5.3 Space-Time Reference Based on Newtonian Mechanics
297
sum of the gravitational forces acting on the object with the mass of m, resulting
from the objects with the mass of m 1 , m 2 , . . . , m n , respectively.
The gravity is considered as a conservative force. At the position vector r, if the
strength g(r) of the gravitational field is defined as the gravitational force acting on
the object with a unit of mass, then
g(r) =
F
m
= −∇(r),
(5.11)
where (r) = −G
n
i=1
m i
r−r i , known as gravitational potential;
∇ =
∂
∂x
i +
∂
∂y
j +
∂
∂z
k, known as Hamiltonian operator.
For continuously distributed matter, the gravitational potential function is
(r) = −G
ρ
r
r − r
d
3 r
,
(5.12)
where ρ
r
is the matter density at position vector r
.
By using Gauss’s theorem, it can be proved that the differential expression of
Eq. (5.12) satisfies Poisson’s equation, namely,
= ∇
2
= 4π Gρ,
(5.13)
where = ∇
2
=
∂
2
∂x 2 +
∂
2
∂y 2 +
∂
2
∂z 2 , known as Laplace operator.
For the gravitational potential of a celestial body like the Earth, its interior satisfies
Poisson’s equation, while its exterior space does Laplace’s equation. Conceptually
speaking, the inertial mass and gravitational mass are essentially two different physical quantities. If the ratio of both is the same for all substances, then it can be
treated as the same quantity in practice. It is the problem of the equivalence between
the inertial mass and the gravitational mass. For this reason, Isaac Newton (1687),
Roland Eötvös (1889), Robert H. Dicke (1964) and Vladimir Braginsky (1971), et al.,
successively designed the special experiments, to measure the small deviation of the
ratio of the inertial mass to the gravitational mass relative to 1. Generally, when the
measurement accuracy is on the level of 10
−12 , the inertial is equal to the gravitational mass. In other words, the ratio of the inertial mass to the gravitational mass has
nothing to do with the size and material of the object, and the two are equal within
the range of experimental accuracy, for which the reason needs further theoretical
research to explain.
For an object, if only gravity acts on it, substituting formula (5.10) into formula
(5.9), then the physical quantity m used to describe the inherent physical properties
of the object will be eliminated. That is to say, no matter what the nature of the object
itself is, as long as the initial velocity is same, any object will move along the same
trajectory. So, there is such question about the Newtonian mechanics: the inertial
mass m of an object was originally introduced as an important dynamics quantity,
297
sum of the gravitational forces acting on the object with the mass of m, resulting
from the objects with the mass of m 1 , m 2 , . . . , m n , respectively.
The gravity is considered as a conservative force. At the position vector r, if the
strength g(r) of the gravitational field is defined as the gravitational force acting on
the object with a unit of mass, then
g(r) =
F
m
= −∇(r),
(5.11)
where (r) = −G
n
i=1
m i
r−r i , known as gravitational potential;
∇ =
∂
∂x
i +
∂
∂y
j +
∂
∂z
k, known as Hamiltonian operator.
For continuously distributed matter, the gravitational potential function is
(r) = −G
ρ
r
r − r
d
3 r
,
(5.12)
where ρ
r
is the matter density at position vector r
.
By using Gauss’s theorem, it can be proved that the differential expression of
Eq. (5.12) satisfies Poisson’s equation, namely,
= ∇
2
= 4π Gρ,
(5.13)
where = ∇
2
=
∂
2
∂x 2 +
∂
2
∂y 2 +
∂
2
∂z 2 , known as Laplace operator.
For the gravitational potential of a celestial body like the Earth, its interior satisfies
Poisson’s equation, while its exterior space does Laplace’s equation. Conceptually
speaking, the inertial mass and gravitational mass are essentially two different physical quantities. If the ratio of both is the same for all substances, then it can be
treated as the same quantity in practice. It is the problem of the equivalence between
the inertial mass and the gravitational mass. For this reason, Isaac Newton (1687),
Roland Eötvös (1889), Robert H. Dicke (1964) and Vladimir Braginsky (1971), et al.,
successively designed the special experiments, to measure the small deviation of the
ratio of the inertial mass to the gravitational mass relative to 1. Generally, when the
measurement accuracy is on the level of 10
−12 , the inertial is equal to the gravitational mass. In other words, the ratio of the inertial mass to the gravitational mass has
nothing to do with the size and material of the object, and the two are equal within
the range of experimental accuracy, for which the reason needs further theoretical
research to explain.
For an object, if only gravity acts on it, substituting formula (5.10) into formula
(5.9), then the physical quantity m used to describe the inherent physical properties
of the object will be eliminated. That is to say, no matter what the nature of the object
itself is, as long as the initial velocity is same, any object will move along the same
trajectory. So, there is such question about the Newtonian mechanics: the inertial
mass m of an object was originally introduced as an important dynamics quantity,