Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry

Stationary Metrics and Optical
Zermelo-Randers-Finsler Geometry

G. W. Gibbons111G.W.Gibbons@damtp.cam.ac.uk ,  C. A. R. Herdeiro222crherdei@fc.up.pt ,  C. M. Warnick333cmw50@cam.ac.uk  and M. C. Werner444mcw36@cam.ac.uk

D.A.M.T.P.
Cambridge University
Wilberforce Road, Cambridge CB3 0WA, U.K.

Departamento de Física e Centro de Física do Porto
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

Institute of Astronomy
Cambridge University
Madingley Road, Cambridge CB3 OHA, U.K.
November 2008
Abstract

We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painlevé-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergo-surface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples.

1 Introduction

There has been a great deal of activity in theoretical physics over the past few years involving the properties of various spacetimes in four and higher dimensions. Apart from their obvious astrophysical significance, there have been applications to string theory, gauge theory (via the AdS/CFT correspondence), and a growing interest in analogue models. Most investigations have centred on particular spacetimes, typically a solution of Einstein’s equations arising in supergravity theory. However, it is important to distinguish particular properties from general ones. In other words, we wish to see to what extent certain results obtained are due to the special properties of the metric used and to what extent they are universal, and valid for all metrics in a general class.

In the case of static metrics a convenient tool for investigating such questions is the well known optical metric (see [1] for an elementary account) which allows one to bring to bear on questions concerning non-rotating Killing horizons, light bending etc., the powerful tools of Riemannian and Projective Geometry. For example, the optical distance diverges as one approaches a static Killing horizon, and in this limit the metric is asymptotic to the metric on hyperbolic space with the radius of curvature being given by the inverse of the surface gravity of the horizon. One may use this fact to gain insight into “No Hair Theorems” and the loss of information arising as matter falls into the associated black hole [2]. One may use the Gauss-Bonnet Theorem and information about the curvature of the optical metric to make general statements about light bending [2, 3, 4] and one may use the Projective Geometry to illuminate the dependence of light bending on the cosmological constant [5].

This leads naturally to the question of what replaces the optical metric when one is dealing with stationary metrics and can the associated geometric structures prove useful in dealing with rotating Killing horizons, ergo-regions and ergo-spheres, the existence or otherwise of closed timelike curves.

The purpose of this paper is to show how, in order to answer this question, one needs to pass from Riemannian to Finsler Geometry, to obtain two spatial geometric objects, a Randers and a Zermelo structure, related by Legendre duality. Taken with the original stationary spacetime, one obtains a triple of structures linked by a kind of triality (see figure 1). This allows one to translate a problem in one language to one of the other two languages, often leading in this way to a dramatic simplification. For example much effort has been expended on understanding constant flag curvature for Randers-Finsler metrics [6]. As we shall see, from the spacetime perspective the spacetime metrics are rather simple: all constant flag curvature Randers-Finsler metrics map to conformally flat spacetimes. In particular, the principal example in [6] lifts to flat Minkowski spacetime in rotating coordinates, whereas the example in [7] maps to an Einstein Static Universe in rotating coordinates.

The link between the three ideas is provided by a 1-form arising, in the spacetime picture, from the cross term in the metric of the spacetime manifold and associated with the dragging of inertial frames. When viewed on the space of orbits of the time translation Killing field , this gives rise to a one-form which, with a metric , endows with a Randers structure, a type of Finsler geometry which may be thought of physically in terms of a fully non-linear and exact version of gravito-magnetism.555In the case of ultra-stationary spacetime metrics this exact version of gravito-magnetism can be understood in terms of an equivalence of magnetic tidal tensors [8]. Light rays in project down on to Finsler geodesics which in turn may be lifted to the cotangent bundle to give a magnetic flow. Such flows have been extensively studied by mathematicians of late and, as we shall show, some of their results have applications in the present setting. Moreover, we show that, in this magnetic picture, for all constant flag curvature Randers-Finsler 3-metrics, the magnetic field is a Killing vector field of the Riemannian part of the geometry .

The alternative Zermelo viewpoint works in terms of a vector field or wind , on , and a different metric . Light rays then project down on to solutions of the Zermelo problem: find the least time trajectory for a ship moving with constant speed in a wind . Physically, this is where one makes contact with fluid dynamical analogue models in which a fluid vortex, sometime thought of in terms of an Aether, gives rise to a Maelstrom-like picture of a rotating black hole. Currently, this idea, which can also work for a non-rotating black hole, is often associated with the use of Painlevé-Gullstrand coordinates. And here we show that, indeed, the Zermelo structure manifests itself in the spacetime picture by expressing the spacetime metric in a Painlevé-Gullstrand form.

This paper is organized as follows. In section 2 we describe the network of relations: the triality between the Zermelo problem, Randers-Finsler geometries and the null geodesic flow in conformally stationary spacetimes. We also discuss the magnetic flow viewpoint, the relation to analogue models and the gauge equivalence from the spacetime picture. In particular, the condition of spacetime conformal flatness is explored in both the Randers and the Zermelo pictures, and the relation to Killing magnetic fields and constant flag curvature is established. In section 3 we provide a variety of examples of increasing complexity. We start with Langevin’s rotating platform in Minkowski spacetime, which maps to a Randers-Finsler geometry of constant flag curvature, physically interpreted as a varying magnetic field in a negatively curved space, and a Zermelo problem of a constant angular velocity tornado (or Shen’s rotating fish tank [6]) in flat space. Let us note that, since the mapping between the spacetime and Randers picture can be seen as a temporal Kaluza-Klein reduction, this example is akin (in fact an analytic continuation) to the Melvin solution in Kaluza-Klein theory [9]. Section 3.2 explores, in contrast to the first one, several examples where the Randers picture is physically simpler than the Zermelo one, and in some cases, even simpler than the spacetime picture: constant or varying magnetic fields in or . The particular example of section 3.2.2 allows us to relate the Finsler condition with isoperimetric inequalities, Novikov’s action functional and ergodic theory. Section 3.3 gives another example of a Randers-Finsler geometry of constant flag curvature which is simply interpreted in the spacetime picture: a conformally flat spacetime – the Einstein Static Universe (ESU) – in rotating coordinates. We then consider a genuinely rotating ESU, the Anti-Mach geometry, and its Randers and Zermelo pictures are exhibited. In section 3.4, we discuss the Randers and Zermelo pictures for the Kerr geometry. While these pictures are valid asymptotically, they break down at the ergo-sphere. To cover the ergo-region, we actually need to introduce co-rotating coordinates, and then the Randers/Zermelo pictures break down asymptotically. Using these co-rotating coordinates, we show that the Riemannian part of the Randers metric is hyperbolic near the horizon for non-extremal Kerr, in similarity with the general non-rotating Killing horizon. The Randers/Zermelo pictures of the near horizon extremal case, the NHEK geometry, are also considered and, in particular, the Zermelo picture turns out to be quite simple: an azimuthal wind in a deformed 3-sphere. We finish in section 4 with some final remarks.

2 A Network of Relations

2.1 The Zermelo problem as a Finslerian flow

The Zermelo navigation problem [10] is a time-optimal control problem, which aims at finding the minimum time trajectories in a Riemannian manifold ( with the Euclidean metric in [10]) under the influence of a drift (wind) represented by a vector field . Assuming a time independent wind, Shen [6] has shown that the trajectories which minimize travel time are exactly the geodesics of a particular Finsler geometry, known as Randers metric [11], whose norm is given by

(1)

For the geodesics of (1) to solve the corresponding Zermelo problem, the Randers data are determined in terms of the Zermelo data as [12]:

(2)
(3)

Note also that

(4)

It has also been shown [13] that every Randers metric arises as the solution to a Zermelo’s problem of navigation, defined by the data . The latter is defined by the former as:

(5)
(6)

Note also that

(7)

Thus, there is a natural identification of Randers metrics with solutions to the Zermelo problem. It should be observed that

The Finsler condition ensures that is positive and the metric is convex, i.e. is positive definite for all non-zero [13].

2.1.1 Zermelo/Randers duality via a Legendre transformation

For a generic Finsler norm, the geodesics of are obtained by taking as the Finsler Lagrangian. However, for a Hamiltonian treatment, we need to consider a Lagrangian which is not homogeneous of degree one in velocities, since the associated Hamiltonian would vanish. It is convenient to consider a Lagrangian of degree two in velocities. Thus, since is a Finsler Lagrangian which is homogeneous of degree one, we define a Lagrangian by

(8)

Hence, the Hamiltonian will also be of degree two in the momenta

(9)

In fact,

(10)

It is convenient to define a function which is homogeneous of degree one in momenta by

(11)

Thus

(12)

Specializing to be given by the Randers metric (1), we have

(13)

since , we find that

(14)

where are the associated Zermelo data. Thus, the Legendre transformation maps the Randers (Lagrangian) data to the Zermelo (Hamiltonian) one.

Note that one may think of as the sum of two moment maps

(15)

where

(16)

generates the geodesic flow and

(17)

generates the lift to the cotangent bundle of the one parameter group of diffeomorphisms of the base manifold generated by the vector field . Observe that these two flows commute

(18)

if and only if is a Killing vector field. In the case that the base manifold is a sphere and a Killing vector field, these are the flows constructed by Katok [14] and discussed by Ziller [15]. They correspond to rigid rotation on the sphere.

2.2 Spacetime picture: Randers and Zermelo form of a stationary metric

The geodesic flow of the Randers metric can be seen as the null geodesic flow in a stationary spacetime with generic form

(19)

so that

(20)

Fermat’s principle arises from the Randers structure given by

(21)
(22)
(23)

Thus we call the form

(24)

the Randers form of a stationary metric. Note that for , is the usual optical metric (see [1] for an elementary introduction).

Using (5), (6), the Randers data are equivalent to a Zermelo structure of the form

(25)
(26)

where

(27)
(28)

and

(29)
(30)

There is actually a more familiar way to extract the Zermelo data from a stationary spacetime. Inverting (21), (23) and using (2), (3) one finds the spacetime metric (19) in the form

(31)

This is the Zermelo form of a stationary spacetime. One recognizes the metric in square brackets as having a Painlevé-Gullstrand form [16, 17]. Since in the spacetime picture of the Zermelo problem/Randers flow one is interested only in the null geodesic flow, the spacetime metric is only defined up to a conformal factor. Thus, the Zermelo data encoded in a stationary spacetime can be read off simply by writing it (conformally) in a Painlevé-Gullstrand form.

It should be noted, however, that for a given spacetime, an apparently different Zermelo structure can be obtained by writing the spacetime metric in Painlevé-Gullstrand coordinates. It is helpful to give an example. In these coordinates , the Schwarzschild metric is

(32)

One finds immediately that the Zermelo structure is

(33)

which is interpreted as a radial wind in flat space. On the other hand, the Zermelo structure derived by writing the Schwarzschild solution in the usual Schwarzschild coordinates , in the form (31), has no wind and the spatial metric is the usual optical metric,

(34)

The two Zermelo structures (33) and (34) are equivalent in the sense that the trajectories of one can be mapped to the trajectories of the other one. This equivalence is obvious in the spacetime picture, since both structures correspond to the same spacetime in different coordinates; but it is not obvious in the Zermelo picture itself. In the Randers picture, on the other hand, the equivalence is again obvious, since is, in both cases, the optical metric and differs only by a gauge transformation. This type of equivalence is one example of the large class of gauge equivalences which will be present in our discussion of the spacetime/Randers/Zermelo triangle and which we will explore more in section 2.5. In section 3.4, we will briefly compare the analogue of these two Zermelo structures in the Kerr case.

Given the simple physical interpretation of the Zermelo problem, it should not come as a surprise at this point that many analogue models of black holes are naturally written in the Painlevé-Gullstrand form (see, for instance [18, 19, 31]), as we shall now discuss briefly.

2.2.1 Analogue Models: waves in moving media

Metrics of Painlevé-Gullstrand form are frequently encountered in discussions of waves propagating in moving media and go back at least to a paper of Gordon [20] on electromagnetic waves in a moving dielectric medium. Similar metrics have arisen when discussing sound waves in a moving compressible medium such as the atmosphere [21, 22]. More recently attempts have been made to construct analogue models of black holes [23, 24] and again similar metrics are used. Typically such metrics are taken to be accurate just to quadratic order in velocities and the underlying spatial metric to be flat. Thus one considers a metric of the form

(35)

where is the local light or sound speed and the velocity of the medium. Both and could, as indicated, be time dependent but from now on we shall assume that both are independent of time. If and were constant, then (35) would be obtained by performing a Galileo transformation on the flat metric

(36)

The scalar wave equation for a scalar field ,

(37)

becomes

(38)

which is indeed the wave equation of (35). One could adopt (38) or some variant of (38) with lower order terms even if and depend on space and time. One would still find that the characteristics, i.e. the “rays ” would be null geodesics of the metric (35).

An interesting example is provided by a cylindrically symmetric vortex flow [25, 26] for which, in suitable units, and

(39)

where , and is a unit vector in the direction. The metric (35) becomes

(40)

The metric is well defined and has Lorentzian signature for all . The surfaces are timelike for all . Thus, there is no Killing horizon, and hence no analogue black hole, in this metric (in agreement with [27]). The surface is a timelike ergo-sphere inside which the Killing vector becomes spacelike. This closely resembles the cylindrical analogues discussed by Zel’dovich when he suggested the possibility of superradience for the Kerr black hole [28, 29].

2.3 The magnetic flow viewpoint

The Randers orbits are generated by the Finsler norm (1) taken as a Lagrangian: . An immediate observation, already pointed out in [11] is that these can be interpreted as motion in a magnetic field. Indeed, the Euler-Lagrange equations for the Finsler norm are

where , and is arc length with respect to the Riemannian metric . In terms of the standard magnetic potential the orbits of a charged particle (mass , charge ) can be written, using the same parametrization, as

where the energy is

and is proper time. Thus, the Randers 1-form relates to the physical magnetic potential 1-form by

(41)

Note, in particular, the dependence on the energy. We shall come back to this point in the example of section 3.2.2.

From a physicists’ perspective, this is actually the most natural way to look at Randers metrics: as describing motion on curved manifolds with a magnetic field

(42)

denoting Hodge duality with respect to by . From the perspective of section (2.2), the spacetime geodesics, when projected down to the base manifold , with metric , are no longer geodesics; rather they satisfy the equations of motion of a charged particle in a magnetic field, which is described by the Maxwell 2-form . The particle paths may be lifted to the cotangent bundle of where they are referred to as magnetic flows. They may be regarded as a Hamiltonian system where the Hamiltonian coincides with the usual energy

(43)

but where the symplectic form is modified

(44)

This magnetic flow then uplifts to one dimension more (in a temporal version of the Kaluza-Klein picture) as the null geodesic flow discussed in section (2.2).

Magnetic flows have been quite extensively studied in the mathematics literature. In particular, some of the phenomena to be discussed here, involving the breakdown of the Randers structure and, at times, the transition between the behaviour with and without closed timelike curves in the spacetime picture, has been noticed, albeit without that interpretation [30]. To discuss them here it is useful to choose a particular “conformal gauge”.

2.4 Spacetime versus magnetic flow pictures in the ultra-stationary gauge

With regard to the Randers/Zermelo structure, conformally related spacetimes form an equivalence class. Thus, for this purpose, we can always take the representative in this equivalence class to be ultra-stationary, that is a spacetime admitting an everywhere timelike Killing vector field with unit length, . Such spacetimes may be considered, as a bundle over a base manifold consisting of the space of orbits of the Killing vector field. These orbits, sometimes called timelines, may either be circles, , in which case time is periodic, or lines . In the former case there are obviously closed timelike curves (ctc’s). If the bundle is a trivial product, , these ctc’s may be eliminated by passing to a covering space. Then the question is: do there exist ctc’s which are homotopically trivial, i.e. which may be shrunk to a point? If the bundle is non-trivial, things may be more complicated. Spacetime may already be simply connected. In that case the ctc’s may not be eliminated by passing to a covering spacetime. This happens in spacetimes of Taub-NUT type. If the bundle is non-trivial it will admit no global section and hence no global time coordinate , say, such that . If the bundle is trivial then there will exist (many) global time coordinates but there may exist no global time function. A time function may be defined to be a function which increases along every timelike curve. Its level sets must therefore be everywhere spacelike. The examples that will be provided in section 3.2 are of this type: the bundle is trivial, a time coordinate exists, but there is no global time function. The line element can be cast in the form (24) with

(45)

where and are coordinates on . The positive definite metric is the projection of the spacetime metric orthogonal to the Killing vector field . The time coordinate is defined only up to a gauge transformation of the form under which . The quantity may be regarded as the pull-back to of the Sagnac connection [32] which governs frame-dragging effects. It depends upon the choice of section of the bundle, however the pull-back of the curvature is gauge-invariant.

Any timelike curve in projects down to . We call its projection and define its element of length with respect to the metric by . The curves will be timelike as long as

(46)

If the right hand side of (46) is always positive then is a time function. On the other hand, suppose that admits a closed timelike curve. Its projection on will be a closed curve and moreover

(47)

where is the length of with respect to the metric . Thus a sufficient condition for the absence of ctc’s is that

(48)

for all closed curves in . Inequality (48) is also a necessary condition for the absence of ctc’s, because if it is not true we can find a timelike curve for which the total change in the coordinate vanishes, in which case it is a closed timelike curve, or for which it is negative. In the latter case we can construct a closed timelike curve by moving along an orbit of .

The metric induced on the surfaces of constant is

(49)

This depends on the choice of gauge. If for some choice of gauge it is positive definite, then the surfaces will be spacelike and and hence will be a time function. In that case no closed timelike curves are possible. But if , where is a sub-manifold of the constant surfaces, (49) is not positive definite on ; this surface is either a null or a singular surface where the Randers structure breaks down.

If is regular, its existence implies the appearance of closed null curves in spacetime. For this reason we call it a Velocity of Light Surface (VLS), from the spacetime perspective. And if beyond the metric induced on constant surfaces becomes timelike, there will be closed timelike curves. Moreover, in the absence of horizons these can be extended all over the spacetime, that is, they are naked ctc’s. In section 3.2 we will give four examples of this situation.

If is singular, on the other hand, the ultrastationary conformal gauge might not be the best gauge to provide a spacetime interpretation. Multiplying the ultrastationary metric by a conformal factor (that vanishes on ) might provide a more physical interpretation. This is illustrated by the examples in sections 3.1 and 3.4. In both cases, a spacetime metric with an appropriate conformal factor renders the interpretation of an ergo-surface for , wherein becomes null. In this gauge is also a VLS, albeit of a different nature than that of the previous case.

Null Geodesic Flow in aClass of Stationary MetricsZermelo ProblemFinslerian Flow in aRanders MetricOptical FormPainlevé-Gullstrand FormTemporal version ofKaluza-Klein FormMagnetic Flow in aRiemannian Metric
Figure 1: A network of relations. Alongside the arrows are the equation numbers which provide the respective map. The dashed lines represent interpretation relations rather than different frameworks.

2.5 Gauge equivalence and Tensorial Relations

In figure 1 we summarise the main relations discussed in the previous sections. In the network of relations described by figure 1 it is possible that a complicated Randers picture is equivalent to a simple Zermelo picture or vice-verse. It is also possible that the spacetime description may illuminate the Randers or Zermelo pictures, or vice-verse. Or even that two Zermelo (or two Randers) pictures are equivalent, one being simple and the other involved. We shall now address the gauge equivalence in the network. In the next section explicit examples will be provided.

The key observation here is that in the spacetime picture of the flows, we have a very large freedom to change the metric by diffeomorphisms (coordinate transformations) and conformal rescalings which do not affect the null geodesics. Because of this, we have access to a large class of transformations which map one problem into another, going far beyond the manifest symmetries of either of the lower dimensional viewpoints.

In the spacetime picture, we consider the null geodesics of a conformal class of metrics . In order to pass to the Randers or Zermelo picture, we require that admits at least one timelike conformal Killing vector , satisfying:

(50)

for some representative of , where is any function on . The conditions are independent of the choice of representative . Locally we may pick so that vanishes. We may choose coordinates so that and the conformal metric takes the form (45). This form of the metric is not uniquely determined: is determined only modulo an exact one-form so that the data

(51)

arise from the same pair . From the Randers point of view, this transformation of does not alter the geodesics, essentially adding an exact differential to the Lagrangian. In the spacetime picture this may be seen as a coordinate transformation to a new time coordinate , as discussed in section 2.4. Whilst this is a natural transformation from the point of view of the Randers picture, in the Zermelo case such a transformation gives rise to a complicated transformation involving both the wind W and the spatial metric .

In the case where the class admits more than one timelike conformal Killing vector, we are also free to make a different choice of . This may lead to very different structures on reduction to Randers or Zermelo data. As an example, we may consider the case where we have a Zermelo structure arising from a conformal Killing vector . If the Zermelo metric admits a Killing vector , then may be seen to be a conformal Killing vector of and provided is sufficiently small this will remain timelike. After the reduction with this new conformal Killing vector, we see that the effect in the Zermelo picture is to introduce a ‘Killing wind’. Once again, this is a natural transformation in the Zermelo picture, but can be very complicated in the Randers picture, as shown in the examples of section 3.1 and 3.3.

There are therefore essentially three possible transformations available to us, which relate Randers or Zermelo problems arising from the same geodesic problem in the spacetime picture.

  • Change of coordinates in the spacetime picture independent of . This descends to a change of coordinates on .

  • Shift of time coordinate in the spacetime picture: . This descends to a gauge transformation in the Randers picture . In the Zermelo picture, this transformation is complicated.

  • Change of timelike conformal Killing vector. This includes the introduction of a “Killing wind”in the Zermelo picture, but may produce other transformations. In the Randers picture, this transformation is complicated.

We thus see that once we allow the more general class of transformations available in the spacetime picture, some problems which from the lower dimensional point of view seem inequivalent arise from choosing different coordinates on the same spacetime, or reducing along different directions within the same spacetime. Such transformations include, but are by no means limited to the manifest symmetries of the lower dimensional pictures, for example under change of coordinates on .

2.5.1 Conformally flat spacetimes

Once we allow this larger set of transformations to act on Randers or Zermelo problems, we are faced with the question of how to identify two equivalent problems. The standard tensorial quantities of the lower dimensional pictures are not preserved under a general higher dimensional transformation, so we must look elsewhere for our answer. The solution is to consider the Weyl tensor of the higher dimensional metric, which is invariant under diffeomorphisms and conformal transformations. A case of particular interest is when the higher dimensional space is conformally flat, so that it may be brought by coordinate transformations to the form , with the Minkowski metric. In this case the null geodesics are simple to find.

If we assume that a spacetime metric in the Randers (24) or Zermelo (31) form is conformally flat, this imposes conditions on the relevant lower dimensional data. In terms of the Randers data, for a four dimensional spacetime, these conditions may be reduced to, using the analysis in [33],

(52)

where the tensorial quantities on the right hand side of the equivalence are all computed from the metric , and is defined by (42), i.e the magnetic field in the magnetic flow interpretation of the Randers picture. Thus, a necessary (but not sufficient) condition for the spacetime Weyl triviality of a Randers structure is that the magnetic field (42) is a Killing vector of the metric . This gives a simple test for examining the Weyl triviality of an apparent complex Randers structure. Examples of this will be given in sections 3.1 and 3.3. In section 3.2 we will give examples of both Killing and non-Killing magnetic fields corresponding, in the spacetime picture, to non-conformally flat geometry.

It is also worthwhile to calculate the Weyl tensor for a metric in the Zermelo or Painlevé-Gullstrand form. To do so it is convenient to first introduce an orthonormal basis of forms for the Zermelo spatial metric together with the dual basis of vector fields . We may then use the following orthonormal basis of forms

(53)

for the space-time metric in Zermelo form, where are the components of the wind vector in the basis , . Letting denote the Levi-Civita connection of in the given basis, we find that the vanishing of the Weyl tensor is equivalent to the following conditions on and

(54)

where we have introduced the quantities

(55)

and all indices are raised and lowered with . The curvatures refer to the metric .

In the case where the ultra-stationary metric is in fact ultra-static, i.e. , we quickly deduce that the Weyl tensor vanishes if is both Einstein and conformally flat, which imply that must in fact be of constant sectional curvature. In other words the only conformally flat ultra-static spacetimes are Minkowski space, or , as shown in [34].

2.5.2 Constant Flag Curvature

A class of spaces of particular interest in the study of Randers geometry are the spaces of ‘constant flag curvature’ which generalise the concept of space of constant sectional curvature to the Finsler regime. It has been shown [13] that the Randers metrics of constant flag curvature have corresponding Zermelo data satisfying

  • has constant sectional curvature ;

  • is a homothety of , i.e.

    (56)

    for a constant .

Substituting into equations (54) one quickly finds that all the terms not involving curvatures of cancel. Since a metric of constant curvature is necessarily both conformally flat and Einstein, the spacetime corresponding to a Randers metric of constant flag curvature is conformally flat.

We conjecture that the converse also holds, in other words we may characterise the Randers metrics of constant flag curvature as those giving rise to a conformally flat spacetime of the form (24). Clearly if we allow ourselves the full freedom to make gauge transformations and change the timelike conformal Killing vector then any conformally flat spacetime may be reduced to a Randers structure of constant flag curvature by taking in standard Minkowski coordinates, which gives the trivial Randers structure of .

If we restrict ourselves to only allowing gauge transformations, so that we have no freedom to change then there are more possibilities. In order to classify the possible Randers structures arising from a conformally flat spacetime, we must classify the timelike conformal Killing vectors of Minkowski space. Any such vector generates a conformal transformation and so may be written as a linear combination of generators of the conformal group. These generators are:

(57)

We claim that any conformal Killing vector which is timelike in some region of Minkowski space may be brought to the following form by a conformal transformation:

(58)

where are constants which we require to be such that the vector remains timelike in some region. For each such , there is a choice of gauge where the Zermelo structure is of the following type:

  • is a metric of constant sectional curvature ;

  • is a Killing vector of , determined by ;

For example, in the case where , give rise to a wind along the rotational Killing vectors and along the translational Killing vectors. The case is less clear cut, but in both cases the give rise to winds along the orbits of some subgroup of the full isometry group.

It would appear that we have not recovered all of the constant flag curvature manifolds of [35], since they find in addition the possibility that is flat and is a homothety. This case may be seen in the Randers form to be gauge equivalent to the hyperbolic metric with no magnetic field, so is included in our classification above. In fact, if we wish to classify constant flag curvature metrics modulo gauge transformations, our list above contains redundancies as any Killing wind which arises from a hypersurface orthogonal Killing vector may be removed by a gauge transformation.

3 Examples

We shall now provide the Randers and Zermelo picture of a number of stationary spacetimes and discuss some physical phenomena using the different viewpoints.

3.1 A rotating platform on : simple Zermelo; non-trivial Randers

An elementary example is obtained by considering Minkowski space in rotating coordinates. Let be the usual azimuthal cylindrical coordinate and a co-rotating coordinate with a rigidly rotating platform (angular velocity ). The spacetime metric, first considered by Langevin [36], is then of the form (24) with and Randers data

(59)

Let us consider briefly the surfaces of this 3-geometry. For simplicity we take . Observe that, introducing , the geometry of these 2-surfaces has the simple line element

(60)

with

(61)

The surface (60) has Gaussian curvature (dot denotes derivative). Thus, our particular case has a negative, but not constant, Gaussian curvature, which diverges at . Surfaces of the form (60) may be embedded in a 3-dimensional Lorentzian/Euclidean space, with metric , using the embedding functions

(62)

For the standard periodicity , regularity at the origin requires . Since, as we depart from , increases (decreases) for negatively (positively) curved surfaces, their embedding, if it exists, must be done in a Lorentzian (Euclidean) 3-dimensional space. Because of the symmetry of (60) one can display the embedding as the following surface

For instance, hyperbolic space, which has constant negative curvature, is described by . Thus, it is embedded in as the surface

For (61) it is straightforward to perform the integral (62); noting that

one can construct the embedding, which is displayed in figure 2.

The impossibility of smoothly and isometrically embedding a 2-surface with a fixed point of a symmetry, whose Gaussian curvature is everywhere negative, into Euclidean 3-space may be understood geometrically as follows. At every point the principal curvatures are opposite in sign; thus any tangent plane cuts the surface into two parts, one lying on one side of the plane and the other one lying on the opposite side. If the smoothly embedded surface is such that it lies entirely on one side or the other of a complete plane (as in the case of a surface of revolution with a fixed point, such as the one displayed in figure 2) we can bring up this plane such that it touches this surface. At this point the plane coincides with the tangent plane, and hence we obtain a contradiction.

Rewriting the metric in a Painlevé-Gullstrand form one obtains the Zermelo data:

(63)

This example illustrates the discussion of section 2.5 concerning “Killing winds”: whereas the Zermelo picture (63) retains the simplicity of the spacetime viewpoint – a rigid rotation in flat space –, the Randers picture (59) appears to reveal a much more complex structure of a non-trivial one-form in a curved space. However, using the tensorial test (52) one could also observe the Weyl triviality of the spacetime description of this Randers structure; in particular the magnetic field

is indeed a (quite simple) Killing vector field of (59). Another instructive conclusion from this example is that the breakdown of the Randers/Zermelo picture for has a clear physical spacetime interpretation: the existence of a Velocity of Light Surface (VLS) at and the consequent spacelike character of the surfaces for larger . Outside this VLS timelike observers are obliged to move in the direction.

Figure 2: Langevin’s rotating platform in Minkowski space and the corresponding Randers and Zermelo pictures. We have chosen positive to depict the tipping of the light-cones (spacetime picture) and the “tornado” direction (Zermelo picture). The direction of the magnetic field presented (Randers picture) is merely illustrative, since the embedding performed excludes the direction.

In figure 2 we summarize the network of relations for this example. In the three pictures we have shown the geometry of a (and in the spacetime picture) 2-surface. In the spacetime picture we have a VLS; both the Randers and the Zermelo picture cover only the region inside the VLS. In the Randers picture the 2-surfaces are negatively curved geometries; the outer surface depicted is the surface with constant negative curvature (and a flat surface) given for comparison; the norm of is represented by the size of the dashed arrows; it diverges, together with the curvature, when the Randers picture breaks down. In the Zermelo picture the 2-surfaces are flat geometries; describes a constant angular velocity “tornado”.

Let us finish this example with the comment that this Randers structure is one of the examples discussed by Shen [6] who envisages it in terms of a fish moving inside a rotating fish tank. The Randers structure has zero flag curvature and zero S-curvature. From the spacetime viewpoint this conclusion is somewhat trivial, since we are dealing with flat Minkowski spacetime.

3.2 Magnetic fields on and : simple Randers; non-trivial Zermelo

We shall now consider four examples of magnetic fields: a constant and a non-constant magnetic field in 3-dimensional flat space and a constant and a non-constant magnetic field in 2-dimensional hyperbolic space times a flat direction . All of these examples are, in the spacetime picture, ultra-stationary metrics of the form (45), so that the discussion of section 2.4 applies.

3.2.1 Heisenberg group manifold

A special case of the Som-Raychaudhuri spacetimes [37] is a homogeneous manifold, which is (up to a trivial -direction) the group manifold of the Nil or Heisenberg group, the Lie group of the Bianchi II Lie algebra. The spacetime metric is (45) with the Randers structure

(64)

The corresponding magnetic field is ; thus the Randers picture has the simple physical interpretation of a constant magnetic field in . This picture breaks down at – see figure 3 – which has the spacetime interpretation of being a VLS, beyond which (i.e. for larger ), the integral curves of are closed timelike curves. Note also that this Randers structure fails to obey the requirements (52), as expected, despite the fact that the magnetic field is actually a Killing vector field of (64).

Figure 3: Surfaces, in , where the Randers structure breaks down. Left: Heisenberg example of section 3.2.1 for (smaller to larger cylinder); the Randers structure is valid inside the cylinder for each case. Right: Störmer example of section 3.2.3 for (smaller to larger surface); the Randers structure is valid outside the surface for each case.

The Zermelo structure, on the other hand, is more complex:

(65)

3.2.2 Squashed and Gödel spacetime

is the group manifold of the Lie group . Introduce the following two sets of 1-forms:

(66)

These are called left and right 1-forms on , respectively. Both sets obey the Cartan-Maurer equations

where are the structure constants of , i.e . The metric can be written in terms of either set of forms; we write it as666For the space (i.e for ) to have unit “radius”, a conformal factor of would have to be included.

(67)

We have introduced a squashing parameter ; corresponds to the case. For we have a family of spacetimes with ctc’s of the Gödel type [38]. To identify the original Gödel universe [39] we introduce a new time coordinate and the vector field . The Einstein tensor of (67) (with a trivial flat direction) reads

which means it is a solution of the Einstein equations with a cosmological constant and a pressure-less perfect fluid with density . Gödel’s original choice was to take which corresponds to , but all spacetimes with are qualitatively similar. Introducing yet another time coordinate the spacetime metric (adding a flat direction to make it 4-dimensional) becomes of the form (45) with the Randers structure

(68)

which can be interpreted as a constant magnetic field on . This Randers structure breaks down at

being valid only for smaller – see figure 4. Thus, the Randers structure is valid everywhere if . When it breaks down there is, in the spacetime picture, a VLS, such that the Killing vector field becomes timelike beyond it.777An illustrative diagram for the light cone structure of the Gödel spacetime is given in [40] (and corrected in [41, 42]). A similar light cone structure applies to the Heisenberg example of section 3.2.1. It is instructive to compare this light cone structure with the one illustrated in figure 2.

The Zermelo structure is, again, less straightforward to interpret:

(69)

This squashed example allows us to make another application of the discussion of section 2.4 together with isoperimetric inequalities. For this purpose, reconsider equation (48), and suppose that the closed curve spans a 2-surface . This will always be true if is simply connected. Using Stokes’s theorem, condition (48) becomes

(70)

Using (68) for a simply connected domain in the hyperbolic plane , with area , condition (70) becomes

(71)

Now the isoperimetric inequality of Schmidt [43, 44] states that for any such domain (with Gaussian curvature )

(72)

In the Euclidean plane the second term in the square root would be absent; it arises from the negative curvature. In fact for any two dimensional connected and simply connected domain with Gauss-curvature everywhere less or equal to , Yau [44, 45] has shown that