5.4 Space-Time Reference Based on General Relativity
307
where x
μ
= (ct, x, y, z).
In order to distinguish in writing conveniently, it is ruled in this book that, under
the general theory of relativity, the superscripts or subscripts for three-dimensional
space are marked with English letters like i, j, k and l, with the values of 1, 2 and 3;
the indicators for four-dimensional space-time are marked with Greek letters like α,
β, μ and ν, with the values of 0, 1, 2 and 3.
The non-inertial system is introduced by a generalized coordinate transformation,
and its four-dimensional space-time coordinate is expressed as
x
μ
= x
μ
x
ν
.
(5.44)
Correspondingly, its inverse transformation is
x
ν
= x
ν
(x
μ
).
(5.45)
Therefore, the dynamical equation that a particle moves freely can be expressed
as
d
2 x
μ
ds 2 = −
μ
να
dx
ν
ds
dx
α
ds
,
(5.46)
where
μ
να =
∂
2 x
μ
∂x ν ∂x α
∂x
μ
∂x μ + β
ν
ν β
α
α β
μ
μ
μ
ν α .
In Eq. (5.46), its left side denotes the particle’s acceleration, and its right side
denotes the gravitational force of the particle per a unit of mass. It can be seen
that the strength of inertial force field is represented by the Riemannian connection.
According to the principle of equivalence, both the gravitation field and inertial force
have the same status in physical laws, so the strength of gravitational field is usually
described by the connection of space, which in turn describes the geometric structure
of space. The basic idea of expressing the gravity through space geometry is called
the gravity as geometry.
When the strength of gravitational field is described by the Riemannian connection, which is the gravitational acceleration, then the metric tensor is equivalent to
the gravitational potential. In Newton’s theory of gravity, the gravitational potential
is a scalar field, and in the general relativity, it is a second-order symmetric tensor.
Supposed that the space-time is flat, a set of Mikowski coordinates whose connection
coefficients are constantly equal to zero can always be found, so that the effects of the
gravitational field are completely disappeared. In fact, the real physical space-time
is a curved Riemannian space, and the curvature tensor must not be zero. As a result,
it is impossible to eliminate all gravitational effects. Certainly, according to the principle of equivalence, it is possible to eliminate the connection of a point in bending
space-time. It means that within an infinitesimal neighborhood of any space-time
point, the gravitational effects can be approximately eliminated to establish a local
inertial frame.
307
where x
μ
= (ct, x, y, z).
In order to distinguish in writing conveniently, it is ruled in this book that, under
the general theory of relativity, the superscripts or subscripts for three-dimensional
space are marked with English letters like i, j, k and l, with the values of 1, 2 and 3;
the indicators for four-dimensional space-time are marked with Greek letters like α,
β, μ and ν, with the values of 0, 1, 2 and 3.
The non-inertial system is introduced by a generalized coordinate transformation,
and its four-dimensional space-time coordinate is expressed as
x
μ
= x
μ
x
ν
.
(5.44)
Correspondingly, its inverse transformation is
x
ν
= x
ν
(x
μ
).
(5.45)
Therefore, the dynamical equation that a particle moves freely can be expressed
as
d
2 x
μ
ds 2 = −
μ
να
dx
ν
ds
dx
α
ds
,
(5.46)
where
μ
να =
∂
2 x
μ
∂x ν ∂x α
∂x
μ
∂x μ + β
ν
ν β
α
α β
μ
μ
μ
ν α .
In Eq. (5.46), its left side denotes the particle’s acceleration, and its right side
denotes the gravitational force of the particle per a unit of mass. It can be seen
that the strength of inertial force field is represented by the Riemannian connection.
According to the principle of equivalence, both the gravitation field and inertial force
have the same status in physical laws, so the strength of gravitational field is usually
described by the connection of space, which in turn describes the geometric structure
of space. The basic idea of expressing the gravity through space geometry is called
the gravity as geometry.
When the strength of gravitational field is described by the Riemannian connection, which is the gravitational acceleration, then the metric tensor is equivalent to
the gravitational potential. In Newton’s theory of gravity, the gravitational potential
is a scalar field, and in the general relativity, it is a second-order symmetric tensor.
Supposed that the space-time is flat, a set of Mikowski coordinates whose connection
coefficients are constantly equal to zero can always be found, so that the effects of the
gravitational field are completely disappeared. In fact, the real physical space-time
is a curved Riemannian space, and the curvature tensor must not be zero. As a result,
it is impossible to eliminate all gravitational effects. Certainly, according to the principle of equivalence, it is possible to eliminate the connection of a point in bending
space-time. It means that within an infinitesimal neighborhood of any space-time
point, the gravitational effects can be approximately eliminated to establish a local
inertial frame.