Spacetimes with changing signature in General Relativity Tzitzimpasis Paraskevas A thesis submitted in partial fulfillment for the bachelor in Physics in the Department of Physics Aristotle University of Thessaloniki Supervisor: Prof. Christos Tsagas July 2016 Contents 1 Introduction5 2 Fundamentals7 2.1 The 1+3 formalism ................................ 7 2.2 The FRW universe................................. 8 2.3 The transition formulation ............................ 10 2.3.1 General formulation............................ 10 2.3.2 Notation in FRW metrics......................... 12 3 Evolution of the Universe 15 3.1 Solution of the Friedmann equations for a fluid with constant barotropic index . 15 3.1.1 Lorentzian Regime............................ 15 3.1.2 Euclidean Regime............................. 17 3.2 General properties of the solutions........................ 18 3.2.1 Existence of an extremum ........................ 18 3.2.2 A simple form of the second junction condition............. 18 3.3 Application for a relativistic fluid......................... 20 3.3.1 Lorentzian regime............................. 20 3.3.2 Euclidean regime............................. 21 3.3.3 Matching the solutions for k=+1 on both sides.............. 21 3.3.4 Matching the solutions for k=-1 on both sides.............. 23 3.4 Application for inflationary fluid......................... 24 3.4.1 Lorentzian regime............................. 25 3.4.2 Euclidean regime............................. 25 3.4.3 Matching the solutions for k=+1..................... 26 3.4.4 Matching the solutions for k=-1 ..................... 26 4 Conclusion 29 3 A The matter tensor 31 B The field equations 33 C The Friedmann equations 35 4 Chapter 1 Introduction Our standard cosmological models describe gravitation in terms of the General Theory of Relativity, formulated in the last century. The theory of Relativity has had a remarkable impact in our understanding of the universe. Its confirmations begin with Einstein’s paper for the precession of Mercury’s perihelium and come all the way to the recent detection of gravitational waves. Although we should feel grateful for having such an effective and elegant tool in our hands, there are some good reasons to be yet unsatisfied. From our current understanding of the physical world, we expect quantum mechanics to have a significant interference at very large energies or very small spacetime scales. Common examples of physical phenomena where quantum mechanics is expected to dominate over our classical picture are black holes singularities or the very beginning of our universe. Our cosmological models can take into account the quantum behaviour of the matter content of the universe but they fail when it comes to the interplay of the quantum theory and gravity. This being said, it should be no surprise that those models become ill-defined at the big bang singularity where they predict infinities in quantities like the curvature or the density. A theory of quantum gravity would be necessary to describe early times preceding inflation. It would also be necessary to eliminate the singularity of the big bang. There have been many attempts to formulate a quantum theory of gravitation sometimes by modifying Classical Relativity and sometimes by building it from zero. One basic difficulty occurs because quantum theory, including modern gauge field theories, usually treats fields on a fixed background, either Euclidean space, in non-relativistic quantum mechanics, or Minkowski spacetime, in relativistic quantum theory. The formulation is thus not well adapted to considering the situation where the background metric is itself a field variable. Even if we had a complete theory of quantum gravity, we would still need to know the initial condition of our universe in order to understand its evolution. A famous proposal due to James Hartle and Stephen Hawking is the so called ”Hartle-Hawking proposal”. One very fascinating aspect of this proposal is the idea that the signature of the metric tensor should change at the very early stages of the universe when quantum effects become so dominant that they affect the very nature of space and time. In particular, it is assumed that spacetime is no longer Lorentzian 5 which means havinga metric signature (-+++) but ”Euclidean” (or sometimes called Riemannian) in the sense that its purely spatial and the signature of the metric tensor is (++++). The use of a Euclidean ”spacetime” allows us to assume that the (purely spatial) universe in its early state had the topology of a 4-sphere. By making this assumption, there is an elegant way to avoid the need of a beginning of our universe in the sense that the universe is compact. A simple way to picture this argument is if we imagine a 2-dimensional sphere, say the Earth. If we start moving in a certain direction, for example south, we will reach a point , the south pole, where there in no meaning in asking ”what lies south of the south pole”. In the same sense, if we turn time into a spatial direction, we can create a universe in which the question ”what was the universe like before that time” has no meaning. The idea of Hartle and Hawking is not necessarily correct but it surely provides us with new possibilities that have both scientific and philosophical significance. This idea of a potential signature change has been adopted by others and applied in Classical Relativity. Now one might be suspicious about a solution of the Einstein field equations with a signature change. Besides, it is very common in General Relativity to assume that the metric signature is constant and then solve the field equations. However, this is more of an assumption we impose, rather than a real physical demand. When we try to apply the idea of signature change in a cosmological model we are faced with a significant problem. Namely the metric at the hypersurface of transition has a zero eigenvalue and thus becomes degenerate. This is clearly a problem considering that in General Relativity the metric is assumed to be non-degenerate. However, this can be tackled by adopting a careful treatment of the transition. The metric is well defined in the rest of the manifold and so are the field equations. What we will attempt to do is solve the field equations in the Lorentzian and Euclidean regime and impose the appropriate junction conditions on the surface of change. This will produce new solutions of the field equations with new degrees of freedom and consequently new, interesting properties. 6 Chapter 2 Fundamentals 2.1 The 1+3 formalism Let O be an open region in spacetime (M; g). A congruence in O is a family of curves such that every P O belongs to one and only one curve from this family. We will be discussing timelike congruences i.e. families of timelike geodesics that fill spacetime and do not intersect. We can imagine such a congruence as a fluid, every point of which corresponds to an observer moving along the curves xa = xa(τ) where τ is the proper time of the observer. Obviously the generator of the congruence will be the tangent vector field dxa ua = (2.1) dτ a which is the 4-velocity of the observer normalized so that u ua = −1. The existence of a preferred direction in spacetime at each point (defined by the 4-velocity mentioned above), implies the existence of simultaneous rest hypersurfaces of dimensionality 3. These hypersurfaces have the structure of a manifold embedded in M and thus there is a natural way to define a metric on them. There is a natural map from the embedded hypersurface Σ to the manifold M by mapping every point of Σ to itself in M. This map can be used to construct the pullback of the metric g to Σ.It is easily verified that the metric hab = gab + uaub (2.2) satisfies the following relations: a b a a a b h bh c = h c; h a = 3; h bu = 0 (2.3) a which indicate that hab is the object we are looking for and that additionally h b is the projection tensor projecting into the three-dimensional tangent plane orthogonal to ua. It is well known in the study of fluid dynamics that the evolution of a fluid is characterized by the tensor uab = ua;b which determines the evolution of a deviation vector of the deformable medium.More specifically, ua;b can be decomposed in the following convenient way: 1 u = σ + ! + Θh − A u (2.4) a;b ab ab 3 ab a b 7 c d where the three first terms are the "spatial derivative" ha hb uc;d while the last term corresponds to the "time derivative" u_ aub. From the above quanities, Θ is the fractional change of volume per unit time.When Θ > 0, the fluid expands while when Θ < 0, the fluid contracts.We shall use Θ to define the scale factor from the equation a_ 1 = Θ (2.5) a 3 The rotation tensor !ab changes the orientation of the deformable medium and being an- tisymmetric, we can define a vector with its independent components in the standard way: 1 ! = !bc (2.6) a 2 abc The shear tensor σab carries the information about the change in shape of the fluid. It is a symmetric tensor with zero trace. The last term is the 4-acceleration of the observer which vanishes in the case of a congruence a of geodesics in light of the geodesic equation u ub;a = 0. Intuitively this is just the statement that non-accelerating observers move on geodesics. 2.2 The FRW universe In this section we will present the FRW model which is the one that will be used in this paper. This model contains two fundamental (and very strong) assumptions: Homogeneity and Isotropy. By isotropy we mean that there is no preferred direction in space.If we want to express this intuitive concept of isotropy in a formal way, we can say that a manifold M is isotropic around a point P M if for any two vectors V; W TP (M) there is an isometry of M such that the pushforward of W is parallel with V.