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Lorentzian Surfaces 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 struc- ture, 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 2 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 geom- etry, 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 r :(X; Y ) 7! rX Y that satisfies the following properties: k • r@i @j = Γij@k • rfX Y = frX Y • rX fY = dfY + frX 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(rX 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 rγ_ (t)X = 0 for all t 2 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.
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