[Paper Review] On the Gravitational Wave in de Sitter Spacetime

Liao Liu|ArXiv.org|Nov 25, 2004

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper proposes that gravitational waves in de Sitter spacetime arise not from metric perturbations—whose standard linearization yields unphysical imaginary graviton masses—but from a conformally invariant, massless tensor field in Minkowski spacetime that transforms into a conformal graviton with effective mass $ m_g = rac{1}{2} horac{2ar{ ho}}{3} $ in de Sitter spacetime. The key result is a consistent, general-covariant, and conformally invariant gravitational wave theory with a physically meaningful, positive effective mass for the graviton.

ABSTRACT

For there is always a wrong sign in the mass of graviton in the so-called perturbation expansion approximation of both Minkowski and de Sitter spacetimes, the existence of gravitational wave from the metric perturbation of de Sitter spacetime is doubtful. We try another way to start from the assumption that the gravitational wave equation should be both general covariant and conformal invariant and find that graviton is no longer a part of metric field, it has an effective mass of $m_g=\sqrt{R/6}=% \sqrt{2Λ/3}$ with correct sign in de Sitter spacetime, though it's intrinsic mass remains zero.

Motivation & Objective

• To resolve the inconsistency in standard metric perturbation theory, where graviton mass in de Sitter spacetime acquires a wrong (negative) sign when $\Lambda > 0$.
• To investigate whether gravitational waves can exist in de Sitter spacetime despite the failure of linearized gravity to yield a physical graviton.
• To develop a new framework for gravitational waves in de Sitter spacetime that preserves general covariance and conformal invariance.
• To show that the graviton, while having zero intrinsic mass, acquires a positive effective mass due to spacetime curvature.
• To provide a consistent quantum field-theoretic description of gravitational waves in a universe with positive cosmological constant.

Proposed method

• Starts from a massless, Lorentz-covariant tensor field equation in Minkowski spacetime: $ \Box_{\eta} \varphi_{\mu\nu}^{(M)} = 0 $.
• Applies a conformal transformation $ ds^2 = \Omega^2 ds_M^2 $ with $ \Omega^2 = e^{2Ht} $, mapping Minkowski to de Sitter spacetime.
• Derives the conformally invariant field equation in de Sitter spacetime: $ \Box \varphi_{\mu\nu} + \frac{2}{3}\Lambda \varphi_{\mu\nu} = 0 $, using $ R = 4\Lambda $.
• Defines the physical field as $ \varphi_{\mu\nu} = \Omega^{-1} \varphi_{\mu\nu}^{(M)} $, ensuring conformal equivalence and shared Fock representation.
• Imposes harmonic gauge to ensure five physical degrees of freedom and general covariance.
• Uses the conformal invariance to preserve the massless nature of the field in Minkowski space while allowing an effective mass in curved spacetime.

Experimental results

Research questions

• RQ1 Can gravitational waves exist in de Sitter spacetime if standard metric perturbation theory leads to a negative or imaginary graviton mass?
• RQ2 Is there an alternative formulation of gravitational waves in de Sitter spacetime that avoids the unphysical mass sign problem?
• RQ3 Can conformal invariance and general covariance be simultaneously preserved in a gravitational wave theory for de Sitter spacetime?
• RQ4 What is the origin of the effective mass of the graviton in de Sitter spacetime, and how does it relate to the cosmological constant?
• RQ5 Does the conformal coupling of the gravitational field to curvature allow for a consistent particle interpretation of the graviton in a de Sitter background?

Key findings

• The standard linearized perturbation approach in de Sitter spacetime leads to a graviton with an unphysical negative mass squared, $ m_g^2 = -2\Lambda/3 $, when $ \Lambda > 0 $, implying an imaginary mass.
• By constructing a conformally invariant tensor field in Minkowski spacetime and mapping it to de Sitter spacetime, a consistent gravitational wave equation is derived: $ \Box \varphi_{\mu\nu} + \frac{2}{3}\Lambda \varphi_{\mu\nu} = 0 $.
• The resulting field in de Sitter spacetime has an effective mass $ m_g = \sqrt{2\Lambda/3} $, which is real and positive, resolving the sign problem.
• The intrinsic mass of the field remains zero, but the effective mass arises solely from the spacetime curvature (via $ R = 4\Lambda $), not from a fundamental mass term.
• The conformal graviton, while not a perturbation of the metric, carries physical degrees of freedom and can be interpreted as a particle with definite quantum field theory properties.
• The theory preserves general covariance and conformal invariance, and the Fock representation of the field is shared between Minkowski and de Sitter spacetimes, supporting a consistent particle interpretation.