### 5.1 Fundamental, or Euler, relation

Consider the standard form of the combined First and Second Laws for a simple, single-species system:
This follows because there is an equation of state, meaning that where
The total energy , entropy , volume , and particle number are said to be extensive if when , , and are doubled, say, then will also double. Conversely, the temperature , pressure , and chemical potential are called intensive if they do not change their values when , , and are doubled. This is the additivity property and we will now show that it implies an Euler relation (also known as the “fundamental relation” [96]) among the thermodynamic variables.

Let a tilde represent the change in thermodynamic variables when , , and are all increased by the same amount , i.e.

Taking to be extensive then means
Of course we have for the intensive variables
Now,
and therefore we find the Euler relation
If we let denote the total energy density, the total entropy density, and the total particle number density, then

The nicest feature of an extensive system is that the number of parameters required for a complete specification of the thermodynamic state can be reduced by one, and in such a way that only intensive thermodynamic variables remain. To see this, let , in which case

The re-scaled energy becomes just the total energy density, i.e. , and moreover since
The first law thus becomes
or
This implies
The Euler relation (79) then yields the pressure as

We can think of a given relation as the equation of state, to be determined in the flat, tangent space at each point of the manifold, or, physically, small enough patches across which the changes in the gravitational field are negligible, but also large enough to contain a large number of particles. For example, for a neutron star Glendenning [48] has reasoned that the relative change in the metric over the size of a nucleon with respect to the change over the entire star is about , and thus one must consider many internucleon spacings before a substantial change in the metric occurs. In other words, it is sufficient to determine the properties of matter in special relativity, neglecting effects due to spacetime curvature. The equation of state is the major link between the microphysics that governs the local fluid behavior and global quantities (such as the mass and radius of a star).

In what follows we will use a thermodynamic formulation that satisfies the fundamental scaling relation, meaning that the local thermodynamic state (modulo entrainment, see later) is a function of the variables , , etc. This is in contrast to the fluid formulation of “MTW” [80]. In their approach one fixes from the outset the total number of particles , meaning that one simply sets in the first law of thermodynamics. Thus without imposing any scaling relation, one can write

This is consistent with our starting point for fluids, because we assume that the extensive variables associated with a fluid element do not change as the fluid element moves through spacetime. However, we feel that the use of scaling is necessary in that the fully conservative, or non-dissipative, fluid formalism presented below can be adapted to non-conservative, or dissipative, situations where cannot be imposed.