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The bimetric spectral Einstein-Hilbert action and the Kastler-Kalau-Walze type theorem for Lorentz warped products

Siyao Liu, Yong Wang

TL;DR

The paper addresses extending the Kastler-Kalau-Walze type theorem to Lorentz warped products by introducing a bimetric spectral Einstein-Hilbert action defined via the noncommutative residue of a product of generalized Laplacians on a warped product and its base. It develops the necessary pseudodifferential calculus for the generalized Laplacian on $_{\\varepsilon}M\\times_f N$ and computes the relevant symbol expansions. The main result provides an explicit expression for the noncommutative residue in terms of base and fiber scalar curvatures $S_M$ and $S_N$, the warping function $f$, and the sign parameter $\\varepsilon$, recapitulating the classical KW theorem when $\\varepsilon=1,f=1$ and yielding a Lorentzian analogue when $\\varepsilon=-1$. This work strengthens the bridge between noncommutative geometry and gravitational actions in non-elliptic, Lorentzian settings and broadens the applicability of spectral action principles to warped-product geometries.

Abstract

In this paper, we define the bimetric spectral Einstein-Hilbert action which generalizes the spectral Einstein-Hilbert action. We compute the bimetric spectral Einstein-Hilbert action for the Lorentz warped product. Thus, we get the Kastler-Kalau-Walze type theorem for the Lorentz warped product.

The bimetric spectral Einstein-Hilbert action and the Kastler-Kalau-Walze type theorem for Lorentz warped products

TL;DR

The paper addresses extending the Kastler-Kalau-Walze type theorem to Lorentz warped products by introducing a bimetric spectral Einstein-Hilbert action defined via the noncommutative residue of a product of generalized Laplacians on a warped product and its base. It develops the necessary pseudodifferential calculus for the generalized Laplacian on and computes the relevant symbol expansions. The main result provides an explicit expression for the noncommutative residue in terms of base and fiber scalar curvatures and , the warping function , and the sign parameter , recapitulating the classical KW theorem when and yielding a Lorentzian analogue when . This work strengthens the bridge between noncommutative geometry and gravitational actions in non-elliptic, Lorentzian settings and broadens the applicability of spectral action principles to warped-product geometries.

Abstract

In this paper, we define the bimetric spectral Einstein-Hilbert action which generalizes the spectral Einstein-Hilbert action. We compute the bimetric spectral Einstein-Hilbert action for the Lorentz warped product. Thus, we get the Kastler-Kalau-Walze type theorem for the Lorentz warped product.
Paper Structure (4 sections, 5 theorems, 35 equations)

This paper contains 4 sections, 5 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The warped product
  3. The generalized Laplace operator
  4. The noncommutative integral

Key Result

Theorem 1.1

Let $M$ be an $m$-dimensional compact Riemannian manifold and $N$ be an $n$-dimensional compact Riemannian manifold, for the definition of $g^{\varepsilon, f}$ and $g$ in Section 2, we get the following equality: where $2\overline{m}=m+n,$$S_{M}$ and $S_{N}$ are scalar curvature of $(M, g^{M})$ and $(N, g^{N}).$

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3