Relativistic thermodynamics, mathematical problems in

The establishment of relationships between quantities that characterize the macroscopic state of bodies (thermodynamic quantities) in the presence of strong gravitational fields and velocities comparable to the velocity of light.

It is normal practice to examine the equilibrium thermodynamics of an ideal fluid with a given chemical composition. The relationships between thermodynamic quantities, established in non-relativistic thermodynamics, are maintained both in a relativistic macroscopic movement of the particles comprising the body, and in the relativistic movement of the body itself, as well as in strong gravitational fields if the thermodynamic quantities are taken in a system of reference at rest with respect to the element of fluid or body in question and if the energy and chemical potential include all forms of energy (in particular the rest energy).

The basic equations of relativistic thermodynamics are formulated as follows:

$(n{u}^i{)}_{,i}=0$( the law of conservation of baryons);

$dϵ=\mu dn+nTd\sigma$( the first law of thermodynamics);

$(\sigma u^i{)}_{,i}=0$( the condition of adiabaticity), where $u^i$ is the four-dimensional velocity. (The quantity $n$ is the baryon density, $ϵ$ is the energy density, $T$ is the temperature, $\mu =(ϵ+p)/n$ is the chemical potential, $p$ is the pressure, and $\sigma$ is the entropy density. These are related to a system of reference at rest with respect to the volume element in question.) In this case the pressure and the energy density are connected by the relation $p\le ϵ/3$. In the transition to a system of reference moving with respect to the volume element, or to a local observer (in the presence of gravitational fields), some quantities (e.g. the baryon proper density $n$ or the entropy $S$) do not change, i.e. they are scalar, but others change, e.g.

$$\tilde{T}=T{u}^0,\tilde{\mu }=\mu u^0,$$

where the component of the four-dimensional velocity $u^0$ is taken along the world line described by a given point of the body. As a result, in a constant gravitational field the condition of thermal equilibrium does not require constancy of the temperature along the body but of the quantity $T\sqrt{g_{00}}=\text{ const }$, where $g_{00}$ is a component of the metric tensor, $g_{00}=1-2\varphi /c^2$ in a weak gravitational field ( $\varphi$ is the gravitational potential, $c$ is the velocity of light). The temperature measured in a frame with respect to which the body moves with a velocity $\nu$ equals

$$\tilde{T}=\frac{T}{(1-{\nu }^2/c^2{)}^{1/2}}.$$

The relativistic invariance of the entropy $S$ permits one to write the second law of thermodynamics in the form that is usual in non-relativistic thermodynamics:

$$dS\ge \frac{\delta Q}{T},$$

where the amount of heat $\delta Q$ supplied to the body and $T$ are transformed in the same way. Equality is achieved for reversible processes.

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Cf. also Thermodynamics, mathematical problems in.

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