[Paper Review] Unimodular relativity and cosmological constant : Comments

S. C. Tiwari|arXiv (Cornell University)|Oct 22, 2003

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper challenges Finkelstein et al.'s claim that the cosmological constant remains a constant of integration and that the matter stress-energy tensor satisfies the standard covariant continuity law in unimodular relativity. By analyzing an ambiguous matter Lagrangian term and its impact on the effective cosmological term, the author shows that these conclusions depend on additional assumptions, not the theory alone, and that the standard continuity law for $ T_{\mu\nu} $ does not hold without imposing extra constraints.

ABSTRACT

We show that the conclusion that matter stress-energy tensor satisfies the usual covariant continuity law, and the cosmological constant is still a constant of integration arrived at by Finkelstein et al (42, 340, 2001) is not valid.

Motivation & Objective

• To assess the validity of Finkelstein et al.'s conclusion that the cosmological constant is a constant of integration and $ T_{\mu\nu} $ satisfies the standard covariant continuity law in unimodular relativity.
• To investigate the consequences of an ambiguous matter Lagrangian term $ \Delta_M L $ introduced in the extended action of unimodular gravity.
• To determine whether the covariant divergence of the energy-momentum tensor $ T_{\mu\nu} $ is truly conserved under the framework proposed by Finkelstein et al.
• To examine the role of the effective cosmological term $ \lambda_{\text{eff}} = \lambda + 8\pi G l_M $ and its implications for the consistency of the theory.

Proposed method

• Analyzes the extended action $ S' $ with an ambiguous matter Lagrangian term $ \Delta_M L = \left[ \frac{\mu(x)}{\sqrt{-g}} - 1 \right] l_M $, where $ l_M $ depends on matter fields and metric.
• Uses the unimodular condition $ \sqrt{-g} = \mu(x) $ with a Lagrange multiplier $ \lambda(x) $ to derive field equations from the variational principle.
• Derives the field equation $ G^{\mu\nu} - \frac{\lambda}{2} g^{\mu\nu} = 8\pi G T'^{\mu\nu} $, where $ T'^{\mu\nu} $ includes an ambiguous contribution from $ \Delta_M L $.
• Calculates the covariant divergence of $ T'^{\mu\nu} $, showing it is not conserved unless additional constraints are imposed.
• Evaluates the effective cosmological term $ \lambda_{\text{eff}} = \lambda + 8\pi G l_M $, demonstrating that $ \lambda $ and $ l_M $ can vary while their sum remains constant.
• Applies the Bianchi identity and traces of field equations to analyze the consistency of $ R + 8\pi G T $ and $ R + 8\pi G T' $ being constant.

Experimental results

Research questions

• RQ1 Does the standard covariant continuity law for $ T_{\mu\nu} $ hold in the unimodular relativity framework proposed by Finkelstein et al.?
• RQ2 Is the cosmological constant truly a constant of integration, or does it depend on additional assumptions?
• RQ3 How does the ambiguous matter Lagrangian term $ \Delta_M L $ affect the energy-momentum tensor and the field equations?
• RQ4 Can the effective cosmological term $ \lambda_{\text{eff}} $ remain constant while $ \lambda $ and $ l_M $ vary?
• RQ5 What conditions are necessary to ensure both $ T_{\mu\nu} $ and $ T'^{\mu\nu} $ are covariantly conserved in this theory?

Key findings

• The standard covariant continuity law for $ T_{\mu\nu} $ does not hold in the unimodular relativity model unless an additional constraint is imposed on $ \lambda + 8\pi G l_M $.
• The effective cosmological term $ \lambda_{\text{eff}} = \lambda + 8\pi G l_M $ must be constant for consistency, but $ \lambda $ and $ l_M $ individually can vary.
• The field equations reduce to the standard Einstein equations without a cosmological constant if $ \lambda + 8\pi G l_M = 0 $, despite the presence of the ambiguity.
• The constancy of $ R + 8\pi G T $ and $ R + 8\pi G T' $ is required for both $ T_{\mu\nu} $ and $ T'^{\mu\nu} $ to be covariantly conserved, which is not guaranteed by the theory alone.
• Finkelstein et al.'s conclusion that $ T_{\mu\nu} $ satisfies the usual continuity law and the cosmological constant is a constant of integration is not a consequence of the theory but relies on unverified assumptions.
• The ambiguity in $ \Delta_M L $ introduces a physical interpretation where $ \lambda $ and $ l_M $ may represent geometric and vacuum energy contributions, respectively, but their physical significance remains unclear.