User:Maschen/Thermodynamics in curved spacetime

In general relativistic physics, and it's numerous applications to astrophysics, thermodynamics in curved spacetime is the mathematical description of classical thermodynamics in curved spacetime.

Thermodynamic laws
Each of the classical thermodynamic laws can be stated in a number of equivalent ways. The fundamental thermodynamic relation combines the first and second laws into one:


 * $$dE = TdS - PdV + \mu dN $$

The second law can be also be stated as:


 * $$ \Delta S_\text{universe} \geq 0 $$

From a more fundamental and modern perspective, a more fundamental law of thermodynamics is baryon number conservation. To formulate this; denote the number density of baryons (number of baryons N per unit 3d volume V) by n in the rest frame, then the proper time derivative of total number is zero:


 * $$\frac{d}{d\tau}N = \frac{d}{d\tau}(nV) = 0 $$

and the changes in volume are given as the four-divergence of the four velocity u of the fluid:


 * $$\frac{d}{d\tau}V = (\boldsymbol{\nabla}\cdot\mathbf{u})V $$

Explicitly rewriting the divergence term leads to a simpler continuity equation:


 * $$\boldsymbol{\nabla}\cdot\mathbf{N} = 0 $$

where the baryon number flux vector is:


 * $$\mathbf{N} = n\mathbf{u} $$

The relativistic generalization of eqn (X) is:


 * $$\frac{d}{d \tau}S \geq 0$$

where the equality holds for thermal equilibrium only, the inequality is strict for non-equilibrium.