Special Relativity as an Emergent Symmetry of Entropy Geometry

1 ENTROPY-STABLE NORMS AND EMERGENT SYMMETRIES

To understand how relativistic structure emerges from entropy geometry, we begin by examining how TEQ quantifies small variations between physical configurations.

In the TEQ framework, each point in configuration space is equipped with an entropy metric $g_{ij}\left(x\right)$, which characterizes how distinguishable a small variation $d{x}^i$ is from nearby configurations, given the system’s entropy structure. Intuitively, this metric assigns greater weight to directions in which changes are more thermodynamically stable and resolvable, and lower weight to directions where distinctions rapidly collapse or decohere.

We can then define an entropy-weighted norm for infinitesimal variations in configuration space:

$$d{s}^2=g_{ij}\left(x\right)d{x}^{i}d{x}^j.$$

(1)

This norm does not represent a spacetime interval in the traditional sense, nor is it derived from kinematic assumptions. It arises from the geometry of distinguishability itself. The quantity $d{s}^2$ represents the entropy-weighted “distance” between two neighboring configurations—a measure of how resolvable one state is from another under entropy flow.

In regions where entropy curvature is negligible—so-called entropy-flat domains—this metric reduces locally to a constant pseudo-Euclidean form:

$$d{s}^2\approx d{x}^2+d{y}^2+d{z}^2-{\sigma }^{2}d{t}^2.$$

(2)

Here, the appearance of a Minkowski-like structure is not imposed but emerges from the symmetry of distinguishability in a flat entropy geometry (see Appendix A). The parameter $\sigma$ appears as a structural conversion factor between space-like and time-like resolution directions. It sets the scale at which spatial and temporal variations contribute equally to entropy-resolved distinguishability. Its empirical value corresponds to the invariant speed c known from Special Relativity [1], but within TEQ it is a derived structural quantity, not a postulate.

Einstein Summation Convention. From this point onward, we adopt the Einstein summation convention: repeated indices—one raised (superscript) and one lowered (subscript)—imply summation over that index. For example,

$$g_{\mu \nu }{x}^{\mu }{x}^{\nu }=g_{00}{x}^0{x}^0+g_{11}{x}^1{x}^1+g_{22}{x}^2{x}^2+g_{33}{x}^3{x}^3.$$

To identify which changes of frame preserve the structure of entropy resolution in a flat regime, we examine the class of coordinate transformations that leave the entropy-resolved norm in Eq. (1) invariant.

Let $x^{\mu }$ denote the local coordinates in configuration space, where $\mu =0,1,2,3$ indexes time and spatial directions (e.g., $x^0=t$, $x^1=x$, etc.). A general linear transformation to a new frame can be written as:

$$x^{\prime \mu }={L^{\mu }}_{\nu }{x}^{\nu },$$

(3)

where ${L^{\mu }}_{\nu }$ is a matrix that expresses how the same configuration is re-expressed in a different observer’s resolution frame.

We now ask: which such transformations leave the entropy-resolved norm

$$d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$$

(4)

unchanged?

In entropy-flat regions, the metric $g_{\mu \nu }$ becomes constant and symmetric. The symmetry of distinguishability implies a fixed ratio of resolution between spatial and temporal directions, encoded in the diagonal structure:

$$g_{\mu \nu }=\mathrm{diag}(1,1,1,-{\sigma }^2).$$

(5)

The set of transformations that preserve this structure—i.e., for which

$$g_{\mu \nu }{x}^{\mu }{x}^{\nu }=g_{\mu \nu }{x}^{\prime \mu }{x}^{\prime \nu }$$

(6)

holds for all $x^{\mu }$—form the Lorentz group relative to the entropy geometry. These are the transformations that maintain entropy-based distinguishability between nearby configurations when changing frames.

In this interpretation, Lorentz invariance arises not from the invariance of light speed, but from a structural requirement: the preservation of resolution geometry under frame transformation. The parameter $\sigma$—later identified empirically with c—sets the scale at which spatial and temporal variations are equally resolvable under entropy-stable motion. The full relativistic symmetry structure thus emerges as a consequence of flat entropy geometry, not as a kinematic input.

2 TIME DILATION AND LENGTH CONTRACTION FROM ENTROPY GEOMETRY

A central insight of the TEQ framework is that physical structure is defined by entropy-resolved distinguishability. What a system can resolve—what it can stably track or distinguish—is determined by the local entropy geometry through which it evolves. As a system moves through configuration space, its resolution rates depend not only on its internal degrees of freedom, but on how those degrees are embedded in the surrounding entropy structure.

In particular, motion through an entropy-flat region skews the direction of entropy flow relative to the resolution axes defined by the entropy metric $g_{\mu \nu }$. This alters both temporal and spatial resolution. Operationally, resolution along a direction measures how many distinguishable entropy gradients a system traverses per unit parameter time. For a stationary system, this rate is maximal along the time axis. When the system moves with velocity v, its effective trajectory through entropy geometry tilts, reducing its resolution rate along time and compressing it along space.

Deriving Time Dilation from Entropy Geometry

Let ${\dot{x}}^{\mu }=\left({\displaystyle \frac{dt}{d\tau }},{\displaystyle \frac{dx}{d\tau }},0,0\right)$ denote the entropy-stable trajectory of a system moving with constant velocity $v=dx/dt$ in an entropy-flat region. In such domains, the entropy-resolved norm introduced in Eq. (2) becomes structurally meaningful along a path: it quantifies the rate at which a system traverses distinguishable configurations. For maximally stable evolution, this rate remains constant, yielding the condition:

$$g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-{\sigma }^2,$$

(7)

where $\sigma$ is the maximal entropy-resolved rate of change—analogous to the invariant speed c in Special Relativity, but here emerging from resolution symmetry. This condition ensures that the system’s motion maintains constant entropy-resolved distinguishability along its trajectory. Using the metric $g_{\mu \nu }=\mathrm{diag}(1,0,0,-{\sigma }^2)$, we find:

$$\begin{array}{cc}\hfill g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }& =-{\sigma }^2{\left({\displaystyle \frac{dt}{d\tau }}\right)}^2+{\left({\displaystyle \frac{dx}{d\tau }}\right)}^2=-{\sigma }^2.\hfill \end{array}$$

(8)

Substituting $dx/dt=v$, and using the chain rule:

$${\displaystyle \frac{dx}{d\tau }}={\displaystyle \frac{dx}{dt}}·{\displaystyle \frac{dt}{d\tau }}=v·{\displaystyle \frac{dt}{d\tau }},$$

we obtain:

$$-{\sigma }^2{\left({\displaystyle \frac{dt}{d\tau }}\right)}^2+v^2{\left({\displaystyle \frac{dt}{d\tau }}\right)}^2=-{\sigma }^2,$$

which simplifies to:

$${\left({\displaystyle \frac{dt}{d\tau }}\right)}^2={\displaystyle \frac{{\sigma }^2}{{\sigma }^2-v^2}}\Rightarrow {\displaystyle \frac{dt}{d\tau }}={\displaystyle \frac{1}{\sqrt{1-\frac{v^2}{{\sigma }^2}}}}.$$

(9)

Thus, the proper time $d\tau$ along the entropy-stable trajectory is:

$$d\tau =dt\sqrt{1-{\displaystyle \frac{v^2}{{\sigma }^2}}},$$

(10)

matching the standard relativistic time dilation formula when $\sigma =c$. But here, the result arises as a structural condition from entropy geometry, not as a postulate.

Deriving Length Contraction from Entropy Curvature

Now consider the entropy-weighted effective action (see §2 of [2]):

$$S_{\mathrm{eff}}\left[\varphi \right]=\int dt\left(L(\varphi ,\dot{\varphi })-i\hslash \beta g(\varphi ,\dot{\varphi })\right),$$

(11)

where the entropy flux functional is (see Appendix B of [2]):

$$g(\varphi ,\dot{\varphi })={\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }.$$

(12)

Using the velocity components above, the entropy flux becomes:

$$g(\varphi ,\dot{\varphi })={\displaystyle \frac{1}{2}}\left(v^2-{\sigma }^2\right){\left({\displaystyle \frac{dt}{d\tau }}\right)}^2.$$

(13)

Entropy curvature $\kappa \left(v\right)$, defined as the second variation of the entropy flux functional $g(\varphi ,\dot{\varphi })$ with respect to path displacement, scales with the square of the time dilation factor:

$$\kappa \left(v\right)\sim {\displaystyle \frac{{\delta }^{2}g}{\delta x^2}}\sim {\left({\displaystyle \frac{dt}{d\tau }}\right)}^2={\displaystyle \frac{1}{1-\frac{v^2}{{\sigma }^2}}}.$$

From TEQ, the resolution scale $\Delta x_{\mathrm{res}}$ is inversely proportional to $\sqrt{\kappa \left(v\right)}$:

$$\Delta x_{\mathrm{res}}\left(v\right)\sim {\displaystyle \frac{1}{\sqrt{\kappa \left(v\right)}}}=\sqrt{1-{\displaystyle \frac{v^2}{{\sigma }^2}}}.$$

(14)

If $L_0$ is the rest length (resolution span at $v=0$), the entropy-resolved length becomes:

$$L\left(v\right)=L_0·\Delta x_{\mathrm{res}}\left(v\right)=L_0\sqrt{1-{\displaystyle \frac{v^2}{{\sigma }^2}}}.$$

(15)

This is the standard Lorentz contraction formula, but here derived from the resolution geometry of the entropy manifold.

Interpretation

Both time dilation and length contraction thus emerge in TEQ as resolution-induced phenomena:

• Time dilation results from motion tilting the entropy trajectory away from the frame’s temporal resolution axis.
• Length contraction results from the entropy curvature induced by motion suppressing spatial resolution.

These effects are not postulates but consequences of a single principle: that motion through entropy geometry deforms the system’s capacity to stably resolve distinct configurations.

Summary. In TEQ, relativistic phenomena emerge from structural constraints on entropy-resolved resolution. The functional form of Lorentz symmetry arises from the geometry of distinguishability—not from spacetime axioms. Motion induces curvature in the entropy manifold, and this curvature defines the contraction and dilation observed in Special Relativity. The mathematical structure is preserved; the interpretation is transformed.

3 MASS–ENERGY EQUIVALENCE FROM ENTROPY GEOMETRY

The iconic equation $E=m{c}^2$, first derived by Einstein in 1905 [1], stands as one of the most celebrated results in physics. Yet it is often misunderstood as a mere assertion of proportionality between mass and energy. In reality, Einstein’s derivation was grounded in careful reasoning from the postulates of Special Relativity and the conservation laws of classical electrodynamics. He considered the recoil of a body emitting light energy and showed, via momentum conservation, that the emission must result in a decrease in mass—thus arriving at the conclusion that mass and energy are physically equivalent.

At the time, however, the modern tools of variational calculus and Noether’s theorem were not yet available. Emmy Noether’s profound insight—that every continuous symmetry of an action principle yields a conserved quantity—would only be formulated over a decade later, in 1915 and formally published in 1918. As a result, Einstein’s route to $E=m{c}^2$ remained kinematically motivated, not variationally derived.

In contrast, the TEQ framework is formulated from the outset as a variational theory: dynamics arise from extremizing an entropy-weighted action. In this setting, Noether’s theorem becomes a central structural tool. Rather than postulating relativistic kinematics or light propagation, we identify conserved quantities by analyzing the symmetries of the entropy geometry that underlies physical evolution.

Entropy-weighted action and time invariance

Consider the entropy-weighted action for a particle moving through a locally entropy-flat region:

$$S_{\mathrm{eff}}=\int dt\left(L-i\hslash \beta g(\varphi ,\dot{\varphi })\right),$$

(16)

where $g(\varphi ,\dot{\varphi })$ encodes the entropy flux rate—i.e., the rate at which the system moves through distinguishable configurations.

In an entropy-flat domain, the entropy metric becomes constant:

$$g_{\mu \nu }=\mathrm{diag}(1,1,1,-{\sigma }^2),$$

(17)

and the system’s trajectory is entropy-stable if its entropy-resolved norm remains constant:

$$g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=\mathrm{const}.$$

(18)

For a system at rest, this reduces to ${\dot{t}}^2=1$ and $\dot{\overrightarrow{x}}=0$, so

$$g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-{\sigma }^2.$$

The temporal part of this structure is invariant under time translations. Applying Noether’s theorem to the action, the conserved quantity associated with this symmetry is the entropy-resolved energy:

$$E={\displaystyle \frac{\partial L_{\mathrm{eff}}}{\partial \dot{t}}}=m{\sigma }^2.$$

(19)

Interpretation

In this formulation:

• Mass m arises as a measure of entropy curvature—how sharply a system resists change under entropy flow.
• The constant $\sigma$ is the resolution scaling factor that balances time-like and space-like distinguishability in entropy-flat regimes.
• Energy E is the Noether charge associated with entropy-resolved temporal stability.

Thus, the relation

$$E=m{\sigma }^2$$

(20)

emerges not from light propagation or relativistic postulates, but from the internal structure of entropy geometry. When $\sigma$ is empirically identified with the invariant speed c, this becomes Einstein’s famous result.

But here, we recognize that what was historically derived from light propagation and inertial frames is, within TEQ, a structural identity—emerging from the coherent relationship between entropy flow, distinguishability, and the stability conditions imposed by entropy geometry.

Summary

In TEQ, the mass–energy relation is not imposed but emerges from the entropy-weighted variational principle and the invariance of resolution structure under time evolution. The conceptual order is reversed: energy is not what mass becomes under motion, but what mass is when viewed through the lens of entropy geometry.

4 DEVIATIONS FROM SR IN ENTROPY-CURVED REGIMES

While Special Relativity (SR) emerges in entropy-flat regions as shown in Eqs. (2), (10), and (15), the TEQ framework predicts deviations in regimes where entropy curvature is non-negligible. In these regions, the local distinguishability structure becomes anisotropic or unstable, and the entropy metric $g_{\mu \nu }\left(x\right)$ varies across configuration space.

Such deviations are not artifacts or corrections to SR but structural consequences of the underlying entropy geometry. Since relativistic structure in TEQ arises from the symmetry of entropy-resolved resolution, any deformation of that symmetry due to curvature directly modifies the effective dynamics.

Key physical regimes where entropy curvature becomes significant include:

• Near black hole horizons: Where local distinguishability collapses due to extreme curvature, standard notions of simultaneity and spatial extension break down.
• During quantum measurement: Where entropy-stable resolution structures become sharply non-symmetric, selecting discrete outcomes as resolution attractors [2].
• In early cosmological epochs: Where large-scale entropy gradients drive non-inertial structure formation beyond the flat approximation.

These deviations imply that SR is not a universally valid framework but a thermodynamic approximation. It holds only in the limiting case where entropy curvature vanishes or remains negligible over relevant scales. Outside these limits, new effects—including quantum, gravitational, and cosmological phenomena—arise from the full geometry of entropy-stable evolution.

This opens the door to novel predictions: where standard SR expects linearity, TEQ anticipates curvature-induced deviation. These differences offer an empirical pathway for distinguishing thermodynamic geometry from traditional kinematic theory.

5 CONCLUSION

Special Relativity is not, in the TEQ framework, a foundational theory. It is an emergent symmetry structure that arises only in regions of negligible entropy curvature—where the geometry of distinguishability becomes symmetric, stable, and flat. In such entropy-flat regimes, TEQ reproduces the full structure of SR:

• The Lorentz transformations (Eq. (6)) emerge as the symmetries preserving the entropy-resolved norm.
• Time dilation and length contraction (Eqs. (10) and (15)) follow from motion-induced deformation of entropy gradients.
• The mass–energy relation (Eq. (20)) is structurally derived from an entropy-weighted variational principle via Noether’s theorem.

These results demonstrate that the relativistic framework familiar from Einstein [1] is not imposed in TEQ but recovered as a special case—a thermodynamic limit—of a more general theory grounded in entropy geometry. The constant c is no longer a postulated speed of light but a resolution-conversion factor emerging from symmetry in entropy-stable dynamics.

Where entropy curvature becomes significant, TEQ predicts structural deviations from SR, as discussed in Section 5. This yields a natural path toward unification: both relativistic and quantum-gravitational phenomena arise from the same underlying principle—the stability of resolution in thermodynamic geometry.

Ultimately, TEQ reframes the logic of modern physics: instead of assuming spacetime and deriving thermodynamic behavior, it begins with entropy and derives spacetime-like structure. Special Relativity, under this lens, becomes not an axiom of nature but a geometric consequence of deep thermodynamic order.

Appendix F.1. Entropy Geometry and Resolution Structure

In TEQ, the entropy-weighted action is given by

$$S_{\mathrm{eff}}\left[\varphi \right]=\int dt\left(L(\varphi ,\dot{\varphi })-i\hslash \beta g(\varphi ,\dot{\varphi })\right),$$

(A21)

where the entropy flux functional has the form

$$g(\varphi ,\dot{\varphi })={\displaystyle \frac{1}{2}}{G}_{ij}\left(\varphi \right){\dot{\varphi }}^i{\dot{\varphi }}^j,$$

(A22)

with $G_{ij}\left(\varphi \right)$ a smooth, symmetric, positive-definite entropy metric over configuration space (see [2], Appendix B).

To analyze resolution flow across configuration-time space, we introduce a metric that quantifies the entropic cost of distinguishing two infinitesimally close states. Since distinguishability degrades over time as entropy accumulates, and across configurations due to entropy curvature, the local structure must reflect both. Because the entropy cost of resolution depends quadratically on the rate of change—as expressed in the entropy flux functional (A22)—the infinitesimal resolution cost must take the form of a quadratic expression in temporal and configurational displacements. This leads to::

$$d{s}^2:=-{\sigma }^{2}d{t}^2+{\alpha }^{-1}{G}_{ij}\left(\varphi \right)d{\varphi }^{i}d{\varphi }^j,$$

(A23)

where $\sigma$ is the maximal resolution-preserving rate, and $\alpha =\hslash \beta$ defines the entropy resolution scale (see [2], §3). The first term penalizes entropy exposure through time; the second term captures the resolution burden of motion through configuration space. The resulting metric quantifies the structural cost of sustaining distinguishable evolution—serving as the entropic analog of a spacetime interval.

Appendix F.2. Resolution Horizon and Light Cones

A key structural feature of TEQ is that not all distinctions are physically sustainable: entropy flow imposes limits on how quickly a system can evolve while remaining resolvable. If a trajectory changes too rapidly in configuration space, or over too long a time, the entropy cost becomes too great, and nearby alternatives become indistinguishable. This defines a resolution horizon: a critical threshold beyond which differences collapse into unresolved structure.

Let ${\dot{\varphi }}^i=d{\varphi }^i/dt$ and consider the condition for marginal resolvability:

$$G_{ij}\left(\varphi \right){\dot{\varphi }}^i{\dot{\varphi }}^j={\sigma }^2.$$

(A24)

This threshold defines the boundary between two qualitatively different types of evolution in configuration-time space:

• Timelike paths evolve slowly enough—both in configuration and over time—that their distinctions remain resolvable.
• Spacelike paths involve changes that are too rapid to be sustained under entropy flow, rendering the resulting configurations indistinguishable.

This boundary occurs precisely where the resolution metric vanishes, $d{s}^2=0$ in (A23). It marks the transition between resolvable and unresolvable change—structurally analogous to a light cone, but defined by entropy geometry rather than spacetime kinematics.

Appendix F.3. Entropy-Flat Limit and Emergent Minkowski Metric

In the regime where:

(a)

Entropy curvature is negligible, $G_{ij}\left(\varphi \right)\approx {\delta }_{ij}$;

(b)

Entropy resolution scale $\alpha$ is constant;

(c)

Time parameter t is affine and uncurved;

the line element (A23) becomes

$$d{s}^2=-{\sigma }^{2}d{t}^2+d{\varphi }^{i}d{\varphi }^i,$$

(A25)

which is precisely the Minkowski metric over $({\varphi }^i,t)$, with signature $(-1,+1,+1,+1)$.

Thus, Minkowski structure emerges as the effective geometry of entropy-flat resolution space.

Appendix F.4. Interpretation

The speed of light c arises not as a fundamental constant, but as the maximal rate at which stable distinctions can propagate under entropy flow. Causal structure—what can affect what—is determined by resolvability, not by kinematic postulates.

Lorentz symmetry is then the invariance group of the entropy-flat limit. It reflects the set of transformations that preserve the resolution metric in regimes where entropy curvature vanishes.

Appendix F.5. Conclusion

Minkowski spacetime is not fundamental in the TEQ framework. It arises as a limiting structure of stable resolution in low-entropy-curvature regimes. The light cone is the geometric expression of the maximal distinguishability boundary, and Special Relativity emerges as the symmetry of entropy-flat resolution flow.