This blog is straight old Newtonian gravity. A goal is to understand where the three kinds of masses in Newton's Universal Law of Gravity live in the context of the action. In the force equation, there is the inertial mass (mi )which sits next to the acceleration term. There is the active gravitational mass (big M) that is the source of the gravitational field. The third amigo is the passive gravitational mass (mp) that sits right next to the active gravitational mass.

What is an action? It is every way a system can exchange energy in a breadbox (per unit volume). Integrate that exchange over volume and time, and the result is a mass times a time. By itself, that is not so interesting because the amount of time one can integrate over is arbitrary. If one can find a symmetry that does not change the integral for all possible time intervals, then that symmetry will have a corresponding conserved quantity for all time. Symmetries and their conserved quantities are keys to understanding physics.

One can vary the action with respect to a potential. The extremum is the field equations. A solution should be found for the field equations.

The action can also be varied with respect to a position and look for an extremum. The path particles follow are these extremums of the action. The equations of motion are the result of this sort of analysis. The solution to the field equations gets plugged into the equations of motion.

It is easy to find a calculation where the action is used to derive the field equations. Likewise, one can find the road between the action and the equations of motion. What one does not often see is one action that does both. That is what I will do because it highlights the relationship between the three masses.

The Action for Newtonian Gravity

Equations of motion calculations are usually done with Lagrangians to follow the path of one test mass. Deriving field equations needs a Lagrange density, mass per unit volume, to avoid technical issues that arise from point sources (I don't understand that issue well, so if someone wishes to explain it better than I can, please feel free to leave a comment). It may be this mismatch that blocks writing one action. The Lagrange density is a more general approach, so that is what will be used. Without further ado here is the action for Newtonian gravity:

$S_{_{NG}} = \int \sqrt{-g} d^4 x \left( -\rho \sqrt{1 - \left(\frac{d x_u}{d t}\right)^2/c^2} - \rho~ \Phi(x) - \frac{c^2}{2 G} (\nabla \Phi)^2 \right)$

[Note: as LagrangiansForBreakfast pointed out in the comments, the first term has Lorentz symmetry, while the next two have Galilean symmetry. Such a mix is not a good thing. I will leave the blog as is, but now the reader is aware of the mismatch which has consequences.]

Let's identify the players:

S - a letter often used for actions, I am not sure why.

$\int \sqrt{-g} d^4 x$a 4D integral that would work even if space-time is curved [Note for advanced readers: I believe this factor should be included even if Newtonian gravity theory requires Galilean transformations.]

$\inline \rho$- one of the mass densities.

$\inline \sqrt{1 - \left(\frac{d x}{d t}\right)^2/c^2}$ - a relativistic velocity stretch factor.

$\inline \Phi(x)$ - a static scalar potential that depends on position.

$\inline \nabla \Phi$ - the gradient of the potential.

How much can you see in this action?

* Name a few symmetries and the corresponding conserved quantities.

* Name a common thing that is not conserved.

* Which terms will be part of the field equations?

* Which terms will be part of the equations of motion?

* Why are there 3 types of mass (inertial, active, and passive), yet only 2 mass densities?

If you can answer these five questions already, you have a fine education and body of experience.  The last issue stumped me for a while, driving me to write this blog.  All these questions will be answered here.

Dimensional Analysis

If I had a job at one of those centers for advanced studies, I would be using natural units, setting c = G = hbar = 1. Since that ain't gonna happen, I like writing out all the constants, checking that everything is consistent. It is a simple task that I am bright enough to do. Every term in the Lagrange density should have units of mass per volume so that when the density gets integrated over space-time the result is mass times time (it is more common to strive for units of energy times time, but I favor mass particularly in this context). Two out of three terms have a mass density, making any other terms next to a rho have no dimensions. The potential must be dimensionless which is not the case in most textbooks. The third term has the square of the gradient of a dimensionless potential. As such, that will bring in an inverse area. What that term needs are constants with units of mass per length. Newton's gravitational constant has units of a cubed distance over mass and time squared. Invert G, multiply by a c2, and the result is a distance per length, just what is needed. The units all work out, which I find comforting.

The Symmetries

There is no dependence on a time t in the action. Time is a symmetry of the action. The conserved quantity is energy. As expected, Newtonian gravity conserves energy. A fancy way of saying this is the action is homogeneous in time.

The action is not homogeneous in space. Different places in space have different strengths of the gravity field. What is not conserved is linear momentum. Both the apple falling off the tree and the Moon falling around the Earth do not conserve linear momentum.

[Note: the claim about not conserving linear momentum is wrong.  The apple does start out with no momentum, then gains some.  This is offset by the Earth moving up to the apple by an amount so small no one could ever see it.  As to why I made the blunder, it was because I wrote the potential term as Phi(x), knowing I would plug in the solution M/R so then the action would have an explicit dependence on position.]

There is no dependence on angles. Rotation is a symmetry of the action. The corresponding conserved quantity is angular momentum. This has been known since Kepler's Third Law about equal areas swept out by the motion of planets in equal amounts of time (I don't know who gets credit for connecting the equal area law to angular momentum). This is called the isotrophy of space.

The Field Equation

Derive the field equation by varying the action with respect to the potential and keeping the position and velocity fixed. When deriving the field equation, there is no test particle involved. The analysis is exclusively about the potential. Only the second and third terms of the action come into play. Use Euler-Lagrange:

\begin{align*} \frac{d \mathcal{L}}{d \Phi} &= \nabla \left( \frac{d \mathcal{L}}{d \nabla \Phi} \right) \\ \rho &= \frac{c^2}{G}\nabla^2 \Phi \end{align*}

The general [sic] solution is: [this is an uncharged, non-rotating, point source]

$\Phi = (-1)^{k_1}\frac{G M}{c^2 R} + k_2$

As I was writing this blog, I wondered if a negative mass over distance solution was valid. At first glance, it looks like a couple of derivatives of phi will bring down exactly two minus signs, so Phi must be positive. What was I forgetting? We are dealing with a source mass that is a point charge. That requires the Dirac delta function (really a distribution). It is the singular nature of the problem that allows k1 to be positive or negative because the distribution could be positive or negative when paired with the right constant. Let's keep both constants along for the ride.

Side note: the blog is now next to a deep subject, known as Green functions. I have not studied the subject enough to blog about it, but it has to do with Poisson's equation with various boundary conditions.

A key thing to note is the mass density rho is all about the active gravitational mass M. There is no place in this expression for either the inertial mass mi which is in the first term of the action, nor the passive gravitational mass mp.

The Equations of Motion

Derive the equations of motion by varying the action with respect to the position and keeping the potential and its derivative fixed. The potential was found in the previous section and contains the active gravitational mass M. Only the first two terms of the action are used in this application of Euler-Lagrange:

\begin{align*} \frac{d}{d t}\left(\frac{d \mathcal{L}}{d \left( \frac{d x_u}{d t}\right)}\right) &= \frac{\partial \mathcal{L}}{\partial x_u} \\ \rho_i \frac{d^2 x_u}{d t^2}/\left(c^2 \sqrt{1 - \left(\frac{d x_u}{d t}\right)^2/c^2}\right) &= - \rho_p \nabla \Phi \\ &= - \rho_p \nabla \left((-1)^{k_1} \frac{G M}{c^2 R} + k_2 \right) \\ &= + \rho_p (-1)^{k_1} \frac{G M}{c^2 R^2} \end{align*}

Gravity is an attractive law, so k1 must be an odd integer. I  count 5 minus signs coming into play. No wonder I struggle to get this right.

Assume the test charge is a point mass. I feel uneasy about the details of how to do this integration over a volume with the potential there. Please feel free to comment about the transition from the mass density to the mass.

Assume only slow speeds are involved. Write the equations of motion with the above assumptions:

$m_i \frac{d^2 x_u}{d t^2} + O\left(\frac{d x_u}{d t}\right) = - m_p \frac{G M}{R^2}$

I was a little surprised by the velocity correction factor. If I had only used ½mv2 in the first place, but I know that is the slow speed approximation. Might as well start things right and approximate at the end.

[Note: if all three terms in the action had Galilean symmetry, then the equations of motion would look exactly like Newton's Universal Law of Gravity. My "surprise" was generated from the symmetry mismatch in the initial action.]

Negative energy is a bad thing that physicists avoid in all cases except for the gravitational potential where after a bit of discussion the issue is dropped. There is a very simple solution: make the constant k2 = 1. The potential will be positive for all but the most super-dense volumes of space-time.

Conclusions

I hope you go back and look at the action like I have a few times. Now I can see what is conserved, what is not, what goes into the field equations, what goes into the equations of motion. I now recognize the twin roles the mass density plays in the current coupling term for the field versus equations of motion.

Might I do a variation on this action? Certainly, but in a future blog.

Note to commentors: please stick to the topic at hand, Newton's model for gravity written as an action. I reserve the right to delete comments and will do so after leaving my own explanation as to why.  If you have a great solution to a problem, congrats, blog about it on this website, just not in this particular blog.