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An impediment to torsion from spectral geometry

Arkadiusz Bochniak, Ludwik Dąbrowski, Andrzej Sitarz, Paweł Zalecki

TL;DR

The paper argues, via spectral geometry and pseudo-differential calculus, that torsion should be excluded from physically acceptable gravity models by showing that no well-defined tensorial extension of the Einstein functional exists once torsion is included. It develops the spectral framework for Laplace-type and Dirac-type operators and computes explicit torsion-induced corrections to the Einstein functional through residue calculations. The main no-go result is supported by explicit computations of the F1(k) and F2(k) densities, which reveal curvature- and torsion-dependent contributions that cannot be assembled into a tensorial Einstein density. Overall, the work provides a rigorous spectral-theoretic argument against incorporating torsion as a dynamical, tensorial degree of freedom in this formalism.

Abstract

Modifications of standard general relativity that bring torsion into a game have a long-standing history. However, no convincing arguments exist for or against its presence in physically acceptable gravity models. In this Letter, we provide an argument based on spectral geometry (using methods of pseudo-differential calculus) that suggests that the torsion shall be excluded from the consideration. We demonstrate that there is no well-defined functional extending to the torsion-full case of the spectral formulation of the Einstein tensor.

An impediment to torsion from spectral geometry

TL;DR

The paper argues, via spectral geometry and pseudo-differential calculus, that torsion should be excluded from physically acceptable gravity models by showing that no well-defined tensorial extension of the Einstein functional exists once torsion is included. It develops the spectral framework for Laplace-type and Dirac-type operators and computes explicit torsion-induced corrections to the Einstein functional through residue calculations. The main no-go result is supported by explicit computations of the F1(k) and F2(k) densities, which reveal curvature- and torsion-dependent contributions that cannot be assembled into a tensorial Einstein density. Overall, the work provides a rigorous spectral-theoretic argument against incorporating torsion as a dynamical, tensorial degree of freedom in this formalism.

Abstract

Modifications of standard general relativity that bring torsion into a game have a long-standing history. However, no convincing arguments exist for or against its presence in physically acceptable gravity models. In this Letter, we provide an argument based on spectral geometry (using methods of pseudo-differential calculus) that suggests that the torsion shall be excluded from the consideration. We demonstrate that there is no well-defined functional extending to the torsion-full case of the spectral formulation of the Einstein tensor.
Paper Structure (5 sections, 11 theorems, 55 equations)

This paper contains 5 sections, 11 theorems, 55 equations.

Table of Contents

  1. Spectral functionals for Laplace-type operators
  2. Einstein Functional for Dirac-type operators
  3. Tensorial Functionals on Differential Forms
  4. $F_1(k)$
  5. $F_2(k)$

Key Result

Proposition 1.2

For a second-order differential operator $O$, with a symbol expressed in normal coordinates around a given point $x$ on $M$, where $F^{ab}=F^{ba}$, $G^a$ and $H$ are endomorphisms of the fibre $V$ at point $x$, we have where $R$ is the scalar curvature and $\mathrm{Ric}$ the Ricci tensor.

Theorems & Definitions (21)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 11 more