5.4 Space-Time Reference Based on General Relativity
323
where w 0 (t, x) =
A
GM A
|x−x A | ;
w
i
(t, x) = G
A
M A v
i
A
|x − x A |
−
[(x − x A ) × S A ]
i
2|x − x A |
3
;
(t, x) =
A
GM A
|x − x A |
⎡
⎣ −2v
2
A +
B =A
GM B
|x B − x A |
+
1
2
((x − x A ) · v A )
2
|x − x A |
2
+ (x − x A ) · a A ] +
2Gv A · [(x − x A ) × S A ]
|x − x A |
3
,
where x A , v A , a A , S A and M A are, respectively, the position vector, velocity vector,
acceleration vector, total angular momentum and gravitational mass of a celestial
body A in the BCRS; w L (t, x) is the expansion of the post-Newtonian multiple
moments involving all celestial bodies; δ ij is the Kronecker symbol.
In most cases, the accuracy requirements can be satisfied for the single mass
approximation metric, and then w L (t, x) = 0. However, in order to keep the continuity
and integrity of the model for all cases, the term w L (t, x) is still remained in formula
(5.82).
In the GCRS, the metric tensor of any space-time coordinate point (cT,X), where
T = TCG can be expressed as
G 00 = −1 +
2W (T ,X)
c 2
−
2W
2 (T ,X)
c 4
G 0a = −
4
c 3 W
a
(T , X)
G ab = δ ab
1 +
2
c 2 W (T , X)
⎫
⎬
⎭
,
(5.83)
where W (T , X) and W
a
(T , X) are, respectively, the scalar and vector gravitational
potentials resulted from the tidal force of the Earth and external celestial bodies.
Since the world-lines of any two observers do not coincide, it is impossible to
directly compare the proper time measured by the clocks carried by the observers.
In the local reference system, the time reading obtained by the observer using the
local clock cannot be directly applied to other local reference systems or global
(background) reference systems. The time difference caused by the relativistic effect
between different reference systems must be considered. It is the problem of timescale
transformation in the general relativity, which will be discussed in detail in Sect. 5.6.2
of this chapter.
323
where w 0 (t, x) =
A
GM A
|x−x A | ;
w
i
(t, x) = G
A
M A v
i
A
|x − x A |
−
[(x − x A ) × S A ]
i
2|x − x A |
3
;
(t, x) =
A
GM A
|x − x A |
⎡
⎣ −2v
2
A +
B =A
GM B
|x B − x A |
+
1
2
((x − x A ) · v A )
2
|x − x A |
2
+ (x − x A ) · a A ] +
2Gv A · [(x − x A ) × S A ]
|x − x A |
3
,
where x A , v A , a A , S A and M A are, respectively, the position vector, velocity vector,
acceleration vector, total angular momentum and gravitational mass of a celestial
body A in the BCRS; w L (t, x) is the expansion of the post-Newtonian multiple
moments involving all celestial bodies; δ ij is the Kronecker symbol.
In most cases, the accuracy requirements can be satisfied for the single mass
approximation metric, and then w L (t, x) = 0. However, in order to keep the continuity
and integrity of the model for all cases, the term w L (t, x) is still remained in formula
(5.82).
In the GCRS, the metric tensor of any space-time coordinate point (cT,X), where
T = TCG can be expressed as
G 00 = −1 +
2W (T ,X)
c 2
−
2W
2 (T ,X)
c 4
G 0a = −
4
c 3 W
a
(T , X)
G ab = δ ab
1 +
2
c 2 W (T , X)
⎫
⎬
⎭
,
(5.83)
where W (T , X) and W
a
(T , X) are, respectively, the scalar and vector gravitational
potentials resulted from the tidal force of the Earth and external celestial bodies.
Since the world-lines of any two observers do not coincide, it is impossible to
directly compare the proper time measured by the clocks carried by the observers.
In the local reference system, the time reading obtained by the observer using the
local clock cannot be directly applied to other local reference systems or global
(background) reference systems. The time difference caused by the relativistic effect
between different reference systems must be considered. It is the problem of timescale
transformation in the general relativity, which will be discussed in detail in Sect. 5.6.2
of this chapter.