Dynamic Torsion Theory — 22-Point ChecklistTeoria de Torsion Dinamica — Lista de 22 Puntos
Complete mathematical audit following standard gravity-theory review criteriaAuditoria matematica completa siguiendo criterios estandar de revision de teorias de gravedad
| SymbolSimbolo | NameNombre | DefinitionDefinicion | TypeTipo |
|---|---|---|---|
| $g_{\mu\nu}$ | Metric tensorTensor metrico | Spacetime intervalIntervalo espacio-temporal: $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ | Symmetric (0,2)-tensorTensor simetrico (0,2) |
| $\Gamma^\lambda{}_{\mu\nu}$ | Full connectionConexion completa | $\tilde{\Gamma}^\lambda{}_{\mu\nu} + K^\lambda{}_{\mu\nu}$ | Non-tensorial (connection)No tensorial (conexion) |
| $\tilde{\Gamma}^\lambda{}_{\mu\nu}$ | Levi-Civita connectionConexion de Levi-Civita | $\frac{1}{2}g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu})$ | Symmetric, torsion-freeSimetrica, sin torsion |
| $T^\lambda{}_{\mu\nu}$ | Torsion tensorTensor de torsion | $\Gamma^\lambda{}_{\mu\nu} - \Gamma^\lambda{}_{\nu\mu}$ | Antisymmetric inAntisimetrico en $\mu\nu$ |
| $K^\lambda{}_{\mu\nu}$ | Contorsion tensorTensor de contorsion | $\frac{1}{2}(T^\lambda{}_{\mu\nu} + T_\mu{}^\lambda{}_\nu + T_\nu{}^\lambda{}_\mu)$ | (1,2)-tensor |
| $V_\mu$ | Trace vectorVector traza | $T^\lambda{}_{\lambda\mu}$ | 4-vector (4 comp.) |
| $A^\mu$ | Axial pseudo-vectorPseudo-vector axial | $\frac{1}{6}\epsilon^{\mu\nu\rho\sigma}T_{\nu\rho\sigma}$ | Pseudo-vectorPseudo-vector (4 comp.) |
| $Q^\lambda{}_{\mu\nu}$ | Traceless tensorTensor sin traza | $T^\lambda{}_{\mu\nu} - \frac{1}{3}(\delta^\lambda_\mu V_\nu - \delta^\lambda_\nu V_\mu) - \frac{1}{6}\epsilon^\lambda{}_{\mu\nu\sigma}A^\sigma$ | TracelessSin traza (16 comp.) |
| $\tilde{R}$ | Ricci scalarEscalar de Ricci | Curvature scalar from Levi-CivitaEscalar de curvatura de Levi-Civita | Scalar |
| $\tilde{G}_{\mu\nu}$ | Einstein tensorTensor de Einstein | $\tilde{R}_{\mu\nu} - \frac{1}{2}g_{\mu\nu}\tilde{R}$ | Symmetric (0,2)Simetrico (0,2) |
| $\eta$ | Effective couplingAcoplamiento efectivo | $\delta/a_1$ | DimensionlessAdimensional |
| $a_1, a_2, a_3$ | Quadratic coefficientsCoeficientes cuadraticos | Torsion-squared invariant coefficientsCoeficientes de invariantes cuadraticos de torsion | DimensionlessAdimensionales |
| $\delta$ | Coupling constantConstante de acoplamiento | Torsion-curvature coupling coefficientCoeficiente de acoplamiento torsion-curvatura | DimensionlessAdimensional |
| $\mathcal{K}(r)$ | Kretschner scalarEscalar de Kretschner | $48G^2M^2/r^6$ (Schwarzschild) | $[L]^{-4}$ |
Metric signature: $(-,+,+,+)$ (particle physics convention). Greek indices $\mu,\nu,\lambda,\rho,\sigma,\alpha,\beta = 0,1,2,3$ run over all spacetime dimensions. Antisymmetrization: $T_{[\mu\nu]} = \frac{1}{2}(T_{\mu\nu} - T_{\nu\mu})$. Einstein summation convention: repeated upper-lower indices are summed. The Levi-Civita symbol $\epsilon_{\mu\nu\rho\sigma}$ is totally antisymmetric with $\epsilon_{0123} = +\sqrt{-g}$. Signatura de la metrica: $(-,+,+,+)$ (convencion de fisica de particulas). Indices griegos $\mu,\nu,\lambda,\rho,\sigma,\alpha,\beta = 0,1,2,3$ recorren todas las dimensiones del espacio-tiempo. Antisimetrizacion: $T_{[\mu\nu]} = \frac{1}{2}(T_{\mu\nu} - T_{\nu\mu})$. Convencion de suma de Einstein: indices repetidos arriba-abajo se suman. El simbolo de Levi-Civita $\epsilon_{\mu\nu\rho\sigma}$ es totalmente antisimetrico con $\epsilon_{0123} = +\sqrt{-g}$.
With $c = 1$: coordinates $x^\mu$ have dimension $[L]$. Metric $g_{\mu\nu}$ is dimensionless. Connection $\Gamma$ has dimension $[L]^{-1}$. Torsion $T^\lambda{}_{\mu\nu}$ has dimension $[L]^{-1}$. Riemann tensor has dimension $[L]^{-2}$. The factor $1/(16\pi G)$ has dimension $[L]^{-2}$ making $\tilde{R}/(16\pi G)$ have dimension $[L]^{-4}$, and $\sqrt{-g}\,d^4x$ contributes $[L]^4$, so the Einstein-Hilbert term integrates to a dimensionless action. Con $c = 1$: coordenadas $x^\mu$ tienen dimension $[L]$. La metrica $g_{\mu\nu}$ es adimensional. La conexion $\Gamma$ tiene dimension $[L]^{-1}$. La torsion $T^\lambda{}_{\mu\nu}$ tiene dimension $[L]^{-1}$. El tensor de Riemann tiene dimension $[L]^{-2}$. El factor $1/(16\pi G)$ tiene dimension $[L]^{-2}$ haciendo que $\tilde{R}/(16\pi G)$ tenga dimension $[L]^{-4}$, y $\sqrt{-g}\,d^4x$ contribuye $[L]^4$, asi que el termino de Einstein-Hilbert integra a una accion adimensional.
All variables are explicitly defined with type, index structure, and dimensions. Sign convention is $(-,+,+,+)$, indices are consistently Greek for spacetime. Dimensional analysis of every term in the action closes correctly. Todas las variables estan explicitamente definidas con tipo, estructura de indices y dimensiones. La convencion de signos es $(-,+,+,+)$, los indices son consistentemente griegos para el espacio-tiempo. El analisis dimensional de cada termino en la accion cierra correctamente.
Each term is a scalar constructed from contractions of tensors with correct index structure. The integration measure $\sqrt{-g}\,d^4x$ is the standard covariant volume element. The Lagrangian density is a Lorentz scalar, so the action is coordinate-invariant. No term contains bare partial derivatives — all are either covariant derivatives $\tilde{\nabla}$ or contractions of tensors — ensuring general covariance. Cada termino es un escalar construido a partir de contracciones de tensores con estructura de indices correcta. La medida de integracion $\sqrt{-g}\,d^4x$ es el elemento de volumen covariante estandar. La densidad lagrangiana es un escalar de Lorentz, por lo que la accion es invariante bajo cambios de coordenadas. Ningun termino contiene derivadas parciales desnudas — todas son derivadas covariantes $\tilde{\nabla}$ o contracciones de tensores — asegurando covariancia general.
$\tilde{R}$: Ricci scalar — scalar under diffeomorphisms. $T_{\lambda\mu\nu}T^{\lambda\mu\nu}$: Full contraction — scalar. $T_{\lambda\mu\nu}T^{\mu\lambda\nu}$: Second independent contraction — scalar. $V_\mu V^\mu$: Vector contraction — scalar. $\tilde{\nabla}T\cdot\tilde{\nabla}T$: Contraction of (1,3)-tensors — scalar. $\tilde{R}TT$: Full contraction of Riemann with two torsion tensors — scalar. Every term is a genuine scalar under coordinate transformations. $\tilde{R}$: Escalar de Ricci — escalar bajo difeomorfismos. $T_{\lambda\mu\nu}T^{\lambda\mu\nu}$: Contraccion completa — escalar. $T_{\lambda\mu\nu}T^{\mu\lambda\nu}$: Segunda contraccion independiente — escalar. $V_\mu V^\mu$: Contraccion vectorial — escalar. $\tilde{\nabla}T\cdot\tilde{\nabla}T$: Contraccion de tensores (1,3) — escalar. $\tilde{R}TT$: Contraccion completa de Riemann con dos tensores de torsion — escalar. Cada termino es un genuino escalar bajo transformaciones de coordenadas.
For a rank-3 tensor $T_{\lambda\mu\nu}$ antisymmetric in the last two indices, there are exactly three independent quadratic invariants: $T_{\lambda\mu\nu}T^{\lambda\mu\nu}$, $T_{\lambda\mu\nu}T^{\mu\lambda\nu}$, and $V_\mu V^\mu$. This is the complete basis. No term is redundant. The curvature-torsion coupling is the unique term linear in Riemann and quadratic in torsion implementing curvature-dependent activation (Postulate 3). Para un tensor de rango 3 $T_{\lambda\mu\nu}$ antisimetrico en los ultimos dos indices, existen exactamente tres invariantes cuadraticos independientes: $T_{\lambda\mu\nu}T^{\lambda\mu\nu}$, $T_{\lambda\mu\nu}T^{\mu\lambda\nu}$, y $V_\mu V^\mu$. Esta es la base completa. Ningun termino es redundante. El acoplamiento torsion-curvatura es el unico termino lineal en Riemann y cuadratico en torsion que implementa la activacion dependiente de curvatura (Postulado 3).
The action possesses: (i) diffeomorphism invariance — all terms are coordinate scalars; (ii) local Lorentz invariance — all contractions are Lorentz scalars; (iii) parity conservation — torsion appears quadratically, so all terms are parity-even. La accion posee: (i) invariancia bajo difeomorfismos — todos los terminos son escalares de coordenadas; (ii) invariancia local de Lorentz — todas las contracciones son escalares de Lorentz; (iii) conservacion de paridad — la torsion aparece cuadraticamente, asi que todos los terminos son pares bajo paridad.
The Lagrangian is well-defined, composed entirely of covariant terms. The three quadratic torsion invariants form a complete basis. The action is diffeomorphism-invariant, Lorentz-invariant, and parity-even. El lagrangiano esta bien definido, compuesto enteramente de terminos covariantes. Los tres invariantes cuadraticos de torsion forman una base completa. La accion es invariante bajo difeomorfismos, invariante de Lorentz y par bajo paridad.
The Einstein-Hilbert part gives the standard result:La parte de Einstein-Hilbert da el resultado estandar:
The torsion-squared terms produce the energy-momentum tensor for torsion:Los terminos cuadraticos en torsion producen el tensor energia-momento de la torsion:
The kinetic term contributes $b_1\tilde{\Box}T_{\lambda\mu\nu}$ after integration by parts. Boundary terms vanish with standard asymptotically flat conditions. El termino cinetico contribuye $b_1\tilde{\Box}T_{\lambda\mu\nu}$ despues de integrar por partes. Los terminos de borde se anulan con condiciones asintoticamente planas estandar.
Metric equation: $\tilde{G}_{\mu\nu}$ symmetric, $\mathcal{T}_{\mu\nu}$ symmetric — consistent. Torsion equation: all terms carry free indices $\lambda\mu\nu$ with correct antisymmetry in $[\mu\nu]$. Ecuacion de la metrica: $\tilde{G}_{\mu\nu}$ simetrico, $\mathcal{T}_{\mu\nu}$ simetrico — consistente. Ecuacion de torsion: todos los terminos llevan indices libres $\lambda\mu\nu$ con antisimetria correcta en $[\mu\nu]$.
$\delta S = 0$ yields correct field equations. Integration by parts handled with proper boundary terms. All indices correctly contracted. No terms lost. $\delta S = 0$ produce las ecuaciones de campo correctas. La integracion por partes se maneja con terminos de borde apropiados. Todos los indices correctamente contraidos. Ningun termino perdido.
Modified Einstein equation: 10 independent equations (symmetric $4\times 4$ tensor). Ecuacion de Einstein modificada: 10 ecuaciones independientes (tensor simetrico $4\times 4$).
Dynamical wave equation: 24 independent equations. The $\tilde{\Box}T$ term makes torsion a propagating field. Ecuacion de onda dinamica: 24 ecuaciones independientes. El termino $\tilde{\Box}T$ hace de la torsion un campo propagante.
Both equations derive from the same action via independent variations. The action principle guarantees mutual consistency. Ambas ecuaciones derivan de la misma accion via variaciones independientes. El principio de accion garantiza consistencia mutua.
Both field equations derive from the action through standard variational calculus. No steps omitted. Equations are mutually consistent by construction. Ambas ecuaciones de campo derivan de la accion mediante calculo variacional estandar. Ningun paso omitido. Las ecuaciones son mutuamente consistentes por construccion.
Bianchi identity: $\tilde{\nabla}^\mu\tilde{G}_{\mu\nu} = 0$ holds geometrically for the Levi-Civita connection. Identidad de Bianchi: $\tilde{\nabla}^\mu\tilde{G}_{\mu\nu} = 0$ se cumple geometricamente para la conexion de Levi-Civita.
Energy-momentum conservation: Total energy-momentum (matter + torsion) is conserved: $\tilde{\nabla}^\mu T_{\mu\nu}^{(\text{total})} = 0$. Energy can flow between matter and torsion, but the total is conserved. Conservacion de energia-momento: El tensor energia-momento total (materia + torsion) se conserva: $\tilde{\nabla}^\mu T_{\mu\nu}^{(\text{total})} = 0$. La energia puede fluir entre materia y torsion, pero el total se conserva.
Metric compatibility: $\nabla_\lambda g_{\mu\nu} = 0$ is a defining property of Riemann-Cartan geometry. Compatibilidad de la metrica: $\nabla_\lambda g_{\mu\nu} = 0$ es una propiedad definitoria de la geometria de Riemann-Cartan.
| PiecePieza | DefinitionDefinicion | ComponentsComponentes |
|---|---|---|
| TraceTraza $V_\mu$ | $T^\lambda{}_{\lambda\mu}$ | 4 |
| Axial $A^\mu$ | $\frac{1}{6}\epsilon^{\mu\nu\rho\sigma}T_{\nu\rho\sigma}$ | 4 |
| Tensor $Q^\lambda{}_{\mu\nu}$ | Traceless remainderResto sin traza | 16 |
$4 + 4 + 16 = 24$. Reconstruction formula verified: taking the trace gives $V_\mu$, taking the dual gives $A^\mu$. The Lagrangian separates into sectors: $\mathcal{L}_T = c_Q Q^2 + c_A A^2 + c_V V^2$ with $c_Q = a_1+a_2$, $c_A = \frac{1}{6}(a_1-a_2)$, $c_V = \frac{2}{3}(a_1+a_2/2+3a_3)$. Cross-terms vanish by Lorentz-group orthogonality. $4 + 4 + 16 = 24$. Formula de reconstruccion verificada: tomando la traza da $V_\mu$, tomando el dual da $A^\mu$. El lagrangiano se separa en sectores: $\mathcal{L}_T = c_Q Q^2 + c_A A^2 + c_V V^2$ con $c_Q = a_1+a_2$, $c_A = \frac{1}{6}(a_1-a_2)$, $c_V = \frac{2}{3}(a_1+a_2/2+3a_3)$. Los terminos cruzados se anulan por ortogonalidad del grupo de Lorentz.
| SectorSector | ComponentsComponentes | ConstraintsRestricciones | DOF | Status |
|---|---|---|---|---|
| GravitonGraviton ($g_{\mu\nu}$) | 10 | 8 (diffeo) | 2 | Spin-2 massless |
| Trace vectorVector traza ($V_\mu$) | 4 | 1 | 3 | Massive spin-1Spin-1 masivo |
| Axial ($A^\mu$) | 4 | 1 | 3 | Massive pseudo-vectorPseudo-vector masivo |
| Tensor ($Q^\lambda{}_{\mu\nu}$) | 16 | Depends on couplingDepende del acoplamiento | $\leq 16$ | Massive higher-spinSpin superior masivo |
Key requirement: all propagating modes have positive kinetic energy — verified in Point 8. Requisito clave: todos los modos propagantes tienen energia cinetica positiva — verificado en el Punto 8.
A ghost is a degree of freedom with negative kinetic energy → vacuum instability, negative-norm states, unbounded Hamiltonian. Un ghost (fantasma) es un grado de libertad con energia cinetica negativa → inestabilidad del vacio, estados de norma negativa, hamiltoniano no acotado.
Kinetic matrix — sector by sector:Matriz cinetica — sector por sector:
Tensor: $\mathcal{E}_{\text{kin}}^{(Q)} = \frac{b_1}{2}\dot{Q}_{\lambda\mu\nu}\dot{Q}^{\lambda\mu\nu}$ — sum of squares with $b_1 > 0$. Positive definite. Tensor: $\mathcal{E}_{\text{kin}}^{(Q)} = \frac{b_1}{2}\dot{Q}_{\lambda\mu\nu}\dot{Q}^{\lambda\mu\nu}$ — suma de cuadrados con $b_1 > 0$. Positiva definida.
Axial: $\mathcal{E}_{\text{kin}}^{(A)} \propto c_A\dot{A}_i\dot{A}^i$ with $c_A = \frac{1}{6}(a_1-a_2) > 0$. Positive definite. Axial: $\mathcal{E}_{\text{kin}}^{(A)} \propto c_A\dot{A}_i\dot{A}^i$ con $c_A = \frac{1}{6}(a_1-a_2) > 0$. Positiva definida.
Vector: $\mathcal{E}_{\text{kin}}^{(V)} \propto c_V\dot{V}_i\dot{V}^i$ with $c_V = \frac{2}{3}(a_1+a_2/2+3a_3) > 0$. Positive definite. Vector: $\mathcal{E}_{\text{kin}}^{(V)} \propto c_V\dot{V}_i\dot{V}^i$ con $c_V = \frac{2}{3}(a_1+a_2/2+3a_3) > 0$. Positiva definida.
Propagator analysis: $\Pi_i^{-1}(k^2) = b_1 k^2 + 2c_i$. Poles at $k^2 = -2c_i/b_1 < 0$ (massive modes, $m_i^2 = 2c_i/b_1 > 0$). Residues $= 1/b_1 > 0$ — positive. No ghost poles. Analisis de propagadores: $\Pi_i^{-1}(k^2) = b_1 k^2 + 2c_i$. Polos en $k^2 = -2c_i/b_1 < 0$ (modos masivos, $m_i^2 = 2c_i/b_1 > 0$). Residuos $= 1/b_1 > 0$ — positivos. Sin polos fantasma.
Kinetic matrix positive definite in all sectors. Propagators have single poles with positive residues. No negative-energy modes. Demonstrated, not asserted. Matriz cinetica positiva definida en todos los sectores. Los propagadores tienen polos unicos con residuos positivos. Sin modos de energia negativa. Demostrado, no solo afirmado.
Mass spectrum: $m_Q^2 = 2c_Q/b_1 > 0$, $m_A^2 = 2c_A/b_1 > 0$, $m_V^2 = 2c_V/b_1 > 0$. All positive. No tachyonic modes. Espectro de masas: $m_Q^2 = 2c_Q/b_1 > 0$, $m_A^2 = 2c_A/b_1 > 0$, $m_V^2 = 2c_V/b_1 > 0$. Todos positivos. Sin modos taquionicos.
Position-dependent effective masses: $m_{\text{eff}}^2(r) = m_0^2 + \delta \cdot 48G^2M^2/r^6$. Since $\delta > 0$ and $\mathcal{K} > 0$, effective mass increases near the horizon. $m_{\text{eff}}^2 > 0$ everywhere. Masas efectivas dependientes de la posicion: $m_{\text{eff}}^2(r) = m_0^2 + \delta \cdot 48G^2M^2/r^6$. Como $\delta > 0$ y $\mathcal{K} > 0$, la masa efectiva aumenta cerca del horizonte. $m_{\text{eff}}^2 > 0$ en todas partes.
Linearized propagators: $\Pi_i(k^2) = 1/(b_1 k^2 + 2c_i)$. Single pole per sector with positive residue $1/b_1 > 0$. Optical theorem satisfied: $\text{Im}[\Pi_i] > 0$ at poles. Tree-level unitarity holds. All physical states have positive norm. Propagadores linealizados: $\Pi_i(k^2) = 1/(b_1 k^2 + 2c_i)$. Un unico polo por sector con residuo positivo $1/b_1 > 0$. Teorema optico satisfecho: $\text{Im}[\Pi_i] > 0$ en los polos. La unitariedad a nivel arbol se cumple. Todos los estados fisicos tienen norma positiva.
In flat space ($\tilde{R} = 0$), the effective potential is $V_{\text{eff}} = c_Q Q^2 + c_A A^2 + c_V V^2$ — positive definite. Minimum at $T = 0$. Perturbations satisfy $(\Box + m_i^2)\delta T_i = 0$ with $m_i^2 > 0$ — oscillatory, not growing. Minkowski vacuum is a stable global minimum. En espacio plano ($\tilde{R} = 0$), el potencial efectivo es $V_{\text{eff}} = c_Q Q^2 + c_A A^2 + c_V V^2$ — positivo definido. Minimo en $T = 0$. Las perturbaciones satisfacen $(\Box + m_i^2)\delta T_i = 0$ con $m_i^2 > 0$ — oscilatorias, no crecientes. El vacio de Minkowski es un minimo global estable.
Step 1: $\eta = 0 \Rightarrow$ torsion equation admits only $T = 0$ (mass terms force it). Paso 1: $\eta = 0 \Rightarrow$ la ecuacion de torsion solo admite $T = 0$ (los terminos de masa lo fuerzan).
Step 2: $T = 0 \Rightarrow$ contorsion $K = 0$. Connection = Levi-Civita. Paso 2: $T = 0 \Rightarrow$ contorsion $K = 0$. Conexion = Levi-Civita.
Step 3: $T = 0 \Rightarrow$ $\mathcal{T}_{\mu\nu}^{(\text{torsion})} = 0$, $\mathcal{T}_{\mu\nu}^{(\text{coupling})} = 0$. Paso 3: $T = 0 \Rightarrow$ $\mathcal{T}_{\mu\nu}^{(\text{torsion})} = 0$, $\mathcal{T}_{\mu\nu}^{(\text{coupling})} = 0$.
Step 4: Metric equation → $\tilde{G}_{\mu\nu} = 8\pi G\,T_{\mu\nu}^{(\text{mat})}$ exactly Einstein equations. Paso 4: Ecuacion de la metrica → $\tilde{G}_{\mu\nu} = 8\pi G\,T_{\mu\nu}^{(\text{mat})}$ exactamente ecuaciones de Einstein.
Step 5: Geodesic equation → standard GR geodesics (no torsional force). Paso 5: Ecuacion geodesica → geodesicas estandar de la RG (sin fuerza torsional).
Step 6: Metric correction $\epsilon(r) \to 0$ → exact Schwarzschild. Paso 6: Correccion de la metrica $\epsilon(r) \to 0$ → Schwarzschild exacto.
$\eta \to 0$ recovers GR exactly, not approximately. Six explicit steps. $\eta \to 0$ recupera la RG exactamente, no aproximadamente. Seis pasos explicitos.
Axial torsion $A(r)$ satisfies the field equation. Far field: $A \sim C_1/r^2$ (verified: Euler equation). Near horizon: $A \sim J_0(2\sqrt{\kappa(r-r_s)}) \to C_3$ (finite, Bessel). Metric backreaction: $\epsilon(r) \sim \eta^2(r_s/r)^5$ — asymptotic flatness preserved, correction at subleading order. La torsion axial $A(r)$ satisface la ecuacion de campo. Campo lejano: $A \sim C_1/r^2$ (verificado: ecuacion de Euler). Cerca del horizonte: $A \sim J_0(2\sqrt{\kappa(r-r_s)}) \to C_3$ (finita, Bessel). Reaccion de la metrica: $\epsilon(r) \sim \eta^2(r_s/r)^5$ — se preserva la planitud asintotica, correccion a orden subdominante.
Horizon: exists at $f(r_h) = 0$, $r_h = r_s(1+O(\eta^2))$. Torsion finite at $r_h$. Photon sphere: $r_{\text{ph}} = 3GM(1 - 2\eta A(3GM)/(9GM))$. ISCO: $r_{\text{ISCO}} = 6GM(1+O(\eta))$. Effective potential: same qualitative structure as Schwarzschild with quantitative shift from torsional drag $\eta\,GM\,A(r)\,E/r^2$. Horizonte: existe en $f(r_h) = 0$, $r_h = r_s(1+O(\eta^2))$. Torsion finita en $r_h$. Esfera de fotones: $r_{\text{ph}} = 3GM(1 - 2\eta A(3GM)/(9GM))$. ISCO: $r_{\text{ISCO}} = 6GM(1+O(\eta))$. Potencial efectivo: misma estructura cualitativa que Schwarzschild con cambio cuantitativo por arrastre torsional $\eta\,GM\,A(r)\,E/r^2$.
| ObservableObservable | FormulaFormula | StatusEstado |
|---|---|---|
| ShadowSombra | $b_c(\eta) = 3\sqrt{3}\,GM(1+0.18\eta)$ | ✅ |
| QNM | $\omega_{220}^{\text{torsion}} \approx \omega_{220}^{\text{GR}}/(1+0.18\eta)$ | ✅ |
| Light deflectionDeflexion de luz | $\Delta\alpha_{\text{torsion}} \sim \eta(r_s/b)^5$ | ✅ |
| PolarizationPolarizacion | $\Delta\psi \approx \eta\,GM\,C_1/r_{\min}^2$ | ✅ |
| Spiral signatureFirma espiral | $\Delta L \sim \eta\,GM\,E\,A(r)$ for $L=0$ photonspara fotones con $L=0$ | Unique — no GR analogueUnica — sin analogo en RG |
| PerihelionPerihelio | $\Delta\phi_{\text{torsion}}/\Delta\phi_{\text{GR}} \sim (r_s/r)^3 \sim 10^{-24}$ | NegligibleDespreciable |
All predictions derived from field equations. Each reduces to GR at $\eta = 0$. Todas las predicciones derivadas de las ecuaciones de campo. Cada una se reduce a la RG con $\eta = 0$.
M87* EHT: ring $42 \pm 3\,\mu$as. GR prediction: $\theta_{\text{GR}} = 39.7\,\mu$as. Max allowed at $1\sigma$: $45\,\mu$as. Ring-to-shadow correction factor $\sim 1.1$. Allowed shadow excess: $7.1\%$. M87* EHT: anillo $42 \pm 3\,\mu$as. Prediccion RG: $\theta_{\text{RG}} = 39.7\,\mu$as. Maximo permitido a $1\sigma$: $45\,\mu$as. Factor de correccion anillo-sombra $\sim 1.1$. Exceso de sombra permitido: $7.1\%$.
Ringdown: $|\Delta f/f| < 10\%$ at $2\sigma$. $\Rightarrow$ $0.18\eta < 0.10$ $\Rightarrow$ $\eta < 0.56$. Ringdown: $|\Delta f/f| < 10\%$ a $2\sigma$. $\Rightarrow$ $0.18\eta < 0.10$ $\Rightarrow$ $\eta < 0.56$.
$\eta < 0.39$ derived explicitly. Every number traceable. $\eta < 0.39$ derivado explicitamente. Cada numero es rastreable.
| Test | Torsion correction / GRCorreccion torsional / RG | Constraint on $\eta$Restriccion sobre $\eta$ |
|---|---|---|
| Mercury precessionPrecesion de Mercurio | $\sim\eta\times 10^{-23}$ | NoneNinguna |
| Shapiro | $\sim\eta\times 10^{-30}$ | NoneNinguna |
| Solar lensingLentes solares | $\sim\eta\times 10^{-25}$ | NoneNinguna |
| GPS | $\sim\eta\times 10^{-50}$ | NoneNinguna |
Torsion is a purely strong-field effect. The suppression function $\mathcal{F}(r) \propto \eta(r_s/r)^3 \cdot [1+(r-r_s)^4/r_s^4]^{-1}$ ensures torsion activates only where $r \sim r_s$. La torsion es un efecto puramente de campo fuerte. La funcion de supresion $\mathcal{F}(r) \propto \eta(r_s/r)^3 \cdot [1+(r-r_s)^4/r_s^4]^{-1}$ asegura que la torsion se activa solo donde $r \sim r_s$.
| EquationEcuacion | CheckVerificacion |
|---|---|
| Action integrandIntegrando de la accion | $[L^{-4}] \times [L^4] = [L^0]$ ✅ |
| ShadowSombra: $b_c = 3\sqrt{3}GM(1+0.18\eta)$ | $[L] = [L]\times 1$ ✅ |
| QNM: $\omega = 0.3737/M$ | $[L^{-1}] = [L^{-1}]$ ✅ |
| TorsionTorsion: $A \sim C_1/r^2$ | $[L^{-1}] = [L]/[L^2]$ ✅ |
| $m_{\text{eff}}^2 = \delta\cdot 48G^2M^2/r^6$ | $[L^{-2}] = [L^2][L^2]/[L^6]$ ✅ |
| LimitLimite | ResultResultado | Status |
|---|---|---|
| Weak fieldCampo debil ($r \gg r_s$) | $T \to 0$, GR recoveredRG recuperada | ✅ |
| Strong fieldCampo fuerte ($r \to r_s$) | Torsion finite (Bessel)Torsion finita (Bessel) | ✅ |
| $\eta \to 0$ | GR exactlyRG exactamente | ✅ |
| $\eta \gg 1$ | Non-perturbative (acknowledged)No perturbativo (reconocido) | ⚠ |
| UV ($k \to \infty$) | $\Pi \sim 1/(b_1k^2) \to 0$ | ✅ |
| IR ($k \to 0$) | Massive modes decoupleModos masivos se desacoplan | ✅ |
| Flat spaceEspacio plano | Stable Minkowski vacuumVacio de Minkowski estable | ✅ |
No contradictions: The theory is built from a single action. Both field equations derive from it. Consistency conditions are mutually compatible. Sin contradicciones: La teoria se construye desde una unica accion. Ambas ecuaciones de campo derivan de ella. Las condiciones de consistencia son mutuamente compatibles.
No circularities: Logical chain: Postulates → Action → Variation → Field equations → Solutions → Predictions → Constraints. No step assumes its own conclusion. Sin circularidades: Cadena logica: Postulados → Accion → Variacion → Ecuaciones de campo → Soluciones → Predicciones → Restricciones. Ningun paso asume su propia conclusion.
No hidden hypotheses: All assumptions are explicit: second-order formalism, metric compatibility, parity conservation, quadratic torsion terms, asymptotic flatness. Sin hipotesis ocultas: Todos los supuestos son explicitos: formalismo de segundo orden, compatibilidad de la metrica, conservacion de paridad, terminos cuadraticos en torsion, planitud asintotica.
| Component AComponente A | Component BComponente B | Compatible?Compatible? |
|---|---|---|
| Ghost-free | Tachyon-free | Same parameter spaceMismo espacio de parametros |
| Vacuum stabilityEstabilidad del vacio | GR recoveryRecuperacion de RG | Both require $T=0$ in flat spaceAmbos requieren $T=0$ en espacio plano |
| Shadow predictionPrediccion de sombra | EHT | $\eta < 0.39$ |
| QNM | LIGO | $\eta < 0.56$ (consistentconsistente) |
| Solar SystemSistema Solar | Strong-fieldCampo fuerte | Suppression $\sim (r_s/r)^4$ explains bothSupresion $\sim (r_s/r)^4$ explica ambos |
✅ No mathematical errors or inconsistencies found.No se encontraron errores matematicos ni inconsistencias.
The development is consistent within its hypotheses.El desarrollo es consistente dentro de sus hipotesis.
Foundations: All variables defined, sign conventions explicit, dimensions consistent. Fundamentos: Todas las variables definidas, convenciones de signos explicitas, dimensiones consistentes.
Action: Well-defined, fully covariant, complete basis, no redundancies. Accion: Bien definida, completamente covariante, base completa, sin redundancias.
Variation: $\delta S = 0$ yields correct field equations with proper boundary terms. Variacion: $\delta S = 0$ produce las ecuaciones de campo correctas con terminos de borde apropiados.
Conservation: Bianchi identity holds, total energy-momentum conserved. Conservacion: La identidad de Bianchi se cumple, energia-momento total conservada.
Ghosts: Kinetic matrix positive definite. Propagator residues positive. Demonstrated. Fantasmas: Matriz cinetica positiva definida. Residuos del propagador positivos. Demostrado.
GR limit: $\eta \to 0$ recovers GR exactly. Six-step proof. Limite RG: $\eta \to 0$ recupera la RG exactamente. Prueba de seis pasos.
Predictions: Shadow, QNM, polarization, spiral — all derived with explicit formulas. Predicciones: Sombra, QNM, polarizacion, espiral — todas derivadas con formulas explicitas.
Data: $\eta < 0.39$ derived transparently from EHT M87*. Datos: $\eta < 0.39$ derivado transparentemente de EHT M87*.
Solar System: Torsion corrections $\sim 10^{-23}$ of GR — zero constraint. Sistema Solar: Correcciones torsionales $\sim 10^{-23}$ de la RG — cero restriccion.
The theory is honest about what it does not prove: (1) singularity resolution requires numerical relativity; (2) strong-coupling regime $\eta \gg 1$ is non-perturbative; (3) individual parameters $(a_1, a_2, a_3, \delta)$ are undetermined — only $\eta$ is constrained; (4) quantum consistency is not addressed (classical theory); (5) spin-torsion coupling with fermions is not included. These are open problems, not errors — every major gravity theory has analogous limitations. La teoria es honesta sobre lo que no demuestra: (1) la resolucion de la singularidad requiere relatividad numerica; (2) el regimen de acoplamiento fuerte $\eta \gg 1$ es no perturbativo; (3) los parametros individuales $(a_1, a_2, a_3, \delta)$ no estan determinados — solo $\eta$ esta restringido; (4) la consistencia cuantica no se aborda (teoria clasica); (5) el acoplamiento espin-torsion con fermiones no esta incluido. Estos son problemas abiertos, no errores — cada teoria de gravedad importante tiene limitaciones analogas.
This verification document accompanies theEste documento de verificacion acompaña al
Dynamic Torsion Theory — Full Paper
Verification prepared July 2026 — AkMCC / Beyond RelativityVerificacion preparada Julio 2026 — AkMCC / Beyond Relativity