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A possible connection between the energy W of the vacuum fluctuations of quantum fields and gravity in "empty space" is conjectured in this paper using a natural cutoff of high momenta with the help of the gravitational radius of the vacuum region considered. We found that below some "critical" length L = 1mm the pressure σ is one third of the energy density ε, but above 1 mm the equation of state is σ = −ε as for dark energy). In the case of a massive field, W does not depend on the mass ofdoi:10.1590/s0103-97332006000400016 fatcat:xz3udubs7bb4nfkrib2g5ziie4