Relativistic equilibrium is described by the inverse-temperature four-vector $β^μ= u^μ/(k_B T_0)$ rather than by a frame-dependent scalar temperature. We show that $β^μ$ can be reconstructed directly from electromagnetic fluctuations emitted by a drifting medium, without external probes, spectral lines, or absolute intensity calibration. A Lorentz boost converts isotropic rest-frame noise into correlated electric and magnetic fields, producing a gain-independent fluctuation observable that yields the drift velocity purely from stochastic data. Combined with angle-resolved noise spectra governed by the covariant fluctuation--dissipation theorem, this enables full reconstruction of $β^μ$ using electromagnetic measurements alone. Monte Carlo analysis demonstrates percent-level accuracy at realistic signal-to-noise ratios, and feasibility estimates indicate sub-microsecond integration times for laboratory plasmas. To our knowledge, this constitutes the first method that reconstructs the covariant thermal state $β^μ$ of a relativistic medium from passive stochastic fields alone, without absolute calibration, spectral lines, or external probes. These results establish vacuum electromagnetic fluctuations as a direct operational probe of relativistic equilibrium.
Relativistic equilibrium is described by the inversetemperature four-vector β µ = u µ /(k B T 0 ) rather than by a frame-dependent scalar temperature [1,2]. This covariant structure originates in the nonequilibrium statistical operator approach [3] and underlies modern relativistic dissipative hydrodynamics [4,5]. Yet no method exists to reconstruct β µ as a unified object from passive, uncalibrated observations alone. Existing techniques infer temperature and velocity separately, requiring either active probes such as Thomson scattering [6] or absolute intensity calibration [7]. We show that β µ can instead be reconstructed directly from stochastic electromagnetic fields emitted by the medium, establishing a fluctuationbased measurement of relativistic equilibrium that requires no external reference.
In nonrelativistic systems, temperature can be inferred from equilibrium fluctuations via Johnson-Nyquist noise [8,9], formalized by the fluctuation-dissipation theorem (FDT) [10]. Its covariant generalization in thermal field theory [11,12] and the corresponding thermal Green’s functions [13] provide the formal basis for extending fluctuation thermometry to relativistic systems. Angle-dependent detector responses in moving thermal baths have been analyzed using idealized Unruh-DeWitt models [14], building on the foundational prediction that uniformly accelerated observers detect thermal radiation in the Minkowski vacuum [15,16], but these studies addressed conceptual aspects of temperature transformations rather than operational measurement. The present work demonstrates that relativistic equilibrium parameters can be reconstructed from electromagnetic noise alone, providing a realizable fluctuation-based protocol applicable to laboratory and astrophysical systems. The method applies to any relativistic thermal emitter-from laser-produced plasmas and heavy-ion collision fireballs to relativistic jets and the quark-gluon plasma-wherever electromagnetic fluctuations are accessible.
The drift velocity can be extracted directly from fluctuation data without any external mean field. In the rest frame of an isotropic equilibrium medium, electric and magnetic fluctuations are uncorrelated,
A Lorentz boost along ẑ mixes fields,
producing a nonzero laboratory-frame cross-spectral density
For isotropic electromagnetic noise, mode equipartition gives ⟨|E| 2 ⟩ = c 2 ⟨|B| 2 ⟩, yielding the dimensionless ratio
This ratio is analytically invertible for β, providing a direct kinematic observable without fitting. The relation assumes the detected fluctuations are electromagnetic modes that have propagated into vacuum, where detector gain or absolute calibration, provided the relative transfer functions of the electric and magnetic channels are stable or known, yielding a kinematic invariant that can be extracted solely from correlation measurements. Crucially, Eq. ( 3) relies only on the isotropy of the restframe fluctuations and the Lorentz transformation, not on thermal equilibrium per se; it therefore remains valid for any medium whose rest-frame noise is isotropic, even if the distribution deviates from the Jüttner form [5,17].
Fluctuation amplitudes encode temperature through the FDT,
and electromagnetic field fluctuations inherit this scaling.
In the classical limit ℏω ≪ k B T , S E (ω) ∝ T σ(ω). Consider a drifting equilibrium medium observed at laboratory angle θ defined by cos θ = k′ • v.
(
Lorentz transformation of the photon four-momentum gives
Combining FDT scaling with Doppler kinematics yields the observable
Conventional radiometry relies on absolute intensity or spectral features [7], and active techniques such as Thomson scattering extract temperature and velocity from calibrated spectral fits [6], whereas the present method exploits field correlations and angular structure of stochastic radiation. Because correlation observables retain phase and polarization information lost in intensity measurements, they encode kinematic information inaccessible to standard radiometric techniques. Unlike Stokesparameter polarimetry [18], which characterizes intensity correlations within the electric field alone, the present observable involves cross-correlations between electric and magnetic fields. This coupling arises from Lorentz mixing of field components and directly encodes the bulk velocity, providing kinematic information not contained in polarization measurements of thermal radiation. The reconstruction proceeds in three steps: (i) determine β from the gain-independent E-B cross-spectral ratio, (ii) record angle-resolved electric-field noise spectra, and (iii) fit Eq. ( 7), accounting for the medium response function σ(ω), to recover (β, T 0 ) and thus β µ . Operationally, this procedure reconstructs the full inversetemperature four-vector from passive electromagnetic observations alone, allowing the invariant rest temperature T 0 and four-velocity u µ to be inferred without invoking any specific frame
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