Relativistic Geodesy Research Articles

Time transfer and significance of vertical land motion in relativistic geodesy applications: a review paper

Determination of the Earth’s gravity field and geopotential value is one of the fundamental topics in physical geodesy. Traditional terrestrial gravity and precise leveling measurements can be used to determine the geopotential values at a local or regional scale. However, recent developments in optical atomic clocks have not only rapidly improved fundamental science but also contributed to applied research. The latest generation of optical clocks is approaching the accuracy level of 10−18 when facilitating atomic clock networks. These systems allow examining fundamental theories and many research applications, such as atomic clocks applications in relativistic geodesy, to precisely determine the Earth’s gravity field parameters (e.g., geopotential values). According to the theory of relativistic geodesy, the frequency difference measured by an optical clock network is related to the gravity potential anomaly, provided that the effects of disturbing signals (i.e., tidal and non-tidal contributions) are filtered out. The relativistic geodesy principle could be used for a practical realization of global geodetic infrastructure, most importantly, a vertical datum unification or realization of height systems. This paper aims to review the background of relativistic (clock-based) geodesy and study the variations of optical atomic clock measurements (e.g., due to hydrology loading and land motion).

Spherical harmonics representation of the gravitational phase shift

We investigate the general relativistic phase of an electromagnetic wave as it propagates in the gravitational field of the Earth, which is modeled as an isolated, weakly aspherical gravitating body. We introduce coordinate systems to describe light propagation in the Earth's vicinity along with the relevant coordinate transformations, and discuss the transformations between proper and coordinate times. We represent the Earth's gravitational field using Cartesian symmetric trace-free (STF) mass multipole moments. The light propagation equation is solvable along the trajectory of a light ray to all STF orders $\ensuremath{\ell}$. Although we focus primarily on the quadrupole ($\ensuremath{\ell}=2$), octupole ($\ensuremath{\ell}=3$), and hexadecapole ($\ensuremath{\ell}=4$) cases, our approach is valid to all orders. We express the STF moments via spherical harmonic coefficients of various degree and order, ${C}_{\ensuremath{\ell}k},{S}_{\ensuremath{\ell}k}$. The result is the gravitational phase shift expressed in terms of the spherical harmonics. These results are new. We also consider contributions due to tides and the Earth's rotation. We estimate the characteristic magnitudes of each term of the resulting overall gravitational phase shift. The terms assessed are either large enough to impact current-generation clocks or will become significant as future-generation clocks offer greater sensitivity. Our formulation is useful for many practical and scientific applications, including space-based time and frequency transfers, relativistic geodesy and navigation, as well as quantum communication links and space-based tests of fundamental physics.

Optical frequency transfer with below 10−21 uncertainty using a DFB–laser-based fiber Brillouin amplifier

Exploiting the outstanding performance of optical atomic clocks for improved timekeeping, relativistic geodesy, and fundamental physics beyond the standard model demands comparing distant state-of-the-art optical clocks. Interferometric optical fiber links have been demonstrated as an eminent method for such frequency comparisons over distances up to thousands of kilometers. However, for such distances, the optical fiber attenuation mandates signal amplification. Fiber Brillouin amplification (FBA) has been proven as an efficient amplification technique for coherent frequency transfer. Demonstrated FBA schemes have been designed based on costly narrow-linewidth pump lasers and analog pump-to-signal phase locking schemes. Furthermore, the high pump power requirement of these FBAs hinders the integration of FBA-based frequency dissemination on fiber connections for shared telecommunication signals in the C-band. In this paper, we propose and experimentally demonstrate a novel FBA module (FBAM) employing cost-effective distributed feedback (DFB) pump lasers assisted by a digital phase locking scheme based on a field programmable gated array. The new FBAM is compact, cost-effective, and directly applicable to different bands, which opens up new opportunities to establish a frequency metrology infrastructure within existing telecommunication fiber networks. Additionally, the small-footprint of the DFB-FBAM allows for frequent amplification stages with lower pump power to reach continental scale optical metrology links with an optimized signal-to-noise ratio. We characterized the DFB-FBAM’s frequency transfer uncertainty using a two-way layout over an in-lab 100 km long optical fiber link and reach a fractional frequency instability of 9.3 × 10−22 at a 10 ks integration time. The DFB-FBAM characterizations show uncertainty contributions of (−2.1 ± 3.3) × 10−22 and below for averaging times >100 ks. For the first time, we assess the temporal Brillouin frequency shift variations in an underground fiber link and implement a scheme to track these changes in a remote FBAM.

A space-based quantum gas laboratory at picokelvin energy scales

Ultracold quantum gases are ideal sources for high-precision space-borne sensing as proposed for Earth observation, relativistic geodesy and tests of fundamental physical laws as well as for studying new phenomena in many-body physics during extended free fall. Here we report on experiments with the Cold Atom Lab aboard the International Space Station, where we have achieved exquisite control over the quantum state of single 87Rb Bose-Einstein condensates paving the way for future high-precision measurements. In particular, we have applied fast transport protocols to shuttle the atomic cloud over a millimeter distance with sub-micrometer accuracy and subsequently drastically reduced the total expansion energy to below 100 pK with matter-wave lensing techniques.

Новые методы и средства подготовки карт для навигации по геофизическим полям Земли

The article deals with the matters of creating an integrated navigation system using measurements of the Earth`s gravitational and magnetic fields’ parameters (EGF and EMF). Such a system is to contain a strapdown inertial navigation system (SINS), as well as sensors for the current parameters of the EGF and EMF. Its simulation was carried out. As a result it was found out that, due to the use of information on the parameters of the EGF and EMF, an error of several hundred meters can be obtained. Based on the results of full-scale tests of the created integrated navigation system for the EGF and EMF, a real error of 900 m was obtained with a correction interval of SINS readings of 4 minutes. According to the results of modeling and testing of the mentioned system, the main tasks and ways to solve them are identified. The requirements for measurers of geophysical fields’ parameters and navigation charts are considered, a number of new navigation meters, new methods and tools for preparing navigation charts are proposed. The ways of developing relativistic geodesy and the possibility of using the gravitational-wave astronomy achievements in gravimetry are considered.

Toward a New Generation of Compact Transportable Yb+ Optical Clocks

Optical atomic clocks are currently one of the most sensitive tools making it possible to precisely test the fundamental symmetry properties of spacetime and Einstein’s theory of relativity. At the same time, the extremely high stability and accuracy of compact transportable optical clocks open new perspectives in important fields, such as satellite navigation, relativistic geodesy, and the global time and frequency network. Our project aimed to develop a compact transportable optical clock based on a single ytterbium ion. We present the first prototype of the Yb+ clock (298 kg in 1 m3) and present several solutions aimed to improve the clock’s robustness to approach the demands of a space-qualified system. We present spectroscopic studies of a 435.5 nm quadrupole clock transition with Fourier-limited spectra of 25 Hz. The estimated instability of the output frequency at 1 GHz, which was down-converted with an optical frequency comb (OFC), is at the level of 9×10−15/τ, and the long-term instability and inaccuracy are at the level of 5×10−16. As the next steps, we present a new design for the clock laser and the OFC.

Efficient trace-free decomposition of symmetric tensors of arbitrary rank

Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well as in the study of gravitational radiation and gravitational lensing. Their use allows integration of the relevant wave propagation equations to arbitrary order. We present an improved iterative method for the trace-free decomposition of symmetric tensors of arbitrary rank. The method can be used both in coordinate-free symbolic derivations using a computer algebra system and in numerical modeling. We obtain a closed-form representation of the trace-free decomposition in arbitrary dimensions. To demonstrate the results, we compute the coordinate combinations representing the symmetric trace-free (STF) mass multipole moments for rank 5 through 8, not readily available in the literature.

High-Accuracy Optical Frequency Atomic Clock

Frequencies have been the most accurately measured physical quantity since the second was defined in 1967 based on the microwave atomic transition of a Cs atom. Recently, atomic clocks using optical frequency transitions have shown an order of magnitude better accuracy than microwave clocks. Thanks to their high accuracy and resolution, atomic clocks have become a new tool for investigations involving fundamental science and technology, such as the search for dark matter, gravitational wave detection, the temporal variation of fundamental constants, relativistic geodesy, quantum metrology, and the advanced Global Navigation Satellite System (GNSS). In addition, a redefinition of the second based on the optical frequency is expected. In this paper, we review the principles and applications of optical clocks.

Arne Bjerhammar- a personal summary of his academic deeds

Abstract Arne Bjerhammar is well known worldwide mainly for his research in physical geodesy but also for introducing a new matrix algebra with generalized inverses applied in geodetic adjustment. Less known are his developments in geodetic engineering and contributions to satellite and relativistic geodesy as well as studies on the relation between the Fennoscandia land uplift and the regional gravity low. Most likely part of his research has contributed to worldwide political relaxation during the cold war, which deed was honored by a certificate of achievement awarded by the Department of Research of the US army as well as the North Star Order by the King of Sweden. Arne Bjerhammar’s pioneer scientific production, in particular on a world geodetic system, towards what would become GPS, as well as relativistic geodesy, is still of great interest among the worldwide geodetic community, while the memories and spirit along his outstanding academic deeds have more or less fainted away from his home university (KTH) only a decade after he passed away.

Problems of creating autonomous navigation systems on geophysical fields

Global navigation satellite systems, which provide high accuracy of navigation, in certain conditions (in tunnels, in closed rooms, in conditions of interference, etc.) have restrictions on their use. In this regard, in order to ensure “seamless” navigation in any conditions of the situation, it becomes necessary to develop new methods and means to increase the stability of navigation definitions. The article is devoted to the consideration of the problems of creating an integrated navigation system using measurements of the parameters of the Earth’s gravitational and magnetic fields. Requirements for meters of parameters of geophysical fields and navigation charts are considered, a number of new navigation meters, new methods and means of preparing navigation charts are proposed. The ways of development of relativistic geodesy and the possibility of using the achievements of gravitational-wave astronomy in gravimetry are considered.

Energy of the 229Th nuclear clock transition.

Owing to its low excitation energy and long radiative lifetime, the first excited isomeric state of thorium-229, 229mTh, can be optically controlled by a laser1,2 and is an ideal candidate for the creation of a nuclear optical clock3, which is expected to complement and outperform current electronic-shell-based atomic clocks4. A nuclear clock will have various applications-such as in relativistic geodesy5, dark matter research6 and the observation of potential temporal variations of fundamental constants7-but its development has so far been impeded by the imprecise knowledge of the energy of 229mTh. Here we report a direct measurement of the transition energy of this isomeric state to the ground state with an uncertainty of0.17 electronvolts (one standard deviation) using spectroscopy of the internal conversion electrons emitted in flight during the decay of neutral 229mTh atoms. The energy of the transition between the ground state and the first excited state corresponds to a wavelength of 149.7±3.1 nanometres, which is accessible by laser spectroscopy through high-harmonic generation. Our method combines nuclear and atomic physics measurements to advance precision metrology, and our findings are expected to facilitate the application of high-resolution laser spectroscopy on nuclei and to enable the development of a nuclear optical clock of unprecedented accuracy.

Gravitational clock compass in general relativity

We show how a suitably prepared set of clocks can be used to determine all components of the gravitational field in General Relativity. We call such an experimental setup a clock compass, in analogy to the usual gravitational compass. Particular attention is paid to the construction of the underlying reference frame. Conceptual differences between the clock compass and the standard gravitational compass, which is based on the measurement of the mutual accelerations between the constituents of a swarm of test bodies, are highlighted. Our results are of direct operational relevance for the setup of networks of clocks, for example in the context of relativistic geodesy.

Single-ion, transportable optical atomic clocks

For the past 15 years, tremendous progress within the fields of laser stabilization, optical frequency combs and atom cooling and trapping have allowed the realization of optical atomic clocks with unrivaled performances. These instruments can perform frequency comparisons with fractional uncertainties well below , finding applications in fundamental physics tests, relativistic geodesy and time and frequency metrology. Even though most optical clocks are currently laboratory setups, several proposals for using these clocks for field measurements or within an optical clock network have been published, and most of time and frequency metrology institutes have started to develop transportable optical clocks. For the purpose of this special issue, we chose to focus on trapped-ion optical clocks. Even though their short-term fractional frequency stability is impaired by a lower signal-to-noise ratio, they offer a high potential for compactness: trapped ions demand low optical powers and simple loading schemes, and can be trapped in small vacuum chambers. We review recent advances on the clock key components, including ion trap and ultra-stable optical cavity, as well as existing projects and experiments which draw the picture of what future transportable, single-ion optical clocks may resemble.

Normal gravity field in relativistic geodesy

Modern geodesy is subject to a dramatic change from the Newtonian paradigm to Einstein's theory of general relativity. This is motivated by the ongoing advance in development of quantum sensors for applications in geodesy including quantum gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of the geoid and multipolar structure of the Earth's gravitational field. At the same time, VLBI, SLR, and GNSS have achieved an unprecedented level of accuracy in measuring coordinates of the reference points of the ITRF and the world height system. The main geodetic reference standard is a normal gravity field represented in the Newtonian gravity by the field of a Maclaurin ellipsoid. The present paper extends the concept of the normal gravity field to the realm of general relativity. We focus our attention on the calculation of the first post-Newtonian approximation of the normal field that is sufficient for applications. We show that in general relativity the level surface of the uniformly rotating fluid is no longer described by the Maclaurin ellipsoid but is an axisymmetric spheroid of the forth order. We parametrize the mass density distribution and derive the post-Newtonian normal gravity field of the rotating spheroid which is given in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce the post-Newtonian multipolar expansion of the metric tensor given in terms of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy. Finally, we derive the post-Newtonian generalization of the Somigliana formula for the gravity field on the reference ellipsoid.

Geodesy and metrology with a transportable optical clock

The advent of novel measurement instrumentation can lead to paradigm shifts in scientific research. Optical atomic clocks, due to their unprecedented stability and uncertainty, are already being used to test physical theories and herald a revision of the International System of units (SI). However, to unlock their potential for cross-disciplinary applications such as relativistic geodesy, a major challenge remains. This is their transformation from highly specialized instruments restricted to national metrology laboratories into flexible devices deployable in different locations. Here we report the first field measurement campaign performed with a ubiquitously applicable $^{87}$Sr optical lattice clock. We use it to determine the gravity potential difference between the middle of a mountain and a location 90 km apart, exploiting both local and remote clock comparisons to eliminate potential clock errors. A local comparison with a $^{171}$Yb lattice clock also serves as an important check on the international consistency of independently developed optical clocks. This campaign demonstrates the exciting prospects for transportable optical clocks.

An optical clock to go

Bringing next-generation atomic clocks out of the lab is not an easy task, but doing so will unlock many new possibilities. As a crucial first step, a portable atomic clock has now been deployed for relativistic geodesy measurements in the Alps.

Structured Doctoral Education in Hannover - Joint Programme IMPRS-GW and geo-Q RTG

Two structured doctoral programmes that we have in Hannover, the IMPRS on Gravitational Wave Astronomy and SFB on relativistic geodesy and gravimetry with quantum sensors geo-Q, have not only become major resources for education in each field but have also started to provide substantial synergy to members of both programmes. Our strong crossdisciplinary approach to create a joint programme has received excellent feedback not only from researchers inside the programme but also from various external committee. Building on experience that we have acquired over the last decade, we propose to set up a common doctoral programme within the international gravitational wave astronomy and physics. We envisage that with a common doctoral programme we will create a strong team of young researchers who will carry on building a strong network of third generation gravitational wave detectors and observatories.

Theoretical Tools for Relativistic Gravimetry, Gradiometry and Chronometric Geodesy and Application to a Parameterized Post-Newtonian Metric

An extensive review of past work on relativistic gravimetry, gradiometry and chronometric geodesy is given. Then, general theoretical tools are presented and applied for the case of a stationary parameterized post-Newtonian metric. The special case of a stationary clock on the surface of the Earth is studied.

Laser controlled atom source for optical clocks

Precision timekeeping has been a driving force in innovation, from defining agricultural seasons to atomic clocks enabling satellite navigation, broadband communication and high-speed trading. We are on the verge of a revolution in atomic timekeeping, where optical clocks promise an over thousand-fold improvement in stability and accuracy. However, complex setups and sensitivity to thermal radiation pose limitations to progress. Here we report on an atom source for a strontium optical lattice clock which circumvents these limitations. We demonstrate fast (sub 100 ms), cold and controlled emission of strontium atomic vapours from bulk strontium oxide irradiated by a simple low power diode laser. Our results demonstrate that millions of strontium atoms from the vapour can be captured in a magneto-optical trap (MOT). Our method enables over an order of magnitude reduction in scale of the apparatus. Future applications range from satellite clocks testing general relativity to portable clocks for inertial navigation systems and relativistic geodesy.

Advanced relativistic VLBI model for geodesy

Our present relativistic part of the geodetic VLBI model for Earthbound antennas is a consensus model which is considered as a standard for processing high-precision VLBI observations. It was created as a compromise between a variety of relativistic VLBI models proposed by different authors as documented in the IERS Conventions 2010. The accuracy of the consensus model is in the picosecond range for the group delay but this is not sufficient for current geodetic pur- poses. This paper provides a fully documented derivation of a new relativistic model having an accuracy substantially higher than one picosecond and based upon a well accepted formalism of relativistic celestial mechanics, astrometry and geodesy. Our new model fully confirms the consensus model at the picosecond level and in several respects goes to a great extent beyond it. More specifically, terms related to the acceleration of the geocenter are considered and kept in the model, the gravitational time-delay due to a massive body (planet, Sun, etc.) with arbitrary mass and spin-multipole moments is derived taking into account the motion of the body, and a new formalism for the time-delay problem of radio sources located at finite distance from VLBI stations is presented. Thus, the paper presents a substantially elaborated theoretical justification of the consensus model and its significant extension that allows researchers to make concrete estimates of the magnitude of residual terms of this model for any conceivable configuration of the source of light, massive bodies, and VLBI stations. The largest terms in the relativistic time delay which can affect the current VLBI observations are from the quadrupole and the angular momentum of the gravitating bodies that are known from the literature. These terms should be included in the new geodetic VLBI model for improving its consistency.