5.4 Space-Time Reference Based on General Relativity
315
any observation result only depends on the first three factors, namely, observation
object, space-time structure and observer, while has nothing to do with the fourth
factor. For example, for the measurement of the space-time interval between two
events, once the first three factors are determined, the observer can measure a definite number, the selection of the coordinate system is arbitrary and the space-time
interval will not be affected due to the transformation of the coordinate system.
In summary, since the space-time coordinate system is arbitrarily selected in the
general relativity, the space-time coordinates are only a marker of an event, used
to distinguish from other events, generally without direct measurement significance.
With the measurement significance is the proper time measured by the standard clock
and the proper length measured by the standard scale, which are the basic observables
of the general relativity. From the perspective of tensor analysis, the basic observable
of the general relativity must be a scalar.
5.4.3.2 Expression of Space-Time Reference
From Einstein’s general relativity, there is a new understanding of the space where
matter moves and the time through which it passes, and the time and space cannot
exist independently without matter. The structure and property of space-time depend
on the distribution of matter, and the motion of matter in the gravitational field is
limited by the geometric characteristics of space-time. Under the general theory of
relativity, the space-time reference (or the reference system) is defined by the local
space-time metric, that is
ds
2
= g μν dx
μ dx
ν
.
(5.61)
The space-time metric g μν is a function of the observer’s space-time position.
Selecting the different space-time metric is equivalent to selecting the different
space-time coordinate system, and thus an array of four parameters, representing
a determined space-time point, is gotten.
For the flat space, when the Euclidean metric δ ij is selected, the commonly used
rectangular coordinate system will be obtained; when the spherical metric is selected,
the spherical coordinate system is obtained; when the cylindrical metric is selected,
the cylindrical coordinate system is obtained. These coordinate systems can be transformed one another and be extended to the whole space to describe the events in
the flat space. For the curved Riemannian space, the coordinate system represented
by arbitrary space-time metric has only local meaning, and generally it cannot be
extended to the whole space. Because under the general theory of relativity, the
timescale and length-scale change with the change of spatial position, there is not
a unified coordinate system. According to the theory of relativity, the changes of
the timescale and length-scale mainly come from two factors: one is the kinematical
effect (also known as the special relativity effect), the moving clock is slowed down,
and the moving scale is shortened, called length contraction; another is the gravitational field effect (also known as the general relativity effect), which slows down the
315
any observation result only depends on the first three factors, namely, observation
object, space-time structure and observer, while has nothing to do with the fourth
factor. For example, for the measurement of the space-time interval between two
events, once the first three factors are determined, the observer can measure a definite number, the selection of the coordinate system is arbitrary and the space-time
interval will not be affected due to the transformation of the coordinate system.
In summary, since the space-time coordinate system is arbitrarily selected in the
general relativity, the space-time coordinates are only a marker of an event, used
to distinguish from other events, generally without direct measurement significance.
With the measurement significance is the proper time measured by the standard clock
and the proper length measured by the standard scale, which are the basic observables
of the general relativity. From the perspective of tensor analysis, the basic observable
of the general relativity must be a scalar.
5.4.3.2 Expression of Space-Time Reference
From Einstein’s general relativity, there is a new understanding of the space where
matter moves and the time through which it passes, and the time and space cannot
exist independently without matter. The structure and property of space-time depend
on the distribution of matter, and the motion of matter in the gravitational field is
limited by the geometric characteristics of space-time. Under the general theory of
relativity, the space-time reference (or the reference system) is defined by the local
space-time metric, that is
ds
2
= g μν dx
μ dx
ν
.
(5.61)
The space-time metric g μν is a function of the observer’s space-time position.
Selecting the different space-time metric is equivalent to selecting the different
space-time coordinate system, and thus an array of four parameters, representing
a determined space-time point, is gotten.
For the flat space, when the Euclidean metric δ ij is selected, the commonly used
rectangular coordinate system will be obtained; when the spherical metric is selected,
the spherical coordinate system is obtained; when the cylindrical metric is selected,
the cylindrical coordinate system is obtained. These coordinate systems can be transformed one another and be extended to the whole space to describe the events in
the flat space. For the curved Riemannian space, the coordinate system represented
by arbitrary space-time metric has only local meaning, and generally it cannot be
extended to the whole space. Because under the general theory of relativity, the
timescale and length-scale change with the change of spatial position, there is not
a unified coordinate system. According to the theory of relativity, the changes of
the timescale and length-scale mainly come from two factors: one is the kinematical
effect (also known as the special relativity effect), the moving clock is slowed down,
and the moving scale is shortened, called length contraction; another is the gravitational field effect (also known as the general relativity effect), which slows down the