A new theory extends Einstein’s relativity to real fluids

The theory of special relativity is rife with counterintuitive and surprising effects, the most famous of which are length contraction and time dilation. If an object travels at a relative speed, which is a non-negligible fraction of the speed of light, with respect to an observer, the length of the object in the travel direction will appear shorter to the observer than it actually is in the object’s rest frame.

In particular, it will appear shorter by a factor equal to one divided by the Lorentz factor. The latter depends only on the relative speed between the object and the observer and on the speed of light, and can only be larger or equal to one, hence the “length contraction” effect.

While length contraction and time dilation are well established relativistic effects, which have been known since even before Einstein’s 1905 paper on special relativity, one may wonder if other relativistic effects concerning other fundamental physical properties can be predicted by special relativity.

For example, in spite of intense research in the field of relativistic hydrodynamics, a relativistic theory of the viscosity of fluids that is also able to recover the limit of classical gases has been missing so far. This is the revealing symptom that the available relativistic theories of viscosity are, possibly, incomplete.

In a new article published in Physical Review E, I derived a general microscopic theory of the viscosity of fluids, based on the recently proposed relativistic Langevin equation (derived from a relativistic microscopic particle-bath Lagrangian), combined with a microscopic nonaffine theory of particle-level displacements under flow. This framework describes the microscopic motion of particles (atoms or ions) as a result of their interactions and collisions with other particles, under an imposed flow field.

While the particles have a tendency to follow the flow field, they also deviate from it due to the interactions with other particles. These “deviations” are called “nonaffine” motions and greatly contribute to the dissipation of momentum in the fluid in motion.

In special relativity, the “momentum” that is relevant for relative motion of the object with respect to an observer is the “proper momentum”, which is the ordinary momentum of the particle multiplied by the Lorentz factor (again, the latter is a number always larger than 1 and a very large number for objects traveling at or near the speed of light).

The new theory that I derived shows that the viscosity of the fluid, which is proportional to the loss of proper momentum for a fluid moving near the speed of light, is thus proportional to the ordinary viscosity of the same fluid moving at ordinary speeds multiplied by the Lorentz factor.

I was quite surprised when I checked whether my microscopic relativistic theory is able to recover, in the non-relativistic limit of low speeds, the viscosity of classical gases as is known from kinetic theory and many aerodynamic experiments. Indeed, I found that the new formula could recover the correct dependencies of viscosity on temperature, particle mass and size, and Boltzmann’s constant which are known for classical gases (e.g., for air flowing near the wings of an airplane).

In the opposite limit of high-energy fluids moving at extremely high speeds (e.g., quark gluon plasma or classical relativistic plasmas), the theory predicts the cubic dependence on temperature in agreement with evidence and yields a new fundamental law of physics which brings together the most important fundamental constants in nature.

Interestingly, I realized that the new theory might unveil a hitherto neglected effect of Einstein’s relativity theory. For example, in analogy with length contraction and time dilation, we can speak of “fluid thickening” as a new relativistic effect that has been overlooked so far and may have important consequences for our understanding of relativistic plasmas in astrophysics and in high-energy physics, including the quark-gluon plasma obtained from high energy nuclear collision reactions.

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