TITLE: Lagrangian of Newtonian gravity
QUESTION [0 upvotes]: In this wiki page we can read:

The Lagrangian density for Newtonian gravity is:
$$\mathcal{L}(\mathbf{x},t)= - \rho (\mathbf{x},t) \Phi (\mathbf{x},t) - {1 \over 8 \pi G} (\nabla \Phi (\mathbf{x},t))^2,$$

while in this other wiki page we can see:

The simplest classical field is a real scalar field — a real number at
every point in space that changes in time. It is denoted as $ϕ(x, t)$,
where $x$ is the position vector, and $t$ is the time. Suppose the
Lagrangian of the field, $L$, is
$$L = \int d^3x\,\mathcal{L} = \int d^3x\,\left[\frac 12 \dot\phi^2 - \frac 12 (\nabla\phi)^2 - \frac 12 m^2\phi^2\right]. $$

I assume both expressions of Lagrangian density must agree(?). I can define $m=4 \pi G \rho$, but even after that there are two important differences:

term in $\dot{\phi}$ not present in first equation.

term in $\rho \phi$ of first equation seems to be $m^2 \phi^2$ in second one.


Must these two equations agree? if they must, how to pass from one to the other?
Related: Lagrangian potential for Newtonian gravity but curiously it talks about an expression not currently found in wiki page.

REPLY [3 votes]: These actions are logicaly different.
First Lagrangian describe nonrelativistic scalar field:
$$
\mathcal{L}(\mathbf{x},t)= - \rho (\mathbf{x},t) \Phi (\mathbf{x},t) - {1 \over 8 \pi G} (\nabla \Phi (\mathbf{x},t))^2
$$
Second Lagrangian describe relativistic scalar field:
$$
L = \int d^3x\,\mathcal{L} = \int d^3x\,\left[\frac 12 \dot\phi^2 - \frac 12 (\nabla\phi)^2 - \frac 12 m^2\phi^2\right]
$$
First action can be deduced not from second action, but as non-relativistic Newtonian limit of Einstein-Hilbert action coupled to matter if one will use metric in leading order (see for example David Tong: Lectures on General Relativity, section 5.1):
$$
ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi) \, d\mathbf{x} \cdot d\mathbf{x}
 $$

REPLY [3 votes]: The first Lagrangian density
$$\mathcal{L}(\mathbf{x},t)=
-\rho(\mathbf{x},t)\Phi(\mathbf{x},t)-\frac{1}{8\pi G}(\nabla\Phi(\mathbf{x},t))^2$$
describes the gravity potential $\Phi(\mathbf{x},t)$ (a scalar field)
in the presence of the density $\rho(\mathbf{x},t)$ (another scalar field).
The second Lagrangian density
(I have added the $(\mathbf{x},t)$ dependencies for more clarity)
$$\mathcal{L}(\mathbf{x},t)=
\frac{1}{2}\dot\phi(\mathbf{x},t)^2-\frac{1}{2}(\nabla\phi(\mathbf{x},t))^2 - \frac 12 m^2\phi(\mathbf{x},t)^2$$
describes a single particle by a scalar field $\phi(\mathbf{x},t)$.
Here $m$ is not a field, but just a constant number (the mass of the single particle).
These are two entirely different phenomena.
Hence there is no reason to expect any similarity between the two.
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