Towards an exact relativistic theory of Earth's geoid undulation

• We apply general relativity to define the exact concept of relativistic geoid.
• We derive relativistic equation of geoid and the reference level surface.
• We employ the manifold perturbation theory to discuss geoid's undulation.

The present paper extends the Newtonian concept of the geoid in classic geodesy towards the realm of general relativity by utilizing the covariant geometric methods of the perturbation theory of curved manifolds. It yields a covariant definition of the anomalous (disturbing) gravity potential and formulates differential equation for it in the form of a covariant Laplace equation. The paper also derives the Bruns equation for calculation of geoid's height with full account for relativistic effects beyond the Newtonian approximation. A brief discussion of the relativistic Bruns formula is provided.