The Dilemma of Vacuum Flatness: An Attempt at a Solution within a Generalized Framework

The flatness of the vacuum has two sources: in special relativity, flat spacetimeis prescribed a priori; in general relativity, the Einstein equation without matterand without a cosmological term also admits a flat solution. Quantum field theory,however, indicates that the vacuum is not empty and possesses zero-point energy.The vacuum zero-point energy is positive energy and, according to the generalrelativistic principle that all energy is a gravitational source, it should enter theright-hand side of the field equation as a source. A dilemma then arises: notconfronting it lacks justification; confronting it makes the field equation yield anon-flat vacuum, and the flatness of the vacuum becomes problematic. Existingapproaches—normal-ordering subtraction and assuming that zero-point energy doesnot gravitate—both give up the physical reality of the vacuum zero-point energyand evade rather than resolve the dilemma.This paper employs a generalized framework, the equal-footing equation proposed in earlier work, in an attempt to provide a solution to this dilemma. Theframework introduces three new hypotheses: the gravitational field is on equal footing with matter; the vacuum zero-point energy-momentum tensor and the homogeneous gravitational-field energy-momentum tensor satisfy Lorentz invariance; andthe cosmic zero-energy hypothesis is inherited. In this framework, the vacuum zeropoint energy-momentum tensor and its gravitational-field counterpart have the samestructure, both being proportional to the metric, and the cosmic zero-energy hypothesis constrains the expectation value of their sum to vanish. This cancellationmechanism can resolve the vacuum zero-point energy dilemma. Owing to the suppression mechanism of its gravitational field, the vacuum zero-point energy has noeffect on spacetime geometry at any scale. The approach provides an independentprinciple explaining why it does not gravitate: the suppression mechanism of itsgravitational field, which makes the vacuum zero-point energy produce no net effecton spacetime geometry. The flatness of the vacuum is thereby maintained. Thecancellation is at the level of expectation values, not point by point, and thereforedoes not affect local vacuum fluctuations; quantum effects such as the Casimir effectand the Lamb shift are unaffected.Keywords: vacuum; vacuum ze