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5 X-ray Pulsar-Based Navigation: Theories and Experiments
For the measurement of the space-time interval of the local-rest observer, the
intuitively physical explanation can be given from another perspective. The spatial
distance between two adjacent points M and N is explained as that half of the traveling
time of the light signal, emitted by the at-rest observer at point M and reflected back to
point M through point N, multiplied by the velocity of light. Thereby, from formulas
(5.69) and (5.70), half of the going and returning time between the M and N is
expressed as
dt =
1
2c
dx
0
(1) + dx
0
(2)
=
g 0i g 0j − g 00 g ij
dx i dx j
cg 00
.
(5.75)
Considering formula (5.64), the spatial distance between two points is expressed
as
dL = cdT = c
−g 00 dt =
g ij −
g 0i g 0j
g 00
dx i dx j .
(5.76)
Apparently, formula (5.76) is completely coincident with formula (5.74). It should
be pointed out that formula (5.76) is derived from the precondition that there is the
local-rest observer. If the precondition is physical realizable, the dL measured by the
observer must be real. It is required that the h ij dx
i dx
j in formula (5.74) is a positivedefinite quadratic form. In other words, the space-time metric h ij is required to meet
the physical coordinate conditions.
For the local region of gravitational field, the Euclidean geometric distance
between two adjacent points can be defined as
dl =
dx
2
+ dy
2
+ dz
2
1
2
=
dr
2
+ r
2 d θ
2
+ r
2 sin
2
θ d ϕ
2
1
2
.
(5.77)
The geometric distance dl is called coordinate length.
Due to the different gravitational intensity at the different points in the gravitational
field, the scale which is not affected by the gravitational field in the at-rest inertial
system is taken as the standard of measurement. The standard scale is placed in the
local inertial system, and when it passes through each point in the gravitational field,
the length of the scale at each point is calibrated to keep the same standard scale
length at every point.
Furthermore, the real spatial distance between two adjacent points in the gravitational field measured by the at-rest observer is defined as the proper length, and in the
Schwarzschild metric with weak gravitational field, there is the following expression:
dL =
1 +
2GM
rc 2
1
2
dl,
(5.78)
5 X-ray Pulsar-Based Navigation: Theories and Experiments
For the measurement of the space-time interval of the local-rest observer, the
intuitively physical explanation can be given from another perspective. The spatial
distance between two adjacent points M and N is explained as that half of the traveling
time of the light signal, emitted by the at-rest observer at point M and reflected back to
point M through point N, multiplied by the velocity of light. Thereby, from formulas
(5.69) and (5.70), half of the going and returning time between the M and N is
expressed as
dt =
1
2c
dx
0
(1) + dx
0
(2)
=
g 0i g 0j − g 00 g ij
dx i dx j
cg 00
.
(5.75)
Considering formula (5.64), the spatial distance between two points is expressed
as
dL = cdT = c
−g 00 dt =
g ij −
g 0i g 0j
g 00
dx i dx j .
(5.76)
Apparently, formula (5.76) is completely coincident with formula (5.74). It should
be pointed out that formula (5.76) is derived from the precondition that there is the
local-rest observer. If the precondition is physical realizable, the dL measured by the
observer must be real. It is required that the h ij dx
i dx
j in formula (5.74) is a positivedefinite quadratic form. In other words, the space-time metric h ij is required to meet
the physical coordinate conditions.
For the local region of gravitational field, the Euclidean geometric distance
between two adjacent points can be defined as
dl =
dx
2
+ dy
2
+ dz
2
1
2
=
dr
2
+ r
2 d θ
2
+ r
2 sin
2
θ d ϕ
2
1
2
.
(5.77)
The geometric distance dl is called coordinate length.
Due to the different gravitational intensity at the different points in the gravitational
field, the scale which is not affected by the gravitational field in the at-rest inertial
system is taken as the standard of measurement. The standard scale is placed in the
local inertial system, and when it passes through each point in the gravitational field,
the length of the scale at each point is calibrated to keep the same standard scale
length at every point.
Furthermore, the real spatial distance between two adjacent points in the gravitational field measured by the at-rest observer is defined as the proper length, and in the
Schwarzschild metric with weak gravitational field, there is the following expression:
dL =
1 +
2GM
rc 2
1
2
dl,
(5.78)