Torsion is assumed to be zero in general relativity as part of the Einstein-Cartan hypothesis and standard Levi-Civita connection.
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The retrieved physics literature and reference materials indicate that standard general relativity employs the Levi-Civita connection where torsion is set to zero, whereas extensions such as the Einstein-Cartan theory incorporate non-zero torsion to account for properties like particle spin.
Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist best known for developing the theory of relativity. Einstein also made important contributions to quantum theory. His mass–energy equivalence formula E = mc2, which arises from special relativity, has been called "the world's most famous equation". He received the 1921 Nobel Prize in Physics for "his services t
At thirteen, when his range of enthusiasms had broadened to include music and philosophy, Talmud introduced Einstein to Kant's Critique of Pure Reason. Kant became his favorite philosopher; according to Talmud, "At the time he was still a child, only thirteen years old, yet Kant's works, incomprehensible to ordinary mortals, seemed to be clear to him."
In 1895, at the age of sixteen, Einstein sat the entrance examination for the federal polytechnic school (later the Eidgenössische Technische Hochschule, ETH) in Zurich, Switzerland. He failed to reach the required standard in the general part of the test, but performed with distinction in physics and mathematics. On the advice of the polytechnic's principal, he completed his secondary education at the Argovian cantonal school (a gymnasium) in Aarau, Switzerland, graduating in 1896. While lodging in Aarau with the family of Jost Winteler, he fell in love with Winteler's daughter, Marie. (His sister, Maja, later married Winteler's son Paul.)
In order to incorporate spinning point particles into general relativity, the affine connection needed to be generalized to include an antisymmetric part, called the torsion. This modification was made by Einstein and Cartan in the 1920s.
This work aims to autonomously revisit some puncta salientia of the Einstein–Cartan (EC) theory, focusing wholly on the mathematical aspect, or, better still, emphasizing the differential geometry underlying the theory under examination, without the burden of sensible experiences (experiments) of Galilean heritage. It is shown that it is possible to describe, or rather, derive an Einsteinian-like gravitational field starting from a Cartan $\mathfrak{h}$-subalgebra, and thus produce a couple of formulæ for a torsioning in a $(1 + 3)$-dimensional manifold. Some Cartan $k$-forms and $\mathcal{J}$
Abstract I construct an Einstein–Cartan ekpyrotic model (ECEM): a homogeneous, nearly Friedmann–Lemaître–Robertson–Walker (FLRW) background in Einstein–Cartan (EC) gravity whose spin–torsion sector, modeled phenomenologically as a Weyssenhoff fluid with stiff scaling $$\rho _s\propto a^{-6}$$ ρ s ∝ a - 6 , is coupled to a scalar field with a steep exponential potential that interpolates between a negative ekpyrotic branch and a positive plateau. Extending the Copeland–Liddle–Wands (CLW) scalar–fluid dynamical system to a six-dimensional phase space including shear, curvature, and spin–torsion, I recast the equations in a compact deceleration-parameter form, compute the full Jacobian, and evaluate maximal Lyapunov exponents. Numerical solutions show that the ekpyrotic branch ( $$w_\phi \gg 1$$ w ϕ ≫ 1 ) exponentially damps homogeneous shear, while the softened branch ( $$w_\phi <1$$ w ϕ < 1 ) allows $$\rho _s$$ ρ s to overtake the scalar during contraction and trigger a torsion-supported bounce at high but finite densities where the EC spin–torsion term becomes dynamically dominant. Scans in a two-parameter softening plane $$(\phi _\textrm{b},\Delta )$$ ( ϕ b , Δ ) identify a finite region of nonsingular trajectories and quantify the required tuning; in the parameter ranges explored the maximal Lyapunov exponent on the constrained phase space is negative, giving no indication of chaotic behavior in this homogeneous truncation even when the usual curvature mode that destabilizes contracting General Relativity (GR) backgrounds is included. The construction is purely phenomenological and confined to homogeneous backgrounds: it does not address entropy accumulation, the cosmological arrow of time, or a complete cyclic cosmology.
⚠️ Notice of Model Update: > The Resonant Lattice Model (RLM) framework presented in this preprint has been officially updated and superseded by the Resonant Vacuum Condensate (RVC) model. To view the complete, unified theoretical framework and its updated cosmological mechanics, please refer to the master RVC publication here: [ https://doi.org/10.5281/zenodo.18869120 ] This document serves as the formal mathematical addendum to the Resonant Lattice Model (RLM). It provides the explicit derivations integrating geometric topological impedance into the Einstein Field Equations via Cartan torsio
Abstract We investigated the cosmology of F ( R ) gravity rebuilt with the Cartan formalism. This is called Cartan F ( R ) gravity. The well-known F ( R ) gravity has been introduced to extend the standard cosmology, e.g., to explain the cosmological accelerated expansion as inflation. Cartan F ( R ) gravity is based on the Riemann-Cartan geometry. The curvature R is separated into two parts, one is derived from the Levi-Civita connection and the other from the torsion. Assuming a matter-independent spin connection, we have successfully rewritten the action of Cartan F ( R ) gravity into the E
We use astrophysical data to shed light on fundamental physics by constraining parametrized theoretical cosmological and gravitational models. Gravitational parameters are those constants that parametrize possible departures from Einstein's general theory of relativity (GR). We develop a general framework to describe torsion in the space time around the Earth, and show that certain observables of the Gravity Probe B (GPB) experiment can be computed in this framework. We examine a toy model showing how a specific theory in this framework can be constrained by GPB data. We also search for viable
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