102
7 Classical Gravity and Geometry
There has been a great deal of both theoretical and experimental work done on the
equivalence principle and a variety of versions have been discussed, most notably the
weak equivalence principle and the Einstein equivalence principle that we discussed
above. We will use here only the most basic, the universality of free fall, or weak
equivalence principle. For a more detailed discussion of the various statements of
the principle and relevant experiments on this important topic see Will (1993, 2014).
7.3 Gravity as a Geometric Phenomenon
As sketched above Newtonian gravitational theory is based on distances in 3dimensional space and an absolute universal time. It is therefore quite remarkable
that the conceptual framework of Chap. 5 for affine and metric spaces combined with
some basic ideas of special relativity leads to classical gravity as an approximation
for slow motion in weak fields.
Let us first note how the concept of vector parallel displacement can be naturally related to the concept of a classical force. Apply the basic vector displacement
expression (5.5) to the 4-vector velocity u
β
= dx
β
/dτ of a body in the spacetime of
special relativity, and consider vector transplantation in the time direction by cdt for
low velocity u
i
u
0 and u
0 ∼ = c. This gives for the approximate change in a space
component of the velocity,
du
j ∼ = −
j
00 c
2 dt,
du
j
dt
=
d
2 x
j
dt 2 = a
j ∼ = −
j
00 c
2
.
(7.13)
This is just Newton’s second law
F = m
a, where the affine connection
j
00 plays
the role of a force per unit mass. Notice that it is important that the transplantation is
done in the 4-dimensional spacetime of relativity rather than the 3-space of classical
physics, and also note that the mass of the body does not explicitly appear so the EP
is implied.
Let us pursue this geometric viewpoint further, but more precisely and explicitly.
We saw in the appendix on classical mechanics in Chap. 5 that fictitious forces are
represented by connections in the equations of motion (5.65). Recall that such forces
are called fictitious because they are proportional to the mass of the test body and
may be transformed away by a different choice of lab system or coordinates. But we
have just seen that the force of gravity also is proportional to the mass of the test
body and may be transformed away by a different choice of lab system. It is natural
that we then try to represent gravity as a fictitious force as in (5.65) (Adler 1975; Zee
1989).
Here are the four rules for this analysis:
(1) We use the Lorentz metric of special relativity, but modify it a small amount
g μν = η μν + h μν , h μν 1.
(7.14)
7 Classical Gravity and Geometry
There has been a great deal of both theoretical and experimental work done on the
equivalence principle and a variety of versions have been discussed, most notably the
weak equivalence principle and the Einstein equivalence principle that we discussed
above. We will use here only the most basic, the universality of free fall, or weak
equivalence principle. For a more detailed discussion of the various statements of
the principle and relevant experiments on this important topic see Will (1993, 2014).
7.3 Gravity as a Geometric Phenomenon
As sketched above Newtonian gravitational theory is based on distances in 3dimensional space and an absolute universal time. It is therefore quite remarkable
that the conceptual framework of Chap. 5 for affine and metric spaces combined with
some basic ideas of special relativity leads to classical gravity as an approximation
for slow motion in weak fields.
Let us first note how the concept of vector parallel displacement can be naturally related to the concept of a classical force. Apply the basic vector displacement
expression (5.5) to the 4-vector velocity u
β
= dx
β
/dτ of a body in the spacetime of
special relativity, and consider vector transplantation in the time direction by cdt for
low velocity u
i
u
0 and u
0 ∼ = c. This gives for the approximate change in a space
component of the velocity,
du
j ∼ = −
j
00 c
2 dt,
du
j
dt
=
d
2 x
j
dt 2 = a
j ∼ = −
j
00 c
2
.
(7.13)
This is just Newton’s second law
F = m
a, where the affine connection
j
00 plays
the role of a force per unit mass. Notice that it is important that the transplantation is
done in the 4-dimensional spacetime of relativity rather than the 3-space of classical
physics, and also note that the mass of the body does not explicitly appear so the EP
is implied.
Let us pursue this geometric viewpoint further, but more precisely and explicitly.
We saw in the appendix on classical mechanics in Chap. 5 that fictitious forces are
represented by connections in the equations of motion (5.65). Recall that such forces
are called fictitious because they are proportional to the mass of the test body and
may be transformed away by a different choice of lab system or coordinates. But we
have just seen that the force of gravity also is proportional to the mass of the test
body and may be transformed away by a different choice of lab system. It is natural
that we then try to represent gravity as a fictitious force as in (5.65) (Adler 1975; Zee
1989).
Here are the four rules for this analysis:
(1) We use the Lorentz metric of special relativity, but modify it a small amount
g μν = η μν + h μν , h μν 1.
(7.14)