Spacetime geometry of a wormhole modified BTZ black hole David Sundelin Supervisor Bo Sundborg Stockholm University October 26, 2018 2 Spacetime geometry of a wormhole modified BTZ black hole David Sundelin Abstract The unitarity problem for the BTZ black hole can possibly be solved by a coordinate transformation in which the event horizon is extended to a wormhole. In a model proposed by S.N. Solodukhin, this is done by the addition of a wormhole parameter λ to the BTZ black hole line element. This thesis studies the changes in the spacetime geometry that comes with the addition of such a parameter. The focus of study are geodesic behaviour and possible bound states of waves. Investigating a possible source of the wormhole, the stress-energy tensor ansatz for a perfect fluid is also tested. The thesis concludes that there are notable changes to the spacetime depending on the presence of either a black hole or a wormhole. These changes includes orbital trajectories of geodesics and localized bound states of waves. The changes are most notable for λ > 1 but also detectable for small parameter corrections. The wormhole spacetime can however not be generated by a simple addition of matter in a perfect fluid form. 3 4 Contents 1 Introduction 7 2 Background 9 2.1 General relativity . 9 2.2 Anti-de Sitter spacetime . 10 2.3 Unitarity and the AdS/CFT correspondence . 11 2.4 Wormhole modification of a BTZ black hole . 12 3 Spacetime geometry of a wormhole modified BTZ black hole 14 3.1 Implications for the stress energy tensor . 14 3.2 Geodesics . 15 3.2.1 Preliminaries . 16 3.2.2 Radial geodesics . 18 3.2.3 General geodesics . 21 3.2.4 Elaboration on conserved quantities . 27 3.2.5 Geodesics in the second modification . 29 3.3 Waves . 33 3.3.1 Variable substitution . 34 3.4 Summary . 39 4 Conclusions 41 5 Appendix 43 5.1 The Einstein tensor . 43 5.2 Christoffel symbols and the Ricci scalar . 43 5.3 Image archive . 45 5 6 1 Introduction In physics, there are four fundamental forces. Three of them, electromag- netism and the atomic weak and strong forces, has, through the process of quantization, already been unified in a single framework known as Quantum Field Theory. The fourth force, gravity, is as of yet a classical theory. The first theory of gravity was put forward by Isaac Newton in his work Principia, published in 1687, and is referred to as Newtonian, or classical, mechanics. It is for many the quintessential picture of physics and is often used as an introductory subject in physics. Classical mechanics, however, is not applicable at certain limits: • At very small distances, e.g. on an atomic level • At very high velocities, e.g. close to the speed of light • At very strong gravitational potentials, e.g. close to black holes All of these limits eventually gave rise to new theories around the be- ginning of the 20th century. The first resulted in Quantum Mechanics. The second was Special Relativity, which stipulates that nothing can travel faster than the speed of light with unintuitive things happening close to that limit, e.g. time dilation and length contraction. The final theory was General Rel- ativity, which changed the whole perspective on gravity. In general relativity, grav- ity is not a force in the classical sense that attracts masses like electromagnetism attracts charges of opposite signs but rather it is a con- squence of the curvature of the spacetime. Spacetime is Figure 1: A vizualisation of curved spacetime; in the fusion of the three spa- the outskirts spacetime is flat, at the center some tial directions and time into heavy object curves it causing the effect of gravity. a single quantity which is then curved by presence of mass and energy often vizualised by a two dimensional soft surface being bent in a third direction by some heavy object in the middle, Fig.1. The governing equations of gen- eral relativity is the Einstein field equations which states the relationship between the curvature of the spacetime and the mass/energy that curves it. The equations are solved for the metric which contains all the relevant information about how the geometry of a certain spacetime behaves in time and space. General relativity solved many of the problems of Newtonian mechanics, while presenting new physical phenomena, e.g. black holes and gravitational waves. These phenomena have since become research subjects of their own 7 and observations support their existence. Problems that emerged with gen- eral relativity includes • How does the overall geometry of our universe looks like? Is the uni- verse as a whole curved or flat? • What is the nature of black holes and gravitiational waves? • How can the theory be quantized to form a unifying theory of all the four fundamental forces? In the process of solving these issues, models that don't nesessarily re- semble our own universe, but rather simplifies calculations are often used. This thesis studies the spacetime geometry resulting from the modification of such a model. The model, the BTZ black hole, is of interest in resolving some of the issues that arises when formulating a theory of quantum gravity. 8 2 Background 2.1 General relativity In general relativity, time is a coordinate direction on the same footing as the spatial directions of ordinary Newtonian mechanics, distinguished from the latter by opposite sign. The resulting concept is referred to as spacetime which is described by the distance between points in the geometry, with the distance given by the line element. In a chosen system of coordinates, the line element can be expressed as 2 µ ν ds = gµνdx dx (1) where gµν is the components of the metric tensor, or simply the metric, which is symmetric and coordinate dependent. All the information about the geometry of the spacetime is contained within the line element. The simplest geometry, Minkowski spacetime, is the spacetime analogue of ordinary flat space and is given by the metric gµν = ηµν where 0 −1 0 0 0 1 B 0 1 0 0 C ηµν = B C : (2) @ 0 0 1 0 A 0 0 0 1 The Minkowski line element then reads ds2 = −dt2 + dx2 + dy2 + dz2: (3) The metric gµν is obtained from the Einstein field equations Gµν + gµνΛ = κTµν; (4) which is ten equations for the ten inputs of the metric. The equations state the relation between the curvature of the spacetime, given by the 1 Einstein tensor Gµν and the cosmological constant Λ on the left hand side, and the source of the curvature, namely matter and energy, given by the stress-energy tensor Tµν on the right hand side. Given some input for the stress-energy tensor, the equations are solved for the metric. The cosmological constant was added at a later stage by Einstein since without it the field equations do not permit static cosmological solutions which was in contrast to contemporary beliefs that the universe was static. After observations of an expanding universe the constant was kept however for theoretical reasons and it is today related with vacuum energy, which for example would be the cause of inflation. The constant on the right hand side, κ = 8πG=c4, contains the gravitational constant G, and the speed of light 1see Appendix, section 5.1 9 c, and is determined by taking the limit of low gravitational potential and velocites v << c and demanding that Eq.(4) then reproduces Newtonian mechanics. Being non-linear, the Einstein field equations are difficult to solve and few exact solutions are known, [1, 2]. 2.2 Anti-de Sitter spacetime The metric gµν obtained from the Einstein field equations, describes space- time. Solutions other than Minkowski spacetime includes de Sitter (dS) and Anti-de Sitter (AdS) spacetime. While Minkowski spacetime is a vac- uum solution (Tµν = 0) with a zero cosmological constant, dS and and AdS spacetime are vacuum solutions with a positive (Λ > 0) and negative (Λ < 0) cosmological constant, respectively. Constructing a spacetime can be done much like a surface of dimension d is defined by its embedding in d + 1 dimensions. In general a sphere is then defined as d+1 2 2 2 2 X 2 2 x1 + x2 + ::: + xd + xd+1 = xi = L ; (5) i=1 where L is the radius of the sphere. The surface is defined as every point that satisfies the defining equation, Eq.(5). By the addition of a time coordinate x0 to the sphere, we obtain dS spacetime d X −(x0)2 + (xi)2 = L2; (6) i=1 where xi are spatial coordinates. Introducting another time coordinate and flipping the sign on the right hand side we obtain AdS spacetime d−1 X −(x0)2 + (xi)2 − (xd)2 = −L2: (7) i=1 In d = 3 dimensions, the line element becomes ds2 = −(dx0)2 + (dx1)2 + (dx2)2 − (dx3)2 (8) with the defining equation (x0)2 − (x1)2 − (x2)2 + (x3)2 = L2: (9) 2+1 AdS spacetime is then all points satisfying Eq.(9). The two time coordi- nates are constrained which can be seen through a coordinate transformation 8 x0 = ρ cos t; > <> x1 = r cos θ; (10) x2 = r sin θ; > : x3 = ρ sin t: 10 The transformation gives the line element ds2 = −(dρ2 + ρ2dt2) + (dr2 + r2dθ2); (11) and the constraint ρ2 − r2 = L2: (12) Eliminating ρ in the line element yields the AdS line element in 2+1 dimen- sions: L2 ds2 = −(L2 + r2)dt2 + dr2 + r2dθ2; (13) L2 + r2 with one time coordinate, t and two spatial coordinates, r and θ.