Relativistic heat conduction

The theory of relativistic heat conduction (RHC)[1] claims to be the only model for heat conduction (and similar diffusion processes) that is compatible with the theory of special relativity, the second law of thermodynamics, electrodynamics, and quantum mechanics, simultaneously. The main features of RHC are:

  1. It admits a finite speed of heat propagation, and allows for relativistic effects when heat flux transients approach that speed.
  2. It removes the possibility of paradoxical situations that may violate the second law of thermodynamics.
  3. It, implicitly, admits the wave–particle duality of the heat-carrying “phonon”.

These outcomes are achieved by (1) upgrading the Fourier equation of heat conduction to the form of a Telegraph equation of electrodynamics, and (2) introducing a new definition of the heat flux vector. Consequently, RHC gives rise to a number of interesting phenomena, such as thermal resonance and thermal shock waves, which are possible during high-frequency pulsed laser heating of thermal insulators. The main appealing feature of the theory is its mathematical elegance and simplicity.


Classical model

For most of the last two centuries, heat conduction has been modelled by the well-known Fourier equation:[2]

where θ is temperature,[3] t is time, α = k/(ρ c) is thermal diffusivity, k is thermal conductivity, ρ is density, and c is specific heat capacity. The Laplace operator,, is defined in Cartesian coordinates as

This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, q, as a function of temperature gradient,

into the first law of thermodynamics

where the del operator, , is defined in 3D as

It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,[4]

where s is specific entropy and σ is entropy production. Alternatively, the second law can be written as

which leads to the condition[1]

which is always true, because k is a non-negative material property.

Hyperbolic model

For most of the last century, it was recognized that Fourier equation (and its more general Fick's law of diffusion) is in contradiction with the theory of relativity,[5] for at least one reason: it admits infinite speed of propagation of heat signals within the continuum field. For example, consider a pulse of heat at the origin; then according to Fourier equation, it is felt (i.e. temperature changes) at any distant point, instantaneously. The speed of information propagation is faster than the speed of light in vacuum, which is physically inadmissible within the framework of relativity.

To overcome this contradiction, workers such as Cattaneo,[6] Vernotte,[7] Chester,[8] and others[9] proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form,

also known as the Telegrapher's equation. Interestingly, the form of this equation traces its origins to Maxwell’s equations of electrodynamics; hence, the wave nature of heat is implied. In this equation, C is called the speed of second sound (i.e. the fictitious quantum particles, phonons) and this equation is known as the Hyperbolic Heat Conduction (HHC) equation.

For the HHC equation to remain compatible with the first law of thermodynamics, it is necessary to modify the definition of heat flux vector, q, to

where is a relaxation time, such that

The most important implication of the hyperbolic equation is that by switching from a parabolic (dissipative) to a hyperbolic (includes a conservative term) partial differential equation, there is the possibility of phenomena such as thermal resonance[10][11][12] and thermal shock waves.[13]

Criticism to the HHC model

The theory of RHC attempts to resolve the controversies surrounding the hyperbolic equation, while maintaining the form of that equation. This is achieved by:

Derivation of the RHC equation


In a Euclidean space, distance between any two points, ds, is measured by

where dx, dy, and dz are displacements along three orthogonal axes.

In a Minkowski space, distance between two events, ds, is measured by

where, τ, is space-like-time and is related to real time, t, by

where C is speed of light in vacuum and . Hence,

Consequently, the 3D del, ,operator is upgraded to the 4D quad, , operator (also known as the Four-gradient)

Likewise, the 3D Laplacian, , operator is upgraded to the 4D d'Alembert operator,[19]

Any physical quantity that is Galilean invariant in Euclidean space can be made Lorentz invariant in a Minkowski space, by upgrading from 3D to 4D operators. Consequently, Fourier’s equation can be upgraded to 4D as

which is called the relativistic heat conduction equation. Likewise, the definition of the heat-flux vector, q, is upgraded to the 4D form as


It can be shown that this definition of q is compatible with the first law of thermodynamics,

as well as the second law of thermodynamics,

in their 4D upgraded form.[1] The imaginary terms in these equations are direct manifestation of the wave nature of heat, and are essential for the heat equation to become compatible with all laws of physics. The real terms in these equations are identical to those in the classical heat model.

The most interesting observation about RHC is that it reduces the second law of thermodynamics to a statement of the form

which is the “no action at a distance” principle of special relativity. Essentially, the RHC asserts that relativity and the second law of thermodynamics are two alternative, but equal statements about the nature of time. Both physical principles are mutually derivable from each other and are complementary.[1]

Criticism to RHC

As far as heat conduction is concerned, the RHC equation is identical in form to the hyperbolic equation, and all analytical and experimental results that are relevant to one are equally applicable to the other. The definition of heat flux vector, however, is different; but the RHC definition is merely a 4D upgrade of the original linear Fourier approximation. The mathematics of RHC is much simpler and more elegant. However, RHC raises some significant conceptual challenges:

  1. This weak interpretation of relativity, in which the speed of second sound plays a role similar to that of the speed of light, can be viewed as downgrading or degrading to the universality of the theory of relativity. Notice how the symbol c in standard relativity theory is replaced with C without much interpretation.
  2. The implied wave nature of heat is controversial. Some workers reject the wave nature of heat on dogmatic grounds. Moreover, RHC implies that a phonon is a full-fledged objective quantum particle whose physical reality is no lesser than that of a photon. Existing experimental evidences are not enough to support for or against such views.
  3. Heat quantities become complex numbers, with values including "imaginary temperature", which are hard to interpret experimentally.
  4. The equivalence of relativity and the second law is shocking, because it implies that one of them can be a derivative of the other.

In summary, while the RHC is mathematically simple and elegant, and experimentally practical and relevant, it raises a number of conceptual issues that are highly controversial.


The RHC theory is applicable for any physical problem in which the hyperbolic equation is relevant: when speed of heat propagation is small, e.g. thermal insulators, or when speed of heat-flux variation is very large, e.g. pulsed-laser heating. Applications for those types of problems are abundant, and there is plenty of published work (see Notes, below). Most of these results remain relevant to RHC, but because the definition of heat flux vector is different, final closed-form solutions may not be the same. In many cases, RHC provides closed-form solutions that are not possible using the HHC model. A number of useful fundamental solutions for 1D and 2D relativistic moving heat sources are available in closed-form.[20]


  1. 1 2 3 4 5 Ali, Y. M.; Zhang, L. C. (2005). "Relativistic heat conduction". Int. J. Heat Mass Trans. 48 (12): 2397. doi:10.1016/j.ijheatmasstransfer.2005.02.003.
  2. Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids (Second ed.). Oxford: University Press.
  3. Some authors also use T, φ,...
  4. Barletta, A.; Zanchini, E. (1997). "Hyperbolic heat conduction and local equilibrium: a second law analysis". Int. J. Heat Mass Trans. 40 (5): 1007–1016. doi:10.1016/0017-9310(96)00211-6.
  5. Eckert, E. R. G.; Drake, R. M. (1972). Analysis of Heat and Mass Transfer. Tokyo: McGraw-Hill, Kogakusha.
  6. Cattaneo, C. R. (1958). "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée". Comptes Rendus. 247 (4): 431.
  7. Vernotte, P. (1958). "Les paradoxes de la theorie continue de l'équation de la chaleur". Comptes Rendus. 246 (22): 3154.
  8. Chester, M. (1963). "Second sound in solids". Phys. Rev. 131 (15): 2013–2015. Bibcode:1963PhRv..131.2013C. doi:10.1103/PhysRev.131.2013.
  9. Morse, P. M.; Feshbach, H. (1953). Methods of Theoretical Physics. New York: McGraw-Hill.
  10. Mandrusiak, G. D. (1997). "Analysis of non-Fourier conduction waves from a reciprocating heat source". J. Thermophys. Heat Trans. 11 (1): 82. doi:10.2514/2.6204.
  11. Xu, M.; Wang, L. (2002). "Thermal oscillation and resonance in dual-phase-lagging heat conduction". Int. J. Heat Mass Trans. 45 (5): 1055. doi:10.1016/S0017-9310(01)00199-5.
  12. Barletta, A.; Zanchini, E. (1996). "Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field". Int. J. Heat Mass Trans. 39 (6): 1307. doi:10.1016/0017-9310(95)00202-2.
  13. Tzou, D. Y. (1989). "Shock wave formation around a moving heat source in a solid with finite speed of heat propagation". Int. J. Heat Mass Trans. 32 (10): 1979. doi:10.1016/0017-9310(89)90166-X.
  14. Grad, H. (1949). "On the kinetic theory of rarefied gases". Commun. Pure Appl. Math. 2 (4): 331–407. doi:10.1002/cpa.3160020403.
  15. Ali, A. H. (1999). "Statistical mechanical derivation of Cattaneo's heat flux law". J. Thermophys. Heat Trans. 13 (4): 544–546. doi:10.2514/2.6474.
  16. Koerner, C.; Bergmann, H. W. (1998). "Physical defects of the hyperbolic heat conduction equation". Appl. Phys. A: Mater. Sci. Process. 67 (4): 397–401. Bibcode:1998ApPhA..67..397K. doi:10.1007/s003390050792.
  17. Bai, C.; Lavine, A. S. (1995). "On hyperbolic heat conduction and the second law of thermodynamics". J. Heat Trans., Trans. ASME. 117 (2): 256. doi:10.1115/1.2822514.
  18. Rubin, M. B. (1992). "Hyperbolic heat conduction and the second law". Int. J. Eng. Sci. 30 (11): 1665–1676. doi:10.1016/0020-7225(92)90134-3.
  19. Some authors use for the d'Alembert operator, but to maintain the analogy with the Laplacian operator, using is more consistent.
  20. Ali, Y. M.; Zhang, L. C. (2005). "Relativistic moving heat source". Int. J. Heat Mass Trans. 48: 2741. doi:10.1016/j.ijheatmasstransfer.2005.02.004.


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