96
7 Classical Gravity and Geometry
A gravitational potential φ is defined by
g = −∇φ and a gravitational potential
energy by V = mφ, so for a point mass
φ = −
G M
r
, V = −
G Mm
r
.
(7.4)
For a continuous distribution of matter we may superpose point masses with a mass
density function ρ and find
φ( r ) = −G
ρ
r
| r − −
r |
d
3 r
.
(7.5)
The last expression may be used to obtain Poisson’s equation for the potential,
∇
2
φ = 4π Gρ.
(7.6)
Alternatively, we may postulate Poisson’s equation and obtain the force laws, just as
in electrostatics, and develop the whole theory on that basis.
Newtonian gravitational theory is extraordinarily accurate, and for over 200 years
was used to study the solar system with no known errors. Despite the success in
predicting empirical observations there are two defects with the theory which led to
its abandonment and the adoption of Einstein’s relativistic theory of gravity. These
are:
(1) Classical gravity is instantaneous: the distance in (7.1) is the relative separation
of the masses when the mass m feels the force exerted by M. But special relativity
is not compatible with such action-at-a-distance or instantaneous propagation,
as we will discuss. Of course, this was only seen to be a defect after special
relativity was developed in 1905.
(2) The masses in (7.1) are the same as the inertial masses: these are defined in
terms of resistance to acceleration via Newton’s second law,
F = m
a. Why
should the same inertial mass produce a gravitational field? The analogy with
electrostatics is useful here; the charge of a particle which produces the electric
field is independent of the inertial mass of the particle, so why should the
“gravitational mass” of the particle which produces the gravitational field be
the same as the inertial mass of the particle?
Notice that defect (1) involves a measurably real physics problem, while (2) only
involves a conceptual quandry, that an important and fundamental equality is not
explained by the theory but merely postulated.
Let us look at defect (1) a little further in the light of special relativity. We may
set up a thought experiment or gedanken experiment, as Einstein was fond of doing.
Suppose we are at the position of m, and a colleague wiggles the mass M; according
to (7.1) we would see the effect immediately, so the force propagates at infinite
velocity over the distance r = x in time t = 0 as in Fig. 7.1.
7 Classical Gravity and Geometry
A gravitational potential φ is defined by
g = −∇φ and a gravitational potential
energy by V = mφ, so for a point mass
φ = −
G M
r
, V = −
G Mm
r
.
(7.4)
For a continuous distribution of matter we may superpose point masses with a mass
density function ρ and find
φ( r ) = −G
ρ
r
| r − −
r |
d
3 r
.
(7.5)
The last expression may be used to obtain Poisson’s equation for the potential,
∇
2
φ = 4π Gρ.
(7.6)
Alternatively, we may postulate Poisson’s equation and obtain the force laws, just as
in electrostatics, and develop the whole theory on that basis.
Newtonian gravitational theory is extraordinarily accurate, and for over 200 years
was used to study the solar system with no known errors. Despite the success in
predicting empirical observations there are two defects with the theory which led to
its abandonment and the adoption of Einstein’s relativistic theory of gravity. These
are:
(1) Classical gravity is instantaneous: the distance in (7.1) is the relative separation
of the masses when the mass m feels the force exerted by M. But special relativity
is not compatible with such action-at-a-distance or instantaneous propagation,
as we will discuss. Of course, this was only seen to be a defect after special
relativity was developed in 1905.
(2) The masses in (7.1) are the same as the inertial masses: these are defined in
terms of resistance to acceleration via Newton’s second law,
F = m
a. Why
should the same inertial mass produce a gravitational field? The analogy with
electrostatics is useful here; the charge of a particle which produces the electric
field is independent of the inertial mass of the particle, so why should the
“gravitational mass” of the particle which produces the gravitational field be
the same as the inertial mass of the particle?
Notice that defect (1) involves a measurably real physics problem, while (2) only
involves a conceptual quandry, that an important and fundamental equality is not
explained by the theory but merely postulated.
Let us look at defect (1) a little further in the light of special relativity. We may
set up a thought experiment or gedanken experiment, as Einstein was fond of doing.
Suppose we are at the position of m, and a colleague wiggles the mass M; according
to (7.1) we would see the effect immediately, so the force propagates at infinite
velocity over the distance r = x in time t = 0 as in Fig. 7.1.