Timelike Funk and Hilbert geometries

Timelike Hilbert and Funk geometries

Athanase Papadopoulos Athanase Papadopoulos, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Navy Nagar, Near Navy Canteen, Mandir Marg, Colaba, Mumbai, Maharashtra 400005, India. athanase.papadopoulos@math.unistra.fr  and  Sumio Yamada Sumio Yamada, Gakushuin University, 1-5-1 Mejiro, Toshima, Tokyo, 171-8588, Japan yamada@math.gakushuin.ac.jp
July 13, 2019
Abstract.

A timelike space is a Hausdorff topological space equipped with a partial order relation and a distance function satisfying a set of axioms including certain compatibility conditions between these two objects. The distance function is defined only on a certain subset (whose definition uses the partial order) of the product of the space with itself that contains the diagonal. Distances between triples of points, whenever they are defined, satisfy the so-called time inequality, which is a reversed triangle inequality. In the 1960s, Herbert Busemann developed an axiomatic theory of timelike spaces and of locally timelike spaces. His motivation comes from the geometry underlying the theory of relativity and the classical example he gives is the -dimensional Lorentzian spaces. Two other interesting classes of examples of timelike spaces introduced by Busemann are the timelike analogues of the Funk and Hilbert geometries. In this paper, we investigate these two geometries, and in doing this, we introduce variants of them, we call timelike relative Hilbert geometries, in the Euclidean and spherical settings. We display new interactions among the Euclidean and spherical timelike geometries. In particular, we characterize the de Sitter geometry as a special case of a timelike spherical Hilbert geometry.

Keywords.— Timelike space, timelike Hilbert geometry, timelike Funk geometry, time inequality, convexity, metric geometry, Busemann geometry, Lorentzian geometry, relativity.

AMS classification.— 53C70, 53C22, 5C10, 53C23, 53C50, 53C45.

1. Introduction

A timelike space is a Hausdorff topological space equipped with a partial order relation and a distance function which plays the role of a metric. This distance function is asymmetric in the sense that is not necessarily equal to when they are both defined, as may be defined whereas is not. More precisely, the distance is defined only for pairs satisfying (that is, either ot ). This distance function satisfies the following three axioms:

  1. for every in ;

  2. for every and such that

  3. for all triples of points satisfying .

Not that the last property is a reversed triangle inequality. It is called a time inequality.

The distance function and the partial order relation satisfy an additional set of axioms including compatibility conditions with respect to each other. For instance, it is required that every neighborhood of a point in contains points and satisfying . This axiom and others are stated precisely in the memoir [4] by Herbert Busemann. We shall not recall them here (there are too many of them) but in all the cases that we shall consider, they will be satisfied. As a matter of fact, in this paper, the topological space will always be a subset of , or the hyperbolic space .

Theories of timelike spaces, timelike -spaces, locally timelike spaces and locally timelike -spaces were initiated by Busemann in [4] as analogues of his geometric theories of metric spaces and of -spaces that he developed in his book [2] and in other papers and monographs. The motivation for the study of timelike spaces comes from the geometry underlying the physical theory of relativity. The classical example is the -dimensional Minkowski space, which Busemann generalized, in his paper [4], to the case of general timelike distance functions on finite-dimensional vector spaces which become, under a terminology that we use, timelike Minkowski spaces. As other interesting examples of timelike spaces, Busemann introduced timelike analogues of the Funk and Hilbert geometries. In the present paper, we investigate various such geometries, to which we give the names of Euclidean timelike Funk geometry, Euclidean timelike relative Funk geometry, Euclidean timelike Hilbert geometry, hyperbolic timelike Funk geometry, timelike relative spherical Funk geometry, and timelike sphercal Hilbert geometry. We establish several results concerning their geodesics, their convexity properties and their infinitesimal structure. We show in particular that they are timelike Finsler spaces. This means that the distance between two points is defined infinitesimally by a timelike norm, that is, that there exists a timelike Minkowski structure on the tangent space at each point of our space such that the distance between two points is the length of the longest path joining them, where the length of a path is defined using the timelike distance function. We also give a description of the usual de Sitter space as a special case of a spherical timelike Hilbert geometry.

Busemann’s interest, as well as the authors’ in the subject, stem from Hilbert’s Forth Problem [8] where Hilbert proposed a systematic study of metric spaces modelled on the Euclidiean space where the geodesics coincide with the Euclidean line segments. The best known, and most important example of such a metric space is the Beltrami-Klein model of the hyperbolic plane. The hyperbolic geometry in that context is very much hinged with convex Euclidean geometry. The aim of the current investigation is to revisit the aspect of convex geometry in the exterior region of convex sets in the constant curvature spaces, which naturally produce timelike geometries as exemplified by the de Sitter geometry.

In what follows, we will set up a set of necessary tools to capture the geometry of the exterior region of convex sets, and consequently reformulate the timelike geometry that differs from Busemann’s approach in [4].

2. The timelike Euclidean Funk geometry

We first introduce some preliminary notions and we establish some basic facts. With some few exceptions, we shall use Busemann’s notation in [4], and we first recall it.

Let be a convex hypersurface in , that is, the boundary of an open (possibly unbounded) convex set . If is not a hyperplane, it bounds a unique open convex set , namely, the unique convex connected component of . If is a hyperplane, the two connected components of are both convex, and in this case we make a choice of one of them, that is, of a half-space bounded by the hyperplane . We call the set associated to the interior of . We denote the closure of by , a notation used in Busemann’s paper [4].

Let be the set of supporting hyperplanes of , that is, the hyperplanes having nonempty intersection with and such that the open convex set is contained in one of the two connected components of .

We let be the set of hyperplanes not intersecting the open convex set . We have .

For every element , we let be the open half-space bounded by the hyperplane and containing , and the open half-space bounded by and not containing . We have:

We set

Then, we also have

Definition 2.1 (Order relation).

We introduce a partial order relation between points of . For any two distinct points and in , we write

if the following three properties are satisfied:

  1. The Euclidean ray from through intersects the hypersurface ;

  2. does not belong to a supporting hyperplane of ;

  3. the closed Euclidean segment does not interesect .

When , we say that lies in the future of . We also say that lies in the past of (see Figure 1). We write if either or .

Figure 1.

We denote by (resp. ) the set of ordered pairs in satisfying (resp. ). The set is disjoint from the diagonal set .

Definition 2.2 (The future set of a point).

For in , we define it future set, which we denote by , to be the set of points that satisfy .

Definition 2.3 (The future set in ).

For in , we define it future set in , which we denote by , to be the set of such that and is not contained in any supporting hyperplane of .

For every point in , its future set is nonempty, open and connected.

For in , we denote by the set of hyperplanes that separate the open convex set from . In other words, we have

(1)

We also introduce the set of supporting hyperplanes separating and ,

(2)

is also the set of supporting hyperplanes to at the points of , the future set of in .

We have the following:

Proposition 2.4.

For any two points and in , we have

Proof.

Suppose . We claim that every is an element of . Indeed, if this does not hold, then there exists such that . For that choice of , lies in and at the same time the ray intersects on the side , implying which contradicts the fact that .

To see the strict inclusion when , choose a hyperplane in that intersects . Such a hyperplane is not in .

Next suppose . Then the following inclusion

follows from the characterization (3) of .

Hence is in the past of , and thus .

Corollary 2.5.

For any two points and in , we have

Proof.

Let . Then, . ∎

Note that the strict inclusion in Corollary 2.5 cannot be expected, as observed from the following example in

where we have .

Corollary 2.6.

Let be three points in . If and , then .

Proof.

Proposition 2.4 gives:

Now we can define the timelike Funk distance on the subset of .

Definition 2.7 (The past set of a point).

For , the past set of , denoted by , is the set of points in such that is in the future of .

The set is an open subset of , which is also characterized by the following:

(3)

where denotes the interior of a set.

We shall also need an equivalent description of the set which we give now.

Let

where is as before the set of hyperplanes that separate the open convex set and . Hence for a hyperplane in , is either contained in or is contained in the open half space containing .

The future set of can now be expressed as

(This should be compared with the expression of the past set given in (3).)

As implies the inclusion

in the light of the representation , we have the inlcusion

This implies the inclusion . In fact, we have the following stronger set theoretic relation:

Proposition 2.8.

For any , we have

Proof.

Suppose the contrary, and assume that . Then there is a supporting hyperplane of so that it contains and . As lies in , the ray from through intersect transversely with and it never intersects the convex hypersurface , which is a contradiction as is contained in some supporting hyperplane of as and hence it intersects . ∎

Thus,

Corollary 2.9.

We have the stronger implication:

Definition 2.10 (The timelike Funk distance).

The function on pairs of distinct points in satisfying is given by the formula

and where is the first point of intersection of the ray with . Here, denotes the Euclidean distance.

Note that the value of is strictly positive.

We extend the definition of to the case where , setting in this case .

Let and be two points in such that . Let be a supporting hyperplane to at . For in , let be the foot of the Euclidean perpendicular from the point onto that hyperplane. In other words, is the Euclidean nearest point projection map. From the similarity of the Euclidean triangles and , we have

Using the convexity of , we now give a variational characterization of the quantity .

For any unit vector in and for any , we set

if this intersection is non-empty.

For in , consider the vector where the norm is the Euclidean one.

We then have .

In the case where is not a supporting hyperplane of at , the point lies outside and, again by the similarity of the Euclidean triangles and , we get

Note that as varies in , the farthest point from on the ray of the form is , and this occurs when supports at . This in turn says that a hyperplane which supports at minimizes the ratio

among all the elements of and thus we obtain

Proposition 2.11.
Remark 2.12.

There is an analogous formula for the classical (non-timelike) Funk metric, where the infimum in the above formula is replaced by a supremum (see [15] Theorem 1.)

Remark 2.13.

The set of future points of a point , that is, the set of points satisfying reminds us of the cone of future points of some point in the ambient space of the physically possible trajectories of this point in the case of Minkowski space, that is, in the geometric setting of spacetime for the theory of (special) relativity. The restriction of the distance function to the cone comes from the fact that a material particle travels at a speed which is less than the speed of light. The set of points on the rays starting at that are on the boundary of the future region becomes an analogue of the “light cone” of spacetime (again using the language of relativity). In our definition of timelike geometry, the points of light cone is excluded and we will postpone further discussion of light cones till §15.

We shall prove that the function satisfies the reverse triangle inequality, which we call in this context, after Busemann, the time inequality. This inequality holds for mutually distinct triples of points and in , satisfying :

Proposition 2.14 (Time inequality).

For any three points and in , satisfying , we have

Proof.

We use the formula given by Proposition 2.11 for the timelike Funk distance. We have, from (Corollary 2.5):

In the rest of this section, we study geodesics and spheres in timelike Funk geometries. We shall prove analogues of results in the paper [9] where the corresponding results are proved in the non-timelike Funk setting. The current setting is motivated by Busemann’s work [2].

First we consider geodesics for the timelike Funk distance. We start with the definition of a geodesic. This definition is the same as in an ordinary metric spaces, except that some care has to be taken so that the distances we need to deal with are always defined.

A geodesic is a path , where may be an arbitrary interval of , such that for every pair in we have and for every triple in we have

It follows easily from the definition that for any the Euclidean segment joining to is the image of a geodesic. This makes the distance function satisfy Hilbert’s Fourth Problem [8] if this problem is generalized in an appropriate way to include timelike spaces. (We recall that one form of this problem asks for a characterization of metrics on subsets of Euclidean space such that the Euclidean lines are geodesics for this metric.) In particular, the time inequality becomes an equality when , and satisfying are collinear in the Euclidean sense.

It is important to note that in all the development of geodesics in timelike spaces, it is understood that geodesics are equipped with a natural orientation. Traversed in the reverse sense, they are not geodesics.

Let us make an observation which concerns the non-uniqueness of geodesics and the case of equality in the time inequality. Assume that the boundary of the convex hypersurface contains a Euclidean segment . Take three points in such that and intersect (Figure 2).

Figure 2. The broken segment is a geodesic

Then, using the Euclidean intercept theorem, we have

Applying the same reasoning to an arbitrary ordered triple on the broken Euclidean segment , we easily see that this segment is an -geodesic. More generally, by the same argument, we see that any oriented arc in such that any ray joining two consecutive points on the arc hits the segment is the image of an -geodesic.

We deduce the following:

Proposition 2.15.

A timelike Funk geometry defined on a set associated to a convex hypersurface in satisfies the following properties:

  1. The Euclidean segments in that are of the form where are -geodesics.

  2. Any Euclidean line from a point in to a point in , equipped with the metric induced from the timelike Funk distance, is isometric to a Euclidean ray.

  3. The Euclidean segments in (1) are the unique -geodesic segments if and only if the convex set is strictly convex.

The proof is the same as that of the equivalence between (1) and (2) in Corollary 8.7 of [10], up to reversing some of the inequalities (i.e. replacing the triangle inequality by the time inequality), therefore we do not include it here.

After the geodesics, we consider spheres.

Definition 2.16.

At each point of , given a real number , the future sphere of radius centered at is the set of points in that are in the future of and situated at -distance from this point.

Proposition 2.17 (Future spheres).

At each point of and for each , the future sphere of center and radius is a piece of a convex hypersurface that is affinely equivalent to , the future of in .

The proof is analogous to that of Proposition 8.11 of [9], and we do not repeat it here.

Proposition 2.17 implies that some affine properties of the hypersurface are local invariants of the metric. One consequence is the following strong local rigidity theorem, which is also an analogue of a property satisfied by the non-timelike Funk metric (cf. the concluding remarks of the paper [9]).

Corollary 2.18.

Let and be two hypersurfaces in and and the set of corresponding pairs of points for which the associated timelike Funk distances and respectively are defined. If there exists subsets and and a map which is distance-preserving, then there is an open subset of which is affinely equivalent to an open subset of .

The proof follows from the fact that an isometry sends a future sphere to a future sphere.

The corollary has interesting consequences. For instance, it implies that if is the boundary of a polyhedron and a strictly convex hypersurface, then there is no local isometry between the associated timelike spaces.

We next show a useful monotonicity result for a pair of timelike Funk geometries which is essentially follows from a remark in convex geometry.

Given our open convex set with associated Funk distance , we let be another open convex set containing and the associated timelike distance defined on the appropriate set of pairs .

Proposition 2.19.

For all and in the domains of definition of both distances and (that is, for with respect to both convex sets and ), we have

Proof.

Using the notation of Definition 2.10, we have

With similar notation, we have

Since , we have and for some . The result follows from the fact that the function defined for by

where are two constants, is increasing. ∎

3. Timelike Minkowski spaces

Consider a finite-dimensional vector space, which we identify without loss of generality with . We introduce on this space a timelike norm function which we also call a timelike Minkowski functional, in analogy with the usual Minkowski functional (or norm function) defined in the non-timelike sense. To be more precise, we start with the following definition (cf. [4] § 5).

Definition 3.1 (Timelike Minkowski functional).

A timelike Minkowski functional is a function satisfying the following:

  1. is defined on , where is an open convex cone of apex the origin , that is, an open convex subset invariant by the action of the positive reals ;

  2. ;

  3. for all in ;

  4. for all in and ;

  5. for all .

It follows from the concavity condition (5) that the closure of the cone possesses a supporting hyperplane which intersects it only at the apex; cf. Busemann [4] p. 30. We shall say that a cone posessing this property is proper.

We shall say that is the cone associated with the timelike Minkowski functional .

Note that by the convexity of , is continuous.

The unit sphere of such a timelike norm function is the set of vectors in satisfying . In general, is a piece of a hypersurface in which is concave when viewed from the origin (see Figure 3). We allow the possibility that is asymptotic to the boundary of the cone . The unit sphere is called the indicatrix of .

Figure 3. The indicatrix in the tangent space to a point in .

The reason of the adjective timelike in the above definition is that in the Lorentzian setting, the Minkowski norm measures the lengths of vectors in the timelike cone, which is the part of spacetime where material particles move. In particular, there is a timelike Minkowski functional for the standard Minkowski space , equipped with the Minkowski metric

It is given by

and it is defined for vectors in satisfying or .

4. Timelike Finsler structures

Definition 4.1 (Timelike Finsler structure).

A timelike Finsler structure on a differentiable manifold is a family where each is a timelike Minkowski functional defined on the tangent space of at . In the tangent space at each point in , there is a cone associated to which plays the role of the cone associated in Definition 3.1 to a general timelike Minkowski functional. We assume that together with its associated cone depend continuously on the point .

In the situations considered in this paper, will be either an open subset of a Euclidean space or of a sphere . (In some rare cases, it will be a subset of a hyperbolic space .)

We say that a piecewise curve , , defined on an interval of , is timelike if at each time the tangent vector is an element of the cone .

Definition 4.2 (The partial order relation).

If and are two points in , we write , and we say that is in the -future of , if there exists a timelike piecewise curve joining to .

Proposition 4.3.

The two order relations and coincide; namely, for any two points and in , we have

Proof.

The implication follows from that fact that for , the parameterized curve

for is a timelike curve from to .

The other implication follows from the claim that given and for any piecewise timelike curve with and ,

for all in . In particular, we have , and hence .

The proof of the claim is as follows.

We start by the observation that the path , being timelike, starts at the point with a right derivative at pointing strictly inside the cone . This implies that the point is strictly inside the set for any sufficiently small . Likewise, from the continuity of and the openness of , the connected component of the set containing is open in . Define by

We want to show that . This will imply that the set is an open and closed subset of . This will give the desired result.

Suppose the contrary; namely and thus lies in the boundary . Then we can find a sequence such that for all and such that

while is uniformly bounded from below by some positive number for all sufficiently large, as

for any , as follows from Corollary 2.9. This gives the desired contradiction.

We define the length of a piecewise timelike curve by the Lebesgue integral

We then define a function on pairs of points satisfying by setting

(4)

where the supremum is taken over all the timelike piecewise curves satisfying and . We shall show that defines a timelike distance function.

It is easy to see from the definition of that it satisfies the timelike inequality, once we show the following

Lemma 4.4.

For any pair and satisfying , we have .

Proof.

To see that the supremum in (4) is finite, we introduce a reference metric on a chart of the manifold modelled on the Minkowski space as follows. Let be a local chart on containing a point so that is an open subset of with . As is a diffeomorphism, each open cone in on which the Minkowski functional is defined is mapped to a proper convex cone in by the linear map . Hence we have a field of proper cones . By the continuity of in , there exists an open neighborhood of so that on there is a field of supporting hypersurfaces of : with all the hyperplanes sharing the same normal vector in .

We introduce a Minkowski metric on where the constant (the speed of light) will be determined below. The -plane is identified with for each . We also consider , the set of future directed timelike vectors with . Then we can choose the constant sufficiently large so that

  1. the light cone properly contains at each ;

  2. each -unit vector in , which is identified with a tangent vector in has norm .

So far, we have defined an auxiliary norm for any with . We denote the distance with respect to the Minkowski metric by . Note that the condition ensures that a timelike curve in with respect to the family of norms is also timelike for the auxiliary family of norms .

Now given a timelike -curve through , we have the following length comparison

with the auxiliary length bounded above;

as the line segment is the length maximizing timelike curve in the Minkowski space . ∎

It follows that the timelike distance function defines a timelike structure on the space . This timelike structure is the analogue of the so-called intrinsic metric in the non-timelike case. We call the timelike intrinsic distance associated with the timelike Finsler structure.

5. The timelike Finsler structure of the timelike Funk distance

In this section, we show that the timelike Funk distance associated to a convex hypersurface in is Finsler in an appropriate sense which we now describe; we shall call such a structure a timelike Finsler structure in the sense of §4. In other words, we show that on the tangent space at each point of , there is a timelike Minkowski functional which makes this space a timelike Minkowski space, such that the timelike Funk distance between two points and is obtained by integrating this norm on tangent vectors along piecewise paths joining to and taking the supremum (instead of the infimum, in the non-timelike case) of the lengths of such piecewise paths. The paths are restricted to those where the tangent vector at each point of belongs to the domain of the timelike Minkowski functional.

For every point in the timelike Funk geometry of a space associated to a convex hypersurface , there is a timelike Minkowski functional defined on the subset of the tangent space of at consisting of the non-zero vectors satisfying

where we recall that is the future of . We denote by the union of vectors that satisfy this property or are the zero vector. We define the function for and by the following formula:

(5)

for where is as in (2) and where is the unit tangent vector at perpendicular to and pointing toward . We define . We shall show that this defines a timelike Minkowski functional and that this functional is associated to a timelike Finsler geometry underlying the timelike Funk distance .

By elementary geometric arguments (see [15] for a detailed discussion in the non-timelike case which can be adapted to the present setting) it is shown that

(6)

for any nonzero vector .

Note that the quantity in the denominator is the Euclidean length of the line segment from to the point where the ray hits the convex set for the first time. A simpler way to write the Minkowski functional in (5) is:

(7)

We have the following;

Proposition 5.1.

The functional defined on the open cone in satisfies all the properties required by a timelike Minkowski functional.

Proof.

It is easy to check the required properties. We make a remark regarding the last property in Definition 3.1. The inequality is a concavity of the linear functional on the tangent space , which follows from the fact that is an infimum over of the linear (and in particular concave) functionals, which is concave. ∎

Now we repeat the argument in §4, to set up a timelike space using the Finsler structure. We say that a piecewise curve , , defined on an interval of , is timelike if at each time the tangent vector is an element of the cone .

Definition 5.2 (The partial order relation).

If and are two points in , we write , and we say that is in the -future of , if there exists a timelike piecewise curve joining to .

The following proposition is proved as was done in the proof of Proposition 4.3.

Proposition 5.3.

The two order relations and coincide; namely, for any two points and in , we have

As we did so in §4, we denote by the timelike intrinsic distance function associated to this timelike Finsler structure:

(8)

where the supremum is taken over all the timelike piecewise curves satisfying and . Like in Lemma 4.4, it is seen that the intrinsic distance for is finite.

We thus have shown that the domain of definition of the set associated with the partial order for the timelike Funk distance and the domain of definition for the timelike distance function coincide. Furthermore, we shall prove the equality for any pair in . We state this as follows:

Theorem 5.4.

The value of the timelike distance for a pair coincides with . That is, we have

In other words, we have the following

Theorem 5.5.

The timelike Funk geometry is a timelike Finsler structure defined by the Minkowski functional .

The timelike Minkowski functional which underlies a timelike Funk geometry has a property which is analogous to the one noticed in [9] which makes that metric the tautological Finsler structure associated with the hypersurface (or the convex body ). The term “tautological” is due to the fact that the indicatrix of the timelike Minkowski functional at , that is, the set

is affinely equivalent to the relative interior (with respect to the topology of ) of the intersection of that hypersurface with , the closure in of the subset .

We also note that with this identification, given a pair of points with , there always exists a distance-realizing (length-maximizing) geodesic from to , since the Euclidean segment is an -geodesic.

Proof of Theorem 5.4.

For a pair of points with , we consider the map

(9)

parametrizing the Euclidean segment parametrized proportionally to arc-length with Then we have

since

By taking the supremum over the set of paths from to , this implies the inequality

(10)

Before continuing the proof of Theorem 5.4, we show a monotonicity property for the intrinsic distance that will be useful.

Let be an open convex set containing and let be its bounding hypersurface. Let be the timelike Funk metric, its associated timelike Minkowski functional, and the associated intrinsic distance. (Note that the domains of definition of and contain those of and respectively.) We have the following:

Lemma 5.6.

For and in the domains of definition of both intrinsic distances and , we have

Proof.

Between the two timelike Minkowski functionals and , we have the following inequality

whenever the two quantities are defined concurrently. This follows from the definition of the Minkowski functional:

(11)

for any nonzero vector in both domains of definition, as is closer to than . Hence by integrating each functional along an admissible path (note that admissible paths for are also admissible paths for ) and taking the supremum over these paths, we obtain

Proof of Theorem 5.4 continued.— Suppose that we have a convex hypersurface bounding an open convex set , and for , let

where is the open half-space bounded by a hyperplane supporting at and containing . The open set is equipped with its intrinsic distance . We now apply Lemma 5.6 to this setting where a convex set contains , and obtain .

For the open half space , the values of , and all coincide. Indeed, under the hypothesis , the set of supporting hyperplanes consists of the single element , and the line segment from to defined in (9) is a length-maximizing path, since every timelike path for is -geodesic. (Such arguments were already used in §2.)

By combining the above observations, we have

(12)