Triangle Nuclear Theory Colloquium - Pressure-Energy Equations of State of Hadrons, Superconductor Vortices, and the Cosmos

The pressure-energy relations in the nucleon are derived from the gravitational form factors, which parameterize matrix elements of the energy-momentum tensor (EMT), together with EMT conservation. The static pressure distribution arising from the Lorentz trace part of the EMT, as manifested in the spatial stress $\frac{1}{3} T^{ii}$, equals minus the corresponding trace part of the energy density, which can be understood in terms of the gluon and quark condensates. It can be shown that this trace-anomaly- and sigma-term-induced pressure plays a fundamental role in the confinement dynamics of QCD. In contrast, the dynamic pressure distribution from the traceless part of the spatial stress tensor equals $1/d$ of the corresponding traceless part of the energy density, where $d$ is the spatial dimension. We point out that the same pressure-energy relations also hold for vortices in type-II superconductors, where the static pressure-energy relation originates from the Cooper-pair condensate. Furthermore, identical equations of motion appear in the $\Lambda{\rm CDM}$ model of cosmology, where the static pressure-energy relation arises from the cosmological constant.