Chapter 7
Classical Gravity and Geometry
Abstract In this chapter we look at the familiar classical gravitational force from a
novel perspective, as a geometric effect. This perspective is motivated by the equivalence principle, the close similarity of gravitational effects to the effects of acceleration. As an application of the geometric view the gravitational redshift can be easily
derived.
7.1 Newtonian Gravity
Classical or Newtonian gravitational theory is well-known to almost all physicists,
so only a short review need be given here. For more detail see Chap. 1 of Ohanian
(1994) and Chap. 12 of Misner (1973). Our review is focused on the troubles with
the theory. The basic postulate is the inverse square law of Newton, in which the
force of attraction between point masses M and m separated by distance r is given
by
F = −
G Mm
r 2 ˆ
r , G = 6.672 × 10
−11 N m
2
/kg, Newtonian gravity.
(7.1)
Notice how similar this is to the Coulomb force law of electrostatics for charges q
and Q,
F =
Qq
4πε o r 2 ˆ
r ,
1
4πε o
= 8.99 × 10
9 N m
2
/C
2
, electrostatics.
(7.2)
The main difference is that in electrostatics the charges Q and q can have either sign
and the force may thus be attractive or repulsive. One may thus develop classical
gravitational theory in close analogy with electrostatics, using the correspondence
mass ↔ charge and G ↔ 1/4πε o . Only the signs require some care. For example,
we define a gravitational vector field
g by
F = m
g, so for a point mass
g = −
G M
r 2 ˆ
r .
(7.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_7
95
Classical Gravity and Geometry
Abstract In this chapter we look at the familiar classical gravitational force from a
novel perspective, as a geometric effect. This perspective is motivated by the equivalence principle, the close similarity of gravitational effects to the effects of acceleration. As an application of the geometric view the gravitational redshift can be easily
derived.
7.1 Newtonian Gravity
Classical or Newtonian gravitational theory is well-known to almost all physicists,
so only a short review need be given here. For more detail see Chap. 1 of Ohanian
(1994) and Chap. 12 of Misner (1973). Our review is focused on the troubles with
the theory. The basic postulate is the inverse square law of Newton, in which the
force of attraction between point masses M and m separated by distance r is given
by
F = −
G Mm
r 2 ˆ
r , G = 6.672 × 10
−11 N m
2
/kg, Newtonian gravity.
(7.1)
Notice how similar this is to the Coulomb force law of electrostatics for charges q
and Q,
F =
4πε o r 2 ˆ
r ,
1
4πε o
= 8.99 × 10
9 N m
2
/C
2
, electrostatics.
(7.2)
The main difference is that in electrostatics the charges Q and q can have either sign
and the force may thus be attractive or repulsive. One may thus develop classical
gravitational theory in close analogy with electrostatics, using the correspondence
mass ↔ charge and G ↔ 1/4πε o . Only the signs require some care. For example,
we define a gravitational vector field
g by
F = m
g, so for a point mass
g = −
G M
r 2 ˆ
r .
(7.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_7
95