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The Effect of Non-Static Metrics on Observables

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The Effect of Non-Static Metrics on Observables Radboud University Master thesis The effect of non-static metrics on observables Renz Bakx Supervisor: Prof. Dr. Wim Beenakker High Energy Physics Contents 1 Theory 5 1.1 The metric and its dynamics . .5 1.2 The Einstein equations . .6 1.3 Komar integrals . .7 2 Dark Matter and MOND 9 2.1 In a nutshell . .9 2.2 The Schwarzschild metric . 12 2.2.1 Dynamics . 13 2.2.2 An extra term . 15 2.2.3 The weak field limit . 17 2.3 Matos et al. 19 3 Pure General Relativity 23 3.1 Frame Dragging . 23 3.2 Cooperstock and Tieu . 25 3.2.1 The method . 25 3.2.2 Criticism . 30 3.3 Rotating Dust . 37 3.3.1 Obtaining the metric . 37 3.3.2 The Newtonian limit . 43 3.3.3 A constant solution . 47 4 Conclusion and Outlook 52 A Integrals for the Ernst Potential 54 B Integrals for the Newtonian seed 58 C Documentation 62 C.1 metric . 62 C.2 calculation funcs . 64 C.3 calculate Ynu . 65 C.4 calculate ErnstPot . 66 1 Introduction For a long time, Newton's theory of mechanics has been an effective tool to describe the motion of all kinds of objects. From small things like apples falling from trees to planets moving around the sun. There are things however, that cannot be explained by this theory. The precession of the orbit of Mercury or the physics of a black hole, for example. In these cases, we need the theory of General Relativity devised by Einstein. (a) (b) Figure 1: The rotation curves of the planets in our solar system (a) and of galaxy NGC 6503 (b) [1]. Another problem that Newtonian physics cannot solve is the motion of stars in galaxies. The theory predicts that the velocity should fall of like r−1=2, which works for the planets in our solar system. This agreement can be seen in figure 1a. However, measurements indicate that the velocity becomes more or less constant at larger radial distance to the center of the galaxy. This can be seen in figure 1b, where the velocity of the matter in the galaxy NGC 6503 is plotted. The \extra gravity" mentioned in figures 1a and 1b refers to the unexplained 2 forces that are needed to correct the theoretical velocity. A common explana- tion is that there is more matter within galaxies than can be observed: Dark Matter [2]. Our telescopes can only see the stars and luminous gas present in NGC 6503, so perhaps there is more than meets the eye in this galaxy. This Dark Matter may consist of large objects like dead stars or exotic particles that we have yet to discover. Another explanation is given by Modified Newtonian Dynamics (MOND), which states that Newtonian physics behaves differently for low accelerations than for large accelerations, given some reference acceleration [3]. In this case no extra dust or exotic particles are needed, we simply adjust Newtonian physics at low accelerations so that the velocities it predicts match the ones we observe. It is commonly assumed that the weakness of the gravitational field of galaxies (especially in the outer regions) allows for a Newtonian approach when calcu- lating the angular velocities of the stars within the galaxy. This approach works well enough within our own solar system, where the sun is by far the heaviest object around. In this case the structure of the spacetime is dominated by the curvature generated by the sun. However, in galaxies the spacetime curvature is not dominated by some giant source in the center. It is generated by vast amounts of smaller objects rotating around the center. This combined move- ment of all the stars within a galactic disk might have a measurable effect on the dynamics of the system. If this effect is big enough, it may help to solve the issue with the rotation curves. That is precisely the topic of this thesis. Our goal is to find out if the Newtonian approach is valid in this regime, or if relativistic effects do have to be taken into account. This thesis is meant as a comprehensive toolbox, in which all the techniques and literature that can be used to achieve our goal are gathered. We will first show methods to obtain the dynamics of a relativistic space-time and will treat Dark Matter and MOND in this context. After that we will discuss the theory of Cooperstock and Tieu, who claim they can explain rotation curves without resorting to Dark Matter or MOND [4]. Finally, we will attempt to model a galaxy using a flat disk of dust. In order to do this we follow the method described by Ansorg [5] to build our own computer program which can calculate the metric of such a disk. Conventions We will use a few conventions in this thesis, which will be listed here. Firstly, we will use geometrized units unless mentioned otherwise. These units are char- acterized by setting G = c = 1, meaning all observables will have a dimension of a power of length. We will also use a metric signature (−; +; +; +), and adopt the Einstein sum- mation convention when using greek indices: 3 µ 0 1 2 3 x xµ = x x0 + x x1 + x x2 + x x3 (1) As for the coordinate systems used in this thesis: (t; r; θ; ') are used when we assume spherical symmetry and (t; ρ, ζ; ') are used in a system with axial symmetry (with θ 2 [0; π] and ' 2 [0; 2π]). Lastly, partial derivatives will @ sometimes be written as @x = @x, in order to save space. 4 Chapter 1 Theory In this chapter we will briefly go over some of the aspects of General Relativity that are used in this thesis. The material in this chapter was obtained from books written by Sean Carroll [6] and Robert Wald [7]. 1.1 The metric and its dynamics A metric gµν is usually given by its line element: 2 µ ν ds = gµν dx dx : (1.1) To study the dynamics of a spacetime generated by such a metric we will look at geodesics. These are the paths of shortest distance between two points, the curved spacetime equivalent of a straight line. A definition of these paths that works in General Relativity is that if a path xµ(λ) is a geodesic, it parallel- d µ transports its own tangent vector dλ x (λ). Parallel transport means that the covariant derivative of the tangent vector along the path vanishes. This is an important requirement, because it ensures that the norm and orthogonality of vectors is preserved along the path. For this demand to be satisfied, the path xµ(λ) needs to obey the geodesic equation: d2xµ dxρ dxσ + Γµ = 0: (1.2) dλ2 ρσ dλ dλ 5 µ In the geodesic equation the Christoffel symbols Γρσ are given by: 1 Γµ = gµν (@ g + @ g − @ g ) (1.3) ρσ 2 ρ σν σ ρν ν ρσ µρ µ with g gρν = δ ν (1.4) µ Here δ ν is the Kronecker-delta. In order to satisfy the geodesic equation λ needs to be an affine parameter. This means that λ = aτ +b for some constants a and b, where τ is the proper time. The proper time can be obtained from the line element: dτ 2 = −ds2. Usually we pick λ = τ, so that trajectories are parameterised by the proper time. When using an affine parameter the four-velocity is normalised, meaning: µ ν ( µ dx dx −1 for massive particles UµU = gµν = (1.5) dτ dτ 0 for massless particles This normalisation condition and the geodesic equations can be combined into a set of differential equations for the path xµ(λ). When there are symmetries present in the metric, conserved quantities can be obtained through Killing vectors that can make solving these differential equations much easier. A Killing vector is the vector in the direction of the symmetry. For example, if a metric has a time-translation symmetry, Kµ = (1; 0; 0; 0) is the appropriate Killing vector. The conserved quantity associated with this symmetry is energy, which can be derived from the following equation. For any Killing vector: dxν dxν K = g Kµ = const: (1.6) ν dλ µν dλ The equations above will aid us in finding the dynamics of any system we might wish to describe, in particular when we look at the angular velocity and the orbits of the metrics we are going to study. 1.2 The Einstein equations Of course, we cannot just pick any function for the line element. To make sure that the metric describes a system where mass and energy are the source of the curvature of spacetime, it has to obey the Einstein field equations. For these equations we first need the Riemann tensor: ρ ρ ρ ρ λ ρ λ Rσµν = @µΓνσ − @ν Γµσ + ΓµλΓνσ − ΓνλΓµσ: (1.7) We can internally contract indices to obtain the Ricci tensor and Ricci scalar, respectively: λ µ µν Rµν = Rµλν and R = R µ = g Rµν (1.8) The Einstein equations can then be written as 6 1 R = 8πG T − T g : (1.9) µν µν 2 µν In these equations G is Newton's constant of gravitation (which will be set to 1), µν Tµν is the energy-momentum tensor and T = g Tµν is its trace. This tensor contains all the information of the matter content of the system we want to describe. We will use it later to describe a flat disk of dust, but it could also be used to represent for example a vacuum.
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