Title page

In memoriam of Jurgen Ehlers

Indice

Preface

Gruppo fotografico dei partecipanti al Corso

Introduction to general relativity

Preface

Special relativity

General relativity

Rotation in relativity and the propagation of light

Introduction

Concepts of rotation from Foucault to Godel

Space-based research

Goal of the paper

Outline

Notation and conventions

Formulation of the general relativistic Sagnac effect

Sagnac's original experiment

Sagnac time delay for a stationary metric

Mathematical description of the arrangement

Null curves of the counterpropagating beams

Final expression for the time delay

Form invariance

Coordinates appropriate for local satellite experiments

Motivation

Construction of coordinates

Building blocks

Exploration of the spatial neighborhood with spacelike geodesics

Caveat emptor

Metric expansion

Leading-order contributions

Special cases of proper reference frame coordinates

Sagnac time delay in a proper reference frame

Framework for the Sagnac time delay measurement

Leading-order contributions of the Sagnac time delay

Measurement strategies

Rotation sensor

Curvature sensor

Double eight-loop interferometer (DELI)

Rotation in general relativity

Rotating frame of reference in flat spacetime

Metric

Light cone diagram for a rotating reference frame

Sagnac time delay

World line, four-velocity and acceleration

Tetrad basis and transport matrix

Sagnac time delays for the DELI

Godel's Universe

Why a rotating universe?

Metric

Light cone diagram

Sagnac effect

World line, four-velocity and acceleration

Tetrad basis and transport matrix

Non-vanishing components of the curvature tensor

Sagnac time delays with the DELI

How things look like in Godel's Universe

Fundamentals of ray tracing

Two examples of visualizations of the Godel Universe

View of the inner surface of a sphere

View on a small object in the Godel Universe

Appendix A. Basic concepts of general relativity: A brief review

Tensors and all that jazz

The Minkowski form of a metric at a fixed spacetime point

Transformation to Minkowski metric

Construction of light cone diagrams

Symmetries and Killing vectors

Killing equations

Symmetries

World lines and geodesics of test particles and of light

Parallel transport versus Fermi-Walker transport

Appendix B. Tetrads and their orthonormal transport

Orthonormal tetrads

Construction

Decomposition into tetrad components

General orthonormal transport of tetrads and the proper transport

Transport matrix

Correspondence between distinct transport laws

Proper transport as natural generalization of the Fermi-Walker transport

Appendix C. Riemann normal coordinates and proper reference frames

Formal solution of the geodesic equation

Riemann normal coordinates

Definition and subtleties

Metric coefficients

Summary

Local coordinates of a proper reference frame

Definition

Metric

Approximate solution of the geodesic equation

Appendix D. Expansion of the Sagnac time delay in proper reference frame coordinates

General expansion

Stokes' theorem

Complete expansion of the Sagnac time delay

Leading-order contributions

Appendix E. Integration of null geodesics in Godel's Universe

Killing vectors

Constants of motion

Explicit expressions for the coordinates

Integration of the radial coordinate

Time coordinate

Polar coordinate

Vertical coordinate

Gravito-magnetism in one-body and two-body systems: Theory and experiments

Introduction

The Kepler problem and its post-Newtonian generalization

One-body system

The GP-B experiment

Determination of Omega so from lunar-laser-ranging measurements

Ciufolini's Lageos experiment

Two-body systems

The binary pulsar PSR 1913+16

The binary pulsar PSR B1534+12

The double binary system PSR J0737-3039

Conclusions

Postscript

Gravity Probe B

Introduction

Gravity Probe B flight hardware

Gyroscope

Gyroscope readout

Electrostatic suspension system

Telescope

Quartz block metrology frame

Low-temperature probe

Superfluid liquid-helium dewar

Spacecraft

Flight operations

Mission operations, communication, and telemetry

Mission phases

Initialization phase

Science data collection

Calibration phase operations

Analysis of flight data

Determination of gyroscope spin axis orientation relative to guide star

Combined gyroscope and telescope signals for a single orbit

Modulation of gyroscope readout scale factor due to trapped magnetic flux

History of the orientation of the gyroscope spin axis relative to the apparent position of the guide star

History of gyroscope spin axis orientation relative to the true position of guide star

Results from the calibration phase

Data analysis in the presence of misalignment torques

Conclusion

Tests of general relativity in the Solar System

Introduction

The basic ideas in general relativity

Post-Newtonian gravity tests in the Solar System

Scale-dependent tests in the Solar System

The Pioneer anomaly

Can the Pioneer signal be a metric anomaly?

Post-Einsteinian metric extensions of GR

Conclusion

Gravitational waves, diffusion and decoherence

Introduction

Tracking observables

Gravitational waves (GW) in linearized general relativity

The effect of GW on tracking observables

Gravitational backgrounds

Gravitational diffusion and decoherence in interferometers

Classical vs. quantum behaviours

Conclusion

How to detect gravitational waves

Introduction

Resonant bar detectors

Interferometric detectors

LISA---a space-borne detector

Conclusion

Detecting gravitational waves on ground

The transducer: a Fabry-Perot cavity

Gravitational waves

A thought experiment

Realization of coordinates

Realization of a clock

Sensitivity and resolution

Sensitivity

Recycled interferometers

Antenna pattern

Network of antennas

Resolution from fundamental noises

Suspension and mirror thermal noise

Shot noise

Resolution from technical noises

Feedback loop on dark fringe and resolution

Non diagonal sensing and control matrices

Scattered light

Power noise and alignment

Frequency noise

Data analysis

Detectors and status

Advanced detectors and challenges

Resolution from fundamental noises

Thermal noise

Quantum noises

Resolution from technical noises

Handling high powers

Parametric instabilities

Suspension r.m.s. motions

Mirror static charges

Conclusion

Variation of fundamental constants: Theory

Introduction

Optical spectra

Comparison of quasar absorption spectra with laboratory spectra

Optical clocks

Enhanced effects of alpha variation in atoms

Enhanced effects of alpha variation in molecules

Variation of the strong interaction

Microwave clocks

Enhanced effect of variation of alpha and strong interaction in UV transition of 229Th nucleus (nuclear clock)

Enhancement of variation of fundamental constants in ultracold atom and molecule systems near Feshbach resonances

Changing physics near massive bodies

Precisely engineered interactions between light and ultracold matter

Introduction---the quest for coherence of light-matter interactions

Optical frequency standard with trapped atoms

Confined neutral atoms and the Stark cancelation technique

Magic wavelength optical lattice

Considerations for the optical lattice

Optical clockwork

Stable optical local oscillator

Optical frequency comb clockwork and precision fiber transfer

High-resolution spectroscopy of confined atoms

Nuclear structure effects

Clock accuracy evaluation

Absolute frequency measurements

Fundamental physics

Precision measurement and control with ultracold molecules

Perspectives of atomic quantum sensors on ground and in space

Introduction

Cold-atom Sagnac interferometry

Coherent beam splitters for atoms

Atom interferometer topologies

Phase shifts

Atomic sources

Source requirements for an atomic Sagnac interferometer

The concept of the atomic source

Atomic launch analysis

State preparation and detection

Coherent beam splitting with Raman processes

Characterization of the performance of the atom interferometer

Beam splitter optimization

Atomic-clock--type interferometer

Mach-Zehnder interferometer

Velocity-selective configuration

HYPER-atom interferometry in space

Mission objectives

Mapping the Lense-Thirring effect around the Earth

The Lense-Thirring rotation

The geodetic effect

The Lense-Thirring effect for microscopic and macroscopic objects

HYPER's mission scenario

The measurement principle

The science requirements

HYPER payload design and structure

Payload module

The orbit

The Field-Electric-Propulsion System (FEEPS) and drag-free sensor

The telescope

The guide stars

The optical bench

The atomic Sagnac interferometer

Gyroscopes for space

QUANTUS

Conclusion

Accelerometer using atomic waves for space applications

Introduction

Experimental set-up

Sensitivity of the interferometer

Sensitivity function

Influence of the phase noise onto the sensitivity of the interferometer

The 100 MHz source oscillator

The frequency chain

Propagation in the fiber

Detection noise

The case of parasitic vibrations

Systematic effects

k-independent phase shifts

k-dependent phase shifts

Conclusion

Cold ions in Space

Introduction

The progress of ion clocks

Quantum computing techniques are improving ion clocks

Long-lived Bell states

Measurement of electric quadrupole shift in Ca+ using entanglement

Heisenberg limited spectroscopy using entangled Be+ ions

Readout of an Al+ clock ion using quantum logic techniques

Modern segmented traps for scalable quantum information processing

Proposed space mission for tests of fundamental physics and exploration of the outer Solar System

Clock segment based on a single trapped and laser cooled ion

Selection criteria for the space ion clock

Technological issues

The 88Sr+ ion clock system design and its sub-components

88Sr+ ion clock performance and critical issues

Mission goals

Test of the gravitational redshift and of Lorentz invariance

Tests of Parameterized Post-Newtonian gravity (PPN)

Exploring large-scale gravity

Exploring outer Solar System masses

Variation of fundamental constants

Upper limits on low-frequency gravitational waves

Conclusion

Interferometry in Plebanski-Demianski space-times

Introduction

Dynamics of the matter field

Model of the interferometer

Stationary congruences

The model of the interferometer

Consequences

Plebanski-Demianski space-time

The phase shift

The general expression

Natural Killing observer

Actively rotating observer

Conclusion

Introduction to 5D optics for space-time sensors

Introduction

Klein-Gordon equation for matter waves

Introduction of the proper time coordinate

WKB solution

Hamiltonian and Lagrangian expressions in the parabolic approximation

Classical derivation

Quantum-mechanical derivation

Schroedinger-like equation in (4+1)D

Weak-field approximation

Propagator and ABCD law

Phase-shift formula for atom interferometers

Appendix A. Hamiltonian and equations of motion for a massive point particle in general relativity

Light-pulse atom interferometry

Introduction

Atom interferometry overview

Phase shift determination

Phase shift formulae

Justification of phase shift formulae

Applications in inertial navigation

Gyroscope

Accelerometer

Gravity gradiometer

Application to tests of the Equivalence Principle

Proposed experiment overview

Error model

Gravity inhomogeneities

Magnetic-field inhomogeneities

Controlling potential systematic errors

Rotation of the Earth

Gravity gradients

Magnetic fields

Conclusion

Gravitational physics experiments with ultracold atoms

Introduction

Determination of G by atom interferometry

Precision gravity measurements at mu m scale with laser-cooled Sr atoms in an optical lattice

Transportable atom gravimeters for geophysics applications and future experiments in space

Geophysics applications

Space applications

Precision gravity measurement

Introduction

Gravimeters

Free-fall absolute gravimeters

Conclusions and outlook

Elenco dei partecipanti