The Volume Element of Space-Time and Scale Invariance

Abstract

Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form S=∫ L 1 Φ d4x+∫ L 2 \(\sqrt{-g}\) d4x where the volume element Φ d4x is independent of the metric. For global scale invariance, a “dilaton” φ has to be introduced, with non-trivial potentials V(φ)=f 1 eαφ in L 1 and U(φ)=f 2 e2αφ in L 2 . This leads to non-trivial mass generation and a potential for φ which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for “Quintessential models,” which are scale invariant but formulated with the use of volume element Φ d4x alone. For closed strings and branes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants.

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Guendelman, E.I. The Volume Element of Space-Time and Scale Invariance. Foundations of Physics 31, 1019–1037 (2001). https://doi.org/10.1023/A:1017572522313

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Keywords

  • Potential Versus
  • Global Scale
  • Late Time
  • Volume Element
  • Integration Constant