Theoretical Semester Project (MSc) – Anisotropy Effects on Atomic Clocks in GR – Nuhro Ego.pdf

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Theoretical Semester Project (MSc) - Anisotropy Effects on Atomic Clocks in GR - Nuhro Ego.pdf

Abstract

The general purpose of this theoretical semester project at the Physik-Institut at the University of Zurich is the theoretical determination whether the anisotropy of the Earth constitutes a measurable effect in the redshift of an extremely accurate atomic clock inside a freely falling satellite orbiting Earth in general relativity. For this, the gravitational redshift for several metrics modeling the space-time geometry of the rotating Earth is determined and expanded in different orders in the context of the post-Newtonian approximation. Since the redshift still depends on the concrete trajectory of the freely falling satellite, the Hamiltonians for the corresponding metrics are determined and also expanded in the same fashion. Together with Hamilton's equations in the Hamiltonian formulation of general relativity, the trajectories for different satellites could be determined in numerical simulations. However, the estimations of the anisotropy effects are compared to other relativistic effects like frame-dragging effects which are due to the rotation of the Earth for satellites on circular orbits at different altitudes. The calculations show that these anisotropy effects are measurable in current-generation atomic clocks with fractional timing inaccuracies of around $sim 10^{-16}$, whereas higher degree terms in the spherical multipole expansion of the gravitational Newtonian potential can be identified with more accurate next-generation atomic clocks, whereby the concrete number of terms highly depends on the orbital altitude range of the specific satellite. These effects will be explained and some interesting observations will be discussed.

Contents

1 Introduction 3 2 Redshift 5 2.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Linearized, rotating metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Linearized metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Kerr metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Hamiltonians 22 3.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Linearized, rotating metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Linearized metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Kerr metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Consistency check of the post-Newtonian expansions 28 4.1 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.1 Linearized, rotating metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.2 Kerr metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Linearized, rotating metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 Kerr metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Contribution of the Earth’s multipole moments 31 5.1 Satellite on a circular LEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Satellite on a circular MEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Satellite on a circular GEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Satellite on a circular HEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6 Conclusion & Summary 36 Acknowledgements 36 A EGM2008 Global Gravitational Model 37 B List of Figures 38 C List of Tables 38 D References 38