  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 co-rotating 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.
