Violation of energy conditions and the gravitational radius of the proton

Abstract

The energy-momentum tensor (EMT) of the proton encodes fundamental information about its mass, pressure, and shear distributions. Using recent lattice QCD data for the gravitational form factors, we show that the Breit-frame Wigner EMT may be of Hawking-Ellis type IV in the proton's core. Such EMT violates all pointwise energy conditions and lacks a causal rest frame so that the usual mechanical picture fails at short distances. We define the gravitational radius -- a new hadronic observable marking the scale where the EMT becomes ordinary (type I) and the classical interpretation is restored. We also derive from the Averaged Null Energy Condition (ANEC) non-perturbative, model-independent QFT constraints on gravitational form factors.

Comments: 11 pages (incl. appendices and references), 1 figure; v2: added a constraint on proton GFFs derived from ANEC; v3: added ANEC for pion, final version to be published in Phys. Rev. D