RELGEO 2016

Titles and abstracts

last update:
2016-05-13

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## A. Bauch

### A status report on time and frequency metrology

Time and frequency metrology comprises several elements:

1. Development of clocks;
2. Operation of clocks;
3. Characterization of the properties of clocks;
4. Realization of time scales, e. g. national legal time;
5. Time and frequency comparison of clocks and time scales;
6. Dissemination of time-of-day, time interval, and standard frequency to the public.
PTB, as Germany's National Metrology Institute (NMI), is engaged in all fields mentioned. The characters of the above elements are quite different: (a) and (c) may be (academic) challenge, sportive competition, excitement, (b) and (f) are often just boring and a mere service to be provided. In the context of Relativistic Geodesy we can safely skip item (f), but it is essential to understand the status and the progress to be expected in the elements (b), (c) and (e).

So the program for my presentation is obvious: I will explain how we realize UTC (PTB) as the national reference for time and frequency, how the 70+ NMIs world-wide collaborate to realize International Atomic Time TAI and Coordinated Universal Time UTC. This will include a brief description of the clocks involved and their properties. Collaboration means: coordinated and disciplined operation of clocks and time transfer systems - exactly what is required when clocks shall be used to determine gravity potential differences: Two clocks have to be operated, one at a known ''height'', one at an unknown ''height''. Their frequency accuracy and stability, and the frequency difference or frequency ratio prevailing when both are operated side-by side need to be known at the outset. They need to be compared with sufficient accuracy in order to determine the height-difference to the desired accuracy, based on the new frequency difference or frequency ratio observed while they are operated at different heights.

The concept is simple, the practical implementation is challenging.

[ slides ]

## I. Ciufolini

### Status of the LARES Space Experiment to Test General Relativity with Satellite Laser-Ranging and GRACE Earth Gravity Models

LARES is a satellite of the Italian Space Agency (ASI) successfully launched on 13 February 2012 by the European Space Agency with the new launch vehicle VEGA (ESA, ASI, ELV and AVIO). LARES is a passive, spherical laser-ranged satellite designed to approach as closely as possible an ideal test particle following geodesic motion. The main objective of the LARES space mission is to test General Relativity by providing an accurate test, with an uncertainty of a few per cent, of the dragging of inertial frames, or frame-dragging, predicted by Einstein's gravitational theory. This goal is achieved by coupling its orbital data with those of the two LAGEOS satellites respectively launched in 1976 and 1992 by NASA and ASI. We report on our orbital analysis of the laser-ranging data of the LARES, LAGEOS and LAGEOS 2 satellites from 26 February 2012 until 6 September 2015 using a prominent state-of-the-art Earth's gravity field model, GGM05S. GGM05S is an Earth gravity model released in 2013, based on approximately 10 years of GRACE data. It describes the Earth's spherical harmonics up to degree 180.

References

1. I. Ciufolini, General Relativity and Dragging of Inertial Frames, in: General Relativity: The most beautiful of Theories, Applications and trends after 100 years, Centennial Jubilee Volume of General Relativity, edited by C. Rovelli (De Gruyter, Berlin) 2015, pp. 125-163.
2. I. Ciufolini et al. A Test of General Relativity Using the LARES and LAGEOS Satellites and a GRACE Earth's Gravity Model, to appear in: The European Physical Journal C (2016).

## B. Coll

### Epistemic relativity: An experimental approach to physics

The recent concept of relativistic positioning system (RPS) has opened the possibility of making Relativity the general standard frame in which to state any physical problem, theoretical or experimental.

Because the velocity of propagation of the information is finite, epistemic relativity proposes to integrate the physicist as a truly component of every physical problem, taking into account explicitly what information, when and where, he is able to know. This leads naturally to the concept of relativistic stereometric system (RSS), allowing measuring the intrinsic properties of physical systems. Together, RPSs and RSSs complete the notion of laboratory in general relativity, allowing performing experiments in finite regions of any space-time.

Epistemic relativity incite the development of relativity in new open directions: advanced studies in RPSs and RSSs, intrinsic characterization of gravitational fields, composition laws for them, construction of a finite-differential geometry adapted to RPSs and RSSs, covariant approximation methods, etc. Some of these directions are sketched here, and some open problems are posed.

[ slides ]

## P. Delva

### Chronometric geodesy: methods and applications

The flow of time depends on the velocity of a clock and on the space-time metric. By comparing the frequencies of two clocks it is therefore possible to directly measure gravity potential differences: this is chronometric geodesy. The chronometric observable is very different in nature than all other classical observables in geodesy (gravimetry, gradiometry,...), and the accuracy of optical clocks begins to be competitive with classical methods in geodesy which have accuracies up to a few centimeters for the static potential.

In the first part of this talk we will introduce the basics of chronometric geodesy in terms of definitions, conventions and operations. Then in the second part we will present two projects that investigate the new possibilities of chronometric geodesy: International Timescales with Optical Clocks (ITOC/EMRP) and Chronometric Geodesy for high resolution Geopotential (ChronoGeo/FirstTF/ERC AdOC).

All these applications foster the need for transportable optical clocks and highly stable links over intercontinental distances.

[ slides ]

## J. Flury

### Relativistic geodesy: perspectives and applications

In time and frequency metrology recent breakthroughs have been achieved, both regarding optical atomic frequency standards and long-distance frequency transfer through phase-stabilized optical fiber links that promise essential benefits for practical geodesy. With a relative frequency accuracy of $10^{-18}$, optical clocks connected by optical fiber links can provide an atomic reference for gravity potential and height differences at the cm level. The atomic reference is inherently more reliable and in the future possibly also more accurate than conventional reference frames for height and gravity potential. A possible perspective is the establishment of a first order precision network, to which gravity models from satellite, airborne and terrestrial gravimetry could be tied in order to provide a both reliable and continuous height reference frame. Near term applications will be links of National Metrology Institutes while the far term perspective is a global atomic clock network for geodesy and time and frequency metrology.

## C. Gerekos

### The time transfer and Synge world functions in relativistic global navigation satellite systems Part 1

Relativistic positioning systems (RPS) are global navigation satellite systems described directly in general relativity [1-4]. In Schwarzschild spacetime, we have devised iterative methods to generate semi-analytical solutions to both the problem of determining the proper time of a satellite for a given reception event (the ''direct problem''), and to the problem of position determination with a satellite constellation (the ''inverse problem''), for satellites on any orbit. These methods use the time transfer function and more specifically, its expansion in the post-Minkowskian formalism, which we have calculated to the fourth order via the method described in [5-9].

We have used these solutions to perform a proof-of-concept study of an actual RPS system. The solution to the direct problem are first used to simulate the proper time of several satellites for a given reception event. Then, using these simulated times as if they were broadcast by actual satellites, the solution to the inverse problem allows us to successfully retrieve the four coordinates of the reception event. Using a series of realistic settings based on the orbital parameters of satellites from the Galileo navigation system, we have found that the coordinates of the receiver can be retrieved with an accuracy of $10^{-20}$ metres, neglecting all ''engineering'' errors.

In this talk, I will outline the concept of relativistic global navigation satellite systems and the benefits of such a system. I will describe the post-Minkowskian time transfer function expansion and how we incorporate it into RPS.

References

1. Delva, P. and Olympio, J., Mapping the Spacetime Metric with GNSS: a preliminary study, 2nd International Colloquium, Scientific and Fundamental Aspects of the Galileo Programme, COSPAR colloquium, Padua, Italy, 2009
2. Cadez, A., Kostic, U. and Delva, P., Mapping the Spacetime Metric with a Global Navigation Satellite System, Ariadna Final Report ID 09/1301, 2010
3. Cadez, A., Kostic, U., Delva, P. and Carloni, S., Mapping the Spacetime Metric with a Global Navigation Satellite System - extension of study: Recovering of orbital constants using inter-satellites links, Ariadna Final Report ID 09/1301 CCN, 2011
4. Gomboc, A., Horvat, M. and Kostic, U., Relativistic GNSS, The PECS Project Final Report, Contract NO. 4000103741/11/NL/KML, 2014
5. Linet, B., Teyssandier, P., Time transfer and frequency shift to the order $1/c^4$ in the field of an axisymmetric rotating body, Phys. Rev. D66 (2002) 024045
6. Le Poncin-Lafitte, C., Linet, B., Teyssandier, P., World function and time transfer: general post-Minkowskian expansions, Class.Quant.Grav. 21 (2004) 4463-4484
7. Teyssandier, P., Le Poncin-Lafitte, C., General post-Minkowskian expansion of time transfer functions, Class.Quant.Grav.25 (2008) 145020
8. Teyssandier, P., Direction of light propagation to order G2 in static, spherically symmetric spacetimes: a new derivation, Class. Quantum Grav. 29 (2012) 245010
9. Bernard Linet and Pierre Teyssandier. New method for determining the light travel time in static, spherically symmetric spacetimes. Calculation of the terms of order G3. Class. Quant. Grav., 30:175008, 2013. [Erratum: Class. Quant. Grav.31,079502(2014)]

## E.W. Grafarend

### The Geodetic Anholonomity Problem

We answer the key question: why is Geodesy physical? Our answer is based on two examples. FIRST, we find the reason in geodetic observables which are basically anholonomic or non-integrable. Indeed Geodesy finds a basis in Grassmann Algebra and Exterior Calculus invented Elie Cartan. Geodesists measure locally by ''Leveling Instruments'' establishing horizontal-vertical relative positioning between neighboring points. In the local frame of reference they measure the spherical coordinates of the gravity vector, the sum of the gravitational force and the centrifugal force, namely of type (modulus of gravity, astronomical longitude, astronomical latitude). Alternatively Geodesists use Geodetic Positioning in an Earth-fixed (or body-fixed) frame of reference used in GPS: it leads to the local anholonomity problem demonstrated by two examples, local networks extending in the average not more than 30 kilometres. They cause a misclosure in the range of 15 meters at maximum. SECOND, we build-up the Lagrange variational equations for a conformal flat metric where the factor of conformability is the modulus of the gravity vector. We refer to the Marussi gauge in terms of gravity gradients subject of the latest GOCE Satellite Mission 2013.

References

1. Grafarend,E. and M.Fujimoto: Spacetime coordinates in the geocentric reference frame, in: Relativity in Celestial Mechanics and Astrometry, Int. Astr. Union, Symp. 114, pp.269-276, Leningrad/UDSSR, eds. J.Kovalevky and V.A. Brumberg, Reidel Publ., Dordrecht-Boston 1986
2. Grafarend, E.: Three-dimensional Geodesy - the holonomity problem, Z. Vermessungswesen 100 (1975) 269-281
3. Grossman, N.: Holonomic measurable in Geodesy, J.Geophys.Res. 79 (1974) 689-694
4. Hehl, F.W. and Obukhov, Y.N.: Foundation of classical electrodynamics - charge, flux and metric, Birkhaeuser Verlag, Boston-Basel-Berlin 2002
5. Hotine, M.: Differential Geodesy, pp. 38-41, Springer Verlag, Berlin-Heidelberg-New York
6. Marussi, A.: Natural reference systems and their reciprocals in geodesy, Publ. T.J. Kukkamaeki 70 Birthday, Finish Geodetic Institute , No. 89, Helsinki 1979
7. Marussi, A.: Intrinsic Geodesy, Springer Verlag, Heidelberg 1985
8. Moritz, H.: The Hamiltonian structure of refraction and of gravity field, manuscripta geodaetica 20(1994) 52-60
9. Zund, J.: Foundation of Differential Geodesy , pp. 373, Springer Verlag, Berlin-Heidelberg-New York

[ slides ]

## G. Grosche

### Frequency transfer through long-distance optical fiber links

Accurate frequency references based on atomic transitions probed with lasers (''optical clocks'') have been realised with a fractional frequency uncertainty $\Delta \nu / \nu$ of only a few parts in 10 to the 18 [1]. Making the ultra-stable and accurate output of these instruments available beyond the walls of the metrology laboratory to enable physics experiments, is a tough challenge for satellite based methods using radio frequency or microwave signals.

In the last decade, frequency transfer techniques using optical fibre have greatly developed: since the landmark paper by Daussy et al. [2], the length of fibre connections increased from 86 km to over 1000 km, and frequency transfer uncertainty (achieved in loop experiments) was improved by almost three orders of magnitude [3]. I will outline some of the techniques of using optical telecommunication fibre (1.55 $\mu$m) for high precision long-distance frequency transfer.

One ingredient is to use the optical carrier near 200 THz [4,5] instead of radio frequency modulation, for transferring the frequency information. Viewed differently, we establish an interferometric optical fibre connection between a local and a remote lab: the optical phase at the remote end is actively stabilised to a reference mirror in the local lab [5]. Other instrumental advances concern low noise signal amplification, to overcome the optical losses of about 20 dB/100 km along the fibre - using erbium doped fibre amplifiers [5], Brillouin amplification [3] and/or remote laser stations [6].

The low uncertainty of clocks and of frequency transfer via optical fibre now makes chronometric levelling (or ''relativistic geodesy'') feasible [6]. We detect the gravitational redshift experienced by light travelling from one clock to a second clock positioned at a different gravitational potential: on earth, the fractional frequency shift is about 10-16 for one meter height difference.

We have recently established an interferometric fibre link between PTB, Braunschweig and Strasbourg (based on the work in [3]). A similar interferometric link exists between Observatoire de Paris in France and Strasbourg [7]. In Strasbourg, we compare the frequency signals from Paris and from Braunschweig, thus connecting the two national metrology institutes of France and Germany. This link was successfully used for the first international comparison of two strontium optical lattice clocks in Paris and Braunschweig [8].

References

1. T.L. Nicholson, S.L. Campbell, R.B. Hutson et al., Systematic evaluation of an atomic clock at 2x10-18 total uncertainty, Nature Comm. 6:6896 (2015)
2. C. Daussy, O. Lopez, A. Amy-Klein et al., Long-Distance Frequency Dissemination with a Resolution of 10-17, Phys. Rev. Lett. 94, 203904 (2005)
3. S. Raupach, A. Koczwara and G. Grosche, Brillouin amplification supports 1x10-20 uncertainty in optical frequency transfer over 1400 km of underground fiber, Phys Rev A 92, 021801(R) (2015)
4. L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path, Opt. Lett. 19 (21), 1777 (1994)
5. G. Grosche, O. Terra, K. Predehl, et al., Optical frequency transfer via 146km fiber link with 10-19 relative accuracy, Opt. Lett. 34 (15), 2270 (2009)
6. M. Vermeer, Chronometric leveling, Report 83:2 of the Finnish Geodetic Institute, Helsinki; ISSN 0355-1962, ISBN 951-711-087-1 (1983)
7. N. Chiodo, N. Quintin, F. Stefani et al., Cascaded optical fiber link using the internet network for remote clocks comparison, Opt. Exp. 23 (26) 33927 (2015)
8. Ch. Lisdat, G. Grosche, N. Quintin et al., A clock network for geodesy and fundamental science, arXiv:1511.07735 (2015)

[ slides ]

## B. Hartmann

### Operationalization of basic measurements

We present a novel approach to the foundation of relativistic physics. We do not presuppose a single word of mathematics, but obtain the mathematical structure directly from the underlying tangible operations of the physicists. Like Einstein for the foundation of relativistic kinematics we start from vivid measurement operations and simple natural principles. Each step (the definition of energy, impulse, mass from physical comparisons, the construction of ''sufficiently constant'' reference devices and of a machinery, which ''functions'' for a basic measurement) follows from practical requirements. We give a tangible definition of the basic observables, the quantification and derive the fundamental equations (e.g. kinetic energy-velocity relation, Einstein's energy-mass equivalence etc.). We demonstrate an operationalization of relativistic kinematics (arXiv:1205.2680) and dynamics (arXiv:1504.03571). The approach can be extended to the foundation of general relativity theory. Future work will develop an intrinsic construction for physical quantities of the stress-energy tensor and the Riemannian curvature.

[ slides ]

## A. Hees

### Solar System tests of gravitation beyond standard formalisms

Searching for deviations from General Relativity and constraining alternative theories of gravitation is of prime importance regarding the development of a quantum theory of gravitation, the development of a theory that would unify gravitation with the standard model of particles and the development of models of Dark Matter and Dark Energy. In this presentation, we will briefly review the standard tests of gravitation performed in the Solar System within standard formalisms: the Parametrized Post-Newtonian and the fifth force frameworks. Then, we will motivate the need to extend these current tests to frameworks beyond the standard ones. We will show how the radioscience tracking of spacecraft, the construction of planetary ephemerides, Lunar Laser Ranging and Very Long Baseline Interferometry can efficiently constrain violations of the Lorentz symmetry and of a MOND (Modified Newtonian Dynamics) theory. The importance of Solar System tests when combined with galactic or cosmological observations will also be mentioned.

[ slides ]

## A. Heffernan

### The time transfer and Synge world functions in relativistic global navigation satellite systems Part 2

In relativistic global navigation satellite systems, several studies have led to the development of simulations with a system of satellites and ''user'' in full general relativity [1-4]. By simulating null coordinates, and solving the inverse problem, it has been shown that the user can obtain their location from the null coordinates. It has also been shown that once in orbit, the satellites can further constrain the metric in which they travel. Such simulations have involved heavy numerical computation and are slowed considerably by localized minima. By attacking the same problem with our semi-analytic methods, we hope to lead to techniques which can further enhance the already existing methods. We have, thus so far, simulated null coordinates and retrieved local coordinates for a relativistic positioning system user in a Schwarzschild space-time for both circular and eccentric satellite orbits.

We use different perturbation methods to expand both the time transfer function and Synge world functions in the calculation of light travel time between two events. In the time transfer function, we use a method developed in a series of papers that uses a Post-Minkowskian expansion [5-9]. For the Synge world function, we assume a small separation between events and expand appropriately. We find agreement with both techniques in simulating null coordinates of an observed event. In this talk, I will outline the Synge world function method and how it was used in our simulated systems. I compare the output of the two methods and highlight possible future developments.

References

1. Delva, P. and Olympio, J., Mapping the Spacetime Metric with GNSS: a preliminary study, 2nd International Colloquium, Scientific and Fundamental Aspects of the Galileo Programme, COSPAR colloquium, Padua, Italy, 2009
2. Cadez, A., Kostic, U. and Delva, P., Mapping the Spacetime Metric with a Global Navigation Satellite System, Ariadna Final Report ID 09/1301, 2010
3. Cadez, A., Kostic, U., Delva, P. and Carloni, S., Mapping the Spacetime Metric with a Global Navigation Satellite System - extension of study: Recovering of orbital constants using inter-satellites links, Ariadna Final Report ID 09/1301 CCN, 2011
4. Gomboc, A., Horvat, M. and Kostic, U., Relativistic GNSS, The PECS Project Final Report, Contract NO. 4000103741/11/NL/KML, 2014
5. Linet, B., Teyssandier, P., Time transfer and frequency shift to the order $1/c^4$ in the field of an axisymmetric rotating body, Phys. Rev. D66 (2002) 024045
6. Le Poncin-Lafitte, C., Linet, B., Teyssandier, P., World function and time transfer: general post-Minkowskian expansions, Class.Quant.Grav. 21 (2004) 4463-4484
7. Teyssandier, P., Le Poncin-Lafitte, C., General post-Minkowskian expansion of time transfer functions, Class.Quant.Grav.25 (2008) 145020
8. Teyssandier, P., Direction of light propagation to order G2 in static, spherically symmetric spacetimes: a new derivation, Class. Quantum Grav. 29 (2012) 245010
9. Bernard Linet and Pierre Teyssandier. New method for determining the light travel time in static, spherically symmetric spacetimes. Calculation of the terms of order G3. Class. Quant. Grav., 30:175008, 2013. [Erratum: Class. Quant. Grav.31,079502(2014)]

## F.W. Hehl

### Real lightlike frames in physics and geodesy

For recording measurements in physics and/or geodesy, we need local reference frames. Conventionally, these frames are chosen orthonormal. If we aim for a relativistic descriptions, we have a frame consisting of four orthonormal vectors $\mathbf{e}_a(x)$, with $a=0,1,2,3$. This is valid in flat Minkowski spacetime of special relativity or in the curved Riemannian spacetime of general relativity. Dual to the frame, we have a so-called coframe $\boldsymbol{f}^b(x)$ consisting of four covectors (or one-forms), with $b=0,1,2,3$. The interior product $\mathbf{e}_a\rfloor\boldsymbol{f}^b=\delta^b_a$ yields the Kronecker delta.

In general, such a frame is anholonomic, that is, it cannot be derived from an underlying coordinate system. The object of anholonomity $\boldsymbol{O}^b$ is defined as exterior derivative of the coframe: $\boldsymbol{O}^b:=d\boldsymbol{f}^b$. Only if $\boldsymbol{O}^b$ vanishes, an underlying coordinate system can be found. Anholonomic frames are commonplace in physical geodesy [1] and are as such of central importance in applications.

So far, with our four-dimensional frames and coframes, we always had one time-like vector $\mathbf{e}_0$ and three spacelike vectors $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$, and correspondingly for $\boldsymbol{f}^b$.

If one uses light (or radar) signals for geodesic measurements, it is much more convenient to use frames and coframes consisting of light-like (or null) vectors and covectors. Often one chooses, following Newman & Penrose, two real and two complex light-like vectors [2]. However, we want to circumvent complex quantities and we introduce 4 real light-like vectors [3,4]. They turn out to be no longer orthonormal. We will find certain tetrahedral frames that will be described in our seminar. These frames have a close relationship to the coordinates of the Global Positioning System (GPS). We derive the corresponding relations.

References

1. E. Grafarend (Stuttgart), private communication; see also E. Grafarend and W. Kuehnel, A minimal atlas for the rotation group SO(3), Int. J. Geomath. 2, 113-122 (2011).
2. R. Penrose and W. Rindler, Spinors and space-time, vol.1, Cambridge (1984).
3. B. Coll and J. A. Morales, Symmetric frames on Lorentzian spaces, J. Math. Phys. 32, 2450-2455 (1991).
4. M. Blagojevic et al., Real null coframes in general relativity and GPS type coordinates, Phys. Rev. A 65, 044018 (2002).

[ slides ]

## G. Heinzel

### The Laser-Ranging Interferometer aboard GRACE Follow-On

The GRACE Follow-On mission, a joint US-German project, is being built now for a launch in 2017. It will contain a novel Laser Ranging interferometer as experimental demonstrator for inter-spacecraft ranging with much lower noise than microwave systems. I will discuss the design, expected performance and present status of that instrument.

References

1. Sheard, B. S. et al, Journal of Geodesy, Vol. 86, pp. 1083-1095, 2012

## P.A. Hogan

### A Snapshot of J. L. Synge

A brief description of the influence of Professor J. L. Synge on Relativity accompanied by some technical illustrations of his style of work and its impact.

[ slides ]

## U. Hugentobler

### Stable Clocks in Satellite Geodesy

Time and frequency play an essential role in geodesy. All space geodetic techniques rely on precise time and time interval measurements. Except the Satellite Laser Ranging (SLR) all currently used techniques such as GNSS (Global Navigation Satellite Systems), DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) and VLBI (Very Long Baseline Interferometry) are one-way techniques. They require, different to two-way techniques, two clocks for a precise measurement of the signal travel time. These clocks thus have to be synchronized at the picosecond level.

GNSS is a special case as the concept of satellite navigation allows for an epoch-wise synchronization of space and ground clocks, as each receiver observes more than one satellite for each observation epoch, and more than one receivers measures signals from same satellites at every epoch. As a consequence a GNSS system, although relying on precise signal travel time measurements, does in principle not require precise clocks. Indeed, for geodetic applications all clock parameters are estimated for every observation epoch either explicitly or implicitly by forming double differences.

Nevertheless new GNSS systems such as Galileo or modernized GPS carry very stable clocks. While Galileo satellites carry ultra-stable passive hydrogen masers, GPS Block IIF satellites carry excellent rubidium oscillators. Main reasons for system operators to equip navigation satellites with very stable clocks are increased robustness and reliability of the system as update frequency of clock corrections broadcast to the users can be increased and standard positioning accuracy be improved.

Exploiting the stability of the existing clocks for geodetic applications is attractive to increase the stability of solutions by reducing the large number of solve-for clock parameters, in particular for kinematic applications and to improve orbit and datum stability by reducing orbit-clock correlations. Modeling of onboard oscillators however requires precise modeling of relativistic effects as well as of all types of hardware delay variations affecting the complete signal chain, as only the apparent clock, i.e., the time-referenced signals emitted by the satellite antenna are accessible to the user.

[ slides ]

## Y. Itin

### Quadratic invariants of the elasticity tensor

Elasticity tensor plays a central role in mechanics of materials, in particular in geophysics. We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of closeness to a pure Cauchy material and a similar isotropic factor is as a measure for closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.

References

1. Y. Itin (2016) Quadratic invariants of the elasticity tensor, J. Elasticity.
2. F. W. Hehl and Y. Itin (2002) The Cauchy relations in linear elasticity theory, J. Elasticity 66, 185-192.
3. Y. Itin and F. W. Hehl (2013) The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation, J. Math. Phys. 54, 042903 (2013).

[ slides ]

## R. Koenig

### Measurement of Frame-Dragging with Geodetic Satellites based on Gravity Field Models from CHAMP, GRACE and Beyond

The experimental measurement of frame-dragging or the Lense-Thirring (LT) effect based on Satellite Laser Ranging (SLR) observations to the LAGEOS twin satellites was firstly successfully demonstrated by [1] with an accuracy of about 10%. Here we look in detail into the effect of the node drift induced by the time variable part of the C(2,0)-term of the gravity field describing the flattening of the Earth. This and the errors in C(2,0) can effectively be taken care of by analyzing two satellites. Then we adopt some recent gravity field models in order to independently repeat the LT measurement experiment. The gravity models themselves are derived either completely independent from observations of dedicated gravity field satellite missions like CHAMP, GRACE and GOCE, or partly dependent from LAGEOS SLR observations. It turns out that all in all the claimed accuracy of 10% is confirmed.

References

1. I. Ciufolini, E. Pavlis, Nature 431, 958-960 (2004).

## S. Kopeikin

### Reference-ellipsoid and the normal gravity field in the post-Newtonian geodesy

We apply general relativity to construct the post-Newtonian background manifold that serves as a reference spacetime in relativistic geodesy for conducting relativistic calculation of the geoid's undulation and the deflection of the plumb line from the vertical. We chose an axisymmetric ellipsoidal body made up of a perfect homogeneous fluid uniformly rotating around a fixed axis, as a source generating the reference geometry. We formulate hydrodynamic equations of rotating fluids defining the shape of the post-Newtonian reference ellipsoid. We explicitly perform all integrals characterizing gravitational field inside and outside the fluid body and represent them in terms of the elementary functions depending on its eccentricity. We fully explore the coordinate freedom of the equations describing the post-Newtonian ellipsoid and evaluate the fractional deviation of the post-Newtonian level surface from the Maclaurin ellipsoid. We also derive the gauge-invariant relations of the post-Newtonian normal gravity field of the rotating fluid with the parameters characterizing the shape of the post-Newtonian ellipsoid including relativistic mass and angular velocity. Finally, we present the exact post-Newtonian expression for physically-measurable normal force of gravity in the external space. It generalizes the normal gravity field adopted in classic geodesy to the realm of general relativity. The normal gravity field of the reference-ellipsoid is compared with the post-Newtonian approximation of the Kerr metric which describes rotating black holes in astrophysics.

[ slides ]

Abstract (tbd)

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Abstract (tbd)

## T. Mayer-Guerr

### From satellites measurements to mass variations in the system Earth - Processing of GRACE data at the TU Graz

The monitoring of mass redistributions of the Earth is important to understand the dynamic system Earth in times of climate change. The ice melting in Greenland, the sea level rise, floods and droughts in major river basins can be quantified by changes in the regional water masses. The satellite mission GRACE enables for more than 13 years to observe the time variable gravity field generated by these mass changes.

Two satellites orbiting around the Earth in the same orbit with a distance of about 250 km. Gravity field variations causes changes in the distance which is measured accurately with an Microwave ranging assembly. The task of analysis centers like the TU Graz is to calculate every month a gravity field from these distant changes. As every single measurement depends on integrated mass distribution, the processing of the data results in a complex inverse modeling of the orbit dynamics including the solution of a large system of equations. Additional the noise behaviour and other systematic effects of the instruments must be taken into account

In this talk the ongoing results of the GRACE processing at the TU Graz are presented.

[ slides ]

## T. Mehlstaeubler

### Towards Optical Clocks based on Ion Coulomb Crystals

In order to exploit their full potential and to resolve frequencies with a fractional frequency instability of $10^{-18}$ and below, optical ion clocks need to integrate over many days to weeks. For the characterisation of systematic shifts of the clock, as well as for applications, such as relativistic geodesy, the long averaging time scales put up fundamental limits. Scaling up the number of ions for optical clock spectroscopy is a natural way to significantly reduce integration times and to relax the requirements on clock laser stability, but was hindered so far by the poor control of the dynamics of coupled many body systems, micromotion and systematic shifts due to interacting ions. However, ion species, such as Yb$^+$, In$^+$ or Al$^+$, with low or zero quadrupole moments of the clock states are interesting candidates for frequency standards based on multiple ions [1,2,3].

In our experiment we will implement linear chains of Yb$^+$ and In$^+$ ions for a first evaluation for optical clock operation and detail on the expected uncertainties. For optimal control of the ion motion, scalable chip-based ion traps are engineered in the clean room facilities of PTB. An operating prototype trap with minimized axial micromotion [4] allows us to trap and cool large ion Coulomb crystals and study many-body physics with trapped ions. To reduce systematic shifts due to blackbody radiation, this trap design is transferred to an AlN based chip trap. We will present a first evaluation of the expected performance of a multi-ion-clock in this new system.

References

1. Huntemann, N., Okhapkin, M., Lipphardt, B., Weyers, S., Tamm, C., Peik, E., High-Accuracy Optical Clock Based on the Octupole Transition in 171Yb+, PRL 108, 090801 (2012).
2. Chou, C.W., Hume, D.B., Koelemeij, J.C.J., Wineland, D.J., Rosenband, T., Frequency Comparison of Two High-Accuracy Al+ Optical Clocks, PRL 104, 070802 (2010).
3. Herschbach, N., Pyka K., Keller J. and Mehlstäubler T.E., Linear Paul trap design for an optical clock with Coulomb crystals, Appl. Phys. B 107, 891 (2012).
4. Pyka K., Herschbach N., Keller J. and Mehlstaeubler T.E., A high-precision segmented Paul trap with minimized micromotion for an optical multiple-ion clock. Appl. Phys. B 114, 231 (2014).

[ slides ]

## J. Mueller

### The benefit of clock measurements for gravity field applications

We will show how the new concept of clock measurements ('relativistic geodesy') is connected to classical geodetic concepts, e.g. geopotential numbers, height systems, vertical reference. We will briefly address the accuracy of recent height systems based on classical measurement approaches and show examples of spatial and temporal variations of the gravitational potential (tidal effects, non-tidal mass changes, etc.) that affect clock measurements.

Clock measurements might be applied for connecting height systems in remote areas, like islands or areas where classical spirit levelling and terrestrial gravimetry is difficult to carry out. Clock measurements might then also be used for defining and realizing height systems in a new way, i.e. based on frequencies and their distribution via fiber links.

In future, clock measurements can be used in combined gravity field solutions. There, the long wavelengths are taken from satellite gravimetry (GRACE and GOCE), the shorter wavelengths of the gravity field from terrestrial gravimetry and the datum, i.e. the anchor points, can be provided by clock readings. The major benefit is that clocks can provide the fine-structure of the geoid in areas where terrestrial gravimetry is limited and the spatial resolution of satellite missions is not sufficient anyway.

We will discuss potential (new) applications of relativistic geodesy, such as the connection of tide gauges, e.g. in South America, that monitor sea level variations along coastlines. In addition when clocks are run on ships, they can be used to check altimetry results along profiles, where the sea-surface topography is determined with respect to the marine geoid.

Moreover (in some future), clocks in space could serve as external reference that could be used not only to realize relativistic geodesy but also to distribute time in a consistent way to and between geodetic observatories where VLBI, GNSS, SLR and DORIS are operated.

A number of open questions (stability of the clocks, effect of averaging time, connection of the clocks, control of errors, size of clock networks, etc.) remain that shall be at least presented in the workshop for future studies.

References

1. Mai, E.: Time, Atomic Clocks, and Relativistic Geodesy. DGK, Reihe A, No. 124, Beck, Muenchen, 2013. http://dgk.badw.de/fileadmin/docs/a-124.pdf.
2. Mai, E., Mueller, J.: General Remarks on the Potential Use of Atomic Clocks in Relativistic Geodesy. zfv, 4/2013, S. 257-266, 2013.

[ slides ]

## Y.N. Obukhov

### Motion of test bodies: moments, symmetries, reference frames

Relativistic geodesy is underlain by the relativistic theory of gravity. The fundamental issues such as the validity of the equivalence principle, definition of the reference frames, precision tests of the gravity models etc require a systematic study of dynamics of massive and massless test particles in an arbitrary gravitational field.

An overview of the problem of motion of extended bodies in the relativistic theory of gravity is presented, with a special attention to clarifying the relation between the physical symmetries, conservation laws and equations of motion within the framework of the multipole expansion approach. As applications we discuss the issues of the precision measurements of the gravitational field that lays down the foundation for the modern relativistic geodesy.

[ slides ]

## V. Perlick

### Using clocks for probing the geometry of spacetime

The notion of standard clocks (i.e., clocks measuring proper time) is mathematically well-defined in general relativity and also in several alternative theories, such as Weyl or Finsler spacetime theories. However, the mathematical definition does not by itself give an operational procedure of how to test a clock for being a standard clock. In the first part of the talk I discuss such an operational procedure, in general relativity and in some alternative theories, which uses light rays and freely falling particles as tools.

In the second part of the talk I discuss the question of how the ticking of standard clocks can be used for probing the geometry of spacetime. In any case, this requires comparing two or more clocks. This can be done (a) by clock transport which is the appropriate method for deriving integrability conditions such as the first or second clock effect; (b) by redshift measurements which allows to investigate the existence of a redshift potential (i.e., in the language of clock geodesy, of isochronometric surfaces); or (c) by comparing two clocks that stay ''infinitesimally close'' to each other which is the relevant method for gradiometry, i.e., for the measurement of curvature.

The part on Finsler geometry is based on joint work with Wolfgang Hasse, Institute of Theoretical Physics, TU Berlin, Germany.

[ slides ]

## D. Puetzfeld

### Determination of the gravitational field by means of deviation equations in General Relativity

In General Relativity, the comparison of test bodies moving along adjacent world lines is of direct operational significance. The observation of a suitably prepared set of test bodies allows for the determination of the components of the curvature.

We present some recent results on generalized deviation equations [1] which allow for the description of adjacent test bodies, and thereby for a measurement of the gravitational field. In particular, we provide explicit solutions for the curvature by using the standard geodesic deviation equation as well as its next order generalization.

Deviation equations, as well as the concept of a ''curvature detector'' or ''gravitational compass'' [2,3] based on such equations, are of direct relevance for several applications in relativistic geodesy.

References

1. D. Puetzfeld, Y.N. Obukhov: Generalized deviation equation and determination of the curvature in General Relativity, arXiv: gr-qc/1511.08465, Phys. Rev. D 93 (2016) 044073
2. J.L. Synge. Relativity: The general theory. North-Holland, Amsterdam (1960)
3. P. Szekeres. The gravitational compass. J. Math. Phys., 6 (1965) 1387

[ article ]

## H. Quevedo

### Determination of the metric from the curvature

Assuming that the components of the curvature tensor in an orthonormal frame are explicitly given, we show that it is possible to integrate Cartan's structure equations in order to obtain the components of the metric tensor. This procedure is performed for particular type D curvature tensors with and without cosmological constant. In this way, we derive the metric components of the Schwarzschild-de-Sitter spacetime and a particular case of the Kasner-de-Sitter spacetime. The possibility of generalizing this method to consider gravitoelectric and gravitomagnetic multipole moments is discussed.

[ slides ]

## S. Schiller

### The Space Optical Clocks mission on the ISS: concept and development of a prototype lattice clock

The ESA candidate mission ''Space Optical Clock'' aims at operating an optical lattice clock on the ISS in approximately 2022. The mission is the natural follow-on of the ACES mission, which will fly in 2017. The scientific goals of the mission are to perform tests of fundamental physics (Einstein's gravitational time dilation, search for dark matter), to enable space-assisted relativistic geodesy at < 1 cm uncertainty level, and to intercompare optical clocks on the ground at the < $1 \times 10^{-18}$ level. Comparison of ground clocks via the ISS will be performed using microwave links and frequency-comb-based optical links. The performance goal of the space clock and the optical link is less than $2 \times 10^{-17}$ inaccuracy and instability.

Within an EU-FP7-funded project, a European consortium of 16 partners is developing a strontium optical lattice clock breadboard [1]. Goal performances are instability below $1 \times 10^{-15} \tau^{-1/2}$ and a fractional inaccuracy at $5 \times 10^{-17}$ level.

For the realization of the clock, techniques and approaches suitable for later space application are used, such as modular layout, diode lasers, low power consumption, and compact dimensions.

For example, a compact, light-weight, and energy-efficient atomics package was developed. A robust frequency stabilization system enables frequency-stabilizing the 461 nm and 689 nm cooling lasers (the latter to sub-kHz linewidth), the 813 nm lattice laser and the 679 and 707 nm repumpers. The fully transportable clock laser is based on a 10 cm long cavity. Its current fractional instability is $3 \times 10^{-15}$.

The Sr clock apparatus is operating reliably with 88Sr, showing good cooling and trapping efficiency. More than 105 atoms can be loaded into the lattice with a temperature of 1.3 $\mu$K. The spectroscopy of the clock transition was performed, showing linewidths as small as 9 Hz. Rabi oscillations as well as the sideband spectrum were acquired, showing a good atom coherence time and cooling efficiency.

The atomics package and laser package were transported by van from Birmingham to PTB (Braunschweig) for integration with the clock laser and for metrological characterization. After reaching its destination, the 1-stage MOT was operational already 2 days after arrival, demonstrating the robustness and reliability of the system.

In parallel to this work, ESA-funded industrial technology development of cooling lasers and of a frequency stabilization unit, aimed at TRL 5, will begin in early 2016.

References

1. K. Bongs et al., C. R. Physique 16, 553 (2015); http://dx.doi.org/10.1016/j.crhy.2015.03.009

[ slides ]

## P. Schmidt

### Transportable optical clocks for relativistic geodesy

Frequency comparisons between optical clocks approaching 18 significant digits are the most accurate measurements we can currently perform. The high accuracy of these devices is a consequence of our capability to isolate atoms from external perturbations. In this regime, special and general relativistic effects start playing a dominant role and their effects have to be taken into account when evaluating a clock or when performing a frequency comparison between clocks [1].

The same relativistic effects define the geoid, the equipotential surface to which orthonormal heights in geodesy are referenced to [2]. Remote frequency comparisons between optical clocks via e.g. a stabilized optical fiber link thus enable the determination of height differences above the geoid between distant points. Prior side-by-side calibration of the clocks turns the frequency ratio measurement into a measurement of the height difference between the clocks. However, this requires transportable optical clocks.

In my presentation, I will introduce the basic concepts of optical clocks, their operation and performance evaluation in terms of instability (statistical uncertainty) and inaccuracy (systematic uncertainty). Strategies for the evaluation of the uncertainty for height measurements that go beyond the combined estimated uncertainty in the frequency comparison between two optical clocks will be discussed. Furthermore, technical challenges in making the performance of state-of-the-art laboratory optical clocks available for transportable systems will be addressed. I will also provide a status report on our activities towards a transportable Aluminium ion quantum logic optical clock.

References

1. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637-701 (2015)
2. A. Bjerhammar, On a relativistic geodesy, Bull. Geodesique 59, 207-220 (1985).

## S. Schoen

### GNSS and Clocks: between Navigation Geometry and Physics

Clocks are the heart of any global navigation satellite system (GNSS). Stable oscillators, such as e.g., cesium or rubidium clocks, or passive hydrogen masers generate the navigation signals in the satellites. On ground low stability TCXO oscillators replicates the signal in the receiver that is correlated with the incoming signal in order determine the signal flight time and subsequently the distance to the satellites. However, GNSS are one-way ranging systems and the transmitter and user clocks are not synchronized. Therefore, only so-called pseudo-ranges and pseudo carrier-phase observations can be measured.

A synchronization is realized by introducing satellite and receiver clock errors w.r.t. GNSS system time. Corrections for the satellite clock errors are made available by the system provider via GNSS navigation message or orbit and clock solutions by the International GNSS Service (IGS). In contrast, due to the limited long-term (>1 s) frequency stability of the receiver's internal quartz oscillator and its generally poor accuracy, the receiver clock error has to be estimated in the GNSS analysis epoch-by-epoch together with the coordinates. The consequences are (i) the need of four satellites for positioning, (ii) bad separability of the height component and the clock parameters, and other elevation-dependent effects, like e.g., delays due to tropospheric refraction, (iii) the height component is typically determined three times worse than the horizontal component and most vulnerable to systematic effects. All these effects restricts the interpretation of GNSS results and may mask valuable geophysical or atmospheric signals.

In this presentation, we will first explain how the treatment of clocks in GNSS is linked to the geometry of the navigation problem. Next, we will discuss how the currently used GNSS analysis concepts can be strengthened in order to benefit from modern ultra-stable atomic clocks in kinematic positioning and navigation. Furthermore, we will question to which extend the physical clock signal is really contained in the attributed parameter "clock error" during GNSS analysis.

References

1. Krawinkel T., Schoen S. Benefits of receiver clock modeling in code-based GNSS navigation, GPS Solutions (2015) DOI: 10.1007/s10291-015-0480-2
2. Weinbach U, Schoen S. GNSS receiver clock modeling when using high-precision oscillators and its impact on PPP. Adv. Space Res. 47:229-238 (2011)

[ slides ]

## J.W. van Holten

### World-line perturbation theory

In curved space-time complete sets of exact solutions of equations of motion are often not available. The method of world-line deviations provides an algorithm for a perturbative construction of solutions. Moreover the method also allows one to investigate the stability of these solutions. I will present the method on some detail and illustrate it with examples and applications to black-hole space-times and gravitational waves

[ slides ]