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Han et al. Satell Navig (2021) 2:18 Page 3 of 10

distance, and the relativity of distance will inevitably lead to carry the same ideal atomic clocks which are always consist-
the correlation between time and space. For any two events ent with the SI second. Terefore, we believe that the gravi-
that occur in the universe, the time intervals and spatial dis- tational feld will change the frequency or clock speed of an
tances measured by observers moving relatively are difer- atomic clock. Due to the local fatness of space, the concept
ent, even if they use same clocks and rulers. Terefore, the of space frame used in Newtonian mechanics is still appli-
theory of relativity considers time and space as a whole and cable, but the diference is that it can only be used locally by
supposes that the space-time interval between two events is the observer and cannot extend outward infnitely.
an invariant quantity that has nothing to do with the observ- A relativistic space-time reference system consists of a
ers (Han, 2017). Under this assumption, the coordinate rela- Four-Dimensional (4D) coordinate system and the cor-
tionship between the two relative moving inertial reference responding metric coefcients. It maps the space-time
systems no longer satisfes the classic Galileo transforma- points, one by one, to the Minkowski Space in which one
tion, but the Lorentz transformation. All the laws of phys- dimension is imaginary and other three are real. Terefore
ics remain unchanged under the Lorentz transformation, every event occurred in the universe has a set of clear and
which is called the principle of special relativity. unique space–time coordinates, while the trajectory of any
Secondly, space–time has non-uniformity or non-Euclid- object and the measurement characteristics of space–time
ean characteristics. Tere is a universal interaction among are determined by the space–time metric. Te space–time
all objects in the universe. Te uneven distribution of mat- metric is a second-order symmetric tensor feld, which is
ter will inevitably lead to the unevenness of the space–time determined by the matter distribution of space–time and
gravitational feld, and hence there is no objective straight satisfes the Einstein feld equation. Te metric tensor has
line in a large-scale space. We know that the basic postu- 10 independent coordinate components, which are the so
late of Euclidean geometry is straight line, so Euclidean called metric coefcients. Obviously, the metric coefcients
geometry does not hold in a large-scale space. Te general are the functions of space–time points and closely related to
theory of relativity considers that space-time including the the basis vectors of coordinates. Diferent basis vectors lead
gravitational feld is a curved four-dimensional pseudo- to diferent metric coefcients. Te Einstein feld equation
Riemann Space. Terefore, it is impossible to construct a has 6 independent nonlinear equations. To solve for 10 met-
Cartesian coordinate system with a large-scale spatial cov- ric coefcients, 4 coordinate conditions are required. In the
erage. Space–time is the basic form of material existence. theory of relativity, the coordinate conditions can be arbi-
Te vacuum or space without matter is only the result of trarily selected. Ten the space–time coordinates in general
artifcial abstraction, and space–time itself does not have relativity have no clear physical or geometric meaning, but
the characteristics of straight or bending. Fundamentally, arbitrariness and equivalence. Terefore, the coordinates of
the curvature of space–time is just the result of gravitation a space–time point in the gravitational feld depend not only
geometrization in general relativity. Terefore, it is easier to on the space–time reference frame located at the coordinate
be understood if saying that space-time is inhomogeneous origin, but also on the space–time metric or the coordinate
rather than that curved (Han, 2017). conditions.
Te inhomogeneity of space–time also leads to no ideal
inertial space in our universe. Both inertia and gravitation Local inertial reference system
are the result of the interaction of substance in the universe. According to the principle of equivalence of relativity, for
It is impossible to separate them completely. In studying any mass point as an observer that moves freely in space–
dynamic and kinematic problems, we cannot take all the time, an inertial space reference frame or a local inertial
celestial bodies into account. An efective way is to separate reference system that is applicable to the observer local
them into two groups, i.e., the near celestial bodies and the space–time can be constructed. Te local inertial reference
far distant ones. Te efect of the former is called gravita- system, which may also be called local Lorentz reference
tion, and that of the later is the inertia, which is the so-called frame, meets the following basic conditions:
Mach principle. Terefore, both gravitational feld and iner-
tial space are relative. Inertial space is not only local but also • Te coordinate origin is a mass point freely moving in
approximate. Te spatial scope of application of the inertia the space–time.
depends on remoteness of the celestial bodies that forms the • Te time reference is the reading of the atomic clock
inertial efect, as well as our requirements for the accuracy coming with the origin.
of space–time measurement. • Te space axes or the coordinate base vectors do not
Te benchmark of space–time metric in relativity is rotate relative to inertial gyroscopes.
essentially the light (Han, 1997). For two determined events,
the time intervals measured by observers at diferent spa- Note that the reason why inertial gyroscopes are used here
tial positions are diferent, even they are relatively static and instead of distant celestial bodies to defne the non-rotating