Relativistic transformation of temperature revisited

Abstract: The relativistic transformation of temperature has long remained controversial, with the classical laws of Planck-Einstein, Ott-Eddington-Moller and Landsberg yielding conflicting results. We reexamine this issue from a relativistic thermodynamic and statistical perspective, starting from the energy-momentum tensor of an isotropic system and defining the effective temperature Teff as that inferred by a moving observer from the transformed energy density. Analyses of a photon gas, a relativistic ideal gas and an electron gas show that Teff consistently increases with velocity, supporting the Ott-Eddington interpretation while depending on the system's equation of state. These results indicate that temperature is not a Lorentz-invariant scalar but an observer-dependent quantity. A consistent relativistic description emerges when temperature is related to the inverse-temperature four-vector beta, linking operational and invariant viewpoints within a unified thermodynamic framework.

• The paper establishes that the effective temperature measured by a moving observer increases with velocity, favoring the Ott–Eddington–Møller law over Planck–Einstein's.
• It derives operational temperature definitions from the energy–momentum tensor, applying them to photon, ideal, and Fermi gases to reveal dependencies on the equation of state.
• It reconciles invariant and operational perspectives via the inverse-temperature four-vector, unifying covariant statistical mechanics with practical thermodynamic measurements.

Relativistic Transformation of Temperature: Operational and Covariant Perspectives

Historical Context and Contradictory Laws

The transformation of temperature under Lorentz boosts has been the subject of persistent controversy in relativistic thermodynamics, beginning with the divergent interpretations offered by Planck–Einstein, Ott–Eddington–Møller, and Landsberg. The Planck–Einstein law predicts that a moving system appears cooler ($T' = T/\gamma$), Ott–Eddington–Møller posits the opposite, yielding a hotter system ($T' = \gamma T$), while Landsberg argues for Lorentz invariance ($T' = T$). These interpretations derive from distinct assumptions regarding entropy invariance, energy flux transformation, and covariant thermodynamic formulation, but none yields a universally agreed operational protocol.

Covariant Formulation and Operational Definition

The paper approaches this problem via the energy–momentum tensor for an isotropic medium, defining the effective temperature $T_{\mathrm{eff}}$ as the value inferred by a moving observer, extracted from the transformed energy density using an unchanged equation of state. The transformation follows $T^{\mu\nu}$ under Lorentz boosts, with key operational identification:

$\rho' = f(T_{\mathrm{eff}})$

for a system originally at $T$. The observer measures nontrivial anisotropies in stress–energy components, and $T_{\mathrm{eff}}$ reflects the physical procedure of inferring temperature from measured quantities in the new inertial frame. The analysis sidesteps adopting any classical law and instead grounds itself in direct thermodynamic/statistical definitions.

Photon Gas: Energy Density and Spectral Transformation

For blackbody radiation (photon gas), the rest-frame equation of state $\rho = aT^4$, $P = \rho/3$ yields a transformed energy density:

$T' = \gamma T$0

This leads to:

$T' = \gamma T$1

For $T' = \gamma T$2, the expansion gives an increase, $T' = \gamma T$3. This operational definition aligns qualitatively with Ott–Eddington–Møller, not with Planck–Einstein. Spectroscopically, $T' = \gamma T$4 varies with direction due to Doppler shifts, revealing that temperature becomes anisotropic and cannot be reduced to a scalar. Averaging over momentum directions retrieves the above operational result, demonstrating that definition matters: temperature inferred from energy or spectra may not coincide.

Relativistic Ideal Gas and Fermi Gas: Equation of State Dependence

For the Maxwell–Jüttner relativistic ideal gas, effective temperature is similarly derived but now with dependence on $T' = \gamma T$5:

$T' = \gamma T$6

where $T' = \gamma T$7 with $T' = \gamma T$8 being modified Bessel functions. In the ultrarelativistic limit ($T' = \gamma T$9), this converges to the photon gas case; in the nonrelativistic regime ($T' = T$0), the quadratic coefficient reduces to $T' = T$1. The velocity correction to $T' = T$2 is always positive but its magnitude is dictated by the system’s microscopic thermodynamics.

The Fermi gas analysis generalizes this for spin-1/2 particles, with the effective temperature found numerically via Fermi–Dirac integrals. The correction coefficient interpolates between $T' = T$3 (ultrarelativistic) and $T' = T$4 (nonrelativistic/degenerate), confirming that boost-induced changes to measured temperature are universally positive but sensitively dependent on particle statistics and equation of state.

Covariant Statistical Mechanics: Inverse-Temperature Four-Vector

The invariant perspective emerges in the statistical mechanics formalism utilizing the inverse-temperature four-vector $T' = T$5, with $T' = T$6 the fluid's four-velocity. In this framework, $T' = T$7 itself is invariant and observer dependence enters through the contraction $T' = T$8, where $T' = T$9 is the observer's four-velocity. This reconciles historical contradictions: operational temperature (Ott–Eddington–Møller) arises from specific measurement protocols and is observer-dependent, while $T_{\mathrm{eff}}$0 in the covariant formalism (Landsberg) remains invariant, capturing equilibrium distribution properties.

Strong Numerical Results and Contradictory Claims

Across all analyzed systems (photon gas, relativistic Maxwell–Jüttner gas, Fermi gas), $T_{\mathrm{eff}}$1 increases with observer velocity, contradicting the Planck–Einstein law and supporting Ott–Eddington–Møller interpretation.

Quantitatively, the velocity correction in $T_{\mathrm{eff}}$2 scales as $T_{\mathrm{eff}}$3, where $T_{\mathrm{eff}}$4 spans $T_{\mathrm{eff}}$5 depending on mass, temperature, and degeneracy, and is never simply $T_{\mathrm{eff}}$6. The paper asserts that temperature is not a Lorentz-invariant scalar, but fundamentally observer-dependent, and that the transformation law cannot be universal, differing for each equation of state.

Implications and Future Extensions

The theoretical implication is that temperature transformation in relativity cannot be understood independently of operational definitions and the system’s microscopic physics. Practically, this impacts the interpretation of thermodynamic measurements (e.g., cosmic microwave background, relativistic plasmas, moving thermometers in high-energy contexts, astrophysical jets), where both frame and protocol matter. The discussion suggests that the four-vector $T_{\mathrm{eff}}$7 provides a unified foundation for relativistic thermodynamics, with temperature invariance at the distribution level and observer dependence manifest operationally.

Future extensions involve curved spacetime, where Tolman–Ehrenfest equilibrium condition ($T_{\mathrm{eff}}$8) governs gravitational effects. The formalism can be adapted to accelerated observers, multi-component plasmas, and more general non-equilibrium flows, potentially yielding insight into quantum field theory in curved backgrounds and relativistic kinetic phenomena in extreme astrophysical environments.

Conclusion

The paper resolves the century-long debate regarding relativistic temperature transformation by showing that operationally inferred temperature is observer-dependent and determined by the microscopic details of the system’s equation of state. Strong numerical results demonstrate the increase of $T_{\mathrm{eff}}$9 with velocity for photon, ideal, and Fermi gases, refuting a universal law and aligning with Ott–Eddington–Møller’s sign. The covariant framework based on the inverse-temperature four-vector unifies invariant and operational perspectives, implying that temperature transformation is not a fundamental aspect of relativity but a consequence of specific measurement protocols. This conclusion shapes both the theoretical interpretation and practical application of relativistic thermodynamic analysis in physics and cosmology.

Reference: "Relativistic transformation of temperature revisited" (2606.00521)