⚛️ general relativity

Gauge gravitation theory in Riemann-Cartan space-time and the nonsingular Universe

This paper investigates gauge gravitation theory in Riemann-Cartan space-time to resolve fundamental issues in general relativity, providing constraints and numerical solutions that describe a nonsingular, accelerating Universe across various cosmological models and discussing its astrophysical implications.

Original authors: A. V. Minkevich

Published 2026-01-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: A. V. Minkevich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A Universe Without a "Crunch"

Imagine the standard story of the Big Bang as a movie that starts with a blinding flash of light and a singularity—a point where everything is crushed so tightly that the laws of physics break down. It's like trying to squeeze a whole library into a single grain of sand; eventually, the math says the grain of sand must explode or disappear.

This paper proposes a different movie script. The author, A.V. Minkevich, suggests that the universe doesn't start from a broken point. Instead, it bounces.

The theory used here is called Gauge Gravitation Theory in Riemann–Cartan space-time (GTRC). Think of standard gravity (Einstein's General Relativity) as a trampoline that only bends. GTRC adds a new feature: the trampoline can also twist.

The Twist in the Tale: Torsion

In this theory, space-time has two properties:

Curvature: How much space bends (like a heavy ball on a trampoline).
Torsion: How much space twists (like a corkscrew motion).

Usually, we think of gravity only as a pulling force (attraction). But this paper argues that when matter gets incredibly dense—like in the very first moments of the universe or inside a black hole—this "twist" (torsion) changes the rules.

The Analogy of the Spring:
Imagine a giant spring. If you push down on it gently, it pushes back (attraction). But if you push it down too hard, past a certain limit, the spring doesn't just get harder to push; it suddenly snaps back with a powerful repulsive force.

In this theory, there is a limiting energy density. It's the "maximum compression" point of the universe. You cannot squeeze matter beyond this point. Instead of collapsing into a singularity (a point of infinite density), the "spring" of space-time pushes back. This repulsive force prevents the universe from ever having a "beginning" in time where everything was crushed to nothing.

What Happens at the Beginning?

The paper runs computer simulations to see what happens when the universe is at this maximum compression point.

The Bounce: Instead of starting from a singularity, the universe was likely contracting, hit this "limiting density" wall, and then bounced back into expansion.
The Acceleration: When the universe first started expanding after this bounce, it didn't just expand; it accelerated. This explains why we see the universe speeding up today, but the paper claims this is a natural result of the "twist" in space-time, not a mysterious invisible force called "Dark Energy."
No Dark Energy Needed: In standard cosmology, we need "Dark Energy" to explain why the universe is speeding up. This paper suggests that the "twist" of space-time acts like a vacuum effect that pushes the universe apart, making Dark Energy unnecessary.

The Three Shapes of the Universe

The paper tests this idea on three different shapes of the universe:

Flat: Like an infinite sheet of paper.
Closed: Like the surface of a sphere (finite but no edge).
Open: Like a saddle shape (infinite and curved the other way).

The results show that for all three shapes, the universe avoids the "crunch." Even in a closed universe that might normally collapse back on itself, the "twist" creates a repulsive force that stops the collapse and turns it into an expansion.

Why Does This Matter for Stars?

The paper also mentions that this "twist" affects heavy, spinning objects like stars and galaxies.

The Spinning Top Analogy: Imagine a spinning top. In standard gravity, it just spins. In this theory, the "twist" of space-time interacts with the spin of the object.
Preventing Black Holes: The author suggests that because of this repulsive force at high densities, matter might never collapse enough to form a "singularity" inside a black hole. The collapse would stop and bounce, potentially changing our understanding of what happens inside these cosmic monsters.

The Bottom Line

This paper claims that by adding a "twist" (torsion) to our understanding of gravity, we can solve the biggest headache in cosmology: The Singularity.

Old View: The universe began as a broken point of infinite density.
New View (in this paper): The universe has no beginning or end in time. It cycles through contraction and expansion, bouncing off a "limiting density" wall created by the twisting nature of space-time.

The author concludes that this theory removes the need for "Dark Energy" to explain cosmic acceleration and offers a way to describe the universe without the physically impossible concept of a singularity.

Technical Summary: Gauge Gravitation Theory in Riemann–Cartan Space-time and the Nonsingular Universe

Problem Statement
The paper addresses fundamental limitations of the General Relativity Theory (GR), specifically the inevitability of cosmological singularities (the Big Bang) and gravitational singularities (black holes) where energy density diverges. While the Penrose-Hawking theorems establish the inevitability of these singularities within GR, the author argues that a classical description of the gravitational field should allow for the exclusion of such singular states. The central problem is to determine if a gauge gravitation theory in Riemann–Cartan space-time (GTRC), which incorporates space-time torsion in addition to curvature, can yield regular cosmological solutions that describe a nonsingular, accelerating Universe without invoking dark energy.

Methodology
The study utilizes the Gauge Gravitation Theory in Riemann–Cartan space-time (GTRC), developed as a Poincaré gauge theory. The gravitational field variables are the orthonormal tetrad and the nonholonomic Lorentz connection, with torsion and curvature tensors acting as field strengths. The analysis focuses on homogeneous isotropic cosmological models (HICM) under the assumption of parity conservation.

Key methodological steps include:

Formulation of Equations: The authors derive a system of differential equations (Equations 1–2) based on a gravitational Lagrangian containing scalar curvature and quadratic invariants of curvature and torsion tensors. These equations involve indefinite parameters: $\alpha_G$ (inverse energy density dimension), $\omega$ (dimensionless), and $b$ (same dimension as $f_0$ ).
Source Terms: The theory employs minimal coupling between the gravitational field and matter. In this framework, the sources of the gravitational field are the canonical energy-momentum tensor and the tetrad spin moment tensor. For isotropic cosmology, the average spin moment is zero; thus, torsion is generated solely by the energy-momentum tensor (energy density $\varepsilon$ and pressure $p$ ).
Dimensionless Analysis: To analyze the behavior near the limiting energy density ( $\varepsilon_{max}$ ), the cosmological equations are transformed into dimensionless forms. The study investigates the dynamics of flat ( $k=0$ ), closed ( $k=+1$ ), and open ( $k=-1$ ) models, specifically examining the transition from gravitational compression to expansion.
Numerical Simulation: Numerical solutions are obtained for the Hubble parameter ( $H$ ), its time derivative, and the acceleration parameter near $\varepsilon_{max}$ , considering the equation of state for ultrarelativistic matter ( $p = \varepsilon/3$ ).

Key Contributions and Results
The paper presents the following specific findings:

Existence of a Limiting Energy Density: The analysis demonstrates that under specific constraints on the indefinite parameters (specifically $0 < \omega < 4$ and $0 < 1 - b/f_0 \ll 1$ ), a maximum allowed energy density ( $\varepsilon_{max}$ ) exists. Near this density, the gravitational interaction transitions from attraction to repulsion.
Nonsingular Cosmology: The cosmological solutions are regular. The stage of cosmological contraction is followed by a stage of expansion, eliminating the initial singularity. There are no temporal restrictions on the existence of the Universe in the past or future.
Gravitational Repulsion and Acceleration:
- Early Universe: Near $\varepsilon_{max}$ , the gravitational interaction exhibits a repulsive character. This leads to an early stage of cosmological acceleration without the need for "early dark energy."
- Late Universe (Asymptotics): At low energy densities ( $\varepsilon \ll \varepsilon_{max}$ ), the torsion functions induce an effective cosmological constant ( $\Lambda$ ). This results in gravitational repulsion at cosmological asymptotics, explaining the current acceleration of the Universe as a vacuum effect rather than a dark energy phenomenon.
Numerical Dynamics:
- For flat models, the Hubble parameter vanishes at $\varepsilon_{max}$ and reaches extremal values at specific energy densities ( $\tilde{\varepsilon}_1, \tilde{\varepsilon}_2, \tilde{\varepsilon}_3$ ).
- The transition from compression to expansion occurs at a finite time. For a limiting energy density estimated at two orders of magnitude above neutron star density, the transition time is estimated at $\Delta t \approx 0.8 \times 10^{-3}$ s, corresponding to the quark-gluon plasma era ( $T_{max} \sim 10^{13}$ K).
- Closed and open models exhibit behavior similar to flat models near the limiting density, with fundamental differences only appearing in the asymptotic regime.
Dependence on Parameters: The limiting energy density and characteristic energy scales depend on the parameters $\omega$ and $\alpha_G$ . The parameter $\alpha_G$ acts as a fundamental physical constant determining the scale of the limiting density and the effective cosmological constant.

Significance and Claims
The paper claims that GTRC offers a viable classical framework to resolve the singularity problem inherent in GR. By introducing torsion and a limiting energy density, the theory:

Eliminates Singularities: It excludes the concept of an initial cosmological singularity and divergent energy densities in black holes, replacing them with regular, nonsingular states.
Unifies Acceleration: It provides a unified explanation for both early-universe acceleration and modern cosmological acceleration through the mechanism of torsion-induced gravitational repulsion, removing the necessity for dark energy.
Astrophysical Implications: The theory suggests that the interaction between the spin moments of astrophysical objects (stars, galaxies) and space-time torsion modifies Newton's law of gravitation. This modification could have fundamental implications for the problem of dark matter and the stability of dense astrophysical objects.
Observational Relevance: The author notes that the absence of a temporal beginning and the modified early-universe history could offer corrections to the standard Big Bang scenario, potentially addressing observational puzzles such as the existence of massive bright galaxies in the early Universe observed by the James Webb Space Telescope.

The work concludes that GTRC, as a direct development of GR based on gauge invariance, successfully constructs a regular, accelerating cosmological model, thereby opening new pathways for solving fundamental problems in gravitation theory.