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5 X-ray Pulsar-Based Navigation: Theories and Experiments
speed of the clock and shortens the scale. So, the clocks and scales placed at different
space points are different completely, which is the locality of the clocks and scales.
The clock and scale are the fundamental elements of establishing the spacetime reference, and hereby it is required that the space-time scale standards at
different space positions are unified, so as to establish a unified space-time scale standard system. The so-called space-time scale standard system refers to selecting a
relatively-stationary reference system or frame within a small-enough area, in which
the relatively-stationary time-keeping system and length measuring system are set
as the standard time and the standard scale, respectively, and then the clocks and
scales in other regions or reference systems are calibrated by taking those in standard reference system as the datum. For example, the clock and scale of the BIPM are
used as the datum to calibrate the clocks and scales deployed in the other places. In
general, the clock’s calibration can be achieved by the Two-Way Satellite Time and
Frequency Transfer (TWSTFT) method, and the scale’s calibration is more complicated because the contraction effects of the scale placed in different directions are
different.
Supposed that a free particle moves along the world-line in the gravitational field,
a point on the world-line as the coordinate origin is selected to establish the locally
orthogonal coordinate frame which moves with the particle. According to the orientations of different coordinate axes, the different names of the coordinate frames are
used. For example, if the coordinate axes point to distant stars, the coordinate system
is called stellar orientation local frame; if the three coordinate axes always point
to the directions of the tangent, main-normal and sub-normal lines of the particle’s
world-line, respectively, the coordinate system is called natural local frame; and
so on. The local coordinate systems established in the general theory of relativity
cannot be extended but can be connected one another through a series of coordinate
transformations.
Although the local coordinate systems are equally weighted and have no advantages or disadvantages, there is always the “superior” coordinate system in practical
application. Generally, a dynamical self-consistent local coordinate system should be
established, which meets two basic properties: one is that the external gravitational
potential appears in the form of tidal potential; another is that the internal gravitational
potential is the gravitational potential of the celestial body under investigation. In fact,
the local reference system is established under the specific space-time background,
so the reference system in the general relativity can be divided into background and
local reference systems.
In summary, the complete processes of establishing the space-time reference under
the general relativity are: to define the coordinate origin of reference system; to
select the world-line of the coordinate origin; to select the rotation of the local
reference system relative to the spatial axis of the background reference system; to
select the coordinate conditions; to finally determine the space-time metric. The IAU
recommends that the celestial mechanics of the general relativity is used to deal with
the mutual transformation between the background and the local reference systems
for the first-order post-Newtonian approximation.
5 X-ray Pulsar-Based Navigation: Theories and Experiments
speed of the clock and shortens the scale. So, the clocks and scales placed at different
space points are different completely, which is the locality of the clocks and scales.
The clock and scale are the fundamental elements of establishing the spacetime reference, and hereby it is required that the space-time scale standards at
different space positions are unified, so as to establish a unified space-time scale standard system. The so-called space-time scale standard system refers to selecting a
relatively-stationary reference system or frame within a small-enough area, in which
the relatively-stationary time-keeping system and length measuring system are set
as the standard time and the standard scale, respectively, and then the clocks and
scales in other regions or reference systems are calibrated by taking those in standard reference system as the datum. For example, the clock and scale of the BIPM are
used as the datum to calibrate the clocks and scales deployed in the other places. In
general, the clock’s calibration can be achieved by the Two-Way Satellite Time and
Frequency Transfer (TWSTFT) method, and the scale’s calibration is more complicated because the contraction effects of the scale placed in different directions are
different.
Supposed that a free particle moves along the world-line in the gravitational field,
a point on the world-line as the coordinate origin is selected to establish the locally
orthogonal coordinate frame which moves with the particle. According to the orientations of different coordinate axes, the different names of the coordinate frames are
used. For example, if the coordinate axes point to distant stars, the coordinate system
is called stellar orientation local frame; if the three coordinate axes always point
to the directions of the tangent, main-normal and sub-normal lines of the particle’s
world-line, respectively, the coordinate system is called natural local frame; and
so on. The local coordinate systems established in the general theory of relativity
cannot be extended but can be connected one another through a series of coordinate
transformations.
Although the local coordinate systems are equally weighted and have no advantages or disadvantages, there is always the “superior” coordinate system in practical
application. Generally, a dynamical self-consistent local coordinate system should be
established, which meets two basic properties: one is that the external gravitational
potential appears in the form of tidal potential; another is that the internal gravitational
potential is the gravitational potential of the celestial body under investigation. In fact,
the local reference system is established under the specific space-time background,
so the reference system in the general relativity can be divided into background and
local reference systems.
In summary, the complete processes of establishing the space-time reference under
the general relativity are: to define the coordinate origin of reference system; to
select the world-line of the coordinate origin; to select the rotation of the local
reference system relative to the spatial axis of the background reference system; to
select the coordinate conditions; to finally determine the space-time metric. The IAU
recommends that the celestial mechanics of the general relativity is used to deal with
the mutual transformation between the background and the local reference systems
for the first-order post-Newtonian approximation.