Relativistic geoid: Gravity potential and relativistic effects

Abstract

The Earth's geoid is one of the most important fundamental concepts to provide a gravity fieldrelated 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.

Quantum-enhanced differential atom interferometers and clocks with spin-squeezing swapping

“…Due to the approximations A1 and A2 introduced in Eqs. (20) and (21), respectively, the estimation uncertainty obtained from our analytical treatment cannot perfectly match the numerical results. At the optimal working point, θ A = θ B = π/2, A1 has no effect, as proved by the fact that the analytical and the numerical approach reach identical shot-noise sensitivity for a vanishing amount of squeezing.…”

“…Correlations between the output signals of the two interferometers can be used to infer φ A − φ B [9][10][11][12][13][14], while cancelling the common-mode noise. Differential AIs configurations have been exploited for gravity gradiometry [15][16][17][18][19], gravity-field curvature measurements [20], relativistic geodesy [21], the measurements of the gravitational constant [22], to probe the law of gravitation through tests of the universality of free-fall [23][24][25][26][27], and are also expected to find applications in the detection of gravitational waves [28][29][30][31]. Furthermore, correlation spectroscopy exploits differential frequency measurements.…”

Constrained Gravity Inversion With Adaptive Inversion Grid Refinement in Spherical Coordinates and Its Application to Mantle Structure Beneath Tibetan Plateau

“…Gravity measurement is an effective means to investigate the Earth's interior structures (Blakely, 1996), whose applications range from mining and hydrological problems (Hinderer et al., 2020; Martinez et al., 2012) to geoid determination (Hofmann‐Wellenhof & Moritz, 2006; Philipp et al., 2020) and plate tectonics (Ebbing et al., 2018; Götze & Pail, 2018; Panet et al., 2014). With the emergence of new gravity satellites in the new millennium (Andersen et al., 2010; Floberghagen et al., 2011; Tapley et al., 2004), static and temporal satellite gravity data with unprecedented data coverage and accuracy became easily accessible to geoscientists (Bucha & Janák, 2013; Ince et al., 2019).…”

On the Equivalence Principle and Relativistic Quantum Mechanics

“…condition (2) is the same in all reference frames). On the other hand, in the existing literature [10][11][12][13][14][15][16][17][18][19][20][21][22][23], one is typically looking for an agreement between the equivalence principle and non-relativistic quantum theory with no electromagnetic field involved (although see [24,25] for a discussion involving relativistic effects). A representative example for this matter is a neutral particle freely falling in a uniform gravitational field.…”