308
5 X-ray Pulsar-Based Navigation: Theories and Experiments
5.4.2.3 The Space-Time Metric
The four-dimensional space-time is a set of one-dimensional time coordinate (x
0 )
and three-dimensional space coordinates (x
1 , x
2 , x
3 ), which is called the “world”
by H. Minkowski. Any point M(x
i ) in the four-dimensional space-time represents
an event, called world-point, and the motion of a particle in space-time is a curve
formed by a series of events, called world-line. In the Newtonian mechanics, the
motion of particles is described by r(t). Because of the absoluteness of the time t in
the Newtonian mechanics, the t is an invariable physical quantity, and thus it can be
used as a basic variable to describe the motion of particles. However, in the general
relativity, the t no longer has this property, and it is conceptually divided into proper
time and coordinate time.
The time τ of a clock moving with a particle is used as the basic quantity to describe
the motion of the particle. Commonly, the τ is known as the proper time, which
is the real record of the time interval between any two events by the same standard
clock in the gravitational field. The τ is an invariant, whose definition does not rely
on the physical process of coordinates, not changed with the change of coordinate
system. Considered that the clocks at rest are not synchronous in the gravitational
field, it is imagined that a reference clock placed in the local inertial frame flies
from the infinity to each point in the gravitation field to calibrate the frequencies of
these clocks, and then the time t recorded by the calibrated clocks is known as the
coordinate time. The coordinate time is related to the selected coordinates, and it is
a variable that changes with the coordinate system. However, in the same coordinate
system, each point in the gravitational field has only a coordinate time, which is a
measure of the time passing at the point.
The world-line of a particle moving in space-time is generally expressed as the
function of the proper time, that is
x
i
= x
i
(τ ).
(5.47)
In the reference frame (or the local inertial frame) which falls freely at point
M(x
i ), the space-time interval between any two world-points M(x
i ) and N(x
i +dx
i ) is
the invariant that has nothing to do with the reference frame, i.e.,
ds
2
= −c
2 d τ
2
= η ij dx
i dx
j
= −c
2 dt
2
+ δ ij dx
i dx
j
,
(5.48)
where ds is the space-time interval between two world-points, c is the velocity of light,
d τ is the proper time interval between two world-points, dt is the coordinate time
interval between two world-points, η ij is the four-dimensional Minkowski metric,
and δ ij is the three-dimensional Euclidean metric in Cartesian coordinate system, an
identity matrix with the order of 3 by 3.
In general, it is impossible that the coordinate transformation from the inertial
system to the non-inertial system is a linear transformation. Let x
i and x
j denote the
coordinates in the inertial and non-inertial systems, respectively, and thus according
to the generalized coordinate transformation, the differential forms of coordinate
5 X-ray Pulsar-Based Navigation: Theories and Experiments
5.4.2.3 The Space-Time Metric
The four-dimensional space-time is a set of one-dimensional time coordinate (x
0 )
and three-dimensional space coordinates (x
1 , x
2 , x
3 ), which is called the “world”
by H. Minkowski. Any point M(x
i ) in the four-dimensional space-time represents
an event, called world-point, and the motion of a particle in space-time is a curve
formed by a series of events, called world-line. In the Newtonian mechanics, the
motion of particles is described by r(t). Because of the absoluteness of the time t in
the Newtonian mechanics, the t is an invariable physical quantity, and thus it can be
used as a basic variable to describe the motion of particles. However, in the general
relativity, the t no longer has this property, and it is conceptually divided into proper
time and coordinate time.
The time τ of a clock moving with a particle is used as the basic quantity to describe
the motion of the particle. Commonly, the τ is known as the proper time, which
is the real record of the time interval between any two events by the same standard
clock in the gravitational field. The τ is an invariant, whose definition does not rely
on the physical process of coordinates, not changed with the change of coordinate
system. Considered that the clocks at rest are not synchronous in the gravitational
field, it is imagined that a reference clock placed in the local inertial frame flies
from the infinity to each point in the gravitation field to calibrate the frequencies of
these clocks, and then the time t recorded by the calibrated clocks is known as the
coordinate time. The coordinate time is related to the selected coordinates, and it is
a variable that changes with the coordinate system. However, in the same coordinate
system, each point in the gravitational field has only a coordinate time, which is a
measure of the time passing at the point.
The world-line of a particle moving in space-time is generally expressed as the
function of the proper time, that is
x
i
= x
i
(τ ).
(5.47)
In the reference frame (or the local inertial frame) which falls freely at point
M(x
i ), the space-time interval between any two world-points M(x
i ) and N(x
i +dx
i ) is
the invariant that has nothing to do with the reference frame, i.e.,
ds
2
= −c
2 d τ
2
= η ij dx
i dx
j
= −c
2 dt
2
+ δ ij dx
i dx
j
,
(5.48)
where ds is the space-time interval between two world-points, c is the velocity of light,
d τ is the proper time interval between two world-points, dt is the coordinate time
interval between two world-points, η ij is the four-dimensional Minkowski metric,
and δ ij is the three-dimensional Euclidean metric in Cartesian coordinate system, an
identity matrix with the order of 3 by 3.
In general, it is impossible that the coordinate transformation from the inertial
system to the non-inertial system is a linear transformation. Let x
i and x
j denote the
coordinates in the inertial and non-inertial systems, respectively, and thus according
to the generalized coordinate transformation, the differential forms of coordinate