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Han et al. Satell Navig (2021) 2:18 Page 4 of 10

characteristics of inertial space is that the infuences of non- coordinates. Due to the in-homogeneity of space–time, the
far-distant celestial bodies need to be taken into account for local inertial system of a space–time point is limited not
the local inertial space. Te spin of a gyro will undergo a so- only in space, but also in time. In practice, we often need
called de Sitter precession relative to the far-distant celestial a local inertial system that is not limited in time, such as a
bodies, which is also named as geodesic precession. local inertial system centered at a spacecraft. For such a local
Tere is no doubt that the gravitation forces acting on the reference system, the scope of adaptation is not a sphere
particles near the origin are almost the same, and free particles but a pipeline in the 4D space–time. Tis unrestricted local
move in a uniform form in the eyes of the observer located inertial system in time is the local Lorentz reference frame
at the origin. Terefore, we need no gravitation but inertia for a free observer.
to describe the particle motions in the local space, in which It is very convenient to use the observer’s local inertial
the tidal forces generated by external celestial bodies can be system to express the events that occur in the space near
ignored. However, the local Lorentz reference frame satisfes the observer. But in most cases, a global coordinate system
the Newtonian inertia condition is just a diferential approxi- must be used which covers the entire space–time range to
mation in mathematics, and its spatial application range is describe the movement of substance in a large-scale space.
very limited. Due to the non-uniform nature of the gravita- Terefore, it is often necessary to give the transformation
tional feld, there is no true Newtonian inertial space in the relationship between the local inertial system and the global
real large-scale space–time. coordinate system.
For a curved space, if the coordinate basis vectors {e α } of According to Eqs. (1) or (2), the relationship between the
′α
α
a coordinate system {x } are orthogonal to a certain space- local inertial system {x } and the global coordinate system

α
time point P x , and its afne connection coefcients are {x } can be expressed as:

A
zero, then the reference frame formed by the coordinate
1
basis vectors of the point is a Lorenz reference frame. Tere- x = x µ + e x − Γ µ x κ A e e x x + ··· (3)
µ
µ ′j
γ ′j ′k
′
t
j
j k
A
fore, it forms a local inertial coordinate system nearby. Te 2 γ
basic conditions can be expressed as: or

� �
 �

 g µν x 0 ′ 1 0 ′j 1 0 � κ � γ ′j ′k

A = η µν  t = e dt + e x − Γ x e e x x + ···


(1) 0 c j 2c γ A k j
� �
µ
 Γ x = 0

αβ A  i i � � i ′j 1 i � κ � γ ′j ′k
′

 x = x t + e x − Γ x e e x x + ···
A j γ A k j
2

where g µν x are the common metric coefcients, η (4)

A µ μν

are the Minkowski ones, Γ αβ x are the afne coef- It can be seen from the coordinate relationship Eq. (4) that
A
cients of connection or expressed in vectors: the relationship is nonlinear between the local inertial sys-
tem and the global coordinate system. Since the coordinate

� � � �
 e µ x · e ν x =η relationship is developed approximately by using the Taylor

A A µν
(2)
� � series of 1/c , the applicable scope of the local inertial system is
 Γ µν x = 0

A determined by the convergence of the series.
Te frst equation of Eq. (1) or Eq. (2) is is the orthogo- The barycentric celestial reference system
nal normalization condition of the coordinate basis vectors. In astronomy, the mass center of the celestial body or system
Although orthogonal normalization is not a necessary con- under study is generally chosen as the origin of the space–time
dition for the inertial reference system, the Cartesian coor- reference system, and its coordinate axes are required to have
dinate system has natural application advantages. Terefore, spatial non-rotating characteristics. Te so-called non-rotating
when establishing a reference system, we always hope that has two meanings. One is that the space coordinate axes have
the coordinate bases can meet the condition of orthogonal no spatial rotation relative to a far-distant celestial body such as
normalization. Te second condition equation is the core of the extragalactic radio sources, which is named as the kinemat-
the local inertial system, which requires the coordinate basis ical non-rotating, and the other is that they are relative to gyro
vectors to satisfy the characteristics of parallel movement at or the inertial space and named as the dynamical non-rotating
the origin. In other words, the time axis of the local inertial (Han, 1997).
system is a time-like geodesic, and the space coordinate axes For an isolated celestial system, the kinematic non-rota-
near the origin are space-like geodesic lines. tion and the dynamic non-rotation are equivalent. How-
Te coordinates that satisfy the geodesic condition are ever, for non-isolated systems, due to the infuence of local
called Fermi coordinates, so the local inertial system is substance on space–time there will be a slight diference
an orthogonal Fermi coordinate system or Fermi normal between them. For example, there is a very slow spatial