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5 X-ray Pulsar-Based Navigation: Theories and Experiments
5.4.3 Physical Interpretation of Space-Time Measurement
5.4.3.1 Reference System and Coordinate System
In order to describe an event or a particle’s motion, the corresponding reference
system and coordinate system must be selected. It is of no practical significance
to discuss the motion law of an event or a particle without the reference system or
coordinate system. According to the general theory of relativity, the four-dimensional
physical space-time is a curved Riemannian space, and generally, there is no global
unified time system and spatial coordinate system. Therefore, the local reference
system and coordinate system are only discussed in this section. The reference system
and coordinate system are two different concepts, which should be distinguished to
better describe the regularity of objective events.
The so-called reference system refers to the clock and local rigid frame (scale)
carried by an observer moving in a certain way in space-time, and focuses on the reference object adopted by the system, rather than the specific mathematical expression.
The so-called coordinate system refers to the establishment of a certain one-to-one
correspondence between the space-time position of a specific physical event and
an array composed of four parameters. In this way, selecting a space-time coordinate system, any definite value of the array represents a certain space-time point.
A reference system can contain many coordinate systems which are essentially the
same, such as the rectangular coordinate system and spherical coordinate system.
The description of objective laws will not be affected by the selection of any coordinate system. In other words, it is essentially equivalent in the same reference system
to select two coordinate systems for the description of the same event. Obviously,
for the objective description of an event, the selection of reference system is very
important. Once a reference system is selected, it does not matter what form of coordinate system is used. When a coordinate system is selected, its specific form is
clearly determined to provide a definite description of the space-time position, but
it may not necessarily represent a physically meaningful reference system. When a
reference system is selected, it is shown that the specific form of coordinate system
is not focused on.
It is well known that any physical quantity must be observable, and any measurement is always related to a certain observer’s local reference system. The theoretical
studies of physical problems in the general relativity need the coordinate system, but
all measurements depend on the reference system. However, the coordinate system
is usually different from the reference system, which leads to a question: what is the
relationship between the tensor’s physical quantities obtained in a coordinate system
and the physical quantities measured by an observer who is observing the same
phenomenon? For a specific measurement problem, there are usually four factors
needed to be considered: the first is the definite observation object, which refers to
the physical quantity in a certain physical phenomenon; the second is the space-time
geometric properties of the event; the third is the selected reference system, which
refers to the observer moving in a certain way; the fourth is the selected coordinate
system, to describe the movement of the observation object and the observer. In fact,
5 X-ray Pulsar-Based Navigation: Theories and Experiments
5.4.3 Physical Interpretation of Space-Time Measurement
5.4.3.1 Reference System and Coordinate System
In order to describe an event or a particle’s motion, the corresponding reference
system and coordinate system must be selected. It is of no practical significance
to discuss the motion law of an event or a particle without the reference system or
coordinate system. According to the general theory of relativity, the four-dimensional
physical space-time is a curved Riemannian space, and generally, there is no global
unified time system and spatial coordinate system. Therefore, the local reference
system and coordinate system are only discussed in this section. The reference system
and coordinate system are two different concepts, which should be distinguished to
better describe the regularity of objective events.
The so-called reference system refers to the clock and local rigid frame (scale)
carried by an observer moving in a certain way in space-time, and focuses on the reference object adopted by the system, rather than the specific mathematical expression.
The so-called coordinate system refers to the establishment of a certain one-to-one
correspondence between the space-time position of a specific physical event and
an array composed of four parameters. In this way, selecting a space-time coordinate system, any definite value of the array represents a certain space-time point.
A reference system can contain many coordinate systems which are essentially the
same, such as the rectangular coordinate system and spherical coordinate system.
The description of objective laws will not be affected by the selection of any coordinate system. In other words, it is essentially equivalent in the same reference system
to select two coordinate systems for the description of the same event. Obviously,
for the objective description of an event, the selection of reference system is very
important. Once a reference system is selected, it does not matter what form of coordinate system is used. When a coordinate system is selected, its specific form is
clearly determined to provide a definite description of the space-time position, but
it may not necessarily represent a physically meaningful reference system. When a
reference system is selected, it is shown that the specific form of coordinate system
is not focused on.
It is well known that any physical quantity must be observable, and any measurement is always related to a certain observer’s local reference system. The theoretical
studies of physical problems in the general relativity need the coordinate system, but
all measurements depend on the reference system. However, the coordinate system
is usually different from the reference system, which leads to a question: what is the
relationship between the tensor’s physical quantities obtained in a coordinate system
and the physical quantities measured by an observer who is observing the same
phenomenon? For a specific measurement problem, there are usually four factors
needed to be considered: the first is the definite observation object, which refers to
the physical quantity in a certain physical phenomenon; the second is the space-time
geometric properties of the event; the third is the selected reference system, which
refers to the observer moving in a certain way; the fourth is the selected coordinate
system, to describe the movement of the observation object and the observer. In fact,