by AkMCC
Complete theory: action, field equations, consistency, and predictions Teoria completa: accion, ecuaciones de campo, consistencia y predicciones
Postulate 1 (Geometric regularity):Postulado 1 (Regularidad geometrica): The physical geometry of spacetime must remain regular for all physically realizable configurations. If a geometric description leads to singularities, this indicates the absence of one or more relevant geometric degrees of freedom. La geometria fisica del espacio-tiempo debe permanecer regular para todas las configuraciones fisicamente realizables. Si una descripcion geometrica conduce a singularidades, esto indica la ausencia de uno o mas grados de libertad geometricos relevantes.
Postulate 2 (Torsion hypothesis):Postulado 2 (Hipotesis de torsion): Dynamic torsion constitutes the missing geometric degree of freedom whose incorporation regulates the extreme-curvature regime. La torsion dinamica constituye el grado de libertad geometrico faltante cuya incorporacion regula el regimen de curvatura extrema.
Postulate 3 (Geometric organization):Postulado 3 (Organizacion geometrica): Curvature concentrates energy; torsion organizes the flow of that energy within the geometry of spacetime. La curvatura concentra energia; la torsion organiza el flujo de esa energia dentro de la geometria del espacio-tiempo.
In a Riemann-Cartan manifold, the affine connection is not necessarily symmetric: En una variedad de Riemann-Cartan, la conexion afin no es necesariamente simetrica:
where $\tilde{\Gamma}$ is the Levi-Civita connection (symmetric, determined by the metric) and $K$ is the contorsion tensor: donde $\tilde{\Gamma}$ es la conexion de Levi-Civita (simetrica, determinada por la metrica) y $K$ es el tensor de contorsion:
The torsion tensor is defined as the antisymmetric part of the connection: El tensor de torsion se define como la parte antisimetrica de la conexion:
Under the Lorentz group, the 24 independent components of $T^\lambda{}_{\mu\nu}$ decompose into three irreducible pieces: Bajo el grupo de Lorentz, las 24 componentes independientes de $T^\lambda{}_{\mu\nu}$ se descomponen en tres piezas irreducibles:
Trace vectorVector traza (4 componentscomponentes): $V_\mu = T^\lambda{}_{\lambda\mu}$
Axial pseudo-vectorPseudo-vector axial (4 componentscomponentes): $A^\mu = \frac{1}{6}\epsilon^{\mu\nu\rho\sigma} T_{\nu\rho\sigma}$
Traceless tensorTensor sin traza (16 componentscomponentes): $Q^\lambda{}_{\mu\nu} = T^\lambda{}_{\mu\nu} - \frac{1}{3}(\delta^\lambda_\mu V_\nu - \delta^\lambda_\nu V_\mu) - \frac{1}{6}\epsilon^\lambda{}_{\mu\nu\sigma} A^\sigma$
The full torsion is reconstructed as:La torsion completa se reconstruye como:
where $\tilde{R}$ is the Ricci scalar constructed with the Levi-Civita connection (torsion-free). donde $\tilde{R}$ es el escalar de Ricci construido con la conexion de Levi-Civita (sin torsion).
The three independent quadratic invariants:Los tres invariantes cuadraticos independientes:
In terms of irreducible pieces:En terminos de las piezas irreducibles:
Effective coefficients for each sector: Tensor: $c_Q = a_1 + a_2$, Axial: $c_A = \frac{1}{6}(a_1 - a_2)$, Vector: $c_V = \frac{2}{3}(a_1 + \frac{a_2}{2} + 3a_3)$ Coeficientes efectivos para cada sector: Tensor: $c_Q = a_1 + a_2$, Axial: $c_A = \frac{1}{6}(a_1 - a_2)$, Vector: $c_V = \frac{2}{3}(a_1 + \frac{a_2}{2} + 3a_3)$
For torsion to be a propagating field (not an algebraic multiplier), we include the derivative kinetic term: Para que la torsion sea un campo propagante (no un multiplicador algebraico), incluimos el termino cinetico derivativo:
This term implements Postulate 3: torsion activates where curvature is extreme. Este termino implementa el Postulado 3: la torsion se activa donde la curvatura es extrema.
A ghost is a degree of freedom with negative kinetic energy, leading to vacuum instability. For the theory to be ghost-free, the kinetic energy of each irreducible sector must be positive definite: Un fantasma es un grado de libertad con energia cinetica negativa, que conduce a inestabilidad del vacio. Para que la teoria este libre de fantasmas, la energia cinetica de cada sector irreducible debe ser positiva definida:
The coupling $\mathcal{L}_{RT}$ generates position-dependent effective masses: El acoplamiento $\mathcal{L}_{RT}$ genera masas efectivas dependientes de la posicion:
No tachyons requires $\delta > 0$, ensuring $m_{\text{eff}}^2 > 0$ everywhere outside the horizon. La ausencia de taquiones requiere $\delta > 0$, asegurando $m_{\text{eff}}^2 > 0$ en todo punto fuera del horizonte.
The effective potential around Minkowski ($R_{\alpha\beta\mu\nu} = 0$) is $V_{\text{eff}} = c_Q Q^2 + c_A A^2 + c_V V^2$. With ghost-free conditions, the minimum is $T = 0$. The Minkowski vacuum is stable: torsion vanishes in flat space. GR is recovered exactly in the weak-field limit. El potencial efectivo alrededor de Minkowski ($R_{\alpha\beta\mu\nu} = 0$) es $V_{\text{eff}} = c_Q Q^2 + c_A A^2 + c_V V^2$. Con las condiciones de ausencia de fantasmas, el minimo es $T = 0$. El vacio de Minkowski es estable: la torsion se anula en espacio plano. La RG se recupera exactamente en el limite de campo debil.
Linearized propagators $\Pi_i^{-1}(k^2) = c_i k^2$ with $c_i > 0$ have positive residues. The theory is unitary at tree level. Los propagadores linealizados $\Pi_i^{-1}(k^2) = c_i k^2$ con $c_i > 0$ tienen residuos positivos. La teoria es unitaria a nivel arbol.
| ConditionCondicion | RequirementRequisito | StatusEstado |
|---|---|---|
| Ghost-free (tensor)Sin fantasmas (tensor) | $a_1 + a_2 > 0$ | SatisfiedCumplido |
| Ghost-free (axial)Sin fantasmas (axial) | $a_1 > a_2$ | SatisfiedCumplido |
| Ghost-free (vector)Sin fantasmas (vector) | $a_1 + a_2/2 + 3a_3 > 0$ | SatisfiedCumplido |
| Tachyon-freeSin taquiones | $\delta > 0$ | SatisfiedCumplido |
| Vacuum stabilityEstabilidad del vacio | $c_Q, c_A, c_V > 0$ | SatisfiedCumplido |
| Unitarity (tree)Unitariedad (arbol) | $c_i > 0$ | SatisfiedCumplido |
| GR recoveryRecuperacion de RG | $T \to 0$ when $R \to 0$ | SatisfiedCumplido |
Quadratic torsion contribution:Contribucion cuadratica de torsion:
Defining $\eta = \delta / a_1$ as the effective coupling parameter: Definiendo $\eta = \delta / a_1$ como el parametro de acoplamiento efectivo:
In the limit $\eta \to 0$: torsion decouples from curvature and the trivial solution $T = 0$ is the only one. GR is recovered. En el limite $\eta \to 0$: la torsion se desacopla de la curvatura y la solucion trivial $T = 0$ es la unica. Se recupera la RG.
In the limit $\eta \gg 1$: torsion is strongly coupled to curvature. The regime is non-perturbative. En el limite $\eta \gg 1$: la torsion esta fuertemente acoplada a la curvatura. El regimen es no perturbativo.
By spherical symmetry: $V_t = V_t(r)$, $V_r = V_r(r)$, and axial component $A^\phi = A(r)/(r\sin\theta)$. The axial component $A(r)$ is responsible for the helical structure. Por simetria esferica: $V_t = V_t(r)$, $V_r = V_r(r)$, y componente axial $A^\phi = A(r)/(r\sin\theta)$. La componente axial $A(r)$ es responsable de la estructura helicoidal.
where $\mathcal{K}(r) = 48\,G^2M^2/r^6$ is the Kretschner scalar. donde $\mathcal{K}(r) = 48\,G^2M^2/r^6$ es el escalar de Kretschner.
Far fieldCampo lejano ($r \to \infty$):
Near the horizonCerca del horizonte ($r \to r_s^+$):
Key result:Resultado clave: Axial torsion is finite and maximal in a shell around the horizon of thickness $\Delta r \sim r_s$, and decays as $1/r^2$ far from it. No divergence at the horizon — consistent with Postulate 1. La torsion axial es finita y maxima en una capa alrededor del horizonte de espesor $\Delta r \sim r_s$, y decae como $1/r^2$ lejos de el. Sin divergencia en el horizonte — consistente con el Postulado 1.
The Schwarzschild metric is modified only at fifth powers of $r_s/r$: undetectable at normal distances, significant near the horizon. La metrica de Schwarzschild se modifica solo a la quinta potencia de $r_s/r$: indetectable a distancias normales, significativa cerca del horizonte.
The right side is an effective force produced by torsion. El lado derecho es una fuerza efectiva producida por la torsion.
The term $\eta\, GM\, A(r)\, E / r^2$ is the torsional drag. It generates effective angular momentum even for photons with $L = 0$. El termino $\eta\, GM\, A(r)\, E / r^2$ es el arrastre torsional. Genera momento angular efectivo incluso para fotones con $L = 0$.
With torsion, the correction is:Con torsion, la correccion es:
| ObservableObservable | GR (Schwarzschild) | Dynamic TorsionTorsion Dinamica |
|---|---|---|
| Shadow shapeForma de sombra | CircularCircular | Spiral / asymmetricEspiral / asimetrica |
| Shadow radiusRadio de sombra | $3\sqrt{3}\,GM$ | $3\sqrt{3}\,GM(1 + \eta\,\xi)$ |
| AsymmetryAsimetria | 0 | $\sim\eta\,A(3GM)$ |
For the two black holes observed by the Event Horizon Telescope: Para los dos agujeros negros observados por el Event Horizon Telescope:
M87*: $M = 6.5 \times 10^9\,M_\odot$, $D = 16.8$ Mpc | Sgr A*: $M = 4.0 \times 10^6\,M_\odot$, $D = 8.178$ kpc
| $\eta$ | $b_c / M$ | $\Delta_{\text{shadow}}$ | $\theta_{\text{M87*}}$ ($\mu$as) | $\theta_{\text{Sgr A*}}$ ($\mu$as) | Status |
|---|---|---|---|---|---|
| 0 | 5.196 | 0% | 39.7 | 50.2 | GR (Einstein) |
| 0.05 | 5.243 | 0.90% | 40.1 | 50.6 | Sub-EHT |
| 0.10 | 5.290 | 1.80% | 40.4 | 51.1 | Sub-EHT |
| 0.20 | 5.383 | 3.60% | 41.1 | 52.0 | Within EHTDentro de EHT $1\sigma$ |
| 0.24 | 5.421 | 4.32% | 41.4 | 52.4 | Simulator defaultDefault del simulador |
| 0.30 | 5.477 | 5.40% | 41.8 | 52.9 | Marginally detectableMarginalmente detectable |
| 0.40 | 5.570 | 7.20% | 42.6 | 53.8 | Excluded by M87*Excluido por M87* |
| 0.50 | 5.664 | 9.00% | 43.3 | 54.7 | Excluded by M87*Excluido por M87* |
| 0.60 | 5.757 | 10.80% | 44.0 | 55.6 | Excluded by M87*Excluido por M87* |
| 0.80 | 5.944 | 14.40% | 45.4 | 57.4 | Excluded by bothExcluido por ambos |
M87* (EHT 2019, updated 2017-2021): Measured bright ring diameter is $42 \pm 3\,\mu$as (first observation), with a multi-epoch average of $43.9 \pm 0.6\,\mu$as. GR prediction for shadow diameter is $\theta_{\text{GR}} = 39.7\,\mu$as. M87* uncertainty imposes $\Delta_{\text{shadow}} < 7.1\%$: M87* (EHT 2019, actualizado 2017-2021): El diametro del anillo brillante medido es $42 \pm 3\,\mu$as (primera observacion), con un promedio multi-epoca de $43.9 \pm 0.6\,\mu$as. La prediccion de la RG para el diametro de la sombra es $\theta_{\text{GR}} = 39.7\,\mu$as. La incertidumbre de M87* impone $\Delta_{\text{shadow}} < 7.1\%$:
Sgr A* (EHT 2022): Shadow diameter $48.7 \pm 7\,\mu$as gives $\eta < 0.80$. The dominant constraint comes from M87*. Sgr A* (EHT 2022): Diametro de sombra $48.7 \pm 7\,\mu$as da $\eta < 0.80$. La restriccion dominante viene de M87*.
Axial torsion rotates the polarization plane of photons passing near the horizon: La torsion axial rota el plano de polarizacion de los fotones que pasan cerca del horizonte:
This rotation is additional to plasma Faraday effect and has a different radial dependence. Esta rotacion es adicional al efecto Faraday del plasma y tiene una dependencia radial diferente.
The fundamental QNM mode ($\ell = 2$, $n = 0$) of Schwarzschild: El modo fundamental QNM ($\ell = 2$, $n = 0$) de Schwarzschild:
With torsion, the real frequency scales as: Con torsion, la frecuencia real escala como:
| $\eta$ | Frequency shiftCambio de frecuencia $\Delta f / f$ | $\delta\omega_{220}$ |
|---|---|---|
| 0.05 | $-0.9\%$ | $-0.0034/M$ |
| 0.10 | $-1.8\%$ | $-0.0066/M$ |
| 0.20 | $-3.5\%$ | $-0.0128/M$ |
| 0.24 | $-4.1\%$ | $-0.0153/M$ |
| 0.40 | $-6.7\%$ | $-0.0243/M$ |
LIGO-Virgo-KAGRA constraint: Recent ringdown analyses (2024-2025) obtain $\delta\omega_{220}/\omega_{220} = -0.05 \pm 0.05$, giving $|\Delta f / f| < 10\%$ at $2\sigma$: Restriccion LIGO-Virgo-KAGRA: Analisis recientes de ringdown (2024-2025) obtienen $\delta\omega_{220}/\omega_{220} = -0.05 \pm 0.05$, dando $|\Delta f / f| < 10\%$ a $2\sigma$:
Strongest prediction of the theory:Prediccion mas fuerte de la teoria: A photon with initial angular momentum $L = 0$ (radial incidence) acquires angular momentum $\Delta L \sim \eta\, GM\, E\, A(r)$ when passing near the horizon. This does not occur in any GR solution, including Kerr. Un foton con momento angular inicial $L = 0$ (incidencia radial) adquiere momento angular $\Delta L \sim \eta\, GM\, E\, A(r)$ al pasar cerca del horizonte. Esto no ocurre en ninguna solucion de la RG, incluyendo Kerr.
If photons initially radial were observed to acquire angular deflection near a non-rotating black hole, this would be direct evidence of dynamic torsion. Si se observara que fotones inicialmente radiales adquieren deflexion angular cerca de un agujero negro sin rotacion, esto seria evidencia directa de torsion dinamica.
The torsional suppression function decays as $1/r^2$, with the curvature coupling introducing additional suppression: La funcion de supresion torsional decae como $1/r^2$, con el acoplamiento de curvatura introduciendo supresion adicional:
For the Sun ($r_s = 2.95$ km) at Mercury's orbit ($r = 5.79 \times 10^7$ km): Para el Sol ($r_s = 2.95$ km) en la orbita de Mercurio ($r = 5.79 \times 10^7$ km):
Even for $\eta = 1$, the torsional correction to Mercury's precession would be $\sim 10^{-30}$ times the GR contribution. Solar System tests do not constrain $\eta$ at all. Torsion is a purely strong-field effect. Incluso para $\eta = 1$, la correccion torsional a la precesion de Mercurio seria $\sim 10^{-30}$ veces la contribucion de la RG. Las pruebas del Sistema Solar no restringen $\eta$ en absoluto. La torsion es un efecto puramente de campo fuerte.
We explicitly demonstrate that $\eta \to 0$ recovers GR: Demostramos explicitamente que $\eta \to 0$ recupera la RG:
1. Torsion field equation (§5.2): With $\eta = 0$, the only solution is $T = 0$.Ecuacion de campo de torsion (§5.2): Con $\eta = 0$, la unica solucion es $T = 0$.
2. Einstein equation (§5.1): With $T = 0$, reduces to $\tilde{G}_{\mu\nu} = 8\pi G\, T_{\mu\nu}^{(\text{mat})}$ exactly.Ecuacion de Einstein (§5.1): Con $T = 0$, se reduce a $\tilde{G}_{\mu\nu} = 8\pi G\, T_{\mu\nu}^{(\text{mat})}$ exactamente.
3. Geodesics (§7.1): With $T = 0$, contorsion $K = 0$ and autoparallels are Levi-Civita geodesics.Geodesicas (§7.1): Con $T = 0$, contorsion $K = 0$ y las autoparalelas son geodesicas de Levi-Civita.
4. Metric (§6.5): With $\epsilon(r) = 0$, exact Schwarzschild is recovered.Metrica (§6.5): Con $\epsilon(r) = 0$, se recupera Schwarzschild exacto.
| Observational sourceFuente observacional | ConstraintRestriccion | ConfidenceConfianza |
|---|---|---|
| Mercury precessionPrecesion de Mercurio | No constraintSin restriccion ($\Delta < 10^{-30}$) | — |
| Solar light deflectionDeflexion de luz solar | No constraintSin restriccion ($\Delta < 10^{-30}$) | — |
| EHT M87* (shadowsombra) | $\eta < 0.39$ | $1\sigma$ |
| EHT Sgr A* (shadowsombra) | $\eta < 0.80$ | $1\sigma$ |
| LIGO-Virgo-KAGRA (QNM) | $\eta < 0.56$ | $2\sigma$ |
| CombinedCombinado | $\eta < 0.39$ | $1\sigma$ |
1. The theory is classical. It does not address torsion quantization. 1. La teoria es clasica. No aborda la cuantizacion de la torsion.
2. Field equations are implicit (torsion appears on both sides). Iterative or numerical methods required. 2. Las ecuaciones de campo son implicitas (la torsion aparece en ambos lados). Se requieren metodos iterativos o numericos.
3. Parameters $(a_1, a_2, a_3, \delta)$ are not determined by the theory — they must be fixed by observation. 3. Los parametros $(a_1, a_2, a_3, \delta)$ no estan determinados por la teoria — deben fijarse por observacion.
4. Full nonlinear stability requires numerical relativity simulations with torsion. 4. La estabilidad no lineal completa requiere simulaciones de relatividad numerica con torsion.
5. Coupling with matter (fermion spin) introduces additional terms not considered here. 5. El acoplamiento con materia (espin fermionico) introduce terminos adicionales no considerados aqui.
Dynamic Torsion Theory:Teoria de Torsion Dinamica:
Is mathematically consistent (ghost-free, tachyon-free, unitary at tree level). Es matematicamente consistente (sin fantasmas, sin taquiones, unitaria a nivel arbol).
Recovers GR exactly in the weak-field limit ($\eta \to 0$) and at astrophysical distances ($r \gg r_s$). Recupera la RG exactamente en el limite de campo debil ($\eta \to 0$) y a distancias astrofisicas ($r \gg r_s$).
Modifies geometry only where curvature is extreme (near horizons), with suppression $\sim (r_s/r)^4$ in weak field. Modifica la geometria solo donde la curvatura es extrema (cerca de horizontes), con supresion $\sim (r_s/r)^4$ en campo debil.
Produces quantitatively testable predictions: shadow grows as $(1 + 0.18\,\eta)$, QNM frequencies shift as $1/(1 + 0.18\,\eta)$. Produce predicciones cuantitativamente verificables: la sombra crece como $(1 + 0.18\,\eta)$, las frecuencias QNM cambian como $1/(1 + 0.18\,\eta)$.
Is compatible with all current observations for $\eta < 0.39$ (EHT M87*, $1\sigma$). Es compatible con todas las observaciones actuales para $\eta < 0.39$ (EHT M87*, $1\sigma$).
The value $\eta = 0.24$ used in the visual simulator predicts a 4.3% deviation in shadow size, within EHT error bars. El valor $\eta = 0.24$ usado en el simulador visual predice una desviacion del 4.3% en el tamanio de la sombra, dentro de las barras de error del EHT.
Offers unique signatures distinguishable from GR: helical geodesic structure, torsional drag on radial photons, additional torsional QNM modes. Ofrece firmas unicas distinguibles de la RG: estructura geodesica helicoidal, arrastre torsional sobre fotones radiales, modos QNM torsionales adicionales.
ngEHT: Improved resolution ($\sim 5\,\mu$as) would detect 1–2% deviations, reaching $\eta \sim 0.06$. Resolucion mejorada ($\sim 5\,\mu$as) detectaria desviaciones del 1–2%, alcanzando $\eta \sim 0.06$.
LISA (2037): Supermassive BH QNMs with $\sim 0.1\%$ precision, sensitive to $\eta \sim 0.006$. QNMs de agujeros negros supermasivos con $\sim 0.1\%$ de precision, sensible a $\eta \sim 0.006$.
Einstein Telescope: High-SNR ringdown to search for additional torsional modes. Ringdown de alto SNR para buscar modos torsionales adicionales.
EHT Polarimetry: Detect torsional polarization rotation, distinguishable from Faraday by radial dependence. Detectar rotacion de polarizacion torsional, distinguible del efecto Faraday por dependencia radial.
The question is no longer whether the idea makes sense.
The question is whether nature implements it — and now we have the exact numbers to check.
La pregunta ya no es si la idea tiene sentido.
La pregunta es si la naturaleza la implementa — y ahora tenemos los numeros exactos para verificarlo.
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