[Paper Review] Variable Light-Cone Theory of Gravity

I.T. Drummond|arXiv (Cornell University)|Aug 20, 1999

Cosmology and Gravitation Theories2 references15 citations

TL;DR

This paper proposes a covariant Variable Light-Cone Theory (VLCT) of gravity by introducing two vierbein bundles—one for gravity and one for matter—linked by an SL(4,R) matrix M that encodes the variable speed of light. The theory maintains standard energy-momentum conservation for matter and reduces to General Relativity when a new coupling constant F vanishes, but post-Newtonian analysis shows F/G_N < 3.2×10⁻⁴ to be consistent with VLBI data, limiting its viability as a departure from GR.

ABSTRACT

We show how to reformulate Variable Speed of Light Theories (VSLT) in a covariant fashion as Variable Light-Cone Theories (VLCT) by introducing two vierbein bundles each associated with a distinct metric. The basic gravitational action relates to one bundle while matter propagates relative to the other in a conventional way. The variability of the speed of light is represented by the variability of the matter light-cone relative to the gravitational light-cone. The two bundles are related locally by an element M, of SL(4,R). The dynamics of the field M is that of a SL(4,R)-sigma model gauged with respect to local (orthochronous) Lorentz transformations on each of the bundles. Only the ``massless'' version of the model with a single new coupling, F, that has the same dimensions as Newton's constant $G_N$, is considered in this paper. When F vanishes the theory reduces to standard General Relativity. We verify that the modified Bianchi identities of the model are consistent with the standard conservation law for the matter energy-momentum tensor in its own background metric. The implications of the model for some simple applications are examined, the Newtonian limit, the flat FRW universe and the spherically symmetric static solution.

Motivation & Objective

• To formulate a geometric, covariant theory of gravity in which the speed of light varies by relating matter and gravitational light-cones via a dynamical SL(4,R) matrix.
• To ensure matter energy-momentum is conserved in its own background metric, avoiding inconsistencies of earlier VSL theories.
• To examine whether variable light speed can provide a mechanism for cosmic homogenisation in the early universe.
• To assess the viability of the model against observational constraints, particularly from post-Newtonian approximations and VLBI measurements.
• To explore the stability and asymptotic behavior of the theory, especially in the presence of new degrees of freedom from the linking action.

Proposed method

• Introduce two vierbein bundles: one for gravity (e_μa) and one for matter (e̅_μā), each associated with their own spacetime metrics.
• Relate the two bundles via a local SL(4,R) transformation M: e̅_μā = e_μa M^a_ā, preserving volume elements (det M = 1).
• Define a linking action I_L based on a SL(4,R)-sigma model that governs the dynamics of M, gauged under local Lorentz transformations.
• Construct the full action I = I_G + I_L + I_M, where I_G is the standard Einstein-Hilbert action for gravity and I_M is the standard matter action in the matter metric.
• Use vierbein formalism and covariant derivatives to ensure local Lorentz invariance and derive field equations from the total action.
• Analyze the Newtonian limit, FRW cosmology, and spherically symmetric solutions to test cosmological and gravitational consistency.

Experimental results

Research questions

• RQ1 Can a variable speed of light be consistently embedded in a generally covariant, geometric theory of gravity without violating energy-momentum conservation?
• RQ2 How does the dynamics of the SL(4,R) matrix M, linking matter and gravitational vierbeins, affect the post-Newtonian approximation and observational constraints?
• RQ3 Can the model provide a mechanism for cosmic homogenisation in the early universe via a varying speed of light?
• RQ4 What are the implications of the new coupling F for the asymptotic behavior and long-distance validity of the theory compared to General Relativity?
• RQ5 Is the theory stable against negative energy modes arising from the new degrees of freedom in M?

Key findings

• The theory provides a fully covariant, geometric formulation of variable speed of light by distinguishing gravitational and matter light-cones via two vierbein bundles.
• Matter energy-momentum is conserved in its own background metric, resolving a key issue in earlier non-covariant VSL theories.
• The model reduces to General Relativity when the new coupling constant F is set to zero, ensuring consistency in the low-F limit.
• Post-Newtonian analysis shows that the parameter γ ≠ 1, but observational constraints from VLBI measurements require F/G_N < 3.2×10⁻⁴, severely limiting deviations from GR.
• A flat FRW solution exists in which the speed of light varies as required for cosmic homogenisation, suggesting a possible mechanism for the observed isotropy of the CMB.
• The stability of the theory remains uncertain, as negative energy modes can arise in certain configurations of the matrix M, though stable solutions exist in the positive-energy sector.