PrimerField Gravity — Paper II

Superposition, Stress Tensor, and Energy Bookkeeping
Plain English Companion

David Allen LaPoint

PrimerField Foundation

January 17, 2026

What This Paper Is About

In Paper I, we showed that gravity in PrimerField Theory comes from differences in energy density across space—like how water flows from high pressure to low pressure. That paper focused on single objects, like one planet or star. This second paper asks: what happens when you have multiple objects? How do their gravitational effects combine? And how do we keep track of the energy and momentum involved?

This paper only deals with objects that aren't moving (or moving very slowly). How things change over time, how energy flows from place to place, and questions about inertia are saved for Paper III.

1. What We're Trying to Do

This paper extends the gravity framework from Paper I to handle multiple objects at once. We're specifically limited to static systems—things that are sitting still or changing very slowly.

We'll cover three main topics: (1) how gravitational effects from multiple sources add together, (2) a mathematical tool called a "stress tensor" that helps us track momentum, and (3) how much energy is stored in a gravitational field.

What we're NOT covering here: anything involving time (like how fields change or energy moves around), anything about inertia (why objects resist being pushed), and any comparison to Einstein's General Relativity. All of that comes later.

2. Quick Review of Paper I

Paper I established that gravity comes from gradients (slopes) in field energy. Think of it like a hillside: objects "roll" toward regions of lower energy, and this rolling is what we experience as gravitational attraction.

The key equation from Paper I says that the gravitational potential (a number that tells you the energy state at each point in space) follows from the distribution of field energy. The connection between them involves a coupling constant κ (kappa), which equals 4πG/c². This links Newton's gravitational constant G to the speed of light c. The c² appears here as a unit-conversion factor that connects energy density to gravitational coupling—it comes from E = mc², not from any claim about spacetime geometry.

Another important result: field energy density and mass are related through Einstein's famous formula E = mc². Wherever you have energy, you have equivalent mass, and that mass acts as a source of gravity.

3. Adding Up Gravitational Effects

Here's good news: gravitational effects from multiple sources simply add together. This is called "superposition." If you have the Earth pulling you down and the Moon pulling you sideways, the total effect is just the sum of both pulls.

Mathematically, if you have many energy sources, you calculate the gravitational potential from each one separately, then add all those potentials together to get the total. The same goes for acceleration: the total gravitational acceleration at any point is the vector sum (adding with direction) of the accelerations from each source.

This matches what Newton figured out centuries ago—no new physics here. We're just confirming that the PF framework gives the same answers as classical gravity for multi-body systems.

Important caveat: This simple addition only works when we treat each energy source independently. If the gravitational field itself contains significant energy that acts as a source (the field generating more field), then the math becomes nonlinear and superposition breaks down. For now, we're ignoring this complication.

4. How Fields Create Forces on Matter

To connect our field description to actual forces on objects, we need a rule. Here's what we propose: the gravitational force on matter at any location equals the matter density times the slope of the gravitational potential.

Think of it this way: if you're standing on a hill, the steeper the slope, the harder gravity pulls you downhill. And the heavier you are (more mass density), the more force you feel. This rule captures both effects.

We're being honest: this is an assumption (what scientists call a "postulate"), not something we derived from deeper principles. It seems reasonable and gives the right answers, but proving it from first principles is work for the future.

When you add up this force over an entire object, you get F = mg for the total force—exactly what Newton said. The mass m is the total matter in the object, and g is the local gravitational acceleration.

5. The Stress Tensor: Tracking Momentum in the Field

A "stress tensor" is a mathematical object that keeps track of momentum and pressure in a system. Think of it like an advanced accounting ledger that tracks not just how much momentum is present, but which direction it's flowing and how it's distributed.

Electromagnetic fields have stress tensors (that's how we calculate radiation pressure and other effects). We're proposing that gravitational fields have them too. The formula looks similar to the electromagnetic version, which isn't surprising—both involve fields that spread through space. The similarity is in mathematical role, not an assumption that gravitational and electromagnetic fields are physically identical.

What does this stress tensor tell us? It describes how momentum flows through the gravitational field configuration. When an object feels a gravitational force, momentum transfers from the field to the object. The stress tensor is the bookkeeping tool that ensures momentum is conserved—what the field loses, the object gains.

In the Appendix, we verify mathematically that this stress tensor correctly accounts for the forces from Section 4. The accounting works out perfectly: momentum lost by matter equals momentum gained by the field (or vice versa).

6. Energy Stored in the Gravitational Field

Gravitational fields contain energy. We define this energy density as proportional to the square of the field gradient—the steeper the slope of the potential, the more energy is stored in that region.

This definition is always positive: wherever there's a gravitational field (wherever the potential is changing), there's positive energy stored there. This might seem odd if you've heard that gravitational binding energy is negative. Both are true—they're talking about different things.

The field energy we're defining is like the energy stored in a stretched spring—the spring configuration itself contains energy. Binding energy (which is negative) is about how much energy you'd need to pull a system apart. You have to be careful not to count the same energy twice when doing calculations.

How this energy moves around over time, and whether there's a gravitational equivalent of electromagnetic radiation, are questions we defer to Paper III.

7. A Hint About Inertia and Gravity

Here's an intriguing possibility that we're not proving in this paper, just mentioning: in PF theory, both gravity (the pull between masses) and inertia (why things resist being pushed) might arise from the same underlying field structure.

Why does a kilogram of mass pull other masses with a certain force AND resist acceleration by exactly the right amount to make inertial mass equal gravitational mass? In Newton's physics, this is a coincidence. In Einstein's General Relativity, it's built into the geometry of spacetime. In PF theory, it might follow from both effects arising from the same field-energy coupling.

This is just a hypothesis at this point—no mechanism is proposed here, only a directional possibility. Making it mathematically precise requires work we haven't done yet, including understanding what happens when things accelerate and how fields resist being reconfigured. That's Paper III material.

8. How This Differs From Other Theories

PrimerField gravity takes a different conceptual approach than Einstein's General Relativity. In GR, gravity IS the curvature of spacetime—mass tells spacetime how to curve, and curved spacetime tells mass how to move. Space and time themselves are the gravitational field.

In PF theory, gravity is energy gradients in a field that exists in flat (uncurved) spacetime. It's conceptually closer to how we think about electric and magnetic fields—they're something that exists in space, not a property of space itself.

Are these approaches compatible? Do they give the same predictions? Those are important questions we're NOT answering in this paper. Proper comparison requires calculating specific predictions (like light bending around the sun, or time running differently in gravitational fields) in both frameworks and checking if they agree. That's future work.

9. Summary

This paper extended PF gravity to handle multiple objects. The key results are: gravitational effects from multiple sources add together (superposition holds for static systems); we proposed a mathematical tool (stress tensor) for tracking momentum in gravitational fields; we defined how much energy is stored in a gravitational field; and we noted that inertia and gravity might have a common origin in PF theory.

We've been careful to label what's proven versus assumed. The superposition result follows mathematically from Paper I. The force rule and stress tensor are proposals (postulates) that give sensible results but haven't been derived from deeper principles yet. The inertia connection is just a hypothesis.

10. What Comes Next

Paper III will tackle everything involving time and motion: how gravitational fields change, how energy flows through them, trying to derive our force rule from basic principles, and making the inertia connection mathematically rigorous.

Later papers will address edge cases (what happens at boundaries?), stability (do these configurations hold together?), strong gravity (do things change when fields get intense?), and the big question: do PF predictions match General Relativity where GR has been tested?

Appendix: Why the Stress Tensor Works

The technical paper includes a mathematical verification that the proposed stress tensor correctly relates to the force rule. Here's what it shows in plain terms:

When you take the divergence of the stress tensor (a mathematical operation that measures how the tensor changes from point to point), you get exactly the negative of the force density from Section 4. This means the stress tensor perfectly accounts for momentum transfer.

The physical meaning: when matter feels a gravitational force, it gains momentum. That momentum has to come from somewhere—it comes from the gravitational field. The stress tensor is what tracks this exchange, ensuring that total momentum (matter plus field) is always conserved.

This kind of verification is important because it shows our mathematical framework is internally consistent. The pieces fit together properly, even if some of them are assumptions rather than derivations.