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Putting Pressure Under Pressure: On the Status of Classical Pressure in Special Relativity

Eugene Y. S. Chua

TL;DR

This work scrutinizes whether the classical pressure concept can be meaningfully extended to special relativity. By analyzing four classical perspectives—hydrostatics, the fundamental relation, equations of state, and continuum mechanics—it demonstrates a breakdown of consilience under Lorentz boosts, yielding no unique relativistic pressure except in the rest frame or in the limit $u\to 0$. It contrasts Planckian and Einsteinian views on energy transformation and highlights how different relativistic treatments of energy and temperature lead to divergent predictions for pressure across frames. The result suggests that the thermodynamic description is inherently frame-relative and that the classical thermodynamic limit alone is insufficient to guarantee a relativistic extension; instead, a rest-frame-centered viewpoint or a velocity-reversion to $u\to 0$ is required for a coherent emergence of thermodynamics from microscopic theory.

Abstract

Much of the century-old debate surrounding the status of thermodynamics in relativity has centered on the search for a suitably relativistic temperature; recent works by Chua (2023) and Chua and Callender (forthcoming) have suggested that the classical temperature concept -- consilient as it is in classical settings -- 'falls apart' in relativity. However, these discussions typically assume an unproblematic Lorentz transformation for -- specifically, the Lorentz invariance of -- the pressure concept. Here I argue that, just like the classical temperature, the classical concept of pressure breaks down in relativistic settings. I discuss how this might suggest a new thermodynamic limit -- a u --> 0 limit -- without which an unambiguous thermodynamic description of systems doesn't emerge.

Putting Pressure Under Pressure: On the Status of Classical Pressure in Special Relativity

TL;DR

This work scrutinizes whether the classical pressure concept can be meaningfully extended to special relativity. By analyzing four classical perspectives—hydrostatics, the fundamental relation, equations of state, and continuum mechanics—it demonstrates a breakdown of consilience under Lorentz boosts, yielding no unique relativistic pressure except in the rest frame or in the limit . It contrasts Planckian and Einsteinian views on energy transformation and highlights how different relativistic treatments of energy and temperature lead to divergent predictions for pressure across frames. The result suggests that the thermodynamic description is inherently frame-relative and that the classical thermodynamic limit alone is insufficient to guarantee a relativistic extension; instead, a rest-frame-centered viewpoint or a velocity-reversion to is required for a coherent emergence of thermodynamics from microscopic theory.

Abstract

Much of the century-old debate surrounding the status of thermodynamics in relativity has centered on the search for a suitably relativistic temperature; recent works by Chua (2023) and Chua and Callender (forthcoming) have suggested that the classical temperature concept -- consilient as it is in classical settings -- 'falls apart' in relativity. However, these discussions typically assume an unproblematic Lorentz transformation for -- specifically, the Lorentz invariance of -- the pressure concept. Here I argue that, just like the classical temperature, the classical concept of pressure breaks down in relativistic settings. I discuss how this might suggest a new thermodynamic limit -- a u --> 0 limit -- without which an unambiguous thermodynamic description of systems doesn't emerge.
Paper Structure (18 sections, 76 equations, 4 figures)

This paper contains 18 sections, 76 equations, 4 figures.

Table of Contents

  1. The question of relativistic thermodynamics
  2. Relativizing thermodynamics: 'T falls apart'
  3. The classical thermodynamic pressure
  4. Pressure in hydrostatics
  5. Pressure in the fundamental relation
  6. Pressure in equations of state
  7. Pressure in continuum mechanics
  8. Putting pressure under pressure
  9. Forces over area: relativistic pressure as invariant
  10. The fundamental equation: relativistic pressure increases for the moving observer
  11. What is the relativistic fundamental relation?
  12. What is the Lorentz transformation for energy?
  13. Equations of state: pressure meets relativistic temperature
  14. Continuum mechanics: anisotropic pressure
  15. Relativistic pressure as rest pressure?
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Newton's argument for the isotropy of pressure, illustrated.
  • Figure 2: Adapted from tadmor_continuum_2012.
  • Figure 3: The ingredients for proving the Lorentz invariance of pressure.
  • Figure 4: Left: a system having equilibrated with its box, without energy flux. Right: the same box from the perspective of a Lorentz-boosted frame, with constant velocity $\textbf{u}$.