DFT-15: Emergent Lorentz Invariance and Inertial Frames

MWells · Post by **MWells** » Wed Dec 03, 2025 8:36 pm

Why the speed limit and Lorentz symmetry follow from projection, not postulates.

In standard relativity, Lorentz invariance is introduced as a starting assumption: all inertial observers measure the same speed of light, and the laws of physics are invariant under Lorentz transformations. From these postulates, one derives the familiar structure of spacetime.

In Dual-Frame Theory (DFT), we invert this logic.

We begin with a simpler ontology:

a three-dimensional Natural Reference System (NRS) of scalar motion, described by a trajectory $\sigma(\lambda)$ , and
two complementary projections of that trajectory:
- the S-frame, which describes geometric trajectories in a spacetime-like embedding, and
- the T-frame, which describes rotational and phase structure.

Within this framework, Lorentz invariance is not an axiom but a consistency condition. It is the requirement that all inertial S-frames give compatible descriptions of the same scalar progression and the same motion budget shared with the T-frame.

This post focuses on special-relativistic structure (flat spacetime). The stronger claims of general relativity (curved spacetime, Einstein equations) are not addressed here and will require additional work in later posts.

1. Scalar progression and a fixed motion budget

In the NRS, a scalar trajectory is a map from the intrinsic parameter lambda into a three-dimensional Euclidean manifold:

$\sigma(\lambda) \in \mathbb{R}^3_{\text{NRS}}$

The Euclidean distance between two scalar states is defined by

$d_{\text{NRS}}(\lambda_1,\lambda_2) = \left\|\sigma(\lambda_2) - \sigma(\lambda_1)\right\|$

Here ∥⋅∥ is the standard Euclidean norm on 𝑅³. This choice encodes the idea that the underlying scalar motion lives in a simple, isotropic 3D manifold before any spacetime-like interpretation is imposed.

By construction, scalar motion has a fixed magnitude per unit progression. In differential form we can express this by

$\left\|\frac{d\sigma}{d\lambda}\right\| = \kappa$

for some constant $\kappa$ that sets the overall scale. This expresses the idea that there is a fixed budget of motion per unit lambda. All physical structure in both frames must ultimately be expressible as different ways of using this same underlying progression.

The numerical value of 𝜅, and its relation to SI units, will be tied to empirical scales (such as 𝑐) in later work. Here we are concerned with the geometric structure that follows from assuming such a fixed norm exists.

Importantly, the NRS is not an observable “preferred frame” inside spacetime; it is a more primitive configuration space. All inertial S-frames are different coordinate descriptions of projections of the same NRS process, none privileged over the others.

2. Embedding scalar motion into the S-frame

The S-frame projection assigns spacetime-like coordinates to each scalar state:

$F_S : \sigma(\lambda) \mapsto x^\mu(\lambda)$

$\mu = 0,1,2,3$

Here x^0 plays the role of a coordinate time, and x^1, x^2, x^3 are spatial coordinates. The S-frame must represent scalar progression in a way that distinguishes between “time-like” and “space-like” contributions while still reflecting the same underlying budget.

To accomplish this, we introduce a spacetime interval

$\Delta s^2 = c^2(\Delta x^0)^2 - (\Delta x^1)^2 - (\Delta x^2)^2 - (\Delta x^3)^2$

The constant 𝑐 here is the projection scale that converts between units of S-frame time and S-frame space. In DFT it is not arbitrary; it is tied back, in principle, to the scalar progression and the choice of unit mapping between NRS and S-frame. The fact that its empirical value is 𝑐≈3×108 m/s is a question of how NRS units are anchored to SI, not of the geometry itself.

The interval must be related to the NRS distance between the corresponding scalar states. For some scale factor $\alpha$ we require, for time-like segments associated with massive motion,

$\Delta s^2 = \alpha^2\, d_{\text{NRS}}(\lambda_1,\lambda_2)^2.$

This expresses the idea that, for any such segment of the trajectory, the amount of spacetime separation in the S-frame is proportional to the amount of scalar distance traversed in the NRS.

The change of signature from the Euclidean NRS norm to a Lorentzian S-frame interval is not a coincidence: one component of the scalar budget is interpreted as “time-like,” three as “space-like,” and the sign choice is fixed by the requirement that proper time remains positive and that causal structure (light cones) matches experiment. At this stage we are taking that Lorentzian form as the unique quadratic form that supports a consistent causal ordering; a deeper derivation of the signature itself would require additional assumptions and will be left for later development.

Different choices of S-frame coordinates that qualify as “inertial” must all maintain this relationship with the NRS distance.

3. Why inertial frames must preserve the interval

Suppose we have two inertial S-frame coordinate systems, x^mu and x'^mu, both describing the same scalar trajectory $\sigma(lambda)$ . For a given pair of events on that trajectory, the first observer computes

$\Delta s^2 = c^2(\Delta x^0)^2 - (\Delta x^1)^2 - (\Delta x^2)^2 - (\Delta x^3)^2$

The second observer computes

$\Delta s'^2 = c^2(\Delta x'^0)^2 - (\Delta x'^1)^2 - (\Delta x'^2)^2 - (\Delta x'^3)^2$

Both of these intervals must be proportional to the same NRS scalar distance for that segment. Therefore they must agree:

$\Delta s^2 = \Delta s'^2$

Any transformation between inertial coordinate systems that preserves this quadratic form is a member of the Lorentz group (up to translations).

In standard relativity, one postulates the invariance of the interval based on symmetry principles (relativity principle + constancy of
𝑐). In DFT, the interval invariance is instead:

the statement that all inertial S-frame observers are describing the same NRS scalar distance for corresponding segments, and
the requirement that S-frame coordinates remain consistent with that scalar constraint.

Under mild assumptions of linearity, isotropy, and homogeneity, it is a standard result that transformations preserving this quadratic form form the Lorentz group. DFT does not change that algebraic result; it provides a deeper reason why S-frame descriptions must preserve this form at all: they are all reparametrizations of a single scalar trajectory with fixed NRS norm.

This is the first key conclusion:

If different S-frame observers are all describing the same scalar motion and agree on its NRS distance per segment, then their coordinate systems must be related by Lorentz transformations that preserve the interval.

4. The speed limit as a projection bound
Now consider the motion of a time-like worldline in the S-frame. Define the spatial velocities by

$v^i = \frac{\Delta x^i}{\Delta x^0}$

$i = 1,2,3$

and let the squared speed be

$v^2 = (v^1)^2 + (v^2)^2 + (v^3)^2$

For a time-like trajectory we have a positive interval:

$\Delta s^2 > 0$

Substituting the definition of the interval and dividing through by $(\Delta x^0)^2$ gives

$\frac{\Delta s^2}{(\Delta x^0)^2} = c^2 - v^2$

The left-hand side is positive for time-like motion, so the right-hand side must be positive as well:

$c^2 - v^2 > 0$

which implies

$v^2 < c^2$

From the DFT point of view, this inequality is not simply a statement about light. It expresses a projection limit: you cannot allocate more translational S-frame motion per unit progression than the scalar budget allows, once the relation between $\Delta s^2$ and the NRS distance is fixed.

The boundary case

$\Delta s^2 = 0$

defines lightlike trajectories, where the S-frame translational component of motion saturates the available budget for spacetime separation. These trajectories correspond to configurations in which the S-frame has no remaining margin to represent internal “time-like” separation; all of the interval is expressed as geometric separation evolving at the limiting speed c.

Thus the usual speed-of-light bound is not an independent dynamical law, but the statement that:

The S-frame cannot project more translational motion out of scalar progression than the fixed NRS norm permits.

In combination with DFT-14, this also defines the causal structure: even though T-frame correlations can connect phase-near states across spacelike S-frame separations (nonlocal correlations), signaling remains limited by timelike/lightlike S-frame trajectories because those are the only ones compatible with the scalar budget and interval structure.

5. The T-frame and the shared motion budget

The T-frame projection assigns phase or rotational coordinates:

$F_T : \sigma(\lambda) \mapsto \theta^i(\lambda)$

$i = 1,2,3$

These phases live on a compact manifold (for example, a three-torus) and encode the rotational and coherence properties of the motion. Their evolution can be written schematically as

$\Delta \theta^i \propto \omega^i \,\Delta\lambda$

where $\omega^i$ are local phase-wind rates.

The key DFT requirement is that the S-frame and T-frame do not evolve independently. They share a common scalar motion budget. In symbolic form we can write a constraint such as

$\mathcal{B}_{\text{total}} = \mathcal{B}_S + \mathcal{B}_T$

where $B_S$ is the portion of the budget used by S-frame translation and $B_T$ is the portion used by T-frame rotation. The details of these quantities are developed elsewhere, but the conceptual point is simple: the more structure appears in one frame, the less is available in the other.

Different inertial observers may disagree about the numerical components $\Delta x^mu$ , but if they are all describing the same scalar progression, they must agree on:

the invariant interval $\Delta s^2$ , and
how the motion budget is split between S-frame translation and T-frame rotation. (i.e., on invariants like rest mass and intrinsic rotational content).

If a transformation between S-frame coordinates were to change the apparent balance between $B_S$ and $B_T$ , then different observers would infer different intrinsic rotational content, different rest masses, and different coherence structures for the same underlying motion. That would contradict the idea that there is a single scalar trajectory being described.

This constraint is what fixes inertial transformations to be Lorentz transformations in practice: only Lorentz transformations preserve both the interval and the decomposition of motion into time-like and space-like parts in a way that remains compatible with a consistent T-frame evolution.

6. Inertial frames as “straightest” S-frame views

From the DFT viewpoint, we can now characterize inertial frames more geometrically.

An S-frame coordinate system is inertial if:

free scalar motions project to straight worldlines in that frame,
the spacetime interval along any segment is proportional to the NRS distance for that segment, and
the coupling to the T-frame motion budget is preserved in form along those worldlines.

In these frames:

a free particle has constant S-frame velocity,
its scalar progression 𝜎(𝜆) advances at constant norm,
its T-frame phase θⁱ(λ) evolves at constant intrinsic rates ωⁱ, subject to the motion budget.

Transformations between such frames must preserve all of these properties. Under reasonable assumptions of linearity and homogeneity, the only transformations that do this are Lorentz transformations.

So instead of treating Lorentz symmetry as a starting axiom, DFT treats it as the answer to a question:

What coordinate changes between S-frames preserve a single scalar progression, a fixed NRS distance per segment, and a consistent coupling to T-frame rotation?

The answer is exactly: Lorentz transformations. The algebra of boosts, time dilation, length contraction, and the associated effects (including phenomena like Thomas precession, in principle) are unchanged; DFT’s contribution is to show why that algebra is the unique one compatible with the underlying scalar-motion picture.

7. Summary: Lorentz invariance as a projection consistency condition

We can now summarize the story in DFT terms.

The NRS provides a scalar trajectory sigma(lambda) with a fixed norm per unit progression.
The S-frame embeds this trajectory into spacetime coordinates x^mu(lambda) and defines an interval Delta s^2 tied to the NRS distance.
Different S-frame observers who are all inertial must preserve this interval for all worldline segments.
Transformations that preserve this quadratic form are Lorentz transformations; thus, the Lorentz group appears as the symmetry group of consistent S-frame embeddings of the same scalar progression.
The bound v^2 < c^2 arises as a projection limit: S-frame translation cannot exceed what the scalar progression budget allows.
The T-frame shares this budget, and requiring all inertial observers to agree on the S/T split further constrains admissible transformations to the Lorentz group.

In this way, Lorentz invariance and the speed limit $c$ are not arbitrary postulates added on top of the theory. They are the geometric consequences of representing a single NRS scalar progression consistently across multiple S-frames while maintaining its coupling to T-frame rotational structure.

What this post has not yet done:

It has not derived the numerical value of 𝑐; that requires linking NRS units to empirical quantities (e.g., via electromagnetic propagation) and lies beyond the scope of this conceptual derivation.
It has not addressed general relativity (curved spacetime, Einstein equations); those predictions (black holes, gravitational waves, strong-field tests) require additional structure not yet introduced.
It has not shown explicit Lorentz transformations as matrices Λ^𝜇_ν; instead, it has identified the interval-preserving group as the only consistent choice and interpreted that group physically.

The point of DFT-15 is therefore interpretive and structural: relativity’s Lorentz symmetry is the unique way to coordinate S-frame descriptions of a deeper scalar-motion process, not an independent layer of axioms.

8. Next: DFT-16 — Probability, Symmetry, and Allowed Rotational Combinations

With Lorentz invariance grounded in projection geometry, we can turn to a different kind of constraint: which rotational combinations of scalar motion are actually allowed.

In DFT-16, we will explore:

why only specific small-integer rotational combinations yield stable atomic structures,
how the familiar 8–18–32 periodic pattern can be understood as a consequence of symmetry on the scalar manifold, and
how a probability measure over scalar trajectories translates into RS rotational rules.