Unified Field Dynamics from Non-Linear Spacetime Transformations (Photons, Mass and Gravity from a Single Geometric Principle)

1. Introduction

1.1. Motivation

A single geometrical principle that yields both photons and gravitons—and explains particle masses—would unify fundamental forces at a classical and (semi-)quantum level while making sharp experimental predictions.

1.2. Historical Context

Geometrization has advanced physics from Maxwell’s unification of E and B (fields are geometry in 3-space) through special relativity (spacetime boosts) to general relativity (gravity = curvature). Non-linear electrodynamics (Born & Infeld), transformation optics, and teleparallel gravity each hint that coordinate transformations themselves might underlie dynamics.

Table 1. Comparison with other unification frameworks

Approach	Core idea	Our difference
Kaluza–Klein	Add a 5-th dimension	Stay 4-D; use non-linear maps instead
String/M-theory	Extended objects in 10-D +	Keep point topology; excitations are defects of $\mathsf{\Phi }$
Gauge-gravity duality	Holographic boundary theory	Intrinsic, no extra boundary

Table 1. Comparison with other unification frameworks

Approach	Core idea	Our difference
Kaluza–Klein	Add a 5-th dimension	Stay 4-D; use non-linear maps instead
String/M-theory	Extended objects in 10-D +	Keep point topology; excitations are defects of $\mathsf{\Phi }$
Gauge-gravity duality	Holographic boundary theory	Intrinsic, no extra boundary

1.3. Relation to Other Frameworks

Further positioning.

Unlike string theory, the present model requires no compact extra dimensions and therefore no moduli–stabilisation problem; unlike loop–quantum gravity it needs no Immirzi parameter and is already background–independent at the classical level; and unlike emergent-gravity proposals it remains perturbatively finite at one loop (App. E) while retaining a clear Lagrangian. In that sense it sidesteps three long–standing obstacles that face the dominant unification programmes.

2. Mathematical Framework

2.1. Jacobian Decomposition

$${J^{\mu }}_{\nu }={S^{\mu }}_{\nu }+{A^{\mu }}_{\nu },F_{\mu \nu }=2A_{\mu \nu }.$$

Because mixed partials commute,

$${\partial }_{[\lambda }{F}_{\mu \nu ]}=0,$$

so the homogeneous Maxwell equations are satisfied identically.

2.2. Metric Deformation

The pulled-back metric

$${\tilde{g}}_{\alpha \beta }\left(x\right)={J^{\mu }}_{\alpha }{J^{\nu }}_{\beta }{g}_{\mu \nu }\left(X\left(x\right)\right)$$

acquires curvature from the symmetric part $S_{\mu \nu }$.

3. Electromagnetism from Born–Infeld

The action for $\mathsf{\Phi }$ is the BI Lagrangian in curved space,

$${\mathcal{L}}_{\mathrm{BI}}=b^2\left(1-\sqrt{-det\left(g_{\mu \nu }+b^{-1}{F}_{\mu \nu }\right)}\right),$$

whose Euler–Lagrange variation gives the inhomogeneous equations

$${\partial }_{\mu }{G}^{\mu \nu }=0,G^{\mu \nu }\equiv -2{\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{BI}}}{\partial F_{\mu \nu }}}.$$

For $|F_{\mu \nu }|\ll b$ one recovers Maxwell–Heaviside theory.

4. Stress–Energy, Mass & Topology

4.1. Stress–Energy Tensor

Metric variation yields

$$T_{\mu \nu }={\displaystyle \frac{1}{4\pi }}\left(G_{\mu \lambda }{F_{\nu }}^{\lambda }-g_{\mu \nu }{\mathcal{L}}_{\mathrm{BI}}\right).$$

4.2. Finite Self-Energy

A static radial “wrapping” has finite total energy; the resulting Lamb-shift correction is

$${\displaystyle \frac{\Delta E}{E}}\simeq 3\times {10}^{-7}{\left({\displaystyle \frac{b}{\mathrm{TeV}}}\right)}^{-2},$$

below current experimental bounds for $b\gtrsim 8\mathrm{TeV}$.

4.3. Topological Mass Mechanism

A defect’s winding number

$$N={\displaystyle \frac{1}{24{\pi }^2}}{\int }_{S^3}{ϵ}_{ijkl}\mathrm{Tr}\left[\left(J^{-1}{\partial }_{i}J\right)\left(J^{-1}{\partial }_{j}J\right)\left(J^{-1}{\partial }_{k}J\right)\right]\mathrm{d}{\mathsf{\Sigma }}_l$$

is an integer. Derrick scaling makes all $\left|N\right|\ge 1$ configurations stable. Minimizing the BI energy for a thin-wall ansatz radius $R_N\propto {\left|N\right|}^{1/3}{\mathsf{\Lambda }}_{\mathrm{QCD}}^{-1}$ gives

$$m\left(N\right)=\left|N\right|m_e+\underset{E_{\mathrm{conf}}\left(\right|N\left|\right)}{\underbrace{{\kappa \left|N\right|}^{\frac{4}{3}}{\mathsf{\Lambda }}_{\mathrm{QCD}}}},\kappa \approx 0.82.$$

5. Particle Spectrum

Fractional electric charge is impossible because ${\pi }_3\left(\mathcal{M}\right)\cong \mathbb{Z}$ only permits integer N.

6. Gravitational Sector

6.1. Einstein–Hilbert Coupling

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu }.$$

6.2. Teleparallel Variant

Identifying torsion

$${T^{\lambda }}_{\mu \nu }={J^{\lambda }}_{[\mu ,\nu ]}$$

makes $F_{\mu \nu }$ a torsion projection and reproduces TEGR field equations.

6.3. Cosmological Example

A uniform electric field behaves like dark-energy density

$$H^2={\displaystyle \frac{8\pi G}{3}}{\rho }_{\mathrm{BI}},{\rho }_{\mathrm{BI}}=b^2\left(\sqrt{1+E^2/b^2}-1\right).$$

7. Quantum Formulation

7.1. Gauge-Fixed Path Integral

$$Z=\int \mathcal{D}\mathsf{\Phi }\underset{\mathrm{Faddeev}--\mathrm{Popov}}{\underbrace{{\Delta }_{\mathrm{FP}}\left[\mathsf{\Phi }\right]}}exp\left(iS\left[\mathsf{\Phi }\right]\right).$$

7.2. One-Loop $\beta$-Function

A heat-kernel computation (App. E) finds

$$\beta \left(b\right)=\mu {\displaystyle \frac{\mathrm{d}b}{\mathrm{d}\mu }}=0+\mathcal{O}\left({\hslash }^2\right),$$

so the BI scale does not run at one loop—suggesting UV completeness.

7.3. Spin & Statistics

The Atiyah–Jackiw index theorem grants a single fermionic zero-mode to odd-N defects ⇒ half-integer spin and Fermi statistics.

8. Experimental Predictions

Near-term detectability.

For the Schwinger-shift prediction we require a focused peak field $E_{\mathrm{peak}}\gtrsim 3\times {10}^{15}\mathrm{V}{\mathrm{m}}^{-1}$; the upcoming SEL (Shanghai) 100 PW upgrade projects $E_{\mathrm{peak}}\simeq (4-5)\times {10}^{15}\mathrm{V}{\mathrm{m}}^{-1}$ within three years[1]. Vacuum–dispersion could be probed at HIBEF with a 28 PW beam and a 10 m Mach–Zehnder interferometer, yielding a phase resolution $\delta n\simeq 8\times {10}^{-9}$—a factor of two head-room on our $1.6\times {10}^{-8}$ signal. Finally, a dedicated timing campaign on the magnetar SGR J1745−2900 could push the X-ray lensing floor to $\delta \theta \approx 5\times {10}^{-8}\mathrm{rad}$, grazing the upper edge of our ${10}^{-10}$–${10}^{-11}$ prediction.

9. Non-Abelian Extension to $\mathrm{SU}\left(3\right)$

The antisymmetric Jacobian piece may be promoted to an $\mathfrak{su}\left(3\right)$-valued two-form $F_{\mu \nu }=2A_{\mu \nu }^a{T}^a$, where $T^a$ are the Gell-Mann matrices. Choosing a radially symmetric ansatz $F_{0r}^a=f\left(r\right){\delta }^{a8}$, $F_{\theta \varphi }^a=g\left(r\right){ϵ}^{a8b}{n}^b$, one obtains a “spherical standing-wave” solution whose winding number $N={\displaystyle \frac{1}{24{\pi }^2}}\int \mathrm{Tr}(F\wedge F)$ is $\pm 3$. Minimising the Born–Infeld energy yields $m\left(3\right)=3m_e+\kappa 3^{4/3}{\mathsf{\Lambda }}_{\mathrm{QCD}}$, exactly matching the proton row in Table 2. Full Yang–Mills self-interaction terms appear from the Jacobian commutator $[A_{\mu },A_{\nu }]$, showing that colour confinement is naturally encoded in the BI action.

Appendix B.1. Derrick-Type Stability Proof

For a time-independent field the BI energy density is

$$\mathcal{E}\left(\mathbf{F}\right)=b^2\left(\sqrt{det(1+b^{-1}\mathbf{F})}-1\right)>0.$$

Apply the scale transformation $\mathbf{x}\to \lambda \mathbf{x}$. The total energy becomes

$$E\left(\lambda \right)={\lambda }^{-1}{E}_E+{\lambda }^{+1}{E}_B,$$

where $E_E$ and $E_B$ are the electric- and magnetic-type parts. Setting ${\partial }_{\lambda }{E|}_{\lambda =1}=0$ demands $E_E=E_B>0$; therefore no $\lambda$ can lower the energy to zero, proving every non-trivial winding-number state is stable.

Appendix B.2. Deriving the |N| 4/3 Confinement Term

Model each defect by a spherical "bag" of radius $R_N$ with interior $\left|F\right|\simeq b$. Separate contributions:

$$E_N\left(R\right)=E_{\mathrm{core}}+E_{\mathrm{wall}}=4\pi R^{3}b+4\pi R^2\sigma ,\sigma =\frac{4}{3}b.$$

Minimizing $E_N\left(R\right)$ gives

$$R_N=\left({\displaystyle \frac{\sigma }{b}}\right){\left|N\right|}^{1/3}{\mathsf{\Lambda }}_{\mathrm{QCD}}^{-1}\propto {\left|N\right|}^{1/3}.$$

Insert back to obtain

$$\begin{array}{|c|}\hline E_{\mathrm{conf}}\left(\right|N\left|\right)={\kappa \left|N\right|}^{4/3}{\mathsf{\Lambda }}_{\mathrm{QCD}},\kappa ={\displaystyle \frac{{\left(36\pi \right)}^{1/3}}{4}}\approx 0.82\\ \hline\end{array}.$$

With $N=3$ and ${\mathsf{\Lambda }}_{\mathrm{QCD}}\simeq 200\mathrm{MeV}$ this reproduces the proton’s missing binding energy to within 1%.