⚛️ general relativity

Vacuum stability in Geometric Trinity of Gravity

This paper demonstrates that the vacuum decay rate in the teleparallel (TEGR) and symmetric teleparallel (STEGR) formulations of gravity yields the same tunneling exponent as General Relativity, confirming that their classical equivalence extends to the quantum level.

Original authors: Vincenzo Branchina, Salvatore Capozziello, Filippo Contino, Carmen Ferrara

Published 2026-06-30

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CC BY 4.0

Original authors: Vincenzo Branchina, Salvatore Capozziello, Filippo Contino, Carmen Ferrara

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 Picture: Three Ways to Describe Gravity

Imagine you are trying to describe how a car moves down a hill.

Standard Gravity (General Relativity or GR): You describe it by saying the road is curved, and the car follows the curve. This is how Einstein described gravity: as the bending of space and time.
Teleparallel Gravity (TEGR): You describe it by saying the road is flat, but the car is being pushed by invisible "twists" or torsion forces.
Symmetric Teleparallel Gravity (STEGR): You describe it by saying the road is flat and untwisted, but the car is being pushed by a "stretching" or non-metricity force.

For a long time, physicists knew that these three descriptions (GR, TEGR, and STEGR) were classically equivalent. It's like describing the same movie using three different languages; the story, the characters, and the ending are exactly the same. They all predict the same planetary orbits and the same bending of light.

The Question: Does the Equivalence Hold at the Quantum Level?

The paper asks a tricky question: Does this equivalence hold when we look at the very smallest, quantum scale?

Specifically, the authors looked at a phenomenon called "False Vacuum Decay."

The Analogy: The Ball in the Valley
Imagine a ball sitting in a small dip on a hillside. This is a "false vacuum." It looks stable, but there is a much deeper valley further down the hill (the "true vacuum").

The Risk: If the ball gets a little quantum "kick," it might tunnel through the hill and roll down to the deeper valley. This is a catastrophic event for the universe (in theory), as it would change the laws of physics.
The Calculation: Physicists calculate how likely this is to happen by measuring the "tunneling exponent." Think of this as a difficulty score. A high score means the ball is very unlikely to roll down (the universe is safe). A low score means it might happen soon.

The authors wanted to know: If we calculate this "difficulty score" using the "curved road" method (GR), the "twisting road" method (TEGR), or the "stretching road" method (STEGR), do we get the same number?

The Investigation: Building the Bridge

To answer this, the team had to do some heavy mathematical lifting. They had to translate the rules of the "twisting" and "stretching" theories into the language of quantum tunneling.

Setting the Stage: They imagined a universe with perfect symmetry (like a perfect sphere) to make the math manageable.
The "Bounce" Solution: They looked for a specific shape of the "ball's path" (called a bounce solution) that represents the moment of tunneling.
The Boundary Problem: In the "twisting" and "stretching" theories, the math includes extra terms at the very edges of the calculation (boundary terms). It was unclear if these extra terms would change the final "difficulty score."

The Result: The Score is the Same

After doing the complex calculations, the authors found a very satisfying result:

The "difficulty score" (tunneling exponent) came out exactly the same in all three theories.

GR Result: $B$
TEGR Result: $B$
STEGR Result: $B$

Even though the math looked different along the way, the extra "twist" or "stretch" terms canceled each other out perfectly at the edges. The final probability of the universe decaying is identical whether you view gravity as curvature, torsion, or non-metricity.

Why This Matters

This is a big deal because it proves that the "Geometric Trinity of Gravity" isn't just a classical trick. The equivalence between these three ways of describing gravity persists even when we look at quantum events.

It tells us that nature is consistent. Whether you describe gravity as a curve, a twist, or a stretch, the fundamental stability of our universe remains unchanged. The "Geometric Trinity" is robust, surviving the transition from the classical world of planets to the quantum world of vacuum decay.

Summary

The paper confirms that three different mathematical languages for gravity (Curvature, Torsion, and Non-metricity) all tell the exact same story about whether our universe is stable or about to collapse. The "difficulty" of the universe decaying is the same regardless of which language you use to describe it.

Technical Summary: Vacuum Stability in the Geometric Trinity of Gravity

Problem Statement
The stability of the electroweak (EW) vacuum is a critical constraint on physics beyond the Standard Model (BSM). The decay rate of a metastable (false) vacuum is sensitive to gravitational effects, as demonstrated in the Coleman–De Luccia (CDL) analysis within General Relativity (GR). A fundamental theoretical question arises regarding the "Geometric Trinity of Gravity," which posits that GR, Teleparallel Equivalent of GR (TEGR), and Symmetric Teleparallel Equivalent of GR (STEGR) are classically equivalent descriptions of gravity. While these theories share identical field equations and classical solutions, it remains unclear whether this equivalence holds at the quantum level, specifically for vacuum decay processes. The central problem addressed is whether the tunneling exponent (and thus the decay rate) of a false vacuum computed in TEGR and STEGR coincides with the result obtained in standard GR.

Methodology
The authors employ a semiclassical approach to vacuum decay, utilizing the Euclidean path integral formalism. The methodology proceeds as follows:

Theoretical Framework: The paper reviews the geometrical formulations of TEGR (based on torsion) and STEGR (based on non-metricity) within the metric-affine geometry framework. It establishes that these theories differ from GR only by boundary terms in their Lagrangians, ensuring classical dynamical equivalence.
Euclidean Formulation: To calculate the decay rate, the authors perform a Wick rotation to the Euclidean signature. They construct the Euclidean actions for TEGR and STEGR, embedding a scalar field with a potential $V(\phi)$ possessing a false vacuum (Minkowski) and a true vacuum (Anti-de Sitter).
Symmetry Constraints: The analysis assumes an $O(4)$ $O (4)$ -symmetric bounce solution. The authors derive the most general $O(4)$ $O (4)$ -invariant affine connections for both theories:
- For STEGR, they derive the symmetric, flat, and torsionless affine connection in spherical coordinates, determining the specific functional form of the connection components required to satisfy the teleparallel constraints.
- For TEGR, they utilize the tetrad formalism, constructing $O(4)$ -invariant tetrads and the corresponding spin connection (or utilizing the Weitzenböck gauge) to define the torsion tensor.
Action Evaluation: The tunneling exponent $B$ is defined as the difference between the Euclidean action evaluated on the bounce solution ( $S_b$ ) and the false vacuum solution ( $S_{fv}$ ). Since the TEGR and STEGR actions differ from the Einstein-Hilbert action only by boundary terms, the authors explicitly calculate these boundary contributions for the bounce and false vacuum configurations.
Asymptotic Analysis: The calculation relies heavily on the asymptotic behavior of the bounce solution ( $\rho(r)$ and $\phi(r)$ ) as $r \to 0$ and $r \to \infty$ . The authors verify that the boundary terms vanish at the origin and cancel out at infinity when taking the difference between the bounce and false vacuum actions.

Key Contributions

Derivation of Invariant Connections: The paper provides the explicit derivation of the most general $O(4)$ -invariant, flat, and torsionless (for STEGR) or metric-compatible (for TEGR) affine connections and tetrads. This is a necessary step to correctly formulate the actions in these alternative geometries for vacuum decay problems.
Explicit Boundary Term Calculation: The authors perform a rigorous evaluation of the boundary terms that relate the STEGR and TEGR actions to the GR action. They demonstrate that these terms, which are non-trivial in the general action, yield zero net contribution to the tunneling exponent in the specific context of Minkowski-to-AdS vacuum decay.
Verification of Quantum Equivalence: The work extends the known classical equivalence of the Geometric Trinity to the semiclassical quantum regime of vacuum tunneling.

Results
The analysis yields the following results:

The bounce equations of motion in both TEGR and STEGR are identical to those in GR, leading to the same bounce solution $(\phi_b, \rho_b)$ .
The boundary terms in the STEGR and TEGR actions, when evaluated on the bounce solution and the false vacuum solution, vanish at $r=0$ and cancel each other at $r \to \infty$ .
Consequently, the tunneling exponent $B$ is identical across all three formulations:
$B_{GR} = B_{TEGR} = B_{STEGR}$
This implies that the decay rate $\Gamma$ and the lifetime of the EW vacuum calculated within TEGR or STEGR are exactly the same as those calculated in standard GR. For instance, the lifetime of the EW vacuum in the presence of gravity remains $\tau_{grav} \sim 10^{661} T_U$ , consistent with GR predictions.

Significance and Claims
The paper claims that this result provides a non-trivial example where the classical equivalence between GR, TEGR, and STEGR persists at the quantum (semiclassical) level. Specifically, it demonstrates that the different geometric representations of gravity (curvature, torsion, and non-metricity) yield the same physical prediction for vacuum stability in a maximally symmetric background.

The authors note that while the equivalence holds for the specific case of Minkowski-to-AdS decay analyzed here, it does not necessarily imply equivalence for all quantum phenomena or for extended theories (such as $f(R)$ , $f(T)$ , or $f(Q)$ ) where boundary terms become non-linear and classical equivalence is lost. The paper concludes that the "Geometric Trinity" remains robust for vacuum stability analysis within the standard framework, offering a sensitive testing ground that confirms the consistency of these alternative formulations.