Relativistic geoid: Gravity potential and relativistic effects

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.

Figure 1

An overview of the relation between Weyl solutions and the Erez-Rosen as well as the Schwarzschild metric as generalizations of Newtonian configurations. An axisymmetric Newtonian gravitational field is generalized by a general Weyl solution with Quevedo moments $q_l$. For $l_{\max }=2$, the Erez-Rosen spacetime is recovered as the generalization of a Newtonian quadrupolar field. If also $q_2=0$, we obtain the Schwarzschild solution as the relativistic monopole analog.

Figure 2

Comparison of the relativistic and Newtonian geoid at leading order for approach (I). We show the geoid radii differences $\mathrm{\Delta }R(\mathrm{\Theta })=R_{\mathrm{PN}}(\mathrm{\Theta })-R_{\mathrm{N}}(\mathrm{\Theta })$ in the embedding space ${\mathbb{R}}^3$ as a function of $\mathrm{\Theta }$. The maximal, minimal, and mean differences are indicated.

Figure 3

Comparison of the relativistic and Newtonian geoid at leading order for approach (II). We show the geoid radii differences $\mathrm{\Delta }R(\mathrm{\Theta })=R_{\mathrm{PN}}(\mathrm{\Theta })-R_{\mathrm{N}}(\mathrm{\Theta })$ in the embedding space ${\mathbb{R}}^3$ as a function of $\mathrm{\Theta }$. The maximal, minimal, and mean differences are indicated.

Figure 4

Comparison of the relativistic and Newtonian geoid at leading order for approach (III). We show the geoid radii differences $\mathrm{\Delta }R(\mathrm{\Theta })=R_{\mathrm{PN}}(\mathrm{\Theta })-R_{\mathrm{N}}(\mathrm{\Theta })$ in the embedding space ${\mathbb{R}}^3$ as a function of $\mathrm{\Theta }$. The maximal, minimal, and mean differences are indicated.

Figure 5

Comparison of the relativistic and Newtonian geoid at leading order for approach (IV), which is the same as approach (I) but without embedding. The Newtonian and post-Newtonian coordinates are identified. We show the geoid radii differences $\mathrm{\Delta }R(\mathrm{\Theta })=R_{\mathrm{PN}}(\mathrm{\Theta })-R_{\mathrm{N}}(\mathrm{\Theta })$ as a function of $\mathrm{\Theta }$. The maximal, minimal, and mean differences are indicated.