Violation of energy conditions and the gravitational radius of the proton

TL;DR

Problem addressed: how the proton's internal mass, pressure, and shear distributions encoded in the gravitational form factors (GFFs) relate to energy conditions in general relativity. Approach: express the proton energy-momentum tensor as the Wigner transform of the QCD EMT using lattice QCD GFFs $A(t)$, $J(t)$, $D(t)$, analyze the Hawking-Ellis classification via the discriminant $\Gamma$, and derive ANEC-based constraints. Findings: the core EMT can be Hawking-Ellis type IV, violating all pointwise ECs, while the outer regions are type I; a gravitational radius is defined as the radius where the EMT transitions to ordinary behavior. Constraints: ANEC yields a model-independent inequality $\int_{-\infty}^0 dt\, \left[ m A(t) - \frac{t}{4m}\left(A(t) - 2 J(t)\right) \right] \ge 0$, with a similar constraint for the pion, testable on the lattice. Significance: guides interpretation of proton mechanical properties and provides nonperturbative QFT constraints on GFFs.

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.

Figures (1)

• Figure 1: Left: Plot of the discriminant $\Gamma(r)$ assuming lattice QCD FFs Hackett:2023rif (with best fit parameters) for momentum transfer below $\Delta^*$, and a transition to the asymptotic FFs at $\Delta^*$. Here, $\vec{x}$ is chosen perpendicular to the direction of the spin axis, so $\vec{M}^2 > 0$. Radial distance is measured in units of the Compton wavelength $\bar{\lambda}=1/m$, where $m$ is the proton mass. Right: $\Gamma(r)$ for $\Delta^* = 2$ GeV obtained by variation of the pole masses in the individual GFFs within the uncertainties quoted in Ref. Hackett:2023rif.