How do you actually use fields?

Question

Note: I'm probably using the wrong letters/notation here. I apologize. I use $omega$ to represent an object, and $mathcal{U}$ is the universe. I'm not sure how else to do it. $m(omega)$ and $x(omega)$ are the mass and position of the object, respectively.
I've stumbled upon a roadblock in my understanding of fields. Namely, it seems to me as if there's a bit of a problem with them.
Take, for instance, the gravitational field $mathbf{G}$.
At every point $x$, multiplying it (in $mathrm{N} over mathrm{kg}$) by the mass of an object gives the force that that object would experience at that point ($F = mathbf{G}_{x(omega)} cdot m(omega)$).   Since $mathbf{G}_{x} = sum_{omega in mathcal{U}}G {m(omega) over {left| x - x(omega) right|}^{2}}$, the value at any point occupied by an object $omega_{1}$  is undefined due to division by zero.
But the force experienced by $omega_{1}$ should be $m(omega_{1}) cdot mathbf{G}_{x(omega_{1})}$!
Of course, assuming non-pointlike objects resolves this. But larger objects are made up out of pointlike objects. Unless you want to use quantum mechanics here, but AFAIK there's no quantum theory of gravity yet (?).
But, for the example of gravity, this is probably resolved by general relativity.
But this problem seems to be a problem in general for fields that increase without a limit as distance to an object gets smaller.
Of course, you could just say "The field for each object is only defined by the other objects", but then, why even have the field? Why not just say that the force on an object is defined this way? Defining a field, especially one that has values elsewhere than objects actually exist, seems unnecessary. Especially since you'd need to have multiple fields / one hyperdimensional field.
How is this resolved?

FGSUZ · Answer

A mass $M$ creates a field everywhere except one point: the place where it is. If only one point fails, I don't think it is dramatic enough to blow down all the wonderful phenomena that can be perfectly explained by fields.

Answered by FGSUZ on February 5, 2021

J. Murray · Answer

Point masses and point charges are inherently singular objects for exactly the reason you state.  They are extremely useful tools for modeling purposes, and that's precisely how you should think of them.
In a classical field theory context, the correct way to understand them is in the limiting sense.  To rigorously define how a point mass will behave, you can replace it with e.g. a continuous sphere with volume $V$ and constant mass density $rho = M/V$, and then take the limit as $Vrightarrow 0$ while holding $M$ constant.
This limiting process often times leads to great simplifications.  For instance, a continuous sphere in a non-uniform gravitational field will experience tidal forces and elastic deformation.  In the limit as $Vrightarrow 0$, these complications disappear, and the gravitational force on the ball converges to some nice average value.  Additionally, extended objects experience torques about their centers, so you need to consider rotational motion as well; if we take the limit and consider only point masses, then this rotational degree of freedom can be ignored as well.
On the other hand, if we ask the wrong questions then this limiting process does not give us meaningful answers.  If we ask for the gravitational field at the surface of our continuous sphere, we get a perfectly finite answer, but in the limit as $Vrightarrow 0$ this quantity diverges.
Field theories such as Newtonian gravity, GR, and electromagnetism  are really only well-behaved when the sources of those fields (mass, energy/momentum, and charge/current) are treated as continuous distributions.  As soon as point sources are introduced, we have to be careful with what questions we ask, or we risk running into spurious infinities which ultimately arise because we're taking an ill-defined limit without realizing it.

Defining a field, especially one that has values elsewhere than objects actually exist, seems unnecessary.

It turns out that those fields actually carry energy and momentum through the empty space between objects, so they are quite necessary unless you are not troubled by energy and momentum vanishing from one region and subsequently reappearing at a later time in another region.
Beyond this, watching small particles respond to an invisible influence in ostensibly empty space makes a compelling case for adopting the field viewpoint.

How do you actually use fields?

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