Operational measurement of relativistic equilibrium from stochastic fields alone

Abstract

The inverse-temperature four-vector $\beta^\mu = u^\mu/(k_B T_0)$ has been the theoretically accepted description of relativistic equilibrium since van Kampen and Israel, yet no experiment has ever reconstructed $\beta^\mu$ as a single observable. All existing methods -- Thomson scattering, spectral fitting, blast-wave models -- infer rest-frame temperature and flow velocity from separate measurements. We propose the first protocol that extracts both components of $\beta^\mu$ from the same passive observable: electromagnetic fluctuation correlations emitted by a drifting medium. A dimensionless $E$-$B$ cross-spectral ratio yields the drift velocity from Lorentz mixing of the field-strength tensor, while angle-resolved noise power governed by the covariant fluctuation-dissipation theorem provides the rest-frame temperature via a ratio method that cancels absolute amplitude. Together, these reconstruct $\beta^\mu$ without external probes, spectral lines, or absolute radiometric calibration. The protocol enables the first direct experimental test of whether the thermal state of a relativistic medium transforms as a four-vector -- a question unresolved since the Planck--Ott-Landsberg controversy of 1907. Monte Carlo simulations parameterized to the HIGGINS dual 100\,TW laser-plasma facility demonstrate sub-percent temperature recovery for $\gamma = 1.05$-$10$, with robustness to additive noise at SNR~$\gtrsim 10$.