Particles with spin in stationary flat spacetimes
We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2+1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse their geometrical properties, introduce a generalised notion of global hyperbolicity and classify all stationary flat spacetimes with singularities that are globally hyperbolic in that sense. We then apply our results to (2+1)-gravity and analyse the causality structure of these spacetimes in terms of measurements by observers. In particular, we derive a condition on observers that excludes causality violating light signals despite the presence of closed timelike curves in these spacetimes.
Key words and phrases:
1991 Mathematics Subject Classification:83C80 (83C57), 57S25
Flat three-dimensional Lorentzian manifolds with conical singularities were first introduced in the physics literature on (2+1)-dimensional gravity, where they model (2+1)-dimensional spacetimes that contain massive point particles with spin.
The first models of (2+1)-gravity with particles were derived in [Sta63] and [SD84]. Their physical properties and their quantisation were studied in the subsequent publications [SD88, Car89, dSG90, tH93b, tH93a, tH96], which led to a large body of work on the classical aspects and quantisation of the models, for an overview see [Car03].
As they are models that include matter and still are amenable to quantisation, these models play an important role in the research subject of quantum gravity. Since they allow one to investigate the quantisation of gravity coupled to matter, they have been studied extensively in the physics literature. Another reason why these models are of interest in quantum gravity is their causality structure. It was shown in [SD84] that the presence of massive point particles with spin leads to the presence of closed timelike curves in these models which, however, can be removed by excising a small cylinder around each particle. Additionally, closed timelike curves can be generated dynamically when two spinless massive particles approach each other with sufficiently high speed (“Gott pairs”). A detailed investigation of this phenomenon has been given in [Got91, SD92b], with the conclusion that these dynamically generated closed, timelike curves are not physically meaningful since they are present only for very short times, and, in particular, “time machines” are excluded [SD92c, Des93].
Despite their relevance, many geometrical properties of the models including massive point particles with spin are not fully understood even on the classical level. Although their properties have been investigated in the physics literature, they are very few results concerning the underlying mathematical structures. The closest treatment in the mathematics literature is the study of geometric Riemannian manifolds, mostly in the case of euclidean, spherical or hyperbolic surfaces with conical singularities ([Tro07], [Mas06, MT02]) sometimes in relation to the work on billiards, and in the 3-dimensional case the work devoted to the Orbifold Theorem ([CHK00, BLP05]).
A similar treatment in the Lorentzian case is still at the beginning. This includes in particular the causality issues arising in these models as well as the lack of systematic definitions and classification. A systematic investigation of the mathematical features of three-dimensional Lorentzian spacetimes with particles has been initiated only recently and is mainly concerned with the case of constant negative curvature [BB09, BS09, LS09, KS07, BBS09].
In this article, we investigate flat stationary Lorentzian spacetimes with a general number of massive particles with spin. We construct examples of stationary flat Lorentzian spacetimes with particles that are based on Euclidean surfaces with cone singularities and closed one-forms on these surfaces. We introduce a generalised notion of global hyperbolicity that can be applied to these models despite the fact that they contain closed timelike curves. Based on this notion of global hyperbolicity, we classify the flat stationary globally hyperbolic Lorentzian spacetimes with particles and give a detailed analysis of their geometrical properties.
The last section of the article is dedicated to a problem that is of high relevance to physics, namely the question, how the presence of particles manifests itself in measurements by observers that probe the geometry of the spacetime by exchanging “test lightrays”111The name “test light rays” is motivated by the fact that they play a role similar to the test masses used in general relativity. They are lightlike geodesics in a given spacetime rather than actual lightlike point sources, and we neglect their impact on the stress energy tensor. This is different from the treatment in [SD92a], which considers solutions of the Einstein equations in (2+1) dimensions with lightlike point sources.. This idea is very natural from the viewpoint of general relativity, whose physical interpretation was formulated in terms of lightrays exchanged by observers from the beginning. It is also of special relevance to quantum gravity in four dimensions, as it is hoped that quantum gravity effects might manifest itself in cosmic microwave background radiation and thus be determined by means of lightrays. The (2+1)-dimensional models considered in this work share many properties with the cosmological models investigated in (3+1) dimensions.
We show that light signals exchanged by observers correspond naturally to piecewise geodesic curves on the underlying Euclidean surfaces with cone singularities. We demonstrate how an observer can construct the relevant parameters that describe the spacetime from such measurements: the positions, masses and spins of such particles as well as their velocity with the respect to the observer.
Building on these results, we investigate the causality issues associated with closed, timelike curves in spacetimes containing particles with spin. In particular, this allows one to establish a condition on the observers that excludes paradoxical signals, i. e. signals that are received before they are emitted. In physical terms, this implies that observers that stay away a sufficient distance from each particle, will not experience paradoxical light signals, even if the light signals themselves enter a region around the particle which contains closed, timelike curves.
Note that these models do not involve the dynamically generated closed, timelike curves that are investigated in [Got91, SD92b, SD92c, Des93], since the spacetimes under consideration are stationary. Instead, the spacetimes investigated in this paper contain closed timelike curves that are due to the presence of massive point particles with spin. Rather than investigating the impact of dynamically generated closed timelike curves and determining the spacetime regions in which they occur, we focus on the impact of these closed timelike curves on observers at a sufficient distance from the particles and on light signals exchanged by such observers.
2. The model: a single particle with spin
In the following we denote by the three-dimensional Minkowski space and by a timelike geodesic in . We choose a suitable coordinate system , in which the Minkowski norm takes the form and is given by the equation . We also introduce spatial polar coordinates , which are given in terms of the spatial coordinates by , .
The model for a single particle introduced in [SD84] depends on two real parameters, an angle , in the following referred to as deficit or apex angle, and a parameter , in the following referred to as spin. The names for these parameters are motivated by their physical interpretation. It is shown in [SD84], see also [Car03], that the metric for a single point particle in is that of a cone with a deficit angle , where is the mass of the particle in units of the Planck mass. Moreover, it is shown there that the resulting spacetime has a non-trivial asymptotic angular momentum that is given by . The parameter which therefore is viewed as an internal angular momentum or spin of the particle in units of [SD84].
The associated flat Lorentzian manifold is constructed as follows. The equations , define two timelike half-planes, which both have the timelike geodesic as their boundary and which we denote, respectively, , . These half-planes bound the region
that we call a wedge of angle . We glue to by the map , which is a restriction of the elliptic isometry . The gluing of the wedge is pictured in Figure 1. The result is a manifold , that is naturally equipped with a flat Lorentz metric and homeomorphic to minus a line. Note that the time orientation of Minkowski space induces a time orientation on , namely the one for which the coordinate increases along future oriented causal curves.
We now demonstrate how a (singular) line can be added to to obtain a manifold homeomorphic to . For this, one is tempted to extend the gluing defined above to the closed wedge in such a way that points of the form are identified with . This is possible if and in that case yields a singular flat spacetime that contains a singular line characterised by the condition . It corresponds to a (2+1)-dimensional spacetime with a single particle of mass and vanishing spin .
However, this procedure does not work in the case . For non-vanishing spin , the quotient of by this isometry is a circle and not a line. When equipped with the quotient topology, it is no longer a manifold. Indeed, an open disc in the -plane that is centred at the point corresponds to a union of infinitely many circular sectors that are identified along the line segments given by and .
A more transparent description of spacetimes containing particles with non-vanishing spin is obtained by introducing a new set of coordinates that includes the radial coordinate as well as
As the coordinate has the range , it induces a map . The pull-back by of the flat metric on to is given by
In the following we denote by the manifold equipped with this metric outside the singular line given by . It contains (an isometric copy of) . Note that this formula can be extended to the case . In geometrical terms, this amounts to the following construction. We consider the wedge not as embedded in Minkowski space, but as embedded in the universal cover of . In other words, we introduce a coordinate system , where is no longer defined modulo but now parametrises the entire real line. The resulting flat singular spacetime is then given as a -branched cover along the singular line over , where is chosen so that is less than . In this description, the mass parameter can become negative or vanish. In particular, the limit case yields a massless particle with non-vanishing spin .
2.2. Closed timelike curves (CTCs) and the CTC surface
In contrast to the coordinate , the coordinate on is not a time function. Introducing the variable , we can rewrite the metric (2) as
For a given value of , the circle of constant radius is spacelike if , timelike if , and null for . This implies in particular that it defines a closed timelike curve (CTC) for and a closed lightlike curve for . In the following we will therefore refer to as the CTC radius, to the surface of constant radius as the CTC surface. We call the domain the CTC region and the region the interior region of the spacetime. The latter is a manifold with boundary, whose boundary is the CTC surface . It is the complement of the CTC region, which is diffeomorphic to , where denotes the open disc in . On the CTC surface the metric (3) degenerates to
Note that does not vanish along spacelike curves in . It follows that a non-timelike curve in cannot close up unless it is contained in a circle in characterised by the condition constant. Such circles are lightlike but they are not geodesics. In the following we will call them null circles on the CTC surface. Note that the future of a point in the CTC surface , i. e. the points in that can be connected to via a future directed timelike curves in , is the region above the null circle containing . The future in of a point on a given null circle therefore coincides with the future in of this null circle.
The CTC region contains many closed timelike curves (CTCs). Note, however, that it does not contain closed timelike geodesics. It follows from the expression for the metric, that in order to close up, timelike curves must have an acceleration, which is related to the spin parameter . The smaller the value of the spin parameter, the bigger the acceleration associated with CTCs must be, and it tends to infinity in the limit of vanishing spin. Due to the presence of CTCs, the CTC region exhibits quite pathological causality relations. The future (or the past) inside of any point in is the entire CTC region. Its future (or past) in is the entire manifold .
In contrast to the CTC region, the causality structure of the interior region is well-behaved. As the coordinate defines a time function on , contains no CTCs. Of course, this does not exclude that a timelike curve starts in the interior region, enters the CTC region and then returns to its starting point in the interior region. However, the absence of CTCs in the interior region implies that any closed timelike curve with a starting point in the interior region must enter the CTC region.
2.3. Killing vector fields
The group of time orientation and orientation preserving isometries of is an abelian group of dimension two. It is generated by rotations and by translations . In particular, is stationary: the translation along induces an isometry between the level sets of . However, if the spin is nonzero, is not static because the lapse term in (3) does not vanish.
The CTC region, CTC surface and the interior region are distinguished by the Killing vector associated with the rotations. The CTC region is characterised by the condition that is timelike, the CTC surface is the locus where is lightlike, and the interior region is the region where is spacelike.
2.4. Cauchy surfaces
As the CTC region around the particles contains closed timelike curves, is far from being globally hyperbolic. However, the level surfaces of the coordinate are Cauchy surfaces for the interior region in the sense that any inextendible causal curve in that is contained in must intersect every level set of . We express this property by saying that is globally hyperbolic relatively to its boundary. As observed in Section 2.2, the only non-timelike loops in the CTC surface are the null circles. The boundary of any Cauchy surface in the interior region therefore must coincide with one of these null circles.
2.5. The developing map
The universal covering of is homeomorphic to the manifold obtained by by taking an infinite number of copies , of the wedge introduced in Section 2.1 and gluing them along the associated planes , via the the elliptic isometry : for all . The covering map is the map induced by the isometry . Denote by the elliptic isometry obtained by applying the elliptic isometry times: . Then the maps together define a (local) isometry . This map is the developing map of the Minkowski structure on . It is equivariant with respect to the natural actions of on and on . The first action is the one that maps every onto and the second action is the one induced by .
Note that the map is never a homeomorphism. When increases, the wedges wrap around the line , and for overlap with the initial wedge . This overlapping is a perfect matching if and only if is rational, in which case might be seen as a finite quotient of . This reflects a general pattern that is also present in the case of Minkowski spacetimes with multiple particles. The developing maps for these spacetimes are not one-to-one. Moreover, as we will see in the following, the developing maps of spacetimes with at least two particles are surjective. The developing maps are thus quite pathological, which reflects the fact that the regular part of these manifolds cannot be obtained as a quotient of a region of the Minkowski space.
To investigate the properties of the geodesics in , it is useful to introduce the Euclidean plane with a cone singularity of cone angle , which in the following will be denoted by . The definition is analogous to the one of the manifold . Consider the wedge of angle in the Euclidean plane : and glue the two sides of this wedge via the identification . Alternatively, the Euclidean plane with a cone singularity is obtained as the completion of the following metric on given in polar coordinates
The vertical projection then induces a map . Denote by , the composition of this projection with the developing map. Let now a geodesic path (timelike, lightlike or spacelike). Then lifts to a geodesic path . As the developing map is a local isometry, the image is a geodesic path in and its projection is a geodesic path in . Note that this path is constant if and only if the geodesic is parallel to .
The path is a geodesic loop in if and only if there exists a timelike geodesic parallel to in such that both its starting and endpoint of lie on . As we will see in the following, a lightlike geodesic with this property corresponds to a returning lightray, i. e. a lightray sent out by an observer with worldline that returns to the observer at a later time. This allows us to conclude that for there can be no returning lightray because does not contain geodesic loops. Any geodesic in lifts to a straight line in the Euclidean wedge . If the angle is greater or equal to , a straight line cannot intersect both sides of the wedge and hence cannot close. More generally, using the developing map for and its identification with rotations in as shown in Figure 2 , one finds that the existence of a geodesic loop in with winding number around the cone singularity implies
2.7. CTC cylinders
In the following section, we will extend our model obtain a more general notion of flat Lorentzian spacetimes with a particles. For this we will need to consider the interior region as a manifold with boundary that is given by the CTC surface. We introduce the following definition.
Let be a positive real number and . A CTC cylinder of height based at is the region in between the two level sets , of . The past (future) complete CTC cylinder based at is the past (future) in of the level set .
Note that all CTC cylinders for a given value of are isometric. In contrast to the quantity , the parameter therefore has no intrinsic geometrical meaning. Similarly, all past and future complete cylinders are isometric to the entire CTC region, which implies in particular that they are complete.
3. Global hyperbolicity
We are now ready to give a general definition of flat Lorentzian manifolds with particles and to define a modified notion of global hyperbolicity, which will allow us to restrict the class of Lorentzian manifolds with particles under consideration.
A flat Lorentzian manifold with particles is a three-dimensional manifold with an embedded closed -submanifold (not necessarily connected), such that is endowed with a flat Lorentzian metric and for every in there exists a neighbourhood of in such that is isometric to the neighbourhood of a point on the singular line (the particle) in with the singular line itself removed222Observe that the map is then necessarily locally constant on ..
This definition provides us with a very general notion of a flat Lorentzian spacetime with particles and thus potentially with a large class of examples. However, there is no hope of obtaining a global understanding of flat spacetimes with particles without suitable additional hypotheses. In Riemannian geometry, it is customary to impose as such an additional hypothesis the compactness of the ambient manifold. However, this condition is not suited to the Lorentzian context, since it implies issues with the causal structure such which are undesirable from both the mathematics and the physics point of view. Such issues arise even in the much simpler situation of flat Lorentzian manifolds without particles (cf. [Gal84, Sán06]).
Instead, the standard condition imposed in Lorentzian geometry is the requirement of global hyperbolicity. This implies the existence of a Cauchy surface, and an especially favourable situation is the case where the Cauchy surfaces are compact. This is the point of view we will adopt in the following. However, the fact that the manifolds under consideration exhibit closed timelike curves in the CTC region requires that we modify our concept of global hyperbolicity in a suitable way. The central idea is to consider the flat Lorentzian manifold as a surface with boundaries that are given by the CTC surfaces associated to particles. The appropriate notion of a Cauchy surface is that of a spacelike surface with lightlike boundaries, the latter corresponding to its intersection with the CTC surfaces.
A globally hyperbolic flat Lorentzian spacetime with particles is a flat Lorentzian manifold with particles such that is the disjoint union of lines and there exist disjoint neighbourhoods , … , of the singular lines , … , such that:
each neighbourhood is isometric to a CTC cylinder of height in .
the complement of the disjoint union is a flat Lorentzian manifold with boundary that admits a Cauchy surface, i. e. an embedded surface with boundary with spacelike interior, such that the boundary components of are null circles in the CTC surfaces , and such that every inextendible causal curve in that is contained in intersects .
If moreover the Cauchy surface can be selected to be compact, then is called spatially compact.
Note that this definition is quite restrictive regarding the CTC region around the particles. This is due to the following reasons. Firstly, we want the particles to be hidden behind a “CTC surface” , and the CTC regions around each particle therefore must be sufficiently big so that they reach the CTC surfaces in the associated one-particle models. Secondly, we need the interior region to be globally hyperbolic and hence foliated by Cauchy surfaces. This induces a foliation of the CTC surfaces around each particle by non-timelike closed curves, and hence by null circles. In order to obtain a notion of globally hyperbolic flat Lorentzian manifold with particles that fulfils each of these requirements, we then have to assume that each surface is the boundary of a CTC cylinder.
Given a flat Lorentzian spacetime with particles that is globally hyperbolic in the sense of Definition 3.2, one can add to each CTC cylinder the entire CTC region in the corresponding one-particle model, and this completion has no impact on the geometry of its interior part . However, in the following, we take the viewpoint that the specific geometry of the CTC region is irrelevant itself and only of interest through its effect on geodesics that enter a connected component of the CTC region from the interior region and then return to . Such a geodesic has to be contained in the CTC cylinder bounded by . What happens outside the CTC cylinder inside the CTC region is therefore not relevant to our situation except through its effects on geodesics outside the CTC region.
In the following we will focus on the situation in which the spins are small compared to the cone angles so that the scale of the CTC radii is small compared with the global geometry of the more classical globally hyperbolic interior region . In the limit case, where one or more spins tend to zero, the associated CTC regions become empty. In that situation, one can extend the notion of causal curves by including curves that contain components of the singular lines. In this setting, our notion of global hyperbolicity requires that there is a closed Cauchy surface intersected by all inextendible curves that are causal in that sense.
3.2. Doubling the spacetime along the CTC surface
Classical results involving global hyperbolicity are not available for spacetimes with boundary such as the interior region in Definition 3.2. However, we can nevertheless relate these spacetimes to the classical framework by employing the following “doubling the spacetime” trick.
Let be a singular flat spacetime satisfying the first condition in Definition 3.2, and denote by its interior region. For each singular line , let be the cone angle around , the spin and let denote the associated CTC radius. The isometry between the boundary components of and the CTC cylinders defines a local coordinate system in a neighbourhood of each boundary component, in which the metric takes the form (2) (with replaced by and by ). Define a new coordinate on through the condition , in the interior region near each surface . In terms of these coordinates the metric (2) for each particle takes the form
When the CTC cylinder is a finite -cylinder one can prescribe the coordinate to vary in . As the number of particles is finite, there exists an such that the subsets of the neighbourhoods characterised by the condition define solid pairwisely disjoint cylinders that contain all surfaces .
Consider two copies of and glue them along their boundaries in the obvious way. More precisely, let and be two identifications and consider the union of and with and identified for every in . We get a manifold , containing a surface (the locus where the glueing has been performed) and two embeddings . In the following, is referred to as the doubling of along . A neighbourhood of every connected component of in can be parametrized by coordinates but where now is allowed to vary in , hence to have negative values. Positive values of correspond to points in the first copy whereas negative values represent points in . The surface is characterised by .
The manifold is equipped with a metric which, however, becomes degenerate on . Nevertheless, it is still reasonable to study the causality properties of such a degenerate cone field. A convenient way to do so is to consider as the limit of non-degenerate Lorentzian metrics. For this we introduce a bump function , which is a non-increasing smooth function that vanishes on the interval and takes constant value on . For every , we define
Then is a Lorentzian metric on , equal to the flat metric on the region , and converges with respect to the -norm to the degenerate flat metric for .
One can allow the coordinate in (5) to take values on the entire real line. In this case, it defines a Lorentzian metric that becomes degenerate for and approximates the doubling of the interior region of .
Every spacetime for is globally hyperbolic, and every level set of is a Cauchy surface.
The level sets of are spacelike for every , hence is a time function. Let be an inextendible -causal curve. Then it is also -causal. The map induces an isometric branched covering that preserves the coordinate . As is globally hyperbolic relatively to its boundary (cf. section 2.4) the image of by must intersect every level set of . The lemma follows. ∎
3.3. A criterion for global hyperbolicity
Let be a singular flat spacetime satisfying the first condition in Definition 3.2. Assume that the closure of the interior region contains no closed causal curves except the null circles in the surface , and that for all , the intersection between the causal future of and the causal past of is either compact or empty. Then admits a Cauchy surface.
1. This lemma is well-known in the non-degenerate case and is at the foundation of the notion of global hyperbolicity. To prove it for the degenerate case, we first observe that the metric has no closed causal curves (CCC) except the null circles. Indeed, consider the map (a branched cover) that sends the points and to . It is an isometry with respect to the metric . If is a CCC in for the metric , its image under is a CCC for the flat metric in and hence, by hypothesis, a null circle. Now observe that weakly dominates all the metrics , in the sense that every causal curve for is also causal for the degenerate metric . A direct calculation shows that the null circles in the CTC surfaces are spacelike for . It follows that the metrics have no closed causal curves.
For every point in , denote by the causal past (-) and future (+) of in with respect to the metric and by its causal past (-) and future (+) with respect to . As every causal curve for is a causal curve for , the intersection is contained in for all . Assume that is not empty. Then maps into a closed subset of . On the other hand, is a proper map. As is compact by hypothesis, the same holds for . As is a closed subset of , it is therefore also compact. This proves that there exists an such that the metric is globally hyperbolic for all .
2. For every let be a Cauchy surface for . Denote by the intersection of the Cauchy surface with the interior region , considered as a subset of . Denote by the region in that is characterised by the condition and by , … , its connected components. Recall that is equal to outside . The intersection of with every connected component is the graph of a map which takes values in .
Claim: There is a compact -spacelike hypersurface and a positive real number such that coincides with a Cauchy surface of in the region .
To prove the claim, we first assume that the spacetime admits only one particle (). We fix a point in the CTC cylinder characterised by the condition , which is the boundary of the interior region . Without loss of generality, we select such that its -coordinate vanishes. Then, we can assume without loss of generality that the Cauchy surfaces have been chosen in such a way that they all contain .
By applying the Ascoli-Arzela Theorem to , one then obtains directly that there is a subsequence of the sequence of surfaces , , which converges to a -spacelike hypersurface in the region . Note, however, that outside the region , these surfaces may escape to infinity when . This issue can be addressed as follows: for sufficiently small, one can extend the part of outside by a surface approximating , which is -spacelike (details are left to the reader).We then obtain a compact surface which, as required, is -spacelike and coincides with outside of (recall that and coincide there). This proves the claim for .
Consider now the case . Fix a point in the CTC cylinder in the first component , and assume that every contains . Reasoning as above, we construct a surface which coincides with (for some ) outside and is -spacelike in . Denote by , the two compact surfaces obtained by pushing in the future (respectively in the past) the surface in such a way that the resulting surfaces are -spacelike in and -spacelike outside . We consider the region between and . As the surfaces are -spacelike for every , is globally hyperbolic for every . It follows that the surfaces can be selected in such a way that they all lie in , with the possible exception of the region .
We now drop the condition and replace it by an analogous condition for the second connected component: we impose that all surfaces contain a given point in the CTC cylinder in . Repeating the argument above, we obtain two disjoint surfaces , which
are chosen in such a way that lies in the future of
are -spacelike in the region ,
are -spacelike in ,
lie between the surfaces and outside of .
We now impose as a condition that the Cauchy surfaces lies between and , with the possible exception of region . Iterating this process, we obtain two compact surfaces , which are -spacelike everywhere. We conclude the proof of the claim as in the case .
3. After proving the claim, we resume our proof of Proposition 3.4. To conclude this proof, we show that the surface is a Cauchy surface for . Let , be an inextendible future oriented causal curve in for the metric . Assume without loss of generality that lies in the past of for the metric . By way of contradiction, assume that never intersects .
Define . By definition of , this implies that for all lies in the region where and are equal. Hence the restriction of to the interval is a causal curve with respect to . As the surface is a Cauchy surface for and coincides with outside , it follows from the assumption that must be finite. Moreover, lies in the past of the Cauchy surface and therefore under the graph of .
However, by hypothesis, , cannot intersect the graph of . It follows that must exit the region and that its exit point it is still in the past of with respect to . Let and let be an increasing sequence such that . Observe that might be infinite. Every point lies in the future of . As is globally hyperbolic, is compact and the sequence converges. As is inextendible, it follows that is finite, and the limit must be . In particular, this implies that is finite and lies in . By hypothesis, is not in because . This implies that for some , is still in the past of . But the argument above, proving that is finite, implies that should meet once more, which is a contradiction.
This implies that every inextensible causal curve with respect to must intersect the surface . Hence is a Cauchy surface, and is globally hyperbolic. ∎
4. Construction of stationary flat spacetimes
4.1. Euclidean surfaces with cone singularities
After discussing the one-particle model and introducing a notion of global hyperbolicity, we will now construct examples of flat Lorentzian spacetimes with particles. The resulting spacetimes are stationary and the construction is based on Euclidean surfaces with cone singularities. In the following, we denote by for the Euclidean plane with one singular point of cone angle , that is with the metric given in (4).
A Euclidean metric with cone singularities on a closed orientable surface consists of a finite number of points , … , (the cone singularities) together with an assignment of positive real numbers (the cone angles) to for , and a flat Riemannian metric on , such that every point admits a neighbourhood in so that is isometric to a ball in centred at the singular point.
Note that the quantities which in (2+1)-gravity are interpreted as masses of the particles, are usually referred to as apex curvatures in the mathematics literature (see for example [Thu98]). They are subject to the relation
where denotes the Euler characteristic of . In particular, if all the cone angles satisfy , then the surface is either the sphere of the torus, and the torus arises only if there is no singularity.
Observe that the flat Euclidean structure naturally defines a conformal structure on and the punctures correspond to the cusps of this conformal structure. Consequently, the flat Euclidean structure equips with the structure of a Riemann surface. That the converse is also true follows from a theorem by Troyanov.
Theorem 4.2 ([Tro86]).
Let , … , be a collection of points in , and , … , positive real numbers such that
Then, for any conformal structure on , there is an Euclidean metric on with cone singularities of cone angles at each that induces the given conformal structure. This singular Euclidean metric is unique up to a rescaling factor - in particular, it is unique if we require the total volume to be equal to .
The study of singular Euclidean surfaces is a very traditional topic in mathematics. For instance, it is related to billiards. A way of investigating a billiard in a polygon in the Euclidean plane is to consider it as the geodesic flow of the singular flat Euclidean metric on the sphere, which is obtained by taking the double of the polygon along its sides (see [MT02]).
An important case is the one in which all cone angles are rational. For instance, if all of these angles are multiples of , the associated singular Euclidean metric is directly related to holomorphic quadratic differentials.
On the other hand, the “positive curvature case”, in which all cone angles are less than and the Euclidean surface is a sphere sphere with conical singularities, there is always a geodesic triangulation of the singular Euclidean surface . This implies that can be obtained by gluing triangles in the Euclidean plane along their sides (see [Thu98, Proposition 3.1]). In particular, when all the cone angles are rational, the associated flat surface is an orbifold. It is obtained as a quotient of a closed Euclidean surface without cone singularities by the action of a finite group of isometries.
4.2. Stationary flat spacetimes with particles
We now construct globally hyperbolic flat spacetimes with particles based on Euclidean surfaces with cone singularities. The simplest and most obvious example are static spacetimes, which are obtained as a direct product of the Euclidean surface with cone singularities with .
Definition 4.3 (Static spacetimes with particles).
Let be a closed Euclidean surface with conical singularities , … of angles , …, . We denote by the flat metric on the regular part . Then the product contains the open subset and can be equipped with the Lorentzian metric , where the coordinate parametrises . This metric is locally flat, and can be considered as the regular part of a flat singular metric on where the lines are spinless particles of cone angle .
Observe that these spacetimes are static: the vertical vector field is a Killing vector field, orthogonal to the level sets of . As the spacetime is static, is a Cauchy time function and the levels of are compact and hence complete. This implies directly that the static singular flat spacetime is globally hyperbolic.
To obtain a more interesting class of examples, we consider a closed -form on , where is the regular part of a singular flat Euclidean surface as in Definition 4.3. We consider again the direct product but now equipped with the metric
instead of . Note that this defines a flat Lorentzian metric on . On any subset where is contractible, the form is exact, i. e. given as the differential of a map . This implies that the metric on is simply the pull-back of under the diffeomorphism and hence a flat Lorentzian metric on . Moreover, this argument shows that only depends, up to isometry, on the cohomology class of .
We now fix open pairwisely disjoint neighbourhoods around every singular point such that is isometric to a ball in centred at the singular point. In a suitable polar coordinate system, the metric on then takes the form
Denote by the closed -form in . As it generates the first cohomology group, the restriction of to is cohomologous to , where and is a loop in that makes one positive turn around . Therefore, there exists a function such that . Let be a function whose restriction to coincides with for all . Such a function can be constructed by means of bump functions , which satisfy and for all . The function is then given by . On every neighbourhood , the one form then coincides with . This implies that the associated metric defined as in (6) and restricted to takes the form
We recognise, up to a rescaling factor for the coordinate , the metric (3) for a particle with spin . Hence