Reference – Quantum Traction Theory: https://doi.org/10.5281/zenodo.17594186
Einstein’s famous equation \[ E = mc^2 \] usually arrives hand‑in‑hand with special relativity: Lorentz transformations, Minkowski spacetime, and thought experiments with fast‑moving boxes and light beams. In Quantum Traction Theory (QTT), something unusual happens:
the same relation drops out of a purely non‑relativistic, Planck‑scale endurance ledger.
No Lorentz group. No postulate about the speed of light being the same for all inertial observers. Just:
- a minimal space quantum,
- a minimal time tick,
- a minimal four‑volume cell, and
- the way those cells feed the endurance current that we perceive as gravity.
This post walks through that ledger and shows where the genuinely new step lives.
1. How E = mc² is usually derived (very briefly)
In standard physics, the mass–energy relation is tied to special relativity. You take:
- the relativistic energy–momentum relation \[ E^2 = (pc)^2 + (mc^2)^2, \]
- look at the case \(p=0\) (particle at rest),
- and you read off the rest energy \[ E_{\rm rest} = mc^2. \]
All of this leans heavily on Lorentz symmetry and the structure of Minkowski spacetime. In that story, \(E=mc^2\) is a relativistic identity.
QTT, instead, starts somewhere very different.
2. QTT’s micro-geometry: space quanta and four-cells
QTT says: space isn’t smooth, it’s built from tiny “pixelletes”—space quanta (SQs). The key quantities are:
- A minimal 3D space quantum: \[ V_{\rm SQ} = 4\pi\,\ell_P^3, \] with \(\ell_P\) the Planck length.
- A microscopic length \(\tilde\ell\) that sets a four-dimensional cell: \[ V_4 = 4\pi\,\tilde\ell^4. \]
- A microscopic tick \[ \tilde t := \frac{\tilde\ell}{c}, \] the time it takes light to cross that cell along the hidden reality direction in QTT’s time plane.
Think of \(V_4\) as the smallest “endurance cell” in which space, time, and reality (\(w\)-direction) are bundled together.
3. Minimal energy bundle: E★ from the four-cell
QTT’s Unified Equilibrium Law (UEL) assigns a minimal energy to a four‑volume cell of thickness \(\tilde t\). By a Planck‑scale uncertainty argument, the smallest energy that can sit coherently on a cell of temporal extent \(\tilde t\) is of order
[
E_\ast \sim \frac{\hbar}{\tilde t}.
]
Using \(\tilde t = \tilde\ell/c\), that becomes
\[ E_\ast = \frac{\hbar c}{\tilde\ell}. \]
This is the QTT energy quantum: the energy carried by a single endurance four‑cell of size \(V_4 = 4\pi\tilde\ell^4\).
- \(E_\ast\) is fixed by the cell size \(\tilde\ell\) and fundamental constants \(\hbar, c\).
- No relativity assumptions are used; we just used the minimal time thickness \(\tilde t\) and the usual \(\hbar /\Delta t\) scaling.
4. Minimal mass bundle: m★ from endurance gravity
On the gravity side, QTT derives Newton’s constant \(G\) from the endurance current instead of inserting it by hand. The key relation (from the sink equations plus the definition of the endurance current) is
[
G = \frac{\tilde\ell^2\,c^3}{\hbar}.
]
This is already non‑standard: \(G\) is not an independent parameter, but a derived combination of \(\tilde\ell\), \(c\), and \(\hbar\).
Now, QTT introduces a minimal bundle mass \(m^\ast\) such that the gravitational coupling of one bundle is the atomic unit of endurance–gravity:
- one endurance cell of size \(\tilde\ell\) carries a bundle mass \(m^\ast\),
- many bundles stack to make up an ordinary mass \(m = N\,m^\ast\).
We choose \(m^\ast\) to be exactly the mass scale naturally associated to this cell:
\[ m^\ast := \frac{\hbar}{c\,\tilde\ell}. \]
Why this choice? Because it synchronises the mass ledger with the same micro‑length \(\tilde\ell\) that appears in both:
- the energy quantum \(E_\ast = \hbar c/\tilde\ell\), and
- the gravitational coupling \(G = \tilde\ell^2 c^3/\hbar\).
With this definition, a single endurance bundle is “the amount of mass that fits into one four‑cell at the QTT equilibrium scale”.
5. The genuinely new step: E★ = m★c² from a single length scale
Now look at the two QTT primitives side by side:
- Minimal energy: \[ E_\ast = \frac{\hbar c}{\tilde\ell}, \]
- Minimal mass: \[ m^\ast = \frac{\hbar}{c\,\tilde\ell}. \]
Eliminate \(\hbar\) and \(\tilde\ell\) between them:
[
E_\ast
= \frac{\hbar c}{\tilde\ell}
= \Bigl(\frac{\hbar}{c\,\tilde\ell}\Bigr)c^2
= m^\ast c^2.
]
\[ E_\ast = m^\ast c^2. \]
This is the heart of the QTT result:
- \(E_\ast\) came from the time thickness of a minimal four‑cell and the quantum of action \(\hbar\).
- \(m^\ast\) came from the endurance–gravity side: how strongly one cell acts as a sink in the endurance current.
- The same micro‑length \(\tilde\ell\) appears in both.
The new piece is that these are defined independently in QTT, but consistency of the endurance ledger forces
the energy of a single endurance bundle to equal its mass times \(c^2\).
No Lorentz transformations, no relativistic kinematics. The relation \(E = mc^2\) is already true at the level of a single endurance cell.
6. From one bundle to any mass: E = mc² as a counting rule
Now take an ordinary mass \(m\). In QTT it is literally a count of bundles:
[
m = N\,m^\ast,
]
for some integer or large rational \(N\) (depending on how you coarse grain). The total endurance energy stored in those bundles is
[
E_{\rm rest} = N\,E_\ast
= N\,m^\ast c^2
= (N m^\ast)c^2
= mc^2.
]
\[ E_{\rm rest} = mc^2 \quad\text{(QTT, non‑relativistic, bundle counting).} \]
So for QTT:
- inertial mass \(m\) = number of endurance bundles,
- gravitational mass = the same count appearing in the endurance sink law for \(G\),
- rest energy = that same count times \(E_\ast\).
All three are locked together by the same micro‑length \(\tilde\ell\) and the same four‑cell ledger. The equality of inertial and gravitational mass and the \(E=mc^2\) relation are not separate axioms; they are different faces of the same endurance bookkeeping.
7. Where this differs from “just redefining m and E”
You might ask: “Is this just a cute re‑definition? Couldn’t I always define \(m^\ast := E_\ast/c^2\) and say I ‘derived’ \(E=mc^2\)?”
The difference in QTT is that:
- \(E_\ast\) is fixed by the four‑cell thickness \(\tilde t\) and the Planckian action quantum: \(E_\ast = \hbar/\tilde t\).
- \(m^\ast\) is fixed independently by the endurance–gravity law: it’s the mass that makes the sink equation give the observed \(G\) with the same \(\tilde\ell\): \[ G = \frac{\tilde\ell^2 c^3}{\hbar}. \]
- Those two sides do not know about each other a priori: one is about capacity of a four‑cell, the other about the strength of the endurance sink that reproduces Newton–Poisson.
The new statement is:
In QTT, the same microscopic length \(\tilde\ell\) that sets the endurance four‑cell also fixes the gravitational constant \(G\). Demanding that the endurance ledger is internally consistent forces the energy per cell and the mass per cell to satisfy \(E_\ast = m^\ast c^2\).
That’s what turns \(E=mc^2\) from a “relativistic postulate” into a bookkeeping identity of the endurance quanta themselves.
8. Why I even I tried to do this?
At the everyday level, QTT doesn’t change the fact that a kilogram of mass has a rest energy of \[ E = (1\ \text{kg}) \times c^2 \approx 9\times 10^{16}\ \text{J}. \] If you run a nuclear reactor or collider, you’ll still see exactly the same numbers as in standard physics.
What changes is the ontology:
- Mass is no longer a primitive. It’s a count of endurance bundles.
- Energy is the time‑thickness cost of those bundles in the four‑cell ledger.
- Gravity is the endurance current those bundles draw from the space‑quanta field.
All three are linked by the single microscopic length \(\tilde\ell\), so \(E = mc^2\) and later more complete Unified Equilibrium Law, becomes an internal consistency condition of the QTT microstructure, not an external relativistic constraint.
That’s the new message: a non‑relativistic, Planck‑scale derivation of \(E=mc^2\) from a unified endurance ledger for mass, energy, and gravity.
QTT - Quantum Traction Theory 