14) we use the mixed tensor . Its eigenvalues satisfy the second order algebraic equation
at each Interestingly, for the symbol (14), the trace and the determinant are real-valued functions of and which are always given in terms of two quadratic forms in the cotangent bundle
Moreover, a closer inspection of (16) gives:
In what follows we shall use the terminology reciprocal effective metric for the quantity defined in (21).
The system will be regular hyperbolic if the reciprocal effective metric satisfies some algebraic conditions. An inescapable (necessary) condition for the system to be well-posed is that constitutes a non-degenerate semi-definite tensor field with a Lorentzian signature. Since depends explicitly on the solution to (12) we are considering, we naturally expect these conditions to imply that not all initial data are admissible for the model, as we now show.
Following Manton Nick () we suppose that can be diagonalized relative to in a given point. The eigenvalues of are necessarily nonnegative and due to rank considerations two of them must vanish identically. We write
We then conclude that the reciprocal effective metric is Lorentzian if its eigenvalues satisfy the inequality
In particular, that will be true if , as is the case for all static solitonic solutions. As we are mainly interested in wave propagation about these solutions, we assume henceforth that is Lorentzian.
Iv Causal Replacements
By a causal replacement we mean the fact that linearized waves propagating about a smooth background solution do not travel with the velocity of light. For the Nicole model, as for quasi-linear hyperbolic field theories in general, wave propagation depends on the particular solution, direction of propagation and ‘wave polarization’ as explained, for instance, in the reference tan (). Accordingly, in order to analyze the causal replacement of our model, we recall the notion of characteristics. Physically, they can be identified with the infinite-momentum limit of the eikonal approximation Perlick () (or, equivalently, with the surfaces of discontinuity obtained via Hadamard’s method).
A hyper-surface , given by , is called characteristic if
with . The set
is called the characteristic set and it consists of the locus of normal covectors to the characteristic surfaces at . Roughly, the existence of characteristics in a region implies that linearized waves have a well-behaved, finite velocity about the background solution.
Using (17), we obtain
with and given by (19) and (20). As a consequence, the wave normals covectors satisfying (26) are determined by the vanishing sets of a multivariate polynomial of fourth order in . Thus, the resulting algebraic variety is always given by a product of quadrics, one of which changes from point to point in a way completely prescribed by the background solution and the nonlinearities present in the Nicole model ().
As we assume that is non-degenerate, we can always define its inverse i.e., . The theory of PDE’s then proceeds by telling us that the characteristic surfaces themselves are given by the zeros of a dual polynomial in the tangent space which also factorizes, i.e.
and , . Thus, the model supports two types of waves: one is governed by the background metric while the other is governed by the effective metric . Note, however, that, since the background solutions carry a topological index Q given by (9), the effective metric will also depend implicitly on .
It is well known that when is Lorentzian, the vectors such that (in a region of spacetime) satisfy the equation of null geodesics w.r.t. the effective metric, see Nov (), i.e.
where denotes covariant differentiation w.r.t. (see, however, Er2 () for a case where this property fails). Conversely, we are led to investigate spacetime trajectories such that the effective line element vanishes, i.e.
Interestingly, if the map (3) is independent of time, the problem of finding null geodesics in the effective space-time reduces to that of finding geodesics in an effective Riemannian manifold of lower dimension. Indeed, choosing a coordinate system , , such that and one reduces (30) and (31) to the equations
with and the Christoffel symbols associated with the spatial part of the effective metric. Therefore, we can qualitatively describe the interaction between background Hopfions and rays by computing the geodesics of an effective three-dimensional manifold (see Low () for a discussion in the context of the Schwarzschild solution).
V Static Hopf solitons
We briefly review here how solutions of Eq. (12) with a nonvanishing Hopf index emerge in the simpler case of static maps (see Gil () for more details). We start by introducing toroidal coordinates in :
with , , and . Note that surfaces of constant are given by non-intersecting toroids of different radii:
In particular, as the tori asymptotically approach the ring , . Conversely, as the tori become infinitely large, approaching asymptotically the line through the origin . In these coordinates, the space-time line element has the form
Now we consider the target manifold . Calculations become easier if we use coordinates in , which can be identified via stereographic projection from the south pole to the equatorial plane. In these coordinates, the curves describe lines of constant latitude (also called parallels) on the sphere while the curves describe its meridians. Particularly, corresponds to while corresponds to . The line element has the simple form
and the non-vanishing Chirstoffel symbols are
We now assume the following ansatz for the map :
being associated with angular windings around the two generating circles of the torus. Roughly, this implies that each torus in is mapped into a parallel in . Adittionaly, in order to match the appropriate boundary conditions we require that (meaning that spatial infinity i.e. the “vacuum” is actually mapped into ) and (meaning that the soliton position is mapped into ).
Now we note that the preimage of a given point in is given by the equations
which define non-intersecting closed loops winding around each torus. As is well known, see Manton (), the linking number between two of such arbitrary preimage curves is given by and uniquely determines the Hopf invariant for the solutions. Given , the collection of all preimage curves define a nontrivial fibration of within the corresponding homotopy class. In terms of the above quantities, we get, for the first symmetric polynomial,
where . With these assumptions, it is an easy task to see that Eq. (12) is identically satisfied for (i.e., in that equation) while the equation for (i.e., ) becomes the nonlinear ODE:
Vi Geodesics about a Hopfion
In this section we present the results of numerical computations in the simpler case . In this case, it is possible to obtain the exact solution for the “profile function” determined by (39), see Adam (), which uniquely determines the effective geometry. Equipped with this solution, we study geodesic motion in this geometric approach by considering the system (32) and (33). We integrate this coupled system with a classical Runge-Kutta method with automatic step-size control for a large number of geodesics emerging from different regions in three-space. For computational reasons, we work here with cartesian coordinates . The reciprocal effective metric then becomes (see the Appendix for the complete geodesic equations)
in which .
It can be checked that is positive definite for all values of and is invariant under rotations about the axis. This last result is a direct consequence of the azymuthal symmetry of the underlying fibration. The fact that this metric is curved may be easily confirmed by calculating its Ricci scalar. Surprisingly, we obtain the simple result
which is bounded and well-behaved everywhere. Interestingly, there are two non-connected surfaces in 3-space on which R vanishes. They work as boundaries in three-dimensional space separating two qualitatively different regions, i.e. and . Close to the core of the fibration (), R is negative and spheres have an excess of area compared to Euclidean spheres. Conversely, as one travels along the -axis, R becomes positive at some point and spheres appear to have a deficit of area compared to Euclidean spheres. A direct calculation shows that two global maxima exist for the points while a global minima exist for the point . In FIG. 1 it is depicted the level sets of the Ricci scalar, the core of the fibration (ring) and the preimage of the north pole (z-axis).
Also, the static Hopfion scatter the rays nontrivially. The closer to the Hopfion, the greater the bending of rays - just like using denser materials to make optical lenses results in a greater amount of refraction. However, there is a striking novelty here: as the soliton carry a topological invariant, the effective metric also carries this number implicitly, ‘warping space’ in a qualitatively different way for different values of . In this sense, different fibrations corresponding to different values of will act as different nontrivial gravitational lenses for magnifying, distorting and refocusing distant ‘objects’ (sources).
Let us investigate how effective geodesics behave in some specific situations. Starting with a ‘source’ located at the point we have analysed several emanating geodesics with initial tangent vectors lying in the plane at time . Solving the system (32) and (33) numerically for geodesics with we obtained the ‘disk’ represented in FIG 2.
We see that, for sufficiently small times, the disk is nearly flat and tangent to the plane. The same is valid for geodesics travelling in the opposite direction of the Hopfion. However, when geodesics approach the core of the fibration, drastic distortions emerge, scattering some of them upwards and others downwards. This is in contrast with gravitational lensing in general relativity due to a spherically symmetric static mass. In the latter case, null geodesics initially lying in the plane stay in the plane forever.
Geometrically speaking, the disks presented in FIG. 2 all have an excess of area (and perimeter) as compared to Euclidean disks with the same radii. This result is expected since they live in the region which tends to defocus geodesic in the average, see FIG. 3.
We have tested this defocusing effect for an extended unidimensional source characterised by . For this case the number of geodesics is while . According to FIG. 4 similar results occur.
Another interesting example is the extended unidimensional source characterised by . This is effectively the same as to rotate by our last ‘experiment’. Here, the initial geodesics lie in the plane and are initially parallel. A subset of these geodesics are shown in FIG 5.
Let us finally consider geodesics emanating from the region . In this region, the curvature of three-dimensional space due to the Hopfion tends to focus the curves instead. Starting with geodesics emanating from the point and initially making an Euclidean angle of with the z-axis we see that they first converge, diverge for a while and converge again to meet at the point , see FIG. 6. Note that for these curves, the Hopfion behaves as a convergent lens for the effective geodesics.
As a last example we consider a pencil of geodesics initially parallel to the z-axis. Interestingly, they pass through the core of the fibration yielding the pattern shown in FIG.7
The net result of this paper is the derivation of an effective metric description of wavy disturbances in the Nicole model. As is well known, this model is a non-polynomial generalization of the -model and supports a Hopf-type fibration as an exact finite-energy solution to the nonlinear equations of motion. In Section III, we have discussed the hyperbolicity properties of the model and showed, in particular, that the evolution of high-frequency excitations about all possible static solutions is well-behaved. Roughly, this means that, as long as the background solutions are static, the linearized equations will entail a definite speed of propagation independently of spacetime position or direction of propagation. Section IV discusses the causal replacement of the model and shows that it is governed by a fouth-order Fresnel-like dispersion relation given by (27). Interestingly, this algebraic equation factorizes and gives rise to two pseudo-Riemannian effective metrics. One of these metrics is given in terms of the background map and depends implicitly on the topological invariant (Hopf index). This last result gives rise to an exotic structure for the causal replacement associated to the model. In order to explore our results further we have studied the behaviour of several effective geodesics about the exact Hopf fibration and showed that they behave in a quite unexpected way. As the mathematical structure of several theories supporting topological solitons is basically the same as that of the Nicole model, our results strongly suggest that similar effects are very likely to occur in effective field theories such as the Skyrme model of pions. It would be interesting to have any deeper insight into this last problem.
Here we give explicit formulas for the geodesic equations and briefly summarize the Runge-Kutta method. In cartesian coordinates, is given by the matrix
which yields the geodesic equations