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Theoretical Aspects of Relativistic Geodesy
Abstract
In this thesis, I show how fundamental geodetic notions can be defined within a general relativistic framework. Among the concepts that are analyzed there are the relativistic gravity potential, the geoid, the normal gravity field and its potential, as well as the genuinely relativistic definition of chronometric height. Moreover, a simple procedure for the operational preservation of a chosen level surface of the relativistic gravity potential is investigated. For all these concepts, the respective Newtonian notions are recovered in the weak-field limit. In the first-order (parametrized) post-Newtonian expansion the results previously published in the literature are obtained and it is shown how they are embedded into the present framework. Magnitudes of leading-order relativistic corrections to the geoid as well as redshift and acceleration measurements are calculated in a simple gravity field model. After the most important geodetic notions are introduced, the theory of General Relativity and the mathematical formalism are briefly discussed. Emphasis lies on some exact solutions to Einstein's vacuum field equation. These spacetimes are used in the following to either estimate relativistic effects or generalize geodetic concepts. Proceeding to a relativistic theory of gravity changes the underlying stage on which all physics takes place. The involved mathematical structure, related to the description of a curved spacetime, causes conventional geodetic notions to become ill-defined in the framework of General Relativity. Here, it is shown how relativistic generalizations of these notions can be constructed, working without any kind of weak-field approximation. The approach is mainly based on a so-called redshift potential of which the level sets foliate a stationary spacetime into isochronometric surfaces. It gives rise to the definition of a relativistic gravity potential which is used intensively. In particular, using a parametrized post-Newtonian spacetime for the Earth, the magnitude of relativistic corrections to the geoid is investigated in a simple Earth model. In the last part, the relation between proper time on the geoid and the defining constant L g in the IAU resolutions is discussed and a consistent relativistic definition for chronometric heights is proposed. Finally, relativistic orbital effects are compared to non-gravitational perturbations of satellite orbits and relativistic gravity gradiometry is investigated to link geodesic deviation to the curvature of spacetime, which is determinable by geodetic measurements.
... The essential steps to do so have already been outlined in Refs. [12,15,16], in which the relativistic geoid is defined in terms of isochronometric surfaces, the level sets of a socalled stationary redshift potential for Killing congruences. Furthermore, the analysis of timelike isometric congruences as the worldlines of observers, or equivalently the Earth's matter constituents, allows to derive an acceleration potential. ...
... In the present formalism, it can be proven that the u-and ageoids generically coincide in General Relativity without any approximation involved; see Refs. [12,16]. ...
... Definition: Let the relativistic gravity potential U * be defined by the following relation to the observers' timeindependent redshift potential φ [16]: ...
The Earth’s geoid is one of the most essential and fundamental concepts to provide a gravity field-related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions (including the geoid) within Einstein’s theory of general relativity is inevitable. Building on the theoretical construction of isochronometric surfaces and the so-called redshift potential for clock comparison, we define a relativistic gravity potential as a generalization of (post-)Newtonian notions. This potential exists in any stationary configuration with rigidly corotating observers, and it is the same as realized by local plumb lines. In a second step, we employ the gravity potential to define the relativistic geoid in direct analogy to the Newtonian understanding. In the respective limit, the framework allows to recover well-known (post-) Newtonian results. For a better illustration and proper interpretation of the general relativistic gravity potential and geoid, some particular examples are considered. Explicit results are derived for exact vacuum solutions to Einstein’s field equation as well as a parametrized post-Newtonian model. Comparing the Earth’s Newtonian geoid to its relativistic generalization is a very subtle problem, but of high interest. An isometric embedding into Euclidean three-dimensional space is an appropriate solution and allows a genuinely intrinsic comparison. With this method, the leading-order differences are determined, which are at the mm level.
... The essential steps to do so have already been outlined in Refs. [12,15,16], in which the relativistic geoid is defined in terms of isochronometric surfaces, the level sets of a so-called stationary redshift potential for Killing congruences. We will, to a large extent, use the results in these references and incorporate them into the definition of a relativistic gravity potential in the next section. ...
... Definition: Let the relativistic gravity potential U * be defined by the following relation to the observers' timeindependent redshift potential φ [16]: ...
The Earth's geoid is one of the most important fundamental concepts to provide a gravity field-related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions (including the geoid) within Einstein's theory of General Relativity is inevitable. Building on the theoretical construction of isochronometric surfaces and the so-called redshift potential for clock comparison, we define a relativistic gravity potential as a generalization of known (post-)Newtonian notions. This potential exists for any stationary configuration and rigidly co-rotating observers. It is the same as realized by local plumb lines. In a second step, we employ the gravity potential to define the relativistic geoid in direct analogy to the Newtonian understanding. In the respective limits, it allows to recover well-known (post-) Newtonian results. For a better illustration and proper interpretation of the general relativistic gravity potential and geoid, we consider some particular examples. Explicit results are derived for exact vacuum solutions to Einstein's field equation as well as a parametrized post-Newtonian model. Comparing the Earth's Newtonian geoid to its relativistic generalization is a very subtle problem. However, an isometric embedding into Euclidean three-dimensional space can solve it and allows a genuinely intrinsic comparison. With this method, the leading-order differences are determined, which are at the mm-level.
The current generation of optical atomic clocks has reached a fractional frequency uncertainty of 1 × 10 − 18 (and beyond) which corresponds to a geopotential difference of 0.1 m 2 /s 2 . Those gravitational potential differences can be observed as gravitational redshift when comparing the frequencies of optical clocks. Even temporal potential variations might be determined with precise novel optical atomic clocks onboard of low-orbiting satellites such as SLR-like (e.g. LAGEOS-1/2) and GRACE-like missions.
In this simulation study, the potential of precise space-borne optical clocks for the determination of temporal variations of low-degree Earth’s gravity field coefficients are investigated. Different configurations of satellite orbits, i.e. at different altitudes (between 400 and 6000 km) and inclinations, are selected as well as certain assumptions on the clock performance are made. A particular focus is put on how well degree-2 coefficients can be estimated from those optical clock measurements and how it compares to results from SLR.
Genel Görelilik teorisinin bir sonucu olarak, kütleçkimsel alanlar gözlemci zaman akışını etkilemektedir. Zaman akışının aynı hıza sahip olduğu yüzeyler, Newton potansiyeli ile tariflenen eşpotansiyel yüzey kavramı ile aynı yüzeyleri tarif etmektedir. Eşpotansiyel yüzeyler klasik anlamda uzun yıllardır gravite ve yükseklik ölçmelerine bağlı olarak belirleniyordu.
Yükseklik belirleme için kullanılan geleneksel yöntemler bütünsel açıdan bakıldığında ölçme sonuçlarını etkileyebilecek önemli hata kaynakları barındırmaktadır. Bunlardan biri nokta üzerindeki potansiyel değerlerin doğrudan ölçülememesi nedeni ile yükseklik taşınması ile artan derecelerde hata birikmesidir. Ortalama deniz seviyeleri farklı olarak belirlenen karasal kütleler arasındaki yükseklik entegrasyonunun zorluğu da ayrı bir sorun oluşturmaktadır. Gözlemlerin yatay düzlemde gerçekleşmesi ve arazi zorlukları nedeniyle engebeli arazilerde işgücü, ekipman gücü ve yol uzunluğunun artması ile klasik yöntemlerde büyük zorluklar yaşanabilmektedir. Bu ciddi olumsuzlukları aşabilmek açısından, son yıllarda uydu teknolojileri önde gelen çözümlerden birisi olarak kullanılmaktadır. Fakat son 10 yıldır, temelleri 20. yüzyılın ikinci yarısına dayanan bir yöntem olan kronometrik nivelman, doğrulukları artan saatler ve ağ teknolojilerinin kullanıldığı test gözlemleri sonucunda önemli sonuçlar ortaya koymaya başlamıştır.
Zaman bilgisi atomik saat teknolojilerindeki gelişmeler ile birlikte atomik frekans standardında optik spektrumda yüksek frekansta gözlemler yapılarak artan ölçüde doğruluklarla belirlenebilmektedir. Bugüne kadar kullanılan mikrodalga atom saatlerinin daha düşük düzeydeki doğruluk ve kararlılıklarına karşı 100 kat daha iyileştirilmiş olan optik atomik saatler yükseklik belirlenmesinde yeni bir yöntem olarak kronometrik nivelman yönteminin önünü açmaktadır. Ayrıca fiber iletim teknolojileri ile birlikte optik atomik saat karşılaştırmaları 10−19 mertebelerinde bir hassasiyetle yapılabilmektedir. Yerçekimi ivmesi g≈10 m/s2 ve c≈ 300 000 000 m/s olmak üzere; 1 santimetrelik yükseklik değişimlerinde Δ𝑣𝑣≈10−18 frekans kayması oranı elde edilebilmektedir. Böylece optik atomik saatlerin 1 santimetrelik yükseklik farklarını belirleyebilecek hassasiyette olduğu söylenebilir.
Bu kapsamda, bu tez çalışmasında atom saatleri arasında yapılan frekans karşılaştırmaları neticesinde kütle-çekimsel Doppler etkisi ile ortaya çıkan farktan yararlanılarak potansiyel farkların belirlenmesi konusundaki teorik temellere, yöntemin genel çerçevesine ve güncel atomik saat test ağlarına değinilmektedir.
Bu bağlamda kronometrik nivelman yönteminin teorik temelleri ve güncel çalışmalar incelenmekte, uluslararası yükseklik referans sistemine olabilecek katkıları, sistemin çalışma mekanizmaları ve geoit belirleme yöntemlerine katkıları tartışılmaktadır.
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