Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics
aa r X i v : . [ m a t h . AG ] M a y ELLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACESAND RELATIVISTIC QUADRICS
VLADIMIR DRAGOVI´C AND MILENA RADNOVI´C
Abstract.
We study geometry of confocal quadrics in pseudo-Euclidean spa-ces of an arbitrary dimension d and any signature, and related billiard dyna-mics. The goal is to give a complete description of periodic billiard trajectorieswithin ellipsoids. The novelty of our approach is based on introduction of anew discrete combinatorial-geometric structure associated to a confocal pencilof quadrics, a colouring in d colours, by which we decompose quadrics of d + 1geometric types of a pencil into new relativistic quadrics of d relativistic types.Deep insight of related geometry and combinatorics comes from our study ofwhat we call discriminat sets of tropical lines Σ + and Σ − and their singulari-ties. All of that enables us to get an analytic criterion describing all periodicbilliard trajectories, including the light-like ones as those of a special interest. Contents
1. Introduction 22. Pseudo-Euclidean spaces and confocal families of quadrics 32.1. Pseudo-Euclidean spaces 32.2. Families of confocal quadrics 43. Elliptical billiard in the Minkowski plane 83.1. Confocal conics in Minkowski plane 83.2. Light-like trajectories of the elliptical billiard 94. Relativistic quadrics 114.1. Confocal quadrics and their types in the Euclidean space 124.2. Confocal conics in the Minkowski plane 134.3. Confocal quadrics in the three-dimensional Minkowski space and theirgeometrical types 144.4. Tropic curves on quadrics in the three-dimensional Minkowski spaceand discriminant sets Σ ± Key words and phrases.
Confocal quadrics, Poncelet theorem, periodic billiard trajectories,Minkowski space, light-like billiard trajectories, tropic curves.The research which led to this paper was partially supported by the Serbian Ministry ofEducation and Science (Project no. 174020:
Geometry and Topology of Manifolds and In-tegrable Dynamical Systems ) and by Mathematical Physics Group of the University of Lis-bon (Project
Probabilistic approach to finite and infinite dimensional dynamical systems,PTDC/MAT/104173/2008 ). M. R. is grateful to the Weizmann Institute of Science (Rehovot,Israel) and
The Abdus Salam
ICTP (Trieste, Italy) for their hospitality and support in variousstages of work on this paper.The authors are grateful to the referee for his useful comments which led us to a significantimprovement of the manuscript. d -dimensionalpseudo-Euclidean space 245. Billiards within quadrics and their periodic trajectories 255.1. Ellipsoidal billiards 255.2. Analytic conditions for periodic trajectories 27References 291. Introduction
Pseudo-Euclidean spaces together with pseudo-Riemannian manifolds occupy avery important position in the science as a geometric background for the generalrelativity. A modern account of the mathematical aspects of the theory of relativityone may find in [CGP2010]. From a mathematical point of view, in comparison withEuclidean and Riemannian cases, apart from a natural similarity which includessome rather technical adjustments, there are some aspects where pseudo-Euclideansetting creates essentially new situations and challenging problems. The aim ofthis paper is to report about such appearances in a study of geometry of confocalquadrics and related billiard dynamics in pseudo-Euclidean spaces. Let us recallthat in the Euclidean d -dimensional space, a general family of confocal quadricscontains exactly d geometrical types of non-degenerate quadrics, and moreover,each point is the intersection of d quadrics of different types. Together with someother properties, see E1–E5 at the beginning of Section 4, these facts are crucial forintroduction of Jacobi coordinates and for applications in the theory of separablesystems, including billiards. A case of d -dimensional pseudo-Euclidean space bringsa striking difference, since a confocal family of quadrics has d + 1 geometric typesof quadrics. In addition, quadrics of the same type have a nonempty intersection.These seem to be impregnable obstacles to an extension of methods of applicationsof Jacobi type coordinates from the Euclidean case (from our paper [DR2006]) tothe pseudo-Euclidean spaces.To overcome this crucial problem, we have been forced to create an essen-tially new feature of confocal pencil of geometric quadrics in pseudo-Euclidean d -dimensional spaces: the novelty of our approach is based on introduction of anew discrete, combinatorial-geometric structure associated to a confocal pencil, acolouring in d colours, which transforms a geometric quadric from the pencil intothe union of several relativistic quadrics . It turns out that these new objects, rela-tivistic quadrics, satisfy the properties PE1–PE5, analogue of E1–E5, and lead usto a new notion of decorated Jacobi coordinates. A decorated Jacobi coordinatenow is a pair of a number, and a type-colour. They allow us to develop methodswe use in further study of billiards within confocal quadrics in pseudo-Euclideanspaces of arbitrary dimension.The study of colouring and relativistic types of quadrics, which is one of themain ingredients of the present paper, is closely related to a study of what we call the discriminant sets Σ + , Σ − attached to a confocal pencil of quadrics in pseudo-Euclidean space, as the sets of the tropical lines of quadrics. They are developable,with light-like generatrices. Their swallowtail type singularities (see [AGZV1985])are placed at the vertices of curvilinear tetrahedra T + and T − . LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 3
Billiards within ellipsoids in pseudo-Euclidean space are introduced in [KT2009],and that paper served as a motivation for our study. Along the first two followingsections, Section 2 and Section 3 some of the properties from [KT2009] are discussed,clarified or slightly improved.In Section 2, we give a necessary account on pseudo-Euclidean spaces and theirconfocal families of geometric quadrics. Our main new result in this Section is The-orem 2.3 where we give a complete description of structures of types of quadricsfrom a confocal pencil in a pseudo-Euclidean space, which are tangent to a givenline. This theorem is going to play an essential role in proving properties PE3-PE5in Section 5. In Section 3 we discuss geometric properties of elliptical billiards indimension 2. An elementary, but complete description of periodic light-like trajec-tories is derived in Theorem 3.3 and Proposition 3.6. In Section 4 we suggest a newsetting of types of confocal quadrics, an essential novelty of the pseudo-Euclideangeometry. In the three-dimensional case, we give a detailed description of discrim-inant surfaces Σ ± , the unions of tropical lines of geometric quadrics from a pencil:see Propositions 4.3, 4.7, 4.9, 4.11. We describe their singularity subsets, curvilin-ear tetrahedra T ± in Proposition 4.5. We introduce decorated Jacobi coordinatesin Section 4.5 for three-dimensional Minkowski space, and we give a detailed de-scription of the colouring in three colours, with a complete descrpition of all threerelativistic types of quadrics. In Section 4.6 we generalize the definition of decoratedJacobi coordinates to arbitrary dimension, and in Proposition 4.21 we prove theproperties PE1-PE2. In Section 5 we apply the technique of relativistic quadricsand decorated Jacobi coordinates to solve the problem of analytic description ofperiodical billiard trajectories. Theorem 5.1 gives an effective criteria to determinea type of a billiard trajectory. In Proposition 5.2 we prove the properties PE3–PE5. Finally, we give an analytic description of all periodic billiard trajectories inpseudo-Euclidean spaces in Theorem 5.3. As a corollary, in Theorem 5.4 we provea full Poncelet-type theorem for the pseudo-Euclidean spaces.2. Pseudo-Euclidean spaces and confocal families of quadrics
In this section, we first give a necessary account of basic notions connected withpseudo-Euclidean spaces, see Section 2.1. After that, in Section 2.2, we review andimprove some basic facts on confocal families of quadrics in such spaces. Our mainresult in this Section is a complete analysis of quadrics from a confocal family thatare touching a given line, as formulated in Theorem 2.3.2.1.
Pseudo-Euclidean spaces.
Pseudo-Euclidean space E k,l is a d -dimensionalspace R d with pseudo-Euclidean scalar product :(2.1) h x, y i k,l = x y + · · · + x k y k − x k +1 y k +1 − · · · − x d y d . Here, k, l ∈ { , . . . , d − } , k + l = d . Pair ( k, l ) is called signature of the space.Denote E k,l = diag(1 , , . . . , , − , . . . , − k l − h x, y i k,l = E k,l x ◦ y, where ◦ is the standard Euclidean product. The pseudo-Euclidean distance between points x , y is:dist k,l ( x, y ) = q h x − y, x − y i k,l . VLADIMIR DRAGOVI´C AND MILENA RADNOVI´C
Since the scalar product can be negative, notice that the pseudo-Euclidean distancecan have imaginary values as well.Let ℓ be a line in the pseudo-Euclidean space, and v its vector. ℓ is called: • space-like if h v, v i k,l > • time-like if h v, v i k,l < • and light-like if h v, v i k,l = 0.Two vectors x , y are orthogonal in the pseudo-Euclidean space if h x, y i k,l = 0. Notethat a light-like line is orthogonal to itself.For a given vector v = 0, consider a hyper-plane v ◦ x = 0. Vector E k,l v isorthogonal the hyper-plane; moreover, all other orthogonal vectors are collinearwith E k,l v . If v is light-like, then so is E k,l v , and E k,l v belongs to the hyper-plane. Billiard reflection in pseudo-Euclidean space.
Let v be a vector and α a hyper-planein the pseudo-Euclidean space. Decompose vector v into the sum v = a + n α of avector n α orthogonal to α and a belonging to α . Then vector v ′ = a − n α is thebilliard reflection of v on α . It is easy to see that then v is also the billiard reflectionof v ′ with respect to α .Moreover, let us note that lines containing vectores v , v ′ , a , n α are harmonicallyconjugated [KT2009].Note that v = v ′ if v is contained in α and v ′ = − v if it is orthogonal to α . If n α is light-like, which means that it belongs to α , then the reflection is not defined.Line ℓ ′ is a billiard reflection of ℓ off a smooth surface S if their intersectionpoint ℓ ∩ ℓ ′ belongs to S and the vectors of ℓ , ℓ ′ are reflections of each other withrespect to the tangent plane of S at this point. Remark 2.1.
It can be seen directly from the definition of reflection that the typeof line is preserved by the billiard reflection. Thus, the lines containing segmentsof a given billiard trajectory within S are all of the same type: they are all eitherspace-like, time-like, or light-like. If S is an ellipsoid, then it is possible to extend the reflection mapping to thosepoints where the tangent planes contain the orthogonal vectors. At such points, avector reflects into the opposite one, i.e. v ′ = − v and ℓ ′ = ℓ . For the explanation,see [KT2009]. As follows from the explanation given there, it is natural to considereach such reflection as two reflections.2.2. Families of confocal quadrics.
For a given set of positive constants a , a ,. . . , a d , an ellipsoid is given by:(2.2) E : x a + x a + · · · + x d a d = 1 . Let us remark that equation of any ellipsoid in the pseudo-Euclidean space can bebrought into the canonical form (2.2) using transformations that preserve the scalarproduct (2.1).The family of quadrics confocal with E is:(2.3) Q λ : x a − λ + · · · + x k a k − λ + x k +1 a k +1 + λ + · · · + x d a d + λ = 1 , λ ∈ R . LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 5
Unless stated differently, we are going to consider the non-degenerate case, whenset { a , . . . , a k , − a k +1 , . . . , − a d } consists of d different values: a > a > · · · > a k > > − a k +1 > · · · > − a d . For λ ∈ { a , . . . , a k , − a k +1 , . . . , − a d } , the quadric Q λ is degenerate and it coin-cides with the corresponding coordinate hyper-plane.It is natural to join one more degenerate quadric to the family (2.3): the onecorresponding to the value λ = ∞ , that is the hyper-plane at the infinity.For each point x in the space, there are exactly d values of λ , such that therelation (2.3) is satisfied. However, not all the values are necessarily real: either all d of them are real or there are d − d or d − x + tv ( t ∈ R ) is tangent to quadric Q λ if quadratic equation:(2.4) A λ ( x + tv ) ◦ ( x + tv ) = 1 , has a double root. Here we denoted: A λ = diag (cid:18) a − λ , · · · , a k − λ , a k +1 + λ , · · · , a d + λ (cid:19) . Now, calculating the discriminant of (2.4), we get:(2.5) ( A λ x ◦ v ) − ( A λ v ◦ v )( A λ x ◦ x −
1) = 0 , which is equivalent to:(2.6) d X i =1 ε i F i ( x, v ) a i − ε i λ = 0 , where(2.7) F i ( x, v ) = ε i v i + X j = i ( x i v j − x j v i ) ε j a i − ε i a j , with ε ’s given by: ε i = ( , ≤ i ≤ k ; − , k + 1 ≤ i ≤ d. The equation (2.6) can be transformed to:(2.8) P ( λ ) Q di =1 ( a i − ε i λ ) = 0 , where the coefficient of λ d − in P ( λ ) is equal to h v, v i k,l . Thus, polynomial P ( λ ) isof degree d − P ( λ ) also as of degree d −
1, taking the corresponding roots to be equalto infinity. So, light-like lines are characterized by being tangent to the quadric Q ∞ .Having this setting in mind, we note that it is proved in [KT2009] that thepolynomial P ( λ ) has at least d − R ∪ {∞} .Thus, we have: VLADIMIR DRAGOVI´C AND MILENA RADNOVI´C
Proposition 2.2.
Any line in the space is tangent to either d − or d − quadricsof the family (2.3). If this number is equal to d − , then there are two conjugatecomplex values of λ , such that the line is tangent also to these two quadrics in C d . This statement with the proof is given in [KT2009]. Let us remark that in[KT2009] is claimed that light-like line have only d − d − Q ∞ is not considered there as a member of the confocal family.As noted in [KT2009], a line having non-empty intersection with an ellipsoidfrom (2.3) will be tangent to d − Theorem 2.3.
In pseudo-Euclidean space E k,l consider a line intersecting ellipsoid E (2.2). Then this line is touching d − quadrics from (2.3). If we denote theirparameters by α , . . . , α d − and take: { b , . . . , b p , c , . . . , c q } = { ε a , . . . , ε d a d , α , . . . , α d − } ,c q ≤ · · · ≤ c ≤ c < < b ≤ b ≤ · · · ≤ b p , p + q = 2 d − , we will additionally have: • if the line is space-like, then p = 2 k − , q = 2 l , a = b p , α i ∈ { b i − , b i } for ≤ i ≤ k − , and α j + k − ∈ { c j − , c j } for ≤ j ≤ l ; • if the line is time-like, then p = 2 k , q = 2 l − , c q = − a d , α i ∈ { b i − , b i } for ≤ i ≤ k , and α j + k ∈ { c j − , c j } for ≤ j ≤ l − ; • if the line is light-like, then p = 2 k , q = 2 l − , b p = ∞ = α k , b p − = a , α i ∈ { b i − , b i } for ≤ i ≤ k − , and α j + k ∈ { c j − , c j } for ≤ j ≤ l − .Moreover, for each point on ℓ inside E , there is exactly d distinct quadrics from(2.3) containing it. More precisely, there is exactly one parameter of these quadricsin each of the intervals: ( c l − , c l − ) , . . . , ( c , c ) , ( c , , (0 , b ) , ( b , b ) , . . . , ( b k − , b k − ) . Proof.
Denote by ℓ a line that intersects E , by x = ( x , . . . , x d ) a point on ℓ which isplaced inside E , and by v = ( v , . . . , v d ) a vector of the line. Then the parameters ofquadrics touching ℓ are solutions of equation (2.5), i.e. they are roots of polynomial P ( λ ).If ℓ is space-like or time-like, that is h v, v i k,l = 0, we have that polynomial P inthen of degree d − d − ℓ is light-like, P is of degree d − d − ∞ as the ( d − A λ v ◦ v = 0, the right-hand sideof (2.5) is positive. So let us first examine roots of A λ v ◦ v .We have: A λ v ◦ v = d X i =1 v i a i − ε i λ = R ( λ ) Q di =1 ( a i − ε i λ ) , LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 7 with R ( λ ) = d X i =1 v i Y j = i ( a j − ε j λ ) . We calculate: sign R ( ε i a i ) = ε i ( − k + i , sign R ( −∞ ) = ( − l sign h v, v i k,l , sign R (+ ∞ ) = ( − k − sign h v, v i k,l . From there, we see that polynomial R , which is of degree d − ℓ and of degree d − ℓ , is changing sign at least d − h v, v i k,l = 0, and at least d − d − ε i a i +1 , ε i a i ) , i ∈ { , . . . , k − , k + 1 , . . . , d − } , and one more in ( −∞ , − a d ) or ( a , + ∞ ) if ℓ is space-like or time-like respectively.Denote roots of R by ζ , ζ , . . . , ζ d − , and order them in the following way: ζ i ∈ ( a i +1 , a i ) , for 1 ≤ i ≤ k − ,ζ j ∈ ( − a j +2 , − a j +1 ) , for k ≤ j ≤ d − ,ζ ∈ ( −∞ , − a d ) for space-like ℓ,ζ ∈ ( a , + ∞ ) for time-like ℓ,ζ = ∞ for light-like ℓ. Note only that the right-hand side of (2.5) is positive for ζ , . . . , ζ d − , it will bealso be positive for λ = 0, because ( A x ◦ v ) ≥ A v ◦ v >
0, and, since x is inside E = Q , A x ◦ x <
1. Notice that (2.5) and the equivalent expression (2.8) changessign at points ε i a i and roots of P only. Thus, these expressions have positive valuesat the endpoints of each of the d − ζ d − , ζ d − ), . . . , ( ζ k +1 , ζ k ), ( ζ k , , ζ k − ), ( ζ k − , ζ k − ), . . . , ( ζ , ζ ), and, in addition, in one of ( ζ , ζ d − ) or ( ζ , ζ ),depending if ℓ is space-like or time-like respectively. Each of these d − { ε i a i } , thus each of them needs to contain at least onemore point where expression (2.8) changes its sign, that is a root of P . Concludethat all roots of P are real, and that they are distributed exactly as stated in thisproposition.Now, let us consider quadrics from the confocal family containing point x . Theirparameters are solutions of the equation A λ x ◦ x = 1. Observe that A λ x ◦ x − − a d , − a d − ) , . . . , ( − a k +2 , − a k +1 ) , ( a k , a k − ) , . . . , ( a , a ) , thus it has one root in each of them. On the other hand, for such solutions, theright-hand side of (2.5) is positive, thus there is one solution in each of the following:( c l − , c l − ) , . . . , ( c , c ) , ( b , b ) , . . . , ( b k − , b k − ) , which makes d − c , b ), because A x ◦ x − < A λ x ◦ x − > (cid:3) The analog of Theorem 2.3 for the Euclidean space, is proved in [Aud1994].
VLADIMIR DRAGOVI´C AND MILENA RADNOVI´C
Corollary 2.4.
For each point placed inside an ellipsoid in the pseudo-Euclideanspace, there are exactly two other ellipsoids from the confocal family containing thispoint. Elliptical billiard in the Minkowski plane
In this part of the paper, we study properties of confocal families of conics inthe Minkowski plane, see Section 3.1. We derive focal properties of such familiesand the corresponding elliptical billiards. Next, in Section 3.2, we study light-liketrajectories of such billiards and derive a periodicity criterion in a simple form, seeTheorem 3.3. It is also proved in Proposition 3.6 that the flow of light-like ellipticalbilliard trajectories is equivalent to a certain rectangular billiard flow.3.1.
Confocal conics in Minkowski plane.
Here, we give a review of basicproperties of families of confocal conics in the Minkowski plane.Denote by(3.1) E : x a + y b = 1an ellipse in the plane, with a , b being fixed positive numbers.The associated family of confocal conics is:(3.2) C λ : x a − λ + y b + λ = 1 , λ ∈ R . The family is shown on Figure 1. We may distinguish the following three sub-
Figure 1.
Family of confocal conics in the Minkowski plane.families in the family (3.2):
LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 9 • for λ ∈ ( − b, a ), conic C λ is an ellipse; • for λ < − b , conic C λ is a hyperbola with x -axis as the major one; • for λ > a , it is a hyperbola again, but now its major axis is y -axis.In addition, there are three degenerated quadrics: C a , C b , C ∞ corresponding to y -axis, x -axis, and the line at the infinity respectively. Note the following threepairs of foci: F ( √ a + b, F ( −√ a + b, G (0 , √ a + b ), G (0 , −√ a + b ); and H (1 : − H (1 : 1 : 0) on the line at the infinity.We notice four distinguished lines: x + y = √ a + b, x + y = −√ a + b,x − y = √ a + b, x − y = −√ a + b. These lines are common tangents to all conics from the confocal family.It is elementary and straightforward to prove the following
Proposition 3.1.
For each point on ellipse C λ , λ ∈ ( − b, a ) , either sum or differenceof its Minkowski distances from the foci F and F is equal to √ a − λ ; either sumor difference of the distances from the other pair of foci G , G is equal to i √ b + λ .Either sum or difference of the Minkowski distances of each point of hyperbola C λ , λ ∈ ( −∞ , − b ) , from the foci F and F is equal to √ a − λ ; for the other pairof foci G , G , it is equal to √− b − λ .Either sum or difference of the Minkowski distances of each point of hyperbola C λ , λ ∈ ( a, + ∞ ) , from the foci F and F is equal to i √ λ − a ; for the other pairof foci G , G , it is equal to i √ b + λ . Billiard within an ellipse also have the famous focal property:
Proposition 3.2.
Consider a billiard trajectory within ellipse E given by equation(3.1) in the Minkowski plane, such that the line containing the initial segment ofthe trajectory passes through a focus of the confocal family (3.2), say F , G , or H .If the tangent line to E at the reflection point of this segment is not light-like, thenthe line containing the next segment will pass through F , G , or H respectively. In other words, the segments of one billiard trajectory will alternately containfoci of one of the pairs ( F , F ), ( G , G ), ( H , H ). The only exception are succes-sive segments obtained by the reflection on the light-like tangent. Such segmentscoincide.3.2. Light-like trajectories of the elliptical billiard.
In this section, we aregoing to study light-like trajectories of elliptical billiard in the Minkowski plane.An example of such a billiard trajectory is shown on Figure 2.Successive segments of such trajectories are orthogonal to each other. Noticethat this implies that a trajectory can close only after even number of reflections.
Periodic light-like trajectories.
The analytic condition for n -periodicity of light-likebilliard trajectory within the ellipse E given by equation (3.1) can be written downapplying the more general Cayley’s condition for closedness of a polygonal lineinscribed in one conic and circumscribed about another one [Cay1853, Cay1854], Figure 2.
Light-like billiard trajectory.see also [Leb1942, GH1978]:(3.3) det B B . . . B m +1 B B . . . B m +2 . . . . . . . . . . . .B m +1 B m +2 . . . B m − = 0 , with n = 2 m. Here, p ( a − λ )( b + λ ) = B + B λ + B λ + . . . is the Taylor expansion around λ = 0.Now, we are going to derive analytic condition for periodic light-like trajectoriesin another way, which will lead to a more compact form of (3.3). Theorem 3.3.
Light-like billiard trajectory within ellipse E is periodic with period n , where n is an even integer if and only if (3.4) arc tan r ab ∈ (cid:26) kπn (cid:12)(cid:12)(cid:12) ≤ k < n , (cid:16) k, n (cid:17) = 1 (cid:27) . Proof.
Applying the following affine transformation:( x, y ) ( x √ b, y √ a ) , the ellipse is transformed into a circle. The light-like lines are transformed into linesparallel to two directions, with the angle between them equal to 2arc tan p a/b .Since the dynamics on the boundary is the rotation by this angle, the proof iscomplete. (cid:3) As an immediate consequence, we get
Corollary 3.4.
For a given even integer n , the number of different ratios of theaxes of ellipses having n -periodic light-like billiard trajectories is equal to: ( ϕ ( n ) / if n is not divisible by ,ϕ ( n ) / if n is divisible by .ϕ is the Euler’s totient function, i.e. the number of positive integers not exceeding n that are relatively prime to n . LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 11
Remark 3.5.
There are four points on E where the tangents are light-like. Thosepoints cut four arcs on E . An n -periodic trajectory within E hits each one of a pairof opposite arcs exactly k times, and n − k times the arcs from the other pair.Light-like trajectories in ellipses and rectangular billiards. Proposition 3.6.
The flow of light-like billiard trajectories within ellipse E is tra-jectorially equivalent to the flow of those billiard trajectories within a rectanglewhose angle with the sides is π . The ratio of the sides of the rectangle is equal to: π r ab − . Proof.
For the ellipses with periodic light-like trajectories, the theorem follows fromTheorem 3.3 and Remark 3.5.In other cases, the number π r ab − (cid:3) Remark 3.7.
The flow of light-light billiard trajectories within a given oval inthe Minkowski plane will be trajectorially equivalent to the flow of certain trajecto-ries within a rectangle whenever invariant measure m on the oval exists such that m ( AB ) = m ( CD ) and m ( BC ) = m ( AD ) , where A , B , C , D are points on the ovalwhere the tangents are light-like. Relativistic quadrics
In this section, we are going to introduce relativistic quadrics , as the main newobject of the present paper, which is going to become a main tool in our furtherstudy of billiard dynamics. The point is that geometric quadrics of a confocalpencil and their types in pseudo-Euclidean spaces do not satisfy usual properties ofconfocal quadrics in Euclidean spaces, including those necessary for applications inbilliard dynamics. For example, we have already mentioned, in the d -dimensionalEuclidean space, there are d geometric types of quadrics, while in d -dimensionalpseudo-Euclidean space, there are d + 1 geometric types of quadrics. Thus, we firstselect those important properties of confocal families in the Euclidean spaces andaxiomatize them as E1–E5 in Section 4.1. Then, in Section 4.2 we consider thetwo-dimensional case, the Minkowski plane, and we study appropriate relativisticconics, where [BM1962] may be seen as a historic origin of ideas of relativistic conics.In Section 4.3 we study geometrical types of quadrics in a confocal family in thethree-dimensional Minkowski space. Next, in Section 4.4, we analyze tropic curveson quadrics in the three-dimensional case and we introduce an important notionof discriminant sets Σ ± corresponding to a confocal family. The main facts aboutdiscriminant sets we prove in Propositions 4.3, 4.7, 4.9, 4.11. Then, we study curvedtetrahedra T ± , which represent singularity sets of Σ ± and we collect related resultsin Proposition 4.5. As the next important step, we introduce decorated Jacobicoordinates in Section 4.5 for three-dimensional Minkowski space, and we give adetailed description of the colouring into three colours. Each colour corresponds to arelativistic type, and we describe decomposition of a geometric quadric of each of the four geometric types into relativistic quadrics. This appears to be a rather involvedcombinatorial-geometric problem, and we solve it by using previous analysis ofdiscriminant surfaces. We give a complete description of all three relativistic typesof quadrics. In Section 4.6, we generalize definition of decorated Jacobi coordinatesin arbitrary dimensions, and, finally, in Proposition 4.21 we prove properties PE1and PE2, the pseudo-Euclidean analogues of E1 and E2.4.1. Confocal quadrics and their types in the Euclidean space.
A generalfamily of confocal quadrics in the d -dimensional Euclidean space is given by:(4.1) x b − λ + · · · + x d b d − λ = 1 , λ ∈ R with b > b > · · · > b d > E d is the intersection of exactly d quadrics from(4.1); moreover, all these quadrics are of different geometrical types;E2 family (4.1) contains exactly d geometrical types of non-degenerate quadrics– each type corresponds to one of the disjoint intervals of the parameter λ :( −∞ , b d ), ( b d , b d − ), . . . , ( b , b ).The parameters ( λ , . . . , λ d ) corresponding to the quadrics of (4.1) that containa given point in E d are called Jacobi coordinates . We order them λ > · · · > λ d .Now, let us consider the motion of a billiard ball within an ellipsoid, denote itby E , of the family (4.1). Without losing generality, take that the parameter λ corresponding to this ellipsoid be equal to 0. Recall that, by Chasles’ theorem,each line in E d is touching some d − d − E has the same d − caustics – denote their parameters by β , . . . , β d − , and introduce the following: { ¯ b , . . . , ¯ b d } = { b , . . . , b d , , β , . . . , β d − } , such that ¯ b ≥ ¯ b ≥ · · · ≥ ¯ b d . In this way, we will have 0 = ¯ b d < ¯ b d − , b = ¯ b > ¯ b . Moreover, it is always: β i ∈ { ¯ b i , ¯ b i +1 } , for each i ∈ { , . . . , d } , see[Aud1994].Now, we can summarize the main properties of the flow of the Jacobi coordinatesalong the billiard trajectories:E3 along a fixed billiard trajectory, the Jacobi coordinate λ i (1 ≤ i ≤ d ) takesvalues in segment [¯ b i − , ¯ b i ];E4 along a trajectory, each λ i achieves local minima and maxima exactly attouching points with corresponding caustics, intersection points with cor-responding coordinate hyper-planes, and, for i = d , at reflection points;E5 values of λ i at those points are ¯ b i − , ¯ b i ; between the critical points, λ i ischanged monotonously.Those properties represent the key in the algebro-geometrical analysis of thebilliard flow.At the first glance, it seems that fine properties like those do not take place inthe pseudo-Euclidean case. In a d -dimensional pseudo-Euclidean space, a general LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 13 confocal family contains d + 1 geometrical types of quadrics and, in addition, quad-rics of the same geometrical type have non-empty intersection. Because of that, itlooks much more complicated to analyze the billiard flow following the Jacobi-typecoordinates.In the rest of this section, we are going to overcome this important problem,by introducing a new notion of relativistic quadrics. In our setting, we equip ageometric pencil of quadrics by an additional structure, a decoration , which de-composes geometric quadrics of the pencil into coloured subsets which form newtypes of relativistic quadrics. The new notion of relativistic quadrics, i.e. colouredgeometric quadrics , is more suitable for the pseudo-Euclidean geometry. In return,we will obtain a possibility to introduce a new system of coordinates, nontrivialpseudo-Euclidean analogue of Jacobi elliptic coordinates, which is going to playa fundamental role in the sequel, as a powerfull tool in the study of separablesystems.4.2. Confocal conics in the Minkowski plane.
Let us consider first the case ofthe 2-dimensional pseudo-Euclidean space E , , namely the Minkowski plane. Wementioned in Section 3.1 that a family of confocal conics in the Minkowski planecontains three geometrical types of conics: ellipses, hyperbolas with x -axis as themajor one, and hyperbolas with y -axis as the major one, as shown on Figure 1.However, it is more natural to consider relativistic conics , which are analysed in[BM1962]. In this section, we give a brief account of that analysis.Consider points F ( √ a + b,
0) and F ( −√ a + b,
0) in the plane.For a given constant c ∈ R + ∪ i R + , a relativistic ellipse is the set of points X satisfying: dist , ( F , X ) + dist , ( F , X ) = 2 c, while a relativistic hyperbola is the union of the sets given by the following equations:dist , ( F , X ) − dist , ( F , X ) = 2 c, dist , ( F , X ) − dist , ( F , X ) = 2 c. Relativistic conics can be described as follows.0 < c < √ a + b : The corresponding relativistic conics lie on ellipse C a − c fromfamily (3.2). The ellipse C a − c is split into four arcs by touching points withthe four common tangent lines; thus, the relativistic ellipse is the union ofthe two arcs intersecting the y -axis, while the relativistic hyperbola is theunion of the other two arcs. c > √ a + b : The relativistic conics lie on C a − c – a hyperbola with x -axis asthe major one. Each branch of the hyperbola is split into three arcs bytouching points with the common tangents; thus, the relativistic ellipse isthe union of the two finite arcs, while the relativistic hyperbola is the unionof the four infinite ones. c is imaginary: The relativistic conics lie on hyperbola C a − c – a hyperbolawith y -axis as the major one. As in the previous case, the branches aresplit into six arcs in total by common points with the four tangents. Therelativistic ellipse is the union of the four infinite arcs, while the relativistichyperbola is the union of the two finite ones.The conics are shown on Figure 3. Figure 3.
Relativistic conics in the Minkowski plane: relativisticellipses are represented by full lines, and hyperbolas by dashedones.
Remark 4.1.
All relativistic ellipses are disjoint with each other, as well as allrelativistic hyperbolas. Moreover, at the intersection point of a relativistic ellipsewhich is a part of the geometric conic C λ from the confocal family (3.2) and arelativistic hyperbola belonging to C λ , it is always λ < λ . This remark will serve as a motivation for introducing relativistic types of quad-rics in higher-dimensional pseudo-Euclidean spaces.4.3.
Confocal quadrics in the three-dimensional Minkowski space andtheir geometrical types.
Let us start with the three-dimensional Minkowskispace E , . A general confocal family of quadrics in this space is given by:(4.2) Q λ : x a − λ + y b − λ + z c + λ = 1 , λ ∈ R , with a > b > c > • z -axis, for λ ∈ ( −∞ , − c ); • ellipsoids, corresponding to λ ∈ ( − c, b ); • y -axis, for λ ∈ ( b, a ); • λ ∈ ( a, + ∞ ) – these hyperboloids are orientedalong z -axis.In addition, there are four degenerated quadrics: Q a , Q b , Q − c , Q ∞ , that is planes x = 0, y = 0, z = 0, and the plane at the infinity respectively. In the coordinateplanes, we single out the following conics: LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 15 • hyperbola C yza : − y a − b + z c + a = 1 in the plane x = 0; • ellipse C xzb : x a − b + z c + b = 1 in the plane y = 0; • ellipse C xy − c : x a + c + y b + c = 1 in the plane z = 0.4.4. Tropic curves on quadrics in the three-dimensional Minkowski spaceand discriminant sets Σ ± . On each quadric, notice the tropic curves – the setof points where the induced metrics on the tangent plane is degenerate.Since the tangent plane at point ( x , y , z ) of Q λ is given by the equation: xx a − λ + yy b − λ + zz c + λ = 1 , and the induced metric is degenerate if and only if the parallel plane that containsthe origin is tangent to the light-like cone x + y − z = 0, i.e.: x ( a − λ ) + y ( b − λ ) − z ( c + λ ) = 0 , we come to the statement formulated in [KT2009]: Proposition 4.2.
The tropic curves on Q λ is the intersection of the quadric withthe cone: x ( a − λ ) + y ( b − λ ) − z ( c + λ ) = 0 . Now, consider the set of the tropic curves on all quadrics of the family (4.2).From Proposition 4.2, we get:
Proposition 4.3.
The union of the tropic curves on all quadrics of (4.2) is a unionof two ruled surfaces Σ + and Σ − which can be parametrically represented as: Σ + : x = a − λ √ a + c cos t, y = b − λ √ b + c sin t, z = ( c + λ ) s cos ta + c + sin tb + c , Σ − : x = a − λ √ a + c cos t, y = b − λ √ b + c sin t, z = − ( c + λ ) s cos ta + c + sin tb + c , with λ ∈ R , t ∈ [0 , π ) . The intersection of these two surfaces is an ellipse in the xy -plane: Σ + ∩ Σ − : x a + c + y b + c = 1 , z = 0 . The two surfaces Σ + , Σ − are developable as embedded into Euclidean space. More-over, their generatrices are all light-like.Proof. Denote by r = ( x, y, z ) an arbitrary point of Σ + ∪ Σ − , and by n the cor-responding unit normal vector, n = r λ × r t / | r λ × r t | . Here, by × we denotedthe vector product in the three-dimensional Euclidean space. Then, the Gaussiancurvature of the surface is K = ( LN − M ) / ( EG − F ), with L = r λλ · n = 0, M = r λt · n = 0, N = r tt · n , E = r λ · r λ , F = r λ · r t , G = r t · r t . Since EG − F = ( a + b − λ + ( b − a ) cos(2 t )) a + c )( b + c ) , the Gaussian curvature K is equal to zero. (cid:3) Surfaces Σ + and Σ − from Proposition 4.3 are represented on Figure 4. Figure 4.
The union of all tropic curves of a confocal family.In [Pei1999], a definition of generalization of Gauss map to surfaces in the three-dimensional Minkowski space is suggested. Namely, the pseudo vector product in-troduced as: x ∧ y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x y y y e e − e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( x y − x y , x y − x y , − ( x y − x y )) = E , ( x × y ) . It is easy to check that h x ∧ y , x i , = h x ∧ y , y i , = 0.Then, for surface S : U → E , , with U ⊂ R , the Minkowski Gauss map isdefined as: G : U → RP , G ( x , x ) = P (cid:18) ∂S∂x ∧ ∂S∂x (cid:19) , where P : R \ { (0 , , } → RP is the usual projectivization. Lemma 4.4.
The Minkowski Gauss map of surfaces Σ ± is singular at all points.Proof. This follows from the fact that r λ ∧ r t is light-like for all λ and t . (cid:3) Since the pseudo-normal vectors to Σ ± are all light-like, these surfaces are light-like developable , as defined in [CI2010]. There, a classification of such surfaces isgiven – each is one part-by-part contained in the following: • a light-like plane; • a light-like cone; • a tangent surface of a light-like curve. LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 17
Since Σ + and Σ − are contained neither in a plane nor in a cone, we expect thatthey will be tangent surfaces of some light-like curve, which is going to be shownin the sequel, see Corollary 4.10 later in this section.On each of the surfaces Σ + , Σ − , we can notice that tropic lines correspondingto 1-sheeted hyperboloids oriented along y -axies form one curved tetrahedron, seeFigure 4. Denote the tetrahedra by T + and T − respectively: they are symmetricwith respect to the xy -plane. On Figure 5, tetrahedron T + ⊂ Σ + is shown. Figure 5.
Curved thetrahedron T + : the union of all tropic curveson Σ + corresponding to λ ∈ ( b, a ).Let us summarize the properties of these tetrahedra. Proposition 4.5.
Consider the subset T + of Σ + determined by the condition λ ∈ [ b, a ] . This set is a curved tetrahedron, with the following properties: • its verteces are: V (cid:18) a − b √ a + c , , b + c √ a + c (cid:19) , V (cid:18) − a − b √ a + c , , b + c √ a + c (cid:19) ,V (cid:18) , a − b √ b + c , a + c √ b + c (cid:19) , V (cid:18) , − a − b √ b + c , a + c √ b + c (cid:19) ; • the shorter arcs of conics C xzb and C yza determined by V , V and V , V respectively are two edges of the tetrahedron; • those two edges represent self-intersection of Σ + ; • other four edges are determined by the relation: (4.3) − a − b + 2 λ + ( a − b ) cos 2 t = 0 , • those four edges are cuspidal edges of Σ + ; • thus, at each vertex of the tetrahedron, a swallowtail singularity of Σ + oc-curs.Proof. Equation (4.3) is obtained from the condition r t × r λ = 0. (cid:3) Lemma 4.6.
The tropic curves of the quadric Q λ represent exactly the locus ofpoints ( x, y, z ) where equation (4.4) x a − λ + y b − λ + z c + λ = 1 has λ as a multiple root.Proof. Without losing generality, take λ = 0. Equation (4.4) is equivalent to:(4.5) λ + q λ + q λ + q = 0 , with q = − x − y + z + a + b − c,q = x ( b − c ) + y ( a − c ) − z ( a + b ) − ab + bc + ac,q = x bc + y ac + z ab − abc. Polynomial (4.5) has λ = 0 as a double zero if and only if q = q = 0. Obviously, q = 0 is equivalent to the condition that ( x, y, z ) belongs to Q . On the otherhand, we have: q = x ( b − c ) + y ( a − c ) − z ( a + b ) − ab + bc + ac = ( ab − bc + ac ) (cid:18) x a + y b + z c − (cid:19) − abc (cid:18) x a + y b − z c (cid:19) , which is needed. (cid:3) Proposition 4.7.
A tangent line to the tropic curve of a non-degenarate quadricof the family (4.2) is always space-like, except on a -sheeted hyperboloid orientedalong y -axis.Tangent lines of a tropic on -sheeted hyperboloids oriented along y -axis are light-like exactely at four points, while at other points of the tropic curve, the tangentsare space-like.Moreover, a tangent line to the tropic of a quadric from (4.2) belongs to thequadric if and only if it is light-like.Proof. The tropic curves on Q λ , similarly as in Proposition 4.3, can be representedas: x ( t ) = a − λ √ a + c cos t, y ( t ) = b − λ √ b + c sin t, z ( t ) = ± ( c + λ ) s cos ta + c + sin tb + c , with t ∈ [0 , π ).We calculate: ˙ x + ˙ y − ˙ z = ( a + b − λ − ( a − b ) cos 2 t ) a + b + 2 c − ( a − b ) cos 2 t ) , which is always non-negative, and may attain zero only if λ ∈ [ b, a ]. For λ ∈ ( b, a ),there are exactly four values of t in [0 , π ) where the expression attains zero.Now, fix t ∈ [0 , π ) and λ ∈ R . The tangent line to the tropic of Q λ at( x ( t ) , y ( t ) , z ( t )) is completely contained in Q λ if and only if, for each τ :( x ( t ) + τ ˙ x ( t )) a − λ + ( y ( t ) + τ ˙ y ( t )) b − λ + ( z ( t ) + τ ˙ z ( t )) c + λ = 1 , LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 19
Figure 6.
The tropic curves and its light-like tangents on a hyperboloid.which is equivalent to: a + b − λ − ( a − b ) cos 2 ta + b + 2 c − ( a − b ) cos 2 t = 0 . (cid:3) Remark 4.8.
In other words, the only quadrics of the family (4.2) that may containa tangent to its tropic curve are -sheeted hyperboloids oriented along y -axis, andthose tangents are always light-like. The tropic curves and their light-like tangentson such an hyperboloid are shown on Figure 6. Notice that equations obtained in the proof of Proposion 4.7 are equivalent toequation (4.3) of Proposition 4.5, which leads to the following:
Proposition 4.9.
Each generatrix of Σ + and Σ − is contained in one -sheetedhyperboloid oriented along y -axis from (4.2). Moreover, such a generatrix is touch-ing at the same point one of the tropic curves of the hyperboloid and one of thecusp-like edges of the corresponding curved tetrahedron. Corollary 4.10.
Surfaces Σ + and Σ − are tangent surfaces of the cuspidal edgesof thetrahedra T + and T − respectively. In next propositions, we give further analysis the light-like tangents to the tropiccurves on an 1-sheeted hyperboloid oriented along y -axis. Proposition 4.11.
For a fixed λ ∈ ( b, a ) , consider a hyperboloid Q λ from (4.2)and an arbitrary point ( x, y, z ) on Q λ . Equation (4.4) has, along with λ , twoother roots in C : denote them by λ and λ . Then λ = λ if and only if ( x, y, z ) isplaced on a light-like tangent to a tropic curve of Q λ . Proof.
Follows from the fact that the light-like tangents are contained in the Σ + ∪ Σ − , see Proposition 4.3, Lemma 4.6, and Proposition 4.9. (cid:3) Proposition 4.12.
Two light-like lines on a one-sheeted hyperboloid oriented along y -axis from (4.2) are either skew or intersect each other on a degenerate quadricfrom (4.2).Proof. Follows from the fact that the hyperboloid is symmetric with respect to thecoordinate planes. (cid:3)
Lemma 4.13.
Consider a non-degenerate quadric Q λ , which is not a hyperboloidoriented along y -axis, i.e. λ [ b, a ] ∪ {− c } . Then each point of Q λ which is noton one of the tropic curves is contained in two additional distinct quadrics from thefamily (4.2).Consider two points A , B of Q λ , which are placed in the same connected compo-nent bounded by the tropic curves, and denote by λ ′ A , λ ′′ A and λ ′ B , λ ′′ B the solutions,different than λ , of equation (4.4) corresponding to A and B respectively. Then,if λ is smaller (resp. bigger, between) than λ ′ A , λ ′′ A , it is also smaller (resp. bigger,between) than λ ′ B , λ ′′ B . Lemma 4.14.
Let Q λ be a hyperboloid oriented along y -axis, λ ∈ ( b, a ) , and A , B two points of Q λ , which are placed in the same connected component boundedby the tropic curves and light-like tangents. Then, if A is contained in two morequadrics from the family (4.2), the same is true for B .In this case, denote by λ ′ A , λ ′′ A and λ ′ B , λ ′′ B the real solutions, different than λ ,of equation (4.4) corresponding to A and B respectively. Then, if λ is smaller(resp. bigger, between) than λ ′ A , λ ′′ A , it is also smaller (resp. bigger, between) than λ ′ B , λ ′′ B .On the other hand, if A is not contained in any other quadric from (4.2), thenthe same is true for all points of its connected component.Proof. The proof of both Lemmae 4.13 and 4.14 follows from the fact that thesolutions of (4.4) are continuously changed through the space and that two of thesolutions coincide exactly on tropic curves and their light-like tangents. (cid:3)
Generalized Jacobi coordinates and relativistic quadrics in the three-dimensional Minkowski space.Definition 4.15.
Generalized Jacobi coordinates of point ( x, y, z ) in the three-di-mensional Minkowski space E , is the unordered triplet of solutions of equation(4.4). Note that any of the following cases may take place: • generalized Jacobi coordinates are real and different; • only one generalized Jacobi coordinate is real; • generalized Jacobi coordinates are real, but two of them coincide; • all three generalized Jacobi coordinates are equal.Lemmae 4.13 and 4.14 will help us to define relativistic types of quadrics inthe 3-dimensional Minkowski space. Consider connected components of quadricsfrom (4.2) bounded by tropic curves and, for 1-sheeted hyperboloids oriented along LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 21 y -axis, their light-light tangent lines. Each connected component will represent arelativistic quadric . Definition 4.16.
A component of quadric Q λ is of relativistic type E if, at eachof its points, λ is smaller than the other two generalized Jacobi coordinates.A component of quadric Q λ is of relativistic type H if, at each of its points, λ is between the other two generalized Jacobi coordinates.A component of quadric Q λ is of relativistic type H if, at each of its points, λ is bigger than the other two generalized Jacobi coordinates.A component of quadric Q λ is of relativistic type 0 if, at each of its points, λ is the only real generalized Jacobi coordinate. Lemmae 4.13 and 4.14 guarantee that types of relativistic quadrics are well-defined, i.e. that to each such a quadric a unique type E , H , H , or 0 can beassigned. Definition 4.17.
Suppose ( x, y, z ) is a point of the three-dimensional Minkowskispace E , where equation 4.4 has real and different solutions. Decorated Jacobicoordinates of that point is the ordered triplet of pairs: ( E, λ ) , ( H , λ ) , ( H , λ ) , of generalized Jacobi coordinates and the corresponding types of relativistic quadrics. Now, we are going to analyze the arrangement of the relativistic quadrics. Letus start with their intersections with the coordinate planes.
Intersection with the xy -plane. In the xy -plane, the Minkowski metrics is reducedto the Euclidean one. The family (4.2) is intersecting this plane by the followingfamily of confocal conics:(4.6) C xyλ : x a − λ + y b − λ = 1 . We conclude that the xy -plane is divided by ellipse C xy − c into two relativisticquadrics: • the region within C xy − c is a relativistic quadric of E -type; • the region outside this ellipse is of H -type.Moreover, the types of relativic quadrics intersecting the xy -plane are: • the components of ellipsoids are of H -type; • the components of 1-sheeted hyperboloids oriented along y -axis of H -type; • the components of 1-sheeted hyperboloids oriented along z -axis of E -type. Intersection with the xz -plane. In the xz -plane, the reduced metrics is the Min-kowski one. The intersection of family (4.2) with this plane is the following familyof confocal conics:(4.7) C xzλ : x a − λ + z c + λ = 1 . The plane is divided by ellipse C xzb and the four joint tangents of (4.7) into 13parts: • the part within C xzb is a relativistic quadric of H -type; • four parts placed outside of C xzb that have non-empty intersection with the x -axis are of H -type; • four parts placed outside of C xzb that have non-empty intersection with the z -axis are of E -type; • the four remaining parts are of 0-type and no quadric from the family (4.2),except the degenerated Q b , is passing through any of their points. Intersection with the yz -plane. As in the previous case, in the yz -plane, the reducedmetrics is the Minkowski one. The intersection of family (4.2) with this plane isthe following family of confocal conics:(4.8) C yzλ : y b − λ + z c + λ = 1 . The plane is divided by hyperbola C yza and joint tangents of (4.8) into 15 parts: • the two convex parts determined by C yza are relativistic quadric of H -type; • five parts placed outside of C yza that have non-empty intersection with thecoordinate axes are of H -type; • four parts, each one placed between C yza and one of the joint tangents of(4.8) are of E -type; • through points of the four remaining parts no quadric from the family (4.2),except the degenerated Q a , is passing.Intersection of relativistic quadrics with the coordinate planes is shown in Figure7. There, the type E quadrics are coloured in dark gray, type H medium gray,type H light gray, while quadrics of type 0 are white. Curves C xy − c , C xzb , C yza arealso white in the figure.Let us notice that from the above analysis, using Lemmae 4.13 and 4.14, we candetermine the type of each relativistic quadric with a non-empty intersection withsome of the coordinate hyper-planes.1 -sheeted hyperboloids oriented along z -axis: λ ∈ ( −∞ , − c ) . Such a hyperboloid isdivided by its tropic curves into three connected components – two of them areunbounded and mutually symmetric with respect to the xy -plane, while the thirdone is the bounded annulus placed between them. The two symmetric ones are of H -type, while the third one is of E -type. Ellipsoids: λ ∈ ( − c, b ) . An ellipsoid is divided by the tropic curves into threebounded connected components – two of them are mutually symmetric with respectto the xy -plane, while the third one is the annulus placed between them. In thiscase, the symmetric components represent relativistic quadrics of E -type. Theannulus is of H -type.1 -sheeted hyperboloids oriented along y -axis: λ ∈ ( b, a ) . The decomposition of thosehyperboloids into relativistic quadrics is more complicated and interesting than forthe other types of quadrics from (4.2). By its two tropic curves and their eight light-like tangent lines, such a hyperboloid is divided into 28 connected components: • two bounded components placed inside the tropic curves are of H -type; • four bounded components placed between the tropic curves and light-liketangents, such that they have non-empty intersections with xz -plane are of H -type; LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 23 x yz
Figure 7.
Intersection of relativistic quadrics with coordinate planes. • four bounded components placed between the tropic curves and light-liketangents, such that they have non-empty intersections with yz -plane are of E -type; • two bounded components, each limited by four light-like tangents, are of H -type; • four unbounded components, each limited by two light-like tangents, suchthat they have non-empty intersections with the xy -plane, are of H -type; • four unbounded components, each limited by two light-like tangents, suchthat they have non-empty intersections with the yz -plane, are of E -type; • eight unbounded components, each limited by four light-like tangents, aresets of points not contained in any other quadric from (4.2). -sheeted hyperboloids: λ ∈ ( a, + ∞ ) . Such a hyperboloid is by its tropic curvesdivided into four connected components: two bounded ones are of H -type, whilethe two unbounded are of H -type.4.6. Decorated Jacobi coordinates and relativistic quadrics in d -dimen-sional pseudo-Euclidean space. Inspired by results obtained in Sections 4.2 and4.5, now we are going to introduce relativistic quadrics and their types in confocalfamily (2.3) in the d -dimensional pseudo-Euclidean space E k,l . Definition 4.18.
Generalized Jacobi coordinates of point x in the d -dimensionalpseudo-Euclidean space E k,l is the unordered d -tuple of solutions λ of equation: (4.9) x a − λ + · · · + x k a k − λ + x k +1 a k +1 + λ + · · · + x d a d + λ = 1 . As already mentioned in Section 2.2, this equation has either d or d − d of points x in R d where equation (4.9) has multiple solutions is analgebraic hyper-suface. Σ d divides each quadric from (2.3) into several connectedcomponents. We call these components relativistic quadrics .Since the generalized Jacobi coordinates depend continuosly on x , the followingdefinition can be made: Definition 4.19.
We say that a relativistic quadric placed on Q λ is of type E if, ateach of its points, λ is smaller than the other d − generalized Jacobi coordinates.We say that a relativistic quadric placed on Q λ is of type H i (1 < i < d − if,at each of its points, λ is greater than other i generalized Jacobi coordinates, andsmaller than d − i − of them.We say that a relativistic quadric placed on Q λ is of type 0 i (0 < i < d − if,at each of its points, λ is greater than other i real generalized Jacobi coordinates,and smaller than d − i − of them. It would be interesting to analyze properties of the discriminant manifold Σ d , aswell as the combinatorial structure of the arrangement of relativistic quadrics, asit is done in Section 4.5 for d = 3. Remark that this description would have [ d/ k and l . Definition 4.20.
Suppose ( x , . . . , x d ) is a point of the d -dimensional Minkowskispace E k,l where equation (4.9) has real and different solutions. Decorated Jacobicoordinates of that point is the ordered d -tuplet of pairs: ( E, λ ) , ( H , λ ) , . . . , ( H d − , λ d ) , of generalized Jacobi coordinates and the corresponding types of relativistic quadrics. Since we will consider billiard system within ellipsoids in the pseudo-Euclideanspace, it is of interest to analyze behaviour of decorated Jacobi coordinates insidean ellipsoid.
Proposition 4.21.
Let E be ellipsoid in E k,l given by (2.2). We have: PE1 each point inside E is the intersection of exactly d quadrics from (2.3);moreover, all these quadrics are of different relativistic types; LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 25
PE2 the types of these quadrics are E , H , . . . , H d − – each type correspondsto one of the disjoint intervals of the parameter λ : ( − a d , − a d − ) , ( − a d − , − a d − ) , . . . , ( − a k +1 , , (0 , a k ) , ( a k , a k − ) , . . . , ( a , a ) . Proof.
The function given by the left-hand side of (4.9) is continous and strictlymonotonous in each interval ( − a d , − a d − ), ( − a d − , − a d − ), . . . , ( − a k +2 , − a k +1 ),( a k , a k − ), . . . , ( a , a ) with infinite values at their endpoints. Thus, equation (4.9)has one solution in each of them. On the other hand, in ( − a k +1 , a k ), the functionis tending to + ∞ at the endpoints, and has only one extreme value – the minimum.Since the value of the function for λ = 0 is less than 1 for a point inside E , it followsthat equation (4.9) will have two solutions in ( − a k +1 , a k ) – one positive and onenegative. (cid:3) In Proposition 4.21 we proved the relativistic analogs of properties E1, E2 fromSection 4.1 for the Euclidean case.5.
Billiards within quadrics and their periodic trajectories
In this section, we are going to derive first further properties of ellipsoidal bil-liards in the pseudo-Euclidean spaces. In Section 5.1 we find in Theorem 5.1 a sim-ple and effective criterion for determining the type of a billiard trajectory, knowingits caustics. Then we derive properties PE3–PE5 in Propostion 5.2. In Section 5.2we prove the generalization of Poncelet theorem for ellipsoidal billiards in pseudo-Euclidean spaces and derive the corresponding Cayley-type conditions, giving acomplete analytical description of periodic billiard trajectories in arbitrary dimen-sion. These results are contained in Theorems 5.3 and 5.4.5.1.
Ellipsoidal billiards.
Ellipsoidal billiard.
Billiard motion within an ellipsoid in the pseudo-Euclideanspace is a motion which is uniformly straightforward inside the ellipsoid, and obeysthe reflection law on the boundary. Further, we will consider billiard motion withinellipsoid E , given by equation (2.2) in E k,l . The family of quadrics confocal with E is (2.3).Since functions F i given by (2.7) are integrals of the billiard motion (see [Mos1980,Aud1994, KT2009]), we have that for each zero λ of the equation (2.6), the corre-sponding quadric Q λ is a caustic of the billiard motion, i.e. it is tangent to eachsegment of the billiard trajectory passing through the point x with the velocityvector v .Note that, according to Theorem 2.3, for a point placed inside E , there are d realsolutions of equation (4.9). In other words, there are d quadrics from the family(2.3) containing such a point, although some of them may be multiple. Also, byProposition 2.2 and Theorem 2.3, a billiard trajectory within an ellipsoid will alwayshave d − E will beof the same type. Now, we can apply the reasoning from Section 2.2 to billiardtrajectories: Theorem 5.1.
In the d -dimensional pseudo-Euclidean space E k,l , consider a bil-liard trajectory within ellipsoid E = Q , and let quadrics Q α , . . . , Q α d − fromthe family (2.3) be its caustics. Then all billiard trajectories within E sharing thesame caustics are of the same type: space-like, time-like, or light-like, as the initialtrajectory. Moreover, the type is determined as follows: • if ∞ ∈ { α , . . . , α d − } , the trajectories are light-like; • if ( − l · α · . . . · α d − > , the trajectories are space-like; • if ( − l · α · . . . · α d − < , the trajectories are time-like.Proof. Since values of functions F i given by (2.7) are preserved by the billiardreflection and d X i =1 F i ( x, v ) = h v, v i k,l , the type of the billiard trajectory depends on the sign of the sum P di =1 F i ( x, v ).From the equivalence of relations (2.6) and (2.8), it follows that the sum dependsonly of the roots of P , i.e. of parameters α , . . . , α d − of the caustics.Notice that the product α · . . . · α d − is changed continuously on the variety oflines in E k,l that intersect E , with infinite singularities at light-like lines. Besides,the subvariety of light-like lines divides the variety of all lines into subsets of space-like and time-like ones. When passing through light-like lines, one of parameters α i will pass through the infinity from positive to the negative part of the reals or viceversa; thus, a change of sign of the product occurs simultaneously with a change ofthe type of line.Now, take α j = − a k + j for 1 ≤ j ≤ l , and notice that all lines placed in the k -dimensional coordinate subspace E k × l will have the corresponding degeneratecaustics. The reduced metrics is Euclidean in this subspace, thus such lines arespace-like. Since α , . . . , α k are positive for those lines of E k × l that intersect E ,the statement is proved. (cid:3) Let us note that, in general, for the fixed d − Proposition 5.2.
Let T be a trajectory of the billiard within ellipsoid E in pseudo-Euclidean space E k,l . Denote by α , . . . , α d − the parameters of the caustics fromthe confocal family (2.3) of T , and take b , . . . , b p , c , . . . , c q as in Theorem 2.3.Then we have: PE3 along T , each generalized Jacobi coordinate takes values in exactly one ofthe segments: [ c l − , c l − ] , . . . , [ c , c ] , [ c , , [0 , b ] , [ b , b ] , . . . , [ b k − , b k − ];PE4 along T , each generalized Jacobi coordinate can achieve local minima andmaxima only at touching points with corresponding caustics, intersectionpoints with corresponding coordinate hyper-planes, and at reflection points; LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 27
PE5 values of generalized Jacobi coordinates at critical points are , b , . . . , b k − , c , . . . , c l − ; between the critical points, the coordinates are changedmonotonously.Proof. Property PE3 follows from Theorem 2.3. Along each line, the generalizedJacobi coordinates are changed continuously. Moreover, they are monotonous atall points where the line has a transversal intersection with a non-degenerate quad-ric. Thus, critical points on a line are exactly touching points with correspondingcaustics and intersection points with corresponding coordinate hyper-planes.Note that reflection points of T are also points of transversal intersection withall quadrics containing those points, except with E . Thus, at such points, 0 will bea critical value of the corresponding generalized Jacobi coordinate, and all othercoordinates are monotonous. This proves PE4 and PE5. (cid:3) The properties we obtained are pseudo-Euclidean analogs of properties E3–E5from Section 4.1, which are true for ellipsoidal billiards in Euclidean spaces.5.2.
Analytic conditions for periodic trajectories.
Now, we are going to de-rive the corresponding analytic conditions of Cayley’s type for periodic trajectoriesof the ellipsoidal billiard in the pseudo-Euclidean space, and therefore to obtain thegeneralization of the Poncelet theorem to pseudo-Euclidean spaces.
Theorem 5.3 (Generalized Cayley-type conditions) . In the pseudo-Euclidean space E k,l ( k + l = d ), consider a billiard trajectory T within ellipsoid E given by equation(2.2). Let Q α , . . . , Q α d − from confocal family (2.3) be caustics of T .Then T is periodic with period n if and only if the following condition is satisfied: rank B d +1 B d +2 . . . B m +1 B d +2 B d +3 . . . B m +2 . . . . . . . . . . . .B d + m − B d + m . . . B m − < m − d + 1 , for n = 2 m ;rank B d B d +1 . . . B m +1 B d +1 B d +2 . . . B m +2 . . . . . . . . . . . .B d + m − B d + m . . . B m < m − d + 2 , for n = 2 m + 1 . Here, p ( α − λ ) · . . . · ( α d − − λ ) · ( a − ε λ ) · . . . · ( a d − ε d λ ) = B + B λ + B λ + . . . is the Taylor expansion around λ = 0 .Proof. Denote: P ( λ ) = ( α − λ ) · . . . · ( α d − − λ ) · ( a − ε λ ) · . . . · ( a d − ε d λ ) . Following Jacobi [Jac1884], along a given billiard trajectory, we consider the inte-grals:(5.1) d X s =1 Z dλ s p P ( λ s ) , d X s =1 Z λ s dλ s p P ( λ s ) , . . . , d X s =1 Z λ d − s dλ s p P ( λ s ) . By PE3 of Proposition 5.2, we may suppose that: λ ∈ [0 , b ] , λ i ∈ [ b i − , b i − ] for 2 ≤ i ≤ k ; λ k +1 ∈ [ c , , λ k + j ∈ [ c j − , c j − ] for 2 ≤ j ≤ l. Along a billiard trajectory, by PE4 and PE5 of Proposition 5.2, each λ s will passthrough the corresponding interval monotonously from one endpoint to anotherand vice versa alternately. Notice also that values b , . . . , b k − , c , . . . , c l − correspond to the Weierstrass points of hyper-elliptic curve:(5.2) µ = P ( λ ) . Thus, calculating integrals (5.1), we get that the billiard trajectory is closed after n reflections if and only if n A ( P ) ≡ A is the Abel-Jacobi map, and P is the pointon the curve corresponding to λ = 0. Further, in the same manner as in [DR1998],we obtain the conditions as stated in the theorem. (cid:3) As an immediate consequence, we get:
Theorem 5.4 (Generalized Poncelet theorem) . In pseudo-Euclidean space E k,l ( k + l = d ), consider a billiard trajectory T within ellipsoid E .If T is periodic and become closed after n reflections on the ellipsoid, then anyother trajectory within E having the same caustics as T is also periodic with period n . Remark 5.5.
The generalization of the Full Poncelet theorem from [CCS1993] topseudo-Euclidean spaces is obtained in [WFS + . However, only space-like andtime-like trajectories were discussed there.A Poncelet-type theorem for light-like geodesics on the ellipsoid in the three-dimensional Minkowski space is proved in [GKT2007] . Remark 5.6.
Theorems 5.3 and 5.4 will also hold in symmetric and degeneratedcases, that is when some of the parameters ε i a i , α j concide, or in the case of light-like trajectories, when ∞ ∈ { α j | ≤ j ≤ d − } . In such cases, we need to applythe desingularisation of the corresponding curve, as explained in detail in our works [DR2006, DR2008] .When we consider light-like trajectories, then the factor containing the infiniteparameter is ommited from polynomial P . Example 5.7.
Let us find all -periodic trajectories within ellipse E given by (3.1)in the Minkowski plane, i.e. all conics C α from the confocal family (3.2) correspond-ing to such trajectories.By Theorem 5.3, the condition is B = 0 , with p ( a − λ )( b + λ )( α − λ ) = B + B λ + B λ + B λ + . . . being the Taylor expansion around λ = 0 . Since B = ( − ab − aα + bα )( − ab + aα + bα )( ab + aα + bα )16( abα ) / , LLIPSOIDAL BILLIARDS IN PSEUDO-EUCLIDEAN SPACES 29 we obtain the following solutions: α = abb − a , α = aba + b , α = − aba + b . Since α ( − b, a ) and α , α ∈ ( − b, a ) , conic C α is a hyperbola, while C α , C α are ellipses. References [AGZV1985] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko,
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