5.4 Space-Time Reference Based on General Relativity
317
5.4.3.3 Time Synchronization Measuring
In the four-dimensional space-time, the coordinate system S
x
i
is selected. When
the observer’s coordinate intervals and four-dimensional velocities are, respectively,
dx
0
= −cdt = 0
dx
i
= 0
.
(5.62)
u
0
= −
1
√ −g 00
c
u
i
= 0
,
(5.63)
then the observer is locally at rest relative to the coordinate system, called local-rest
observer. Therefore, with the passing time of the clock carried by the local-rest
observer as the investigated object, the time interval measured by the observer is
expressed as
dT = −
1
c 2 g 00 u
0 dx
0
=
−g 00 dt.
(5.64)
From the above formula, it can be seen that the time interval recorded by the
observer’s clock is directly proportional to the coordinate time interval, and its
proportional coefficient
√
−g 00 is a function of the observer’s position. Even if the
coordinate time intervals dt between the observers at different points in space are the
same, the proper time intervals are also different. In other words, the time measured
by the different observers is different, which is the effect of gravitational field on
the clocks. Would the gravitational field disappear, the proper time recorded by the
standard clock at different positions is the coordinate time.
The time synchronization refers to the mutual calibration of two remote clocks,
with the simultaneity of time measurement, and belongs to the basic concept of time
system keeping. In the special relativity, the time synchronization is related to the
choice of coordinates, but in the same coordinate system, it can always be established.
In the general relativity, it is generally difficult to synchronize two remote clocks,
or only when certain conditions are satisfied, the simultaneity of two remote clocks
can be discussed. In the bending space-time, the alignment of the at-rest clocks at
two adjacent points M and N can be achieved by optical signal propagation. When
the coordinate time indicated by the clock’s face is x
0
M (1) , the at-rest observer at
point M, sends out a light signal; when the signal arrives at point N, its coordinate
time is x
0
N ; finally, when the signal reflects back to point M, its coordinate time is
x
0
M (2) , as shown in Fig. 5.5, where l M and l N represent the world-lines of the at-rest
observers at points M and N, respectively, and both are parallel to the time axis. As a
result, the coordinate differences between the light signals going and returning are,
respectively,
dx
0
(1) = x
0
N − x
0
M (1) ,
(5.65)
317
5.4.3.3 Time Synchronization Measuring
In the four-dimensional space-time, the coordinate system S
x
i
is selected. When
the observer’s coordinate intervals and four-dimensional velocities are, respectively,
dx
0
= −cdt = 0
dx
i
= 0
.
(5.62)
u
0
= −
1
√ −g 00
c
u
i
= 0
,
(5.63)
then the observer is locally at rest relative to the coordinate system, called local-rest
observer. Therefore, with the passing time of the clock carried by the local-rest
observer as the investigated object, the time interval measured by the observer is
expressed as
dT = −
1
c 2 g 00 u
0 dx
0
=
−g 00 dt.
(5.64)
From the above formula, it can be seen that the time interval recorded by the
observer’s clock is directly proportional to the coordinate time interval, and its
proportional coefficient
√
−g 00 is a function of the observer’s position. Even if the
coordinate time intervals dt between the observers at different points in space are the
same, the proper time intervals are also different. In other words, the time measured
by the different observers is different, which is the effect of gravitational field on
the clocks. Would the gravitational field disappear, the proper time recorded by the
standard clock at different positions is the coordinate time.
The time synchronization refers to the mutual calibration of two remote clocks,
with the simultaneity of time measurement, and belongs to the basic concept of time
system keeping. In the special relativity, the time synchronization is related to the
choice of coordinates, but in the same coordinate system, it can always be established.
In the general relativity, it is generally difficult to synchronize two remote clocks,
or only when certain conditions are satisfied, the simultaneity of two remote clocks
can be discussed. In the bending space-time, the alignment of the at-rest clocks at
two adjacent points M and N can be achieved by optical signal propagation. When
the coordinate time indicated by the clock’s face is x
0
M (1) , the at-rest observer at
point M, sends out a light signal; when the signal arrives at point N, its coordinate
time is x
0
N ; finally, when the signal reflects back to point M, its coordinate time is
x
0
M (2) , as shown in Fig. 5.5, where l M and l N represent the world-lines of the at-rest
observers at points M and N, respectively, and both are parallel to the time axis. As a
result, the coordinate differences between the light signals going and returning are,
respectively,
dx
0
(1) = x
0
N − x
0
M (1) ,
(5.65)