Editing Classical field theory (section)

===Newtonian gravitation===
The first [[field theory (physics)|field theory]] of gravity was [[Newton's theory of gravitation]] in which the mutual interaction between two [[mass]]es obeys an [[inverse square law]]. This was very useful for predicting the motion of planets around the Sun.

Any massive body ''M'' has a [[gravitational field]] '''g''' which describes its influence on other massive bodies. The gravitational field of ''M'' at a point '''r''' in space is found by determining the force '''F''' that ''M'' exerts on a small [[test mass]] ''m'' located at '''r''', and then dividing by ''m'':<ref name="kleppner85">{{cite book|last1=Kleppner|first1=David|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|page=85}}</ref>
<math display="block"> \mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.</math>
Stipulating that ''m'' is much smaller than ''M'' ensures that the presence of ''m'' has a negligible influence on the behavior of ''M''.

According to [[Newton's law of universal gravitation]], '''F'''('''r''') is given by<ref name="kleppner85" />
<math display="block">\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},</math>
where <math>\hat{\mathbf{r}}</math> is a [[unit vector]] pointing along the line from ''M'' to ''m'', and ''G'' is Newton's [[gravitational constant]]. Therefore, the gravitational field of ''M'' is<ref name="kleppner85" />
<math display="block">\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.</math>

The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the [[equivalence principle]], which leads to [[general relativity]].

For a discrete collection of masses, ''M<sub>i</sub>'', located at points, '''r'''<sub>''i''</sub>, the gravitational field at a point '''r''' due to the masses is
<math display="block">\mathbf{g}(\mathbf{r})=-G\sum_i \frac{M_i(\mathbf{r}-\mathbf{r_i})}{|\mathbf{r}-\mathbf{r}_i|^3} \,, </math>

If we have a continuous mass distribution ''ρ'' instead, the sum is replaced by an integral,
<math display="block">\mathbf{g}(\mathbf{r})=-G \iiint_V \frac{\rho(\mathbf{x})d^3\mathbf{x}(\mathbf{r}-\mathbf{x})}{|\mathbf{r}-\mathbf{x}|^3} \, , </math>

Note that the direction of the field points from the position '''r''' to the position of the masses '''r'''<sub>''i''</sub>; this is ensured by the minus sign. In a nutshell, this means all masses attract.

In the integral form [[Gauss's law for gravity]] is
<math display="block">\iint\mathbf{g}\cdot d \mathbf{S} = -4\pi G M</math>
while in differential form it is
<math display="block">\nabla \cdot\mathbf{g} = -4\pi G\rho_m </math>

Therefore, the gravitational field '''g''' can be written in terms of the [[gradient]] of a [[gravitational potential]] {{math|''φ''('''r''')}}:
<math display="block">\mathbf{g}(\mathbf{r}) = -\nabla \phi(\mathbf{r}).</math>
This is a consequence of the gravitational force '''F''' being [[conservative field|conservative]].