Huygens' envelope principle in Finsler spaces and analogue gravity
HHuygens’ envelope principle in Finsler spacesand analogue gravity
Hengameh R. Dehkordi Instituto de Matemática e Estatística, Universidade de São Paulo,05508-090 São Paulo, SP, BrazilE-mail: [email protected], [email protected]
Alberto Saa Departmento de Matemática Aplicada, Universidade Estadual de Campinas,13083-859 Campinas, SP, BrazilE-mail: [email protected]
Abstract.
We extend to the n -dimensional case a recent theorem establishing the validityof the Huygens’ envelope principle for wavefronts in Finsler spaces. Our results havedirect applications in analogue gravity models, for which the Fermat’s principle of least timenaturally gives origin to an underlying Finslerian geometry. For the sake of illustration, weconsider two explicit examples motivated by recent experimental results: surface waves influmes and vortices. For both examples, we have distinctive directional spacetime structures,namely horizons and ergospheres, respectively. We show that both structures are associatedwith certain directional divergences in the underlying Finslerian (Randers) geometry. Ourresults show that Finsler geometry may provide a fresh view on the causal structure ofspacetime, not only in analogue models but also for General Relativity. Keywords : Finsler geometry, Huygens’ Principle, causal structure, analogue gravity
Submitted to:
Class. Quantum Grav.
1. Introduction
Wave propagation in non-homogeneous and anisotropic media has attracted a lot of attentionrecently in the context of analogue gravity. (For a comprehensive review on the subject, see[1].) Many interesting results have been obtained, for instance, by observing surface waves insome specific fluid flows, especially those ones corresponding to analogue black holes, i.e. ,flows exhibiting an effective horizon for wave propagation [2, 3, 4, 5]. Fluid configurationsinvolving vortices, which could in some situations exhibit effective ergospheres, have alsobeen investigated [6, 7, 8, 9]. The key idea, which can be traced back to the seminal work[10] of W. Unruh in the early eighties, is the observation that generic perturbations φ in a a r X i v : . [ g r- q c ] M a r uygens’ envelope principle in Finsler spaces ρ and with a velocity field V = ( v , v , v ) are effectively governed bythe Klein-Gordon equation1 √− g ∂ a √− gg ab ∂ b φ = , (1)where a , b = , , ,
3, with the effective metric g ab given by ds = g ab dx a dx b = ρ c (cid:0) − c dt + δ i j (cid:0) dx i + v i dt (cid:1) (cid:0) dx j + v j dt (cid:1)(cid:1) , (2)where i , j = , ,
3, and δ i j stands for the usual Kronecker symbol. In general configurations,both the perturbation propagation velocity c and the flow velocity field V can indeed dependupon space and time, but we are only concerned here with the stationary situations, i.e. , thecases c = c ( x ) and V = V ( x ) . The spacetime hypersurfaces corresponding to c = V , where V = v i v i = δ i j v i v j , mimic, from the kinematic point of view, many distinctive properties ofthe Killing horizons in General Relativity (GR) [1], and this fact is precisely the starting pointof many interesting analogue gravity studies. For a review on the causal structure of analoguegravity models, see [11]. The region where c > V is the analogue of the exterior regionof a black hole in GR, where the observers are expected to live. The null geodesics of (2)correspond to the characteristic curves of the hyperbolic partial differential equation (1) and,hence, they play a central role in the time evolution of their solutions.For the null geodesics of (2). i.e. the curves such that ds =
0, we have in the exteriorregion dt = F ( x , dx i ) = (cid:113) a i j ( x ) dx i dx j + b i ( x ) dx i , (3)where a i j = ( c − V ) δ i j + v i v j ( c − V ) and b i = v i c − V . (4)Notice that the formulation (3) of the null geodesics is, in fact, equivalent to the Fermat’sprinciple of least time, in the sense that the x i ( s ) spatial curves minimizing the time interval (cid:82) dt correspond to null geodesics of the original four-dimensional spacetime metric (2). Onthe other hand, the metric defined by (3) is an explicit example of a well-known structurein Finsler geometry called Randers metric. Its striking difference when compared with theusual Riemannian metric is that, for b i (cid:54) = dt is not a quadratic form in dx i , implying manydistinctive properties for the underlying geometry as, for instance, that F ( x , dx i ) (cid:54) = F ( x , − dx i ) ,leading to widespread assertion that, in general, distances do depend on directions in Finslergeometries. For some interesting historical notes on this matter, see [12]. General Relativityhas some emblematic examples of directional spacetime structure as, for instance, eventhorizons, i.e. (null)-hypersurfaces which can be crossed only in one direction. It is hardlya surprise that Finsler geometry turns out to be relevant for these issues, but the Finsleriandescription of such spacetime structures from the physical point of view is still a ratherincipient program. The present paper is a small step towards such a wider goal.Finsler geometry is a centenary topic in Mathematics [13], with a quite large accumulatedliterature. The recente review [14] covers all pertinent concepts for our purposes here. Sinceits early days, Finsler geometry has been applied in several contexts, ranging from the already uygens’ envelope principle in Finsler spaces Huygens Theorem.
Let φ p ( t ) be a wavefront, which started at the point p, after time t. Forevery point q of this wavefront, consider the wavefront after time s, i.e. φ q ( s ) . Then, thewavefront of the point p after time s + t, φ p ( s + t ) , will be the envelope of the wavefronts φ q ( s ) , for every q ∈ φ p ( t ) . Such property is rather generic and it is valid, for instance, for any kind of linear waves in flatspacetime. Indeed, it was proved in [24] for all waves obeying the Fermat’s principle of leasttime in Euclidean space. It is also verified, in particular, for the solutions of the Klein-Gordonequation (1) in flat spacetimes of any dimension. The very fundamental concept of light conein Relativity is heuristically constructed from this kind of wavefront propagation, for whichthe Huygens’ envelope principle is expected to hold on physical grounds. However, suchprinciple should not be confused with the more stringent and restrictive Huygens’ principlewhich implies that, besides of the property of the envelope of the wavefronts, the wavepropagation occurs sharply only along the characteristic curves, implying, in particular, theabsence of wave tails. Such more restrictive Huygens’ principle is verified, for instance,for the solutions of the Klein-Gordon equation (1) in flat spacetimes only for odd spatialdimensions. For further details on this issue, see [25]. Provided that the wavefronts satisfythe Huygens’ envelope principle, one can determine the time evolution of the wavefronts for t > t once we know the wavefront at t = t . In this sense, the behavior of the propagationis completely predictable solely with the information of the wavefront at a given time. Itis important to stress that one cannot take for granted Huygens’ theorem in a Finslerianframework due to inherent intricacies of the geodesic flow.In this paper, we will explore some recent mathematical results [26, 27] to present a novelproof extending, for the n -dimensional case, the Markvorsen result on the Huygens’ envelopeprinciple in generic Finsler spaces. Moreover, we show, by means of some explicit examplesin analogue gravity, that the Finslerian formulation of the wavefront propagation in terms of aRanders metric can provide useful insights on the causal structure of the underlying spacetime.In particular, we show that the distinctive directional properties of analogue horizons andergospheres have a very natural description in terms of Finsler geometry. In principle, thesame Finslerian description would be also available for General Relativity.We will start, in the next section, with a brief review on the main mathematical definitionsand properties of Finsler spaces and Randers metrics. Section 3 is devoted to the new proofof the Huygens’ theorem and to the discussion of some generic properties of the geodesic uygens’ envelope principle in Finsler spaces n -dimensional Finsler spaces. The two explicit examplesmotivated by the common hydrodynamic analogue models, the cases of surface waves influmes and vortices, are presented in Section 4. The last section is left for some concludingremarks on the relation between the causal structure of spacetimes and the Finslerian structureof the underlying geometry associated with the Fermat’s principle of least time.
2. Geometrical Preliminaries
For the sake of completeness, we will present here a brief review on Finsler geometry and theRanders metric, with emphasis on the notion of transnormality[26, 27], which will be centralin our proof of Huygens’ theorem. For further definitions and references, see [14].Let V be a real finite-dimensional vector space. A non-negative function F : V → [ , ∞ ) is called a Minkowski norm if the following properties hold:(i) F is smooth on V \{ } ,(ii) F is positive homogeneous of degree 1, that is F ( λ y ) = λ F ( y ) for every λ > y ∈ V \{ } , the fundamental tensor g y , which is the symmetric bilinear formdefined as g y ( u , v ) = (cid:18) ∂ ∂ t ∂ s F ( y + tu + sv ) (cid:19) s = t = , (5)is positive definite on V .The pair ( V , F ) is usually called a Minkowski space in Finsler geometry literature, and this,in principle, might cause some confusion with the distinct notion of Minkowski spacetime.Here, we will adopt the Finsler geometry standard denomination and no confusion shouldarise since we do not mix the two different spaces. Given a Minkowski space, the indicatrixof F is the unitary geometric sphere in ( V , F ) , i.e. , the subset I = { v ∈ V | F ( v ) = } . (6)The indicatrix I defines a hypersurface (co-dimension 1) in ( V , F ) consisting of the collectionof the endpoints of unit tangent vectors. In contrast with the Euclidean case, where the I isalways a sphere, it can be a rather generic surface in a Minkowski space. We are now readyto introduce the notion of a Finslerian structure on a manifold. Let M be an n -dimensionaldifferentiable manifold and T M its tangent bundle. A Finsler structure on M is a function F : T M → [ , ∞ ) with the following properties:(i) F is smooth on T M \{ } ,(ii) For each x ∈ M , F x = F | T x M is a Minkowski norm on T x M . The pair ( M , F ) is called a Finsler space. Suppose now that M is a Riemannian manifoldendowed with metric α : T M × T M → [ , ∞ ) and a 1-form β : T M → R such that α ( y β , y β ) <
1, with y β standing for the vector dual of β . In this case, F = α + β is a particular Finslerstructure called Randers metric on M , and in this case the pair ( M , F ) is called a Randersspace. uygens’ envelope principle in Finsler spaces ( M , h ) witha smooth vector field (wind) W such that h ( W , W ) <
1. The associated Randers metriccorresponding to the solution of a Zermelo’s navigation problem is given by F ( y ) = α ( y ) + β ( y ) = (cid:112) h ( W , y ) + λ h ( y , y ) λ − h ( W , y ) λ (7)where λ = − h ( W , W ) . Comparing with (3), one can easily establish a conversion betweenthe so-called Zermelo data ( M , h , W ) of a Randers space and the analogue gravity quantities c , V , and δ i j .Given a Finsler space ( M , F ) , the gradient ∇ f p of a smooth function f : M → R at point p ∈ M is defined as d f p ( v ) = g ∇ f p ( ∇ f p , v ) , (8)where v ∈ T p M and g y ( y , v ) = (cid:18) ∂∂ s F ( y + sv ) (cid:19) s = , (9)which is the fundamental tensor of F at y ∈ T p M (see [28] for more details). It is importantto stress that, in Randers spaces, where a Riemannian structure is also always available, thegradient (8) differs from the usual Riemannian gradient ˜ ∇ f p at p ∈ M , unless the vector field W vanishes. The following Lemma, which proof can be found in [27], connects the twogradients in a very useful way, since the direct calculation of ∇ f p is sometimes rather tricky. Lemma 1.
Let f : U ⊂ M → R be a smooth function without critical points, ( M , F ) a Randersspace with Zermelo data ( M , h , W ) , and ∇ f p and ˜ ∇ f p , respectively, the gradients with respectto F and to h at p ∈ M. Then(i) || ˜ ∇ f p || F ( ∇ f p ) ( ∇ f p − F ( ∇ f p ) W ) = ˜ ∇ f p ,(ii) F ( ∇ f p ) = || ˜ ∇ f p || + d f ( W ) ,where || y || = h ( y , y ) . If L is a submanifold of a Finsler space ( M , F ) , a non-zero vector y ∈ T p M will beorthogonal to L at p if g y ( y , v ) = v ∈ T p L . Notice that, for the case of a Randersspace with Zermelo’s data ( M , h , W ) , for every non-zero vectors u and y in T p M , we will have g y ( y , u ) = h (cid:18) u , yF ( y ) − W (cid:19) = . (10)The following Lemma, which proof follows straightforwardly from the previous definitions(see also [14]), will be useful in the next section. Lemma 2.
Let ( M , F ) be a Finsler space, U an open subset of M, and f a smooth functionon U with d f (cid:54) = . Then, n = ∇ fF ( ∇ f ) (cid:12)(cid:12)(cid:12) f − ( c ) is orthogonal to f − ( c ) with respect to g n .uygens’ envelope principle in Finsler spaces f : M → R be a smooth function. If there exists a continuous function b : f ( M ) −→ R such that F ( ∇ f ) = b ◦ f , (11)with ∇ f given by (8), then f is called a Finsler transnormal (shortly F -transnormal) function.When transnormal functions are available, some properties of the geodesic flow in a Finslerspace can be easily determined. Geodesics in Finsler geometry are defined in the same way ofRiemannian spaces. First, notice that the length of a piecewise smooth curve γ : [ a , b ] −→ M with respect to F is defined as L ( γ ) = (cid:90) ba F ( γ ( t ) , γ (cid:48) ( t )) dt . (12)Analogously to the Riemannian case, the distance from a point p ∈ M to another point q ∈ M in the Finsler space ( M , F ) is given by d F ( p , q ) = inf γ (cid:90) ba F ( γ ( t ) , γ (cid:48) ( t )) dt , (13)where the infimum is meant to be taken over all piecewise smooth curves γ : [ a , b ] −→ M joining p to q . For a Finsler space ( M , F ) , the geodesics of F are the length (12) minimizingcurves. Notice that, when we are dealing with Randers spaces derived from the null geodesicof a Lorentzian manifold, as it was discussed in Section 1, the geodesics of ( M , F ) correspondto a realization of Fermat’s principle of least time for the original null geodesics. For a moregeneral mathematical discussion on the Fermat’s principle in Finsler geometry, see [23]. Itis worth mentioning that, for some special vectors W , there is a useful relation betweengeodesics in a Randers space with Zermelo data ( M , h , W ) and the usual geodesics in theRiemannian space ( M , h ) . Such relation is expressed by the following Lemma, which followsdirectly as a Corollary of Theorem 2 in [31]. Lemma 3.
Let ( M , h ) be a Riemannian manifold endowed with a Killing vector field W .Given a unitary geodesic γ h : ( − ε , ε ) → M of ( M , h ) , the curve γ F ( t ) = ϕ W ( t , γ h ( t )) , where ϕ W : ( − ε , ε ) × U → M is the flow of W , will be a F-unitary geodesic of the Randers space ( M , F ) with Zermelo data ( M , h , W ) . The distance from a given compact subset A of a manifold M to any point p ∈ M isdefined as ρ : M → R with ρ ( p ) = d F ( A , p ) . If for every p , q ∈ M there exists a shortest unitspeed curve from p to q , then F ( ∇ ρ ) =
1, indicating that ρ is F -transnormal with b = Proposition 4.
Let f : M → R be a F -transnormal function with f ( M ) = [ a , b ] . If c < d ∈ f ( M ) , then for every q ∈ f − ( d ) ,d F ( f − ( c ) , q ) = d F ( f − ( c ) , f − ( d )) = (cid:90) dc ds (cid:112) b ( s ) = L ( α ) , where α is a reparametrization of (an extension of) the integral curve of ∇ f .uygens’ envelope principle in Finsler spaces f − ( c ) ⊆ ρ − ( r ) where ρ ( p ) = d F ( f − ( a ) , p ) = r .We say that two submanifolds C and D of a Finsler space are equidistant if, for every p ∈ C and q ∈ D , d F ( p , D ) = d F ( C , D ) and d F ( D , C ) = d F ( q , C ) (or, equivalently, d F ( C , D ) = d F ( C , q ) and d F ( D , C ) = d F ( D , p ) ). Theorem 5.
Let M be a compact manifold and f : M → R be a F -transnormal and analyticfunction such that f ( M ) = [ a , b ] . Suppose that the level sets of f are connected and a and bare the only critical values of f in [ a , b ] . Then, for every c , d ∈ [ a , b ] , f − ( c ) is equidistant tof − ( d ) . Finally, the cut loci of the point p associated to the distance function ρ is defined analogouslyto the Riemannian case: it consists in the set of all points q ∈ M with two or more differentlength (12) minimizing curves γ : [ a , b ] → M joining p to q .
3. The Huygens’ envelope principle in Finsler spaces
Throughout this section, it is assumed that, on some part of a Finsler space ( M , F ) , a wavefrontis spreading and sweeping the domain U ⊂ M in the interval of time from t = t = r . Itis also assumed that U is a smooth manifold. Given a wavefront φ p ( t ) , we call the waveray at q ∈ φ p ( t ) the shortest time path connecting p to q . Again, due to the intricacies ofFinslerian metrics, one cannot take for granted many properties of wave rays in Euclideanspaces as, for instance, the fact that they are orthogonal to the wavefronts. Let us startby considering the Huygens’ theorem for more general situations. The following theoremgeneralizes Markvorsen’s result [22] for any Finsler space. Theorem 6.
Let ρ : M → R with ρ ( p ) = d F ( A , p ) , where A is a compact subset of M and ρ ( U ) = [ s , r ] , where < s < r. Suppose that ρ − ( s ) is the wavefront at time t = and thatthere are no cut loci in ρ − ([ s , r ]) . Then, for each t ∈ [ s , r ] , ρ − ( t ) is the wavefront at timet − s and the Huygens’ envelope principle is satisfied by all the wavefronts { ρ − ( t ) } t ∈ [ s , r ] . Furthermore, the wave rays are geodesics of F and they are also orthogonal to each wavefront ρ − ( t ) at time t − s.Proof. Since ρ is a transnormal function with b =
1, from Proposition 4 we have that, forevery t > s and q ∈ ρ − ( t ) , d F (cid:0) ρ − ( s ) , q (cid:1) = t − s , (14)meaning that the wavefront reaches ρ − ( t ) after time t − s . The relation d F ( ρ − ( s ) , q ) = d F ( ρ − ( s ) , ρ − ( t )) implies that no part of the wavefront meets ρ − ( t ) before time t − s , andthus ρ − ( t ) is indeed the wavefront at this time.Now, in order to verify the Huygens’ envelope principle, let us assume that e ( δ ) is theenvelope of radius δ of the wavefront ρ − ( t ) for some time t ≥ s . It implies that for every p ∈ e ( δ ) , d F ( ρ − ( t ) , p ) = d F ( ρ − ( t ) , e ( δ )) = δ , (15) uygens’ envelope principle in Finsler spaces p ∈ e ( δ ) and q ∈ ρ − ( t ) such that d F ( ρ − ( t ) , p ) = d F ( q , p ) = r < δ , then the wavefront centeredat q and radius δ would intersect the envelope, which is a contradiction and consequentlyrelation 15 is indeed valid. So, as p ∈ e ( δ ) , there exists a path from a unique point q ∈ ρ − ( t ) to the point p along which the wavefront time of travel is precisely δ . Since ρ − ( t ) is thewavefront, this wave ray has emanated from some point in ρ − ( s ) and reached point q at time t − s . Therefore, we have d F ( ρ − ( s ) , p ) ≤ t − s + δ . (16)Notice that, if d F ( ρ − ( s ) , p ) < t − s + δ , there would exist a path from ρ − ( s ) to p throughwhich wave ray travels in a time shorter than t − s + δ . As d F ( ρ − ( s ) , ρ − ( t )) = t − s , (17)this ray meets ρ − ( t ) at exactly time t − s . As a result, the inequality would hold only whenthis ray travels from ρ − ( t ) to p at a time less than δ which is a contradiction by Eq. (15).Finally, we have d F ( ρ − ( s ) , p ) = t − s + δ = t − s (18)which means p belong to the wavefront ρ − ( t ) , and hence e ( δ ) ⊂ ρ − ( t ) . Now, we canestablish that ρ − ( t ) ⊂ e ( δ ) . Assume that p ∈ ρ − ( t ) . Since ρ − ( t ) is the wavefront, eachwave ray from ρ − ( s ) reaches ρ − ( t ) and ρ − ( t ) , at times t − s and t − s , respectively. UsingProposition 4, one has d F ( ρ − ( t ) , p ) = t − t = δ , (19)and consequently p ∈ e ( δ ) .To accomplish the proof, observe that each wave ray emanates from a point in ρ − ( s ) and reaches ρ − ( t ) in the shortest time, implying that its traveled path is a geodesic of theFinsler space. Furthermore, assuming that α is the unit speed geodesic such that d F ( ρ − ( s ) , ρ − ( t )) = d F ( ρ − ( s ) , p ) = L ( α | [ , t ] ) = t , (20)we have, according to Proposition 4, that α | [ , t ] is an extension of the integral curve of ∇ ρ .Hence, α | ( , t ) is the integral curve of ∇ ρ , and by Lemma 2 it is orthogonal to each ρ − ( t ) .Notice that the cut loci in ρ − ([ s , r ]) are associated with singularities in the wavefronts, anextremely interesting topic [32], but which is out of the scope of the present paper.If a transnormal function f is available, one can determine the wavefronts without dealingwith the Randers metric and/or the distance function. The following proposition, which prooffollows in the same way of Theorem 6, summarize this point. Proposition 7.
Suppose that f : M → R is a F -transnormal function with F ( ∇ f ) = b ( f ) andf ( M ) = [ a , b ] . Assuming that f − ( a ) is a wavefront at time t = , we havea ) for every c ∈ [ a , b ] , f − ( c ) is the wavefront at timer a , c = (cid:90) ca ds (cid:112) b ( s ) , (21) uygens’ envelope principle in Finsler spaces b ) { f − ( c ) } c ∈ [ a , b ] satisfies Huygens’ envelope principle,c ) the wave rays are geodesics of F joining f − ( a ) to f − ( b ) , and they are also orthogonalto each wavefront.
4. Analogue Gravity Examples
In this section, we will present two explicit examples, in the context of analogue gravity, ofwavefront propagation determined from the Huygens’ envelope principle in Randers spaces,whose validity for any space dimension was established by our mathematical results. Theexamples, motivated by very recent experimental results, are namely the cases of surfacewaves in flumes and vortices. Of course, we are assuming that for such realistic cases thesurface waves indeed obey a Klein-Gordon equation (1), for which the Huygens’ envelopeprinciple is expected to hold on physical grounds. Nevertheless, in realistic experiments,typically, the wave propagation speed c may depend on the wave frequency, a situationcommonly dubbed in General Relativity as rainbow spacetimes, a situation which can beindeed also described from a Finslerian perspective [33]. Our present approach and, inparticular, our Huygens’ envelope principle for wavefronts, should be considered as the firststep towards the description of these more realistic configurations. The literature on theexperiments [2, 3, 4, 5, 6, 7, 8, 9] discusses in details all these points. The first, and still more common, type of hydrodynamic analogue gravity model is the caseof surface waves in a long and shallow channel flow, a situation having effectively only onespatial dimension. Typically, the flow is stationary but its velocity V depends on the positiondue to the presence of certain obstacles in the channel bottom, see [2, 3, 4, 5] for some concreterealizations of this kind of experiment. The surface waves propagation velocity c also dependson the position along the channel. Horizons for the surface waves can be produced by selectingobstacles such that c < V on some regions along the channel.We will consider here the simplest case consisting of ( R , h ) , i.e. the real line withthe standard metric h , and the Zermelo vector field W ( x ) , where x is coordinate along thechannel, with W <
1. Let ( R , F ) be the associated Randers space, where the Randersmetric F is given by (7). Since the Randers space is one-dimensional in this case, thewavefronts will correspond to a set of two points, and we do not need to worry aboutwave rays and their orthogonality to the wavefronts. For the sake of simplicity, supposethe waves are emitted at t = q . The wavefront at t = r will be given by ρ − ( r ) = { p ∈ R : d F ( q , p ) = r } . Assuming that γ : [ , r ] → R is the unit speed geodesic thatrealizes this distance, we have1 = F ( γ , ˙ γ ) = | ˙ x | − W ˙ x − W , (22)where (7) was used. From equation (22), we have that the right ( x + ) and left-moving ( x − ) uygens’ envelope principle in Finsler spaces x t W = a x a a a a Figure 1.
Geodesics (25) and (26) for some specific values of a . The blue (traced) curve is theaspect of the Zermelo vector (24), without any scale. The red (continuous) lines correspondsto two sets of right and left-moving geodesics which start at t = x = − x = x + are rather insensitive to the value of a . On the other hand,the left-moving ones that cross the maximum of W at x = a close to 1. The depicted curves correspond to the following values of a : a = . a = . a = .
94, and a = . x = a → x >
0, see themain text. wavefronts are governed by the equations˙ x ± = ± + W ( x ) , (23)and the wavefront at t = r will be simply ρ − ( r ) = { x − ( r ) , x + ( r ) } . Notice that both equations(23) are separable and could be solved straightforwardly by quadrature, but for our purposehere a dynamical analysis for general W typically suffices. Since | W | <
1, there are no fixedpoints in (23), meaning that x + and x − move continuously towards right and left, respectively.Let us consider the explicit example of the Zermelo vector W ( x ) = a + x , (24)with 0 ≤ a <
1. Its aspect is quite simple (see Fig. 1), it corresponds to a flume moving to theright-handed direction, with a smooth and non-homogeneous velocity attaining its maximum W = a at the origin, which might be caused, for instance, due the presence of a smoothobstacle in the channel. We could also add a positive constant to W which would correspondto the flume velocity far from the origin, but for our purposes here this constant is irrelevantand we set it to zero, without any loss of generality. For this choice of W , equations (23) canbe exactly solved as t = x + − x − a √ + a (cid:18) arctan x + √ + a − arctan x √ + a (cid:19) (25) uygens’ envelope principle in Finsler spaces t = x − x − + a √ − a (cid:18) arctan x √ − a − arctan x − √ − a (cid:19) , (26)where we assume, for sake of simplicity and also without loss of generality, that the wavefrontstarted at t = x = x . From (25) and (26), we can draw a ( x , t ) diagram for the geodesics,see Fig. 1. The behavior of the right ( x + ) and left-moving ( x − ) geodesics depends strictlyon the value of x . For x >
0, the right moving geodesics depart from the maximum of W located at x = a . Exactly thesame occurs for the left-moving geodesics starting at x <
0. The situation for the geodesicscrossing x = x > x = W and depict a strong sensibility on a . In particular, for a veryclose to 1, they tend to stay close to x = x <
0, and move in the same direction of W , exhibitlow sensitive on a , they cross x = F ( x , y ) = ( + x )( + x ) − a (cid:0)(cid:0) + x (cid:1) | y | − ay (cid:1) . (27)Notice that for y >
0, the metric near the origin x = F ( x , y ) = y + a + O ( x ) (28)whereas F ( x , − y ) = − y − a + O ( x ) . (29)It is clear that we will have for a → i.e. an hypersurface which can becrossed only in one direction. We will return to this important point in the last section. The one-dimensional flows of the first example are not sufficient to appreciate all thesubtleties of the Finslerian analysis of the wavefronts. Flows involving vortices are verygood candidates for our study, since besides of being intrinsically higher dimensional, theyare indeed important from the experimental point of view in analogue gravity, see [6, 7, 8, 9]for some recent results. We will consider here the simplest possible vortex configuration: afluid in a long cylindrical tank M of radius R >
0. We will assume cylindrical symmetry,so the vertical direction can be neglected and we are left with an effective two-dimensionalspatial problem. The pertinent manifold for our flow will be M = (cid:110) ( x , x ) ∈ R : ( x ) + ( x ) ≤ R (cid:111) . (30)It is important to stress that our manifold in this case has a boundary ∂ M and that someboundary conditions will be needed for wavefronts and geodesics reaching ∂ M . Theassociated Randers space will be ( M , h ) , where h is the usual Euclidean two-dimensional uygens’ envelope principle in Finsler spaces W = ( w , w ) corresponding to a rotation flow aroundthe origin. If one wants to keep the cylindrical symmetry, the more general Zermelo vectorfield in this case will be of the type W = w ( r ) X A , where X = ( x , x ) , r = X X t , w ( r ) is asmooth function, and A is the two-dimensional rotation generator matrix A = (cid:32) − (cid:33) . (31)The case of constant angular fluid velocity (rigid rotation) corresponds to w = a constant,whereas the constant tangential velocity is w = ar − . Notice that the dynamical flowassociated with such a vector field is given by ϕ W ( t , X ) = X Rot r ( t ) , whereRot r ( t ) = (cid:32) cos ( tw ( r )) sin ( tw ( r )) − sin ( tw ( r )) cos ( tw ( r )) (cid:33) . (32)The Randers metric (7) in this case is given by F ( X , Y ) = | WY t | − WW t (cid:115) + ( − WW t ) YY t ( WY t ) − WY t − WW t , (33)where Y = ( y , y ) ∈ T M is an arbitrary vector and λ = − WW t = − r w , (34)from where we have the restriction max | rw | <
1. Of course, we have also assumed WY t (cid:54) = rw close to 1 for some r = r , we will have in the neighborhoodof this hypersurface F ( X , Y ) = | WY t | − WY t − WW t + YY t | WY t | + O ( r − r ) , (35)and that it is clear that for WY t > Y pointing in the samedirection of the Zermelo “wind” W ), the metric is insensitive to the term ( − r w ) − , insharp contrast with the situations where WY t < Y “against” W ). The hypersurface r = r in this case is not exactly an horizon, since it could indeed be crossed in both directionby, for instance, having WY t > Y .This kind of hypersurface mimics the main properties of a black hole ergosphere, sinceit practically favors co-rotating directions for Y , as the counter-rotating ones are stronglyaffected by the singularity arising from ( − r w ) − for rw →
1. An explicit example for w ( r ) will help to illustrate such results. Before that, however, let us notice that the cylindricalsymmetry has an important consequence for the wavefronts. Let us consider the function f : M → R with f ( x ) = r . Since d f ( W ) =
0, we have from Lemma (1) that F ( X , ∇ f ) = f ,implying that f is F-transnormal. Hence, by Proposition 7, we have that the circumferences f − ( t ) = { x ∈ M : r = ( x ) + ( x ) = t } correspond to wavefronts in this Randers space.Of course, due to the cylindrical symmetry, such wavefronts originated form a source at theorigin ( , ) at t =
0. The evaluation of the wavefronts emitted from an arbitrary points forgeneral w ( r ) is much more intricate and involve the Finslerian geodesic flow. uygens’ envelope principle in Finsler spaces w = a . The mainadvantage of this choice is that the Zermelo vector W is a Killing vector of the Euclideanmetric, and hence we can use Lemma 3 to obtain the Finsler geodesics explicitly as γ F ( t ) = ϕ W ( t , γ h ( t )) = γ h ( t ) Rot a ( t ) , (36)where γ h are the usual unit speed Euclidean geodesics and Rot a ( t ) is the matrix (32) for w = a .Since the Euclidean geodesics are γ h ( t ) = ( x , x ) + tV , (37)where ( x , x ) ∈ M is an arbitrary point and V a unit vector, one can write (cid:0) γ F ( t ) − ( x , x ) Rot a ( t ) (cid:1) (cid:0) γ F ( t ) − ( x , x ) Rot a ( t ) (cid:1) t = t . (38)Recalling that the geodesics γ F are the wave rays of the wavefronts, we have from (38)that the wavefront emitted at t = ( x , x ) ∈ M is an expanding circle withrotating center ( x , x ) Rot a ( t ) . Moreover, from (10) one can say that the geodesics γ F areorthogonal to each of these circles, as one can see by observing that F ( γ (cid:48) ) = γ (cid:48) F − W = γ (cid:48) h ( t ) Rot a ( t ) . Fig. 2 depicts a typical example of wavefronts and geodesicsfor this system. Since our manifold has a boundary ∂ M , one needs to specify boundaryconditions for geodesics and wavefronts on ∂ M . We choose to impose perfect reflection on theboundary. The situation for the wavefronts is completely analogous, up to the rigid rotation,to the classical optical problem of the reflection of spherical waves on a spherical mirror. Inparticular, after the reflection, our circular wavefronts will form a caustic, see [34] for a recentapproach for the problem. The animations available in the Supplementary Material depict thetypical dynamics of wavefronts and geodesics with perfect reflection boundary conditions inthis Randers space. Before the reflection occurs, the wavefronts are circles with increasingradius and which centers rotate around the origin ( , ) of M with constant angular velocity a . After the reflection, they will correspond to a circular segment and a caustic, which centersalso rotate around the origin. The caustic can be determined by the classical formula [34] C ( s ) = P + ( s − | P − X | ) (cid:0) ( PX t ) P − R ( P + X ) (cid:1) | P − X | R , (39)where P is the point of reflection on the boundary and X the emitting point. The reflectiontakes place for s > | P − X | , i.e. , for a fixed s > | P − X | , C ( s ) corresponds to the the reflectedwavefront (the caustic). Due to the reflection boundary conditions, the wavefronts eventuallywill contract and give origin to some caustic singularities, see the Supplementary Materialand [32] for further references on these phenomena.We are still left with the ergosphere properties of the hypersurface ar →
1. Theyare illustrated in Fig. 3. Several geodesics starting in different points are depicted. Allgeodesics reach the hypersurface ar → uygens’ envelope principle in Finsler spaces qt t t t t Figure 2.
A wave pulse is emitted from the point q at t = W . The red (solid) curves are some wave rays (Finslergeodesics), plotted for 0 ≤ t ≤ t . The green (traced) curves are the wavesfronts at differenttimes 0 < t < t < t < t < t . Perfect reflection on the boundary is assumed. Before thereflection, the wavefronts are circles with increasing radius and centers rotating on the blue(dot-traced) circle. After the reflection, they correspond to a circle segment and a caustic(see the text), which centers also rotate along the blue (dot-traced) circle. The wave raysare always orthogonal, with respect to the Finlerian structure, to the wavefronts. Due to theperfect reflection boundary condition, the wavefronts eventually evolve some singularities, seethe animations available in the Supplementary Material. is intimately connected with superradiant scattering, a phenomenon already described anddetected in analogue models involving surface waves in vortex flows, see [7], for instance.
5. Final remarks
We have extended to the n -dimensional case a recent theorem due to Markvorsen [22]establishing the validity of the Huygens’ envelope principle in Finsler spaces. We thenapply our results to two explicit cases motivated by recent results in analogue gravity:the propagation of surface waves in flumes and vortex flows. The Finslerian descriptionassociated with the Fermat’s principle of least time for the wave propagation, in both cases,gives rise to an underlying Randers geometry and provides a useful framework for the study ofwave rays and wavefronts propagation. Interestingly, the spatial regions where h ( W , W ) → uygens’ envelope principle in Finsler spaces q q q Figure 3.
For the case of a rigid clockwise rotating vector field W , several Finsler geodesics,starting in three different points q , q , and q , are depicted. The inner circle, whichcorresponds to ar →
1, mimics an ergosphere. In the outermost region (shadowed), whichrigorously does not belong to our Randers space, no counter-rotating wave rays would beallowed. No geodesic is allowed to reach the hypersurface ar → namely a Killing horizon for the uni-dimensional flume and an ergosphere for the two-dimensional vortex. However, from the Randers space point of view, we are confined byconstruction into the regions where the so-called mild Zermelo wind condition h ( W , W ) < h ( W , W ) > h ( W , W ) =
1. This unified description of Randers and Kropina spaces is still a quite recentprogram in Mathematics [35].The analogue gravity examples provide a rather direct application for the Finslerianapproach since the use of the Fermat’s principle manifestly originates an underlying Randersgeometry. However, the same results would also hold for General Relativity. Consider, forinstance, the Schwarzschild metric in the Gullstrand-Painlevé stationary coordinates ds = − (cid:18) − Mr (cid:19) dt + (cid:114) Mr dtdr + dr + r d Ω , (40)where d Ω stands for the usual metric on the unit sphere. Such a metric is, indeed, the uygens’ envelope principle in Finsler spaces dt = F ( r , dr ) = | dr | + (cid:113) Mr dr − Mr , (41)and this is precisely a Randers metric of the type (22) with Zermelo vector W = − (cid:113) Mr .Exactly as in the flume case, we have two qualitative different behaviors for ingoing ( dr = dr − <
0) and outgoing ( dr = dr + >
0) null rays, namely dt = dr ± ∓ (cid:113) Mr . (42)The directional properties of such Randers metric indicate the presence of a horizon at r = M , since ingoing null rays can cross it smoothly, while outgoing rays experiment ametric divergence. It is hardly a surprise that Finsler geometry turns out to be relevant forthese directional properties of a spacetime causal structure. In fact, some recent mathematicalresults [21, 23] show that most of causality results are also valid in a Finslerian framework,under rather weak regularity hypotheses. However, the application of Finsler geometry inphysical studies of causal structures is still a rather incipient program. Dropping the mildwind condition, which in this case should allow for a unified description for the exterior andinterior region of the black hole (40), and the study of the Finslerian curvatures associated tothe divergence in (42), should be the first steps towards a physical Finslerian description ofspacetime causal structures. These topics are now under investigation. Supplementary material
The animations available as Supplementary material at [37] show the continuous timeevolution of the wavefronts and geodesics of Fig. 2. One can appreciate the eventualformation of singularities in the caustic associated with the reflection of the wavefronts inthe boundary ∂ M . Acknowledgment
The authors acknowledge the financial support of CNPq, CAPES, and FAPESP (Grant2013/09357-9). They also wish to thank M.M. Alexandrino, B.O. Alves, M.A. Javaloyes,and E. Minguzzi for enlightening discussions.
References [1] C. Barcelo, S. Liberati, S. and M. Visser, Living Rev. Relativ. , 3 (2011). Available on-line at https://doi.org/10.12942/lrr-2011-3 [2] G. Rousseaux, C. Mathis, P. Maissa, T.G. Philbin, and U. Leonhardt, New J. Phys. , 053015 (2008).[arXiv:0711.4767] uygens’ envelope principle in Finsler spaces [3] G. Rousseaux, P. Maissa, C. Mathis, P. Coullet, T.G. Philbin, and U. Leonhardt, New J. Phys. ,021302 (2011). [arXiv:1008.1911][5] L.P. Euve, F. Michel, R. Parentani, T.G. Philbin, and G. Rousseaux, Phys. Rev. Lett. , 121301 (2016).[arXiv:1511.08145][6] V. Cardoso, A. Coutant, M. Richartz, and S. Weinfurtner, Phys. Rev. Lett. , 271101 (2016).[arXiv:1607.01378][7] T. Torres, S. Patrick, A. Coutant, M. Richartz, E.W. Tedford, and S. Weinfurtner, Nature Physics , 833(2017). [arXiv:1612.06180][8] S. Patrick, A. Coutant, M. Richartz, and S. Weinfurtner, Phys. Rev. Lett. , 061101 (2018).[arXiv:1801.08473][9] T. Torres, S. Patrick, M. Richartz, and S. Weinfurtner, Application of the black hole-fluid analogy:identification of a vortex flow through its characteristic waves . [arXiv:1811.07858][10] W. G. Unruh, Phys. Rev. Lett. , 1351 (1981).[11] M. Visser, Class. Quantum Grav. , 1767 (1998). [arXiv:gr-qc/9712010])[12] S.S. Chern, Notices AMS , 959 (1996).[13] P. Finsler, Über Kurven und Flächen in allgemeinen Räumen , Göttingen University Dissertation (1918).Reprinted by Birkhäuser (1951).[14] Z. Shen,
Lectures on Finsler geometry , World Scientific (2001)[15] D. Bao, C. Robles, and Z. Shen, J. Diff. Geom. , 377 (2004). [arXiv:math/0311233][16] M. Cvetiˇc and G.W. Gibbons, Ann. Phys. , 2617 (2012). [arXiv:1202.2938][17] D.H. Anderson, E.A. Catchpole, N.J. De Mestre, and T. Parkes, J. Austral. Math. Soc. (Series B) , 451(1982).[18] G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, and M. C. Werner, Phys. Rev. D , 044022 (2009).[arXiv:0811.2877][19] G.W. Gibbons and C.M. Warnick. Contemp. Phys. , 197 (2011). [arXiv:1102.2409][20] M.A. Javaloyes, Conformally standard stationary spacetimes and Fermat metrics , In
Recent Trendsin Lorentzian Geometry , M. Sanchez, M. Ortega, Miguel, and A. Romero (Eds.) Springer (2012).[arXiv:1201.1841][21] E. Minguzzi, Monatsh. Math. , 569 (2015). [arXiv:1308.6675][22] S. Markvorsen, Nonl. Anal. , 208 (2016).[23] E. Minguzzi, Rev. Math. Phys. , 1930001 (2019). [arXiv:1709.06494][24] V.I. Arnold, Mathematical Methods of Classical Mechanics , 2nd edition, Springer (1999).[25] P. Gunther,
Huygens’ Principle and Hyperbolic Equations , Academic Press (1988).[26] H.R. Dehkordi,
Finsler Transnormal Functions and Singular Foliations of Codimension 1 , Universidadede São Paulo PhD Thesis (2018). Available on-line here.[27] M.M. Alexandrino, B.O. Alves, and H.R. Dehkordi,
On finsler transnormal functions . [arXiv:1807.08398][28] M.A. Javaloyes and M. Sanchez, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
XIII , 813 (2014).[arXiv:1111.5066][29] M.M. Alexandrino, B.O. Alves, and M.A. Javaloyes,
On singular Finsler foliation , to appear in Annali diMatematica (2018). [arXiv:1708.05457][30] Q. He, S. T Yin, and Y. Shen, Diff. Geom. Appl. , 133 (2016) [arXiv:1507.04219].[31] C. Robles, Trans. American Math. Soc. , 1633 (2007). [arXiv:math/0501358][32] V.I. Arnold, Singularities of caustics and wave fronts , Springer (1990).[33] F. Girelli, S. Liberati, and L. Sindoni, Phys. Rev. D , 064015 (2007). [arXiv:gr-qc/0611024][34] J. Castro-Ramos, et al. , J. Opt. Soc. Am. A , 177 (2013).[35] M.A. Javaloyes and M. Sanchez, Eur. J. Math. , 1225 (2017). [arXiv:1701.01273][36] A.J.S. Hamilton and J.P. Lisle, Am. J. Phys. , 519 (2008). [arXiv:gr-qc/0411060][37] See http://vigo.ime.unicamp.br/Huygens/http://vigo.ime.unicamp.br/Huygens/