[Paper Review] Deriving a complete set of eigendistributions for a gravitational wave equation describing the quantized interaction of gravity with a Yang-Mills field in case the Cauchy hypersurface is non-compact

Claus Gerhardt|arXiv (Cornell University)|May 11, 2016

Advanced Differential Geometry Research5 references3 citations

TL;DR

This paper extends spectral resolution of a gravitational wave equation with quantized gravity-Yang-Mills interactions to non-compact Cauchy hypersurfaces by employing Gelfand triplets and the nuclear spectral theorem. It establishes a complete set of eigendistributions for the spatial operator, proves their smoothness and positivity of eigenvalues, and derives a spectral decomposition of solutions via separation of variables with generalized eigenfunctions, ensuring finite-energy solutions and a basis in $ L^2(\mathbb{R}^*_+, \mathbb{C}) $.

ABSTRACT

In a recent paper we quantized the interaction of gravity with a Yang-Mills and Higgs field and obtained as a result a gravitational wave equation in a globally hyperbolic spacetime. Assuming that the Cauchy hypersurfaces are compact we proved a spectral resolution for the wave equation by applying the method of separation of variables. In this paper we extend the results to the case when the Cauchy hypersurfaces are non-compact by considering a Gelfand triplet and applying the nuclear spectral theorem.

Motivation & Objective

• Extend the spectral resolution of a quantized gravitational wave equation from compact to non-compact Cauchy hypersurfaces.
• Establish a complete set of eigendistributions for the spatial elliptic operator in the wave equation using distribution theory.
• Prove that these eigendistributions are smooth and correspond to strictly positive eigenvalues.
• Generalize the separation of variables method to include distributional eigenfunctions in the non-compact case.
• Ensure the resulting solutions have finite temporal and locally bounded spatial energy, preserving physical consistency.

Proposed method

• Apply the nuclear spectral theorem of Gelfand-Maurin to a self-adjoint elliptic operator $ A $ on a non-compact Riemannian manifold $ \mathcal{S}_0 $, under smooth, bounded coefficient conditions.
• Construct a Gelfand triplet $ \mathscr{S} \subset L^2(\mathcal{S}_0) \subset \mathscr{S}' $ to handle distributions and ensure the existence of a complete set of eigendistributions $ f(\lambda) \in \mathscr{S}' $.
• Prove that eigendistributions $ f(\lambda) $ are smooth and tempered due to the uniform ellipticity and smoothness of $ A $, and that their eigenvalues $ a(\lambda) > 0 $ almost everywhere.
• Use separation of variables by setting $ u(x,t) = w(t)f(x) $, reducing the hyperbolic wave equation to an ODE eigenvalue problem in time: $ -\frac{1}{32}\frac{n^2}{n-1}\ddot{w} - \mu t^{2-\frac{4}{n}}w - n t^2 \varLambda w = 0 $, where $ \mu = a(\lambda) $.
• Apply results from prior work to show that this ODE has countably many solutions $ (w_i, \varLambda_i) $ with $ \varLambda_i < \varLambda_{i+1} < 0 $, $ \lim \varLambda_i = 0 $, and finite energy in $ H $.
• Transform the temporal eigenfunctions via $ \tilde{w}_i(t) = w_i(\lambda_i^{n/(4(n-1))} t) $, showing they form a basis in $ L^2(\mathbb{R}^*_+, \mathbb{C}) $ and in the Hilbert space $ H $ defined by the $ H^1 $-type norm.

Experimental results

Research questions

• RQ1 How can spectral resolution of a gravitational wave equation with Yang-Mills and Higgs interactions be extended from compact to non-compact Cauchy hypersurfaces?
• RQ2 What is the appropriate functional analytic framework to define and characterize eigendistributions for elliptic operators on non-compact manifolds?
• RQ3 How do the eigenvalues of the spatial operator behave in the non-compact case, and can positivity be established for almost every eigenvalue?
• RQ4 Can the separation of variables method be generalized to include distributional eigenfunctions while preserving finite-energy solutions?
• RQ5 Under what strengthened conditions can a mass gap be proven in the spectrum of the spatial operator on non-compact $ \mathcal{S}_0 $?

Key findings

• The eigendistributions $ f(\lambda) \in \mathscr{S}' $ form a complete set for the elliptic operator $ A $ on a non-compact Cauchy hypersurface $ \mathcal{S}_0 $, satisfying $ A f(\lambda) = a(\lambda) f(\lambda) $ with $ a(\lambda) > 0 $ almost everywhere.
• All eigendistributions $ f(\lambda) $ are smooth functions due to the uniform ellipticity and smooth coefficients of $ A $, and are tempered distributions.
• The eigenvalues $ a(\lambda) $ of the spatial operator are strictly positive for almost every $ \lambda $, ensuring the physical relevance of the spectral decomposition.
• Classical solutions $ u_i = w_i f $ of the hyperbolic wave equation (1.1) are constructed via separation of variables, with $ w_i $ solving the time-dependent ODE (4.22) and having finite energy.
• The transformed eigenfunctions $ \tilde{w}_i(t) = w_i(\lambda_i^{n/(4(n-1))} t) $ form a basis in both $ L^2(\mathbb{R}^*_+, \mathbb{C}) $ and the Hilbert space $ H $, ensuring completeness of the spectral resolution.
• Under stronger assumptions, such as $ -\frac{n}{2}R + G + \alpha_2 \frac{n}{2} m_0 V \geq \epsilon_0 r^\delta $ for $ \delta > 0 $, the operator $ A $ would have pure point spectrum, recovering the compact case behavior.