Spinors with torsion and matter$-$antimatter asymmetry

The paper proposes that torsion in the Einstein–Cartan theory of gravity creates a mass difference between matter and antimatter, potentially explaining the observed matter–antimatter asymmetry by suggesting that more massive antimatter particles were preferentially captured by primordial black holes in the early Universe.

The Gist

The Cosmic Filter: How Spacetime’s "Twist" Saved the Universe

Imagine you are at a massive, high-speed cosmic dance party in the very early universe. At this party, everything is created in perfect pairs: for every "Matter" dancer, there is an "Antimatter" dancer. They are identical twins, moving with the same energy and rhythm.

In a perfect world, these twins would eventually bump into each other, cancel each other out in a flash of light, and leave the universe empty and dark. But our universe isn't empty; it’s full of stars, planets, and people. This means something happened to break the tie. Something "filtered" the antimatter out of the crowd.

Nikodem Popławski’s paper proposes a fascinating explanation for this mystery, and the secret lies in a hidden property of space itself called Torsion.

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1. The Secret Ingredient: Torsion (The "Twist" in the Fabric)

In standard Einsteinian gravity, space is like a smooth, flexible trampoline. If you put a bowling ball on it, it curves. This curvature is what we call gravity.

However, Popławski uses a more advanced version called Einstein–Cartan theory. In this version, space isn't just a smooth trampoline; it’s more like a piece of fabric that can also be twisted. This "twistiness" is called Torsion.

While normal gravity is caused by mass, torsion is caused by spin. Since every subatomic particle (like electrons and quarks) is spinning, they actually "twist" the fabric of spacetime around them.

2. The Weight Difference: The "Heavy Twin" Problem

Here is where the magic happens. Popławski shows that when you add this "twist" (torsion) into the math of the Dirac equation (the rulebook for particles), the rules change for matter and antimatter.

In a "twisted" spacetime, matter and antimatter no longer feel the same. Specifically, antimatter becomes slightly heavier than matter.

The Analogy: Imagine two identical runners on a track. One is wearing lightweight sneakers (Matter), and the other is wearing heavy lead boots (Antimatter). They were born at the same time, but because of their "gear" (the torsion interaction), one is much harder to move than the other.

3. The Great Capture: The Cosmic Vacuum Cleaners

Now, let's go back to that early universe dance party. The universe was filled with tiny, growing "Black Holes"—think of them as cosmic vacuum cleaners.

Because the antimatter particles were "heavier" due to the torsion, they moved more slowly than the lighter matter particles. In the world of gravity, slower objects are much easier to catch.

The Analogy: Imagine a leaf and a pebble being blown toward a vacuum cleaner. The light, fast-moving leaf might zip past the nozzle, but the heavy, slow-moving pebble is much more likely to get sucked straight in.

Because the antimatter was "heavy and slow," the primordial black holes sucked them up at a much higher rate. The matter particles, being "light and fast," managed to zip past the black holes and survive.

4. The Result: A Matter-Filled Universe

Once the black holes had "cleaned up" most of the antimatter, the remaining antimatter collided with the surviving matter, creating a massive burst of light (photons). What was left behind was a universe dominated by matter—the stuff we are made of.

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Summary: The Big Picture

Popławski’s theory provides a "one-stop shop" for several massive cosmic mysteries:

• Why is there stuff? Because torsion made antimatter heavy and easy for black holes to swallow.
• Why did the universe expand so fast (Inflation)? The "twistiness" of space at high densities acts like a repulsive force, pushing the universe outward like a spring being released.
• Where did we come from? He suggests our universe might actually be the "inside" of a black hole from a "parent" universe, born from a "Big Bounce" rather than a "Big Bang."

In short: The universe exists because spacetime has a twist, and that twist turned the antimatter into heavy targets for the universe's first black holes.

Technical Summary

Technical Summary: Spinors with Torsion and Matter–Antimatter Asymmetry

Author: Nikodem Popławski
Core Framework: Einstein–Cartan (EC) Theory of Gravity

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1. The Problem: The Matter–Antimatter Asymmetry

A fundamental mystery in cosmology is why the observable Universe is composed almost entirely of matter, despite the fact that the Standard Model suggests matter and antimatter should have been produced in equal amounts during the Big Bang. Most theories attempt to explain this via "Baryogenesis" through new physics (violating baryon number conservation). This paper seeks to explain the asymmetry using existing gravitational frameworks—specifically by addressing how spacetime torsion affects the fundamental properties of fermions.

2. Methodology: Einstein–Cartan Theory and Nonlinear Dirac Equations

The author moves beyond General Relativity (GR) by employing Einstein–Cartan theory, which extends GR to include torsion ($S^k_{ij}$). In GR, the affine connection is symmetric; in EC theory, the antisymmetric part (torsion) is coupled to the intrinsic spin angular momentum of matter.

Mathematical Derivation:

• Nonlinear Dirac Equation: The author derives that in the presence of torsion, the Dirac equation becomes a nonlinear, cubic equation in the spinor field $\psi$. This is because the torsion tensor is proportional to the spin tensor, which is itself quadratic in the spinor field.
• Hamiltonian Analysis: By expressing the Dirac equation in Hamiltonian form and diagonalizing the resulting $4 \times 4$ Hermitian matrix, the author derives a quartic equation for the energy eigenvalues $\lambda$.
• Rest Frame Analysis: By setting momentum $p=0$, the author solves for the energy states, revealing that the energy eigenvalues for a particle at rest are split by a term involving the coupling constant $k$ and the spin pseudovector $j$.

3. Key Contributions and Results

A. Violation of C and CP Symmetry:
The most significant mathematical result is that the energy eigenvalues (and thus the effective masses) of fermions and antifermions are no longer identical.

• Matter particles prefer a lower energy state with an effective mass $M_- = m - k|j|/c^2$.
• Antimatter particles prefer a state with an effective mass $M_+ = m + k|j|/c^2$.
  Because the mass depends on whether the particle is matter or antimatter, Charge Conjugation (C) symmetry is violated. Furthermore, because the energy depends on helicity, CP symmetry is also violated.

B. Mass-Velocity-Capture Relation:
The author demonstrates that this mass difference is density-dependent, becoming significant near the Cartan density ($\rho_C \approx 10^{51} \text{ kg/m}^3$) present in the early Universe.

• Since antimatter particles are effectively more massive ($M_+ > M_-$), they move more slowly than matter particles during pair production in the early Universe.
• In gravitational physics, slower-moving particles have a higher gravitational capture cross-section ($\sigma$) when interacting with black holes.

C. The Mechanism of Asymmetry:
The author calculates the relative difference in capture cross-sections ($\Delta\sigma/\sigma_q$). The result suggests that in a Universe populated by primordial black holes, antimatter would be preferentially captured by these black holes due to its higher mass and lower velocity. The "missing" antimatter is thus sequestered within black holes, leaving a surplus of matter to form the observable Universe.

4. Significance and Implications

• Baryogenesis without New Physics: Unlike many models that require the introduction of new particles or forces to violate baryon number, this mechanism relies on the geometric properties of spacetime (torsion) and the existing Standard Model of particle physics. It satisfies all three Sakharov conditions:
  1. Baryon number violation: Effectively achieved by the asymmetric removal of antimatter via black holes.
  2. C and CP violation: Provided by the torsion-spin coupling.
  3. Departure from thermal equilibrium: Provided by the unidirectional flow of particles through black hole event horizons.
• Cosmological Evolution: The paper links this asymmetry to other cosmological phenomena. The repulsive nature of the spin-torsion coupling provides a mechanism for cosmic inflation and suggests a "Big Bounce" origin for the Universe (where our Universe originated from a black hole in a parent Universe), avoiding the singularity problem of standard GR.
• Observational Potential: The author suggests that the mass asymmetry might be detectable in high-density experiments or through the observed spin asymmetry in protons.