Lorentzian Surfaces

John McCarthy Supervised by Thomas Leistner The University of Adelaide

Contents

1 Introduction 2

2 Preliminaries 2 2.1 Lorentzian Surfaces and their Topology ...... 2 2.2 Conformal Metrics ...... 3

3 Geodesics 5 3.1 Geodesic Equation ...... 5 3.2 Levi-Civita Connection ...... 5 3.3 Geodesic Completeness ...... 6

4 Examples 9 4.1 Cliton-Pohl Torus ...... 9 4.2 Misner Space ...... 10 4.3 Metrics on R2 ...... 11 5 References 13

1 1 Introduction

Lorentzian Geometry is the study of n-dimensional spacetimes. These are spaces with a single time dimension and (n − 1) spacial dimensions. Such spaces form the mathematical basis for modern physical theories. For example:

• Special Relativity is the study of R4 where one axis represents time, and the others are the familiar three spacial dimensions

• General Relativity is the study of 4-dimensional spacetimes which are curved due to gravity according to the Einstein Field Equations • String Theory involves the study of spacetimes of higher dimension, typically 10 or 11 up to 26.

In addition to its physical applications, Lorentzian geometry forms a rich area of mathematical investigation, containing many results that contrast with and distinguish it from the Riemannian case.

In this research project we consider the particular case of 2-dimensional spacetimes, Lorentzian surfaces. Such surfaces exhibit many of the interesting properties of Lorentzian geometry, while being simple to work with and easy to visualize. We consider the local conformal geometry of such surfaces and their geodesic completeness, and investigate the question of when a Lorentzian metric on a surface is conformal to a geodesically complete metric.

The author wishes to thank Thomas Leistner for the patience and supervision, as well as AMSI for providing the funding and the opportunity to partake in this project.

2 Preliminaries 2.1 Lorentzian Surfaces and their Topology Lorentzian surfaces are the simplest cases of Lorentzian manifolds with an interesting causal structure, sometimes exhibiting closed timelike curves, as well as singularities and interesting conformal geometry. To begin our investigation of such surfaces, we require some preliminaries in the language of diﬀerential geometry.

Definition 2.1. An n-dimensional Lorentzian Manifold is a smooth manifold M equipped with a smoothly varying symmetric non-degenerate bilinear form g on the tangent space at each point, with constant (n − 1, 1) signature. Tangent vectors with positive length are called spacelike, negative length are timelike, and zero length, null or lightlike. This is called a vector’s causal character. Paths on a Lorentzian surface inherit the causal character of their tangent vectors. We also note the definition of a Riemannian manifold is identical except that the metric signature has form (n, 0) and so is positive-definite, similar to the familiar dot product in Rn. Manifolds make rigorous the concept of a space that “looks like” Rn around each point, but globally may have a very different structure. In the Lorentzian case these manifolds also come equipped with a distinction between space and time.

2 Example 2.2. Possible examples of Lorentzian manifolds include n-dimensional Minkowski space n R1 (ﬂat spacetime), the odd dimensional spheres for n ≥ 3, or cartesian products of an existing Lorentzian manifold with a Riemmanian manifold. Deﬁnition 2.3. A Lorentzian Surface is a 2-dimensional Lorentzian Manifold.

We will see that the added structure of a Lorentzian metric restricts the topology of a Lorentzian surface. In contrast to the Riemannian case, there does not always exist a Lorentzian metric on a given smooth surface. Deﬁnition 2.4. A Lorentzian manifold is said to be time-orientable if one can make a smoothly varying choice of future and past in the timecone on the tangent space at each point on M.

A useful useful characterisation of this is that a Lorentzian manifold is time-orientable if and only if there is a global timelike vector ﬁeld. We now see when a smooth surface admits a Lorentzian metric. Theorem 2.5. For a smooth surface M, the following statements are equivalent:

1. There exists a Lorentzian metric on M. 2. There exists a time-orientable Lorentzian metric on M. 3. There is a non-vanishing vector ﬁeld on M. 4. M is non-compact, or is the Torus or Klein bottle.

Proof. For a proof we refer to O’Neill [1, pg. 149]. For example, by the Hairy Ball theorem the sphere S2 does not admit any Lorentzian metric, despite being a particularly illustrative example of a Riemmanian manifold.

2.2 Conformal Metrics The previous theorem provides topological restrictions on the surfaces that can be Lorentzian. Given a surface that admits a Lorentzian metric, it turns out there may be many choices of such a metric. A natural example of this is the case of conformal metrics. Deﬁnition 2.6. Given a Lorentzian surface (M, g), a metric g∗ is said to be conformal to g if there is an everywhere positive smooth function Ω on M such that g∗ = Ωg. As in the Riemannian case, conformal changes preserve angles between vectors in the tangent space, even if the notion of length is transformed. We also have the following obvious result. Proposition 2.7. Conformally changing a Lorentzian metric preserves the causal character of tangent vectors. Proof. Given a vector v in the tangent space, the sign of g(v, v) is preserved after being multiplied a positive factor Ω.

3 In general a Lorentzian metric may not be simple to work with. In the case where there is a smooth conformal factor Ω that changes the metric to something “simpler”, the previous proposition tells us that we can still gleam useful information about the original Lorentzian surface after this change – namely results about its causal structure.

In the Riemannian case we have the result that every surface is locally conformally flat, in the sense that about each point has an open neighbourhood on which there exists a smooth function that conformally changes the metric to the flat metric g = dx2 + dy2. This result is also true in the case of Lorentzian surfaces. Theorem 2.8. Every Lorentzian surface is locally conformally flat. About each point p ∈ M there always exists an open neighbourhood with coordinates x, y and a smooth conformal factor Ω > 0 such that g∗ = Ωg = (−dx)2 + dy2.

Proof. We give a sketch of the proof – see Weinstein [2, pg. 13] for a comprehensive proof. Consider two linearly independent lightlike vector ﬁelds X,Y on some neighbourhood of p. Choose scaling factors µ, λ such that the bracket [µX, λY ] = 0. Then we can always ﬁnd coordinates x, y such that ∂x = µX and ∂y = λY . In these coordinates the metric takes the form g = 2Bdxdy for some B > 0. After a π/4 rotation of the coordinate system, this looks like g = 2C((−dx0)2 + (dy0)2) for some C > 0 and we choose Ω = 1/(2C). This theorem shows that locally any Lorentzian surface has the trivial causal structure of Minkowksi 1 space R1. However, clearly this is not the case globally, as for example a compact Lorentzian surface will necessarily have closed timelike curves.

4 3 Geodesics

In Semi-Riemannian geometry, a path γ :(a, b) → M is called a geodesic if it is of locally minimal length, in the sense that along any small segment of the geodesic, a variation in the path will always increase the arclength along this segment. Geodesics generalise the notion of a straight line in Euclidean space to arbitrary surfaces.

3.1 Geodesic Equation One can deﬁne an energy functional on paths γ :[a, b] → M:

1 Z b E(γ) = gγ(t)(γ, ˙ γ˙ )dt 2 a By the calculus of variations, minimizing this energy functional by the Euler-Lagrange equation gives the following system of diﬀerential equations for geodesic paths on M, γ(t) = (x1(t), ..., xn(t)):

d2xl dxi dxj = −Γl (1) dt2 ij dt dt

k Here the Γij are called the Christoffel symbols of the metric g, and can be computed in terms of its components: 1 Γk = gkl(∂ (g ) + ∂ (g ) − ∂ (g )) (2) ij 2 i jl j il l ij 3.2 Levi-Civita Connection The Levi-Civita Connection is a fundamental object of study in Riemannian and Lorentzian geometry, that defines a notion of parallel transport of vectors on a manifold. Definition 3.1. The Levi-Civita connection on a Lorentzian manifold (M, g) is the vector-valued bilinear map on tangent vectors ∇ :(X,Y ) 7→ ∇X Y that satisfies the following properties:

k • ∇∂i ∂j = Γij∂k

• ∇fX Y = f∇X Y

• ∇X fY = dfY + f∇X Y where f is an arbitrary smooth function, X,Y are smooth vector ﬁelds and ∂i are the coordinate vectors.

We also have a second useful characterisation of the Levi-Civita connection: Deﬁnition 3.2. The Levi-Civita is the unique connection whose values are determined by the Koszul Formula: Given any vector ﬁelds X,Y,Z on a Lorentzian manifold (M, g)

2g(∇X Y,Z) = X(g(Y,Z))+Y (g(X,Z))−Z(g(X,Y ))+g([X,Y ],Z)+g([Z,X],Y )+g([Z,Y ],X) (3)

5 In fact, the statement that the Levi-Civita connection satisﬁes the Koszul Formula serves as a proof of the Fundamental Theorem of Semi-Riemannian Geometry. In practice (3) provides a relatively simple method for computing the value of the Levi-Civita connection in most instances.

A vector field X along a curve γ : I → M is called parallel if ∇γ˙ (t)X = 0 for all t ∈ I. Given a curve and an initial condition Xγ(0) = X0, this specifies an ordinary differential equation for the vector field with a unique solution that depends smoothly on X0. We define the parallel transport of the vector V along the curve γ(t) at t to be the vector Xγ(t) where Xγ(0) = V . Parallel transport due to the connection provides us with another way of defining a geodesic. Geodesic paths are those such that parallel transport along the curve preserves the tangent vector:

∇γ˙ γ˙ = 0

From this perspective, geodesics gain a physically intuitive interpretation: They are the paths a particle travels when under no acceleration except due to gravity. For example the elliptical orbits or hyperbolic trajectories of celestial bodies as they travel through space.

Deﬁnition 3.3. A path γ(t) is called a pre-geodesic if ∇γ˙ γ˙ = f · γ˙ , for some smooth function f. Example 3.4. All geodesics are pre-geodesics, where the function f = 0. Example 3.5. On a Lorentzian surface, lightlike geodesics must fall along lightlike coordinate lines, and so any null curve on M 2 is a pre-geodesic. This fails in higher dimensions, where lightlike geodesics may spiral around lightcones rather than falling along the coordinate lines. We also have the following result:

Proposition 3.6. If γ : I → M is a pre-geodesic deﬁned on some interval I ⊆ R, then γ can be re-parametrised to a geodesic on M. Proof. Let τ : J → I be the re-parametrisation factor, such thatγ ˜ = γ ◦ τ. By the chain rule, γ˜0(s) = τ 0(s)γ0(τ(s)).

0 00 0 0 2 0 ∇γ˜0(s)γ˜ (s) = τ (s)γ (τ(s)) + (τ (s)) ∇γ0(τ(s))γ (τ(s))

Ifγ ˜ is to be a geodesic, we require this expression to be 0, so we obtain an ordinary diﬀerential equation for τ:

τ 00(s) + τ 0(s)2f(τ(s)) = 0 This diﬀerential equation has a unique smooth solution if f is smooth, which smoothly depends on the initial conditions, so the re-parametrisationγ ˜(s) is a geodesic on M.

3.3 Geodesic Completeness Geodesics provide one of the most important ways to investigate the geometric and topological structure of a surface. By existence theorems for ordinary differential equations, a geodesic always exists starting at a point with some initial velocity. However each of these geodesics may not necessarily be defined for all time. If a geodesic is defined on some closed interval [a, b] then it can always be extended to some half open interval [a, b + ) by local existence of solutions to the

6 geodesic equations. It may even be possible to extend such a geodesic to a larger open interval. A geodesic γ :(a, b) → M is said to be maximal or inextendible if its affine parameter t can not be extended before a or beyond b (a and b need not be finite). Definition 3.7. A maximal geodesic on a Lorentzian manifold is called complete if its affine parameter is defined on all of R (a = −∞, b = ∞). It is said to be incomplete if one or both of a and b are finite. We also refer to a geodesic as future (rsp. past) incomplete if it can not be extended beyond b (rsp. before a). If every geodesic in a given space is complete we say the space is geodesically complete, otherwise we say it is geodesically incomplete. Example 3.8. Surfaces with holes will in general be geodesically incomplete, as geodesics will “fall down” the hole in finite time. For example consider the geodesic travelling to the right from x = −1 on the punctured plane. Example 3.9. If a surface has a boundary, geodesics may fail to be complete by falling off the edge of the surface.

These are somewhat simple examples. More interesting is when incompleteness comes from the non- linearity of the geodesic equations. Consider again the punctured plane. This space is geodesically incomplete as it has a hole, however its universal cover, the helicoid, is simply connected. An incomplete geodesic in R2\(0, 0) lifts to the cover, which is therefore incomplete, but contains no holes. We will see similar examples where a space fails to be complete due to reasons other than holes or boundaries.

Geodesic paths are intimately linked to the notion of distance on the surface, the metric. It is natural to ask whether a conformal change to the metric can help rectify the failure of geodesics to be complete. To understand how geodesics transform under a conformal change, we consider the corresponding transformation of the Levi-Civita connection.

Suppose (M, g) is a compact Lorentzian surface and g∗ = e2ug for some u ∈ C∞ is a conformally changed metric. From the Koszul Formula, we can compute the relation between ∇ and ∇∗: ∗ ∇X Y = ∇X Y + X(u)Y + Y (u)X − g(X,Y )∇u From this expression one obtains the relation between the corresponding covariant derivatives D/dt, D∗/dt: D∗γ0 Dγ0 = + 2du(γ0(t))γ0 − g(γ0, γ0)∇u (4) dt dt Note that these second derivatives along the curve actually require one to extend γ0 to a vector ﬁeld on some open neighbourhood of each point of γ in M. It can be shown however that in this case the covariant derivative is independent of the choice of extension. For lightlike geodesics, (4) becomes D∗γ0 d(u ◦ γ) = 2 γ0 dt dt So after a conformal change, lightlike geodesics of g become lightlike pre-geodesics of g∗. From 3.6 we know that such a pre-geodesic can be re-parametrised to a geodesic for g∗. Solving the ordinary diﬀerential equation for this re-parametrisation factor gives us that ifγ ˜ = γ ◦ τ, then s0(τ) = Ce2u(γ(t))

7 In particular, if u is bounded, the derivative of the re-parametrisation factor is also. This gives us the following theorem, proved in [3, pg. 11]: Theorem 3.10. Let g, g∗ = Ωg, Ω = e2u be two Lorentzian metrics on a surface M. Let γ : I → M be a maximal lightlike geodesics for g, and γ˜ : J → M be the reparametrised geodesic for g∗:

(1) If inf(Ω) > 0 and γ is complete, then γ˜ is complete. (2) If sup(Ω) < ∞ and γ is incomplete, then γ˜ is incomplete. Namely, if M is compact then Ω is bounded, so (M, g) is geodesically complete if and only if (M, g∗) is.

8 4 Examples 4.1 Cliton-Pohl Torus The Clifton-Pohl Torus is an example of a compact Lorentzian surface that is geodesically incomplete. We construct the Clifton-Pohl Torus as follows: Let M˜ be the surface R2\{(0, 0)} with Lorentzian metric 2 g = dxdy x2 + y2 We can compute the geodesic equations on M˜ using (1) and (2) and obtain

2xx02 2yy02 x00 = , y00 = x2 + y2 x2 + y2

1 Along the x axis geodesics take the form γ(t) = ( , 0); c1, c2 ∈ . Namely, the lightlike geodesic c1t+c2 R 1 γ(t) = ( 1−t , 0) satisfies these differential equations, but is incomplete as the affine parameter t is only defined on (−∞, 1).

Figure 1: The geodesic γ on the punctured plane travels outwards from the origin towards +∞

Deﬁnition 4.1. We say a group action Γ on a manifold M is properly discontinuous if it satisﬁes the following two conditions: (PD1) Every p ∈ M has a neighbourhood U such that φ(U) ∩ U = ∅ whenever φ 6= e ∈ Γ (PD2) If p, q ∈ M such that q is not in the orbit of p under Γ, then there exists neighbourhoods U, V of p, q such that φ(U) ∩ V = ∅ for every φ ∈ Γ.

If M˜ is a smooth manifold and Γ is a properly discontinous group action on M˜ , then the topological quotient M/˜ Γ = M is also a smooth manifold. PD1 implies M is locally Euclidean, while PD2 implies it is Hausdorﬀ.

We now consider the group action by the integers on M˜ deﬁned above, where Γ(k, (x, y)) = (λkx, λky) for some λ > 0. It can be veriﬁed that Γ is properly discontinuous, and hence the

9 quotient M/˜ Γ = M is a smooth manifold, topologically equivalent to the torus. Furthermore, the map (x, y) 7→ (λx, λy) is an isometry on M˜ , so by [1, pg. 191], M is also Lorentzian.

The manifold M˜ is known as a Lorentzian covering of M. That is, there is a locally isometric surjective projection map π : M˜ → M such that π−1(U) ⊆ M˜ is the disjoint union of open sets in M˜ , each mapped diﬀeomorphically onto U by π. For this covering space, paths upstairs descend uniquely to paths downstairs. Due to the action being a local isometry, ifγ ˜ is a geodesic upstairs, it descends uniquely to a geodesic γ = π ◦ γ˜ downstairs. Proposition 4.2. If π : M˜ → M is a Lorentzian covering and γ˜ is a maximal geodesic in M˜ , then the geodesic γ = π ◦ γ˜ in M is maximal. It follows that an incomplete geodesic on M˜ is still incomplete after descending to M.

Proof. If not, the maximal extension of γ :[a, b) → M, say γ∗ :[a, c) → M; c > b, would lift uniquely to a path based atγ ˜(0) in M˜ . The lifted path on the interval [a, b) would agree withγ ˜ wherever it is deﬁned, but the extension [b, c) would lift to smoothly extendγ ˜ to a longer geodesic. But this contradicts the assumption thatγ ˜ is maximal. Thus we have constructed a Lorentzian surface that is topologically the torus (i.e. compact) on which there exist incomplete geodesics. This is in stark contrast to the Riemannian case, where the Hopf-Rinow theorem, combined with the well known result that a metric space is compact if and only if it is complete shows that all compact Riemannian surfaces are geodesically complete. In addition, from 3.10 we have that the Clifton-Pohl Torus can not even be conformally geodesically complete.

4.2 Misner Space The Clifton-Pohl Torus gives an example of a compact Lorentzian surface that is geodesically incomplete. We now investigate a non-compact surface which is also incomplete.

Consider the cylinder M = R × S1 parametrized by (t, ψ). Take the metric g = 2dtdψ + tdψ2 This Lorentzian surface (M, g) is called Misner space. Using (1) and (2) we compute the geodesic equations on M: Given a geodesic γ(s) = (u(s), eiv(s)) we have 1 u¨ +u ˙v˙ + uv˙ 2 = 0 (5) 2 1 v¨ − v˙ 2 = 0 (6) 2 Solving (5) and (6) will give all possible geodesics on M, but we restrict ourselves to the case of lightlike geodesics, for which (5) Figure 2: Misner Space [4]. 1 2 simpliﬁes greatly, as g(γ, ˙ γ˙ ) =u ˙v˙ + 2 uv˙ = 0. First we consider when the horizontal component of velocityv ˙ = 0. Clearly we get a solution

γ(s) = (u ˙(0)s + u(0), eiv(0))

10 This is a vertical path on M of constant velocity, starting at (u(0), eiv(0)). More interesting however is the case wherev ˙ 6= 0. We solve the equations explicitly in this case.

v˙(0) We note that for lightlike geodesics,u ˙(0) = − 2 u(0). Thus, given initial conditions u(0) = W0 U0, u˙(0) = − 2 U0, v(0) = V0, v˙(0) = W0:

iV0 W0 e γ(s) = (− U0s + U0, ) (7) 2 W0 2i (1 − 2 s)

2 If U0 < 0 and W0 > 0, the geodesic γ(s) is only deﬁned on s ∈ [0, ), and is therefore an incomplete W0 null geodesic on M. Such null geodesics spiral faster and faster around the cylinder, but are future 1 trapped in the compact set [U0, 0] × S , as seen in Figure 3.

Figure 3: The compact set from U0 up to 0.

Once again we can ask the question of whether Misner space is conformally geodesically complete. Because geodesics starting below t = 0 with transverse velocity become trapped in a compact set, we can apply the argument of Theorem 3.10 to this set, and see that these lightlike geodesics must still be incomplete after any conformal change of g.

4.3 Metrics on R2 The previous two examples have shown that the geodesic completeness of Lorentzian surfaces is distinct from its Riemannian counterpart, and that even after a conformal change, it is not generally true that a surface will become geodesically complete. We now state a result for the case where the underlying surface is R2, but with Lorentzian metrics other than the standard Minkowski metric.

Theorem 4.3. Let g be any Lorentzian metric on R2. Then g is conformal to a geodesically complete metric g∗. Proof. We omit the full proof, which is quite technical and can be found in [5], and only provide a rough sketch. The proof relies on the existence of a cosmic time function. This is a scalar function

11 f on R2 with everywhere timelike gradient. One finds coordinates about each point for which this scalar function is the first, and then chooses a conformal factor that everywhere scales up this first coordinate, with the effect of blowing up the affine length of timelike or lightlike geodesics. The process is repeated using a cosmic space function, with everywhere spacelike gradient, resulting in the affine parameter of all geodesics on R2 extending to all R.

12 5 References

[1] B. O’Neill. Semi-Riemannian Geometry. Pure and Applied Mathematics. Academic Press, 1983. [2] T. Weinstein. An Introduction to Lorentz Surfaces. De Gruyter Expositions in Mathematics 22. Berlin, 1996 [3] A. M. Candela and M. Sanchez, Recent developments in pseudo-Riemannian Geometry. (D.V. Alekseevsky & H. Baum Eds), Special Volume, ESI-Series on Mathematics and Physics, EMS Publishing House, 2008, pp. 359-418 [math/0610144 [math.DG]]. [4] S. W. Hawking and G. F. R. Ellis. The large scale structure of space-time. Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. [5] Beem, John K. Conformal changes and geodesic completeness. Comm. Math. Phys. 49 (1976), no. 2, 179186.