2.1. The Balance Equations
The balance equations of energy-momentum and spin of phenomenological GCCT in a curved space-time (the comma denotes partial and the semicolon covariant derivatives, round brackets the symmetric part of a tensor, square brackets its asymmetric part) are [
2]:
Here,
is the, in general, non-symmetric energy-momentum tensor of CT and
the current of spin density, often denoted in brief as spin tensor. The
and
are internal source terms, the Geo-SMEC-terms (geometry-spin-momentum-energy-coupling) [
2], which are caused by the choice of a special space-time geometry and by a possible coupling between energy-momentum, spin and geometry.
For non-isolated systems,
denotes an external force density, and
is an external momentum density. As in the continuum theory of irreversible processes [
3,
4], the balance Equation (
1) must be supplemented by those of particle number and entropy density:
(
particle flux density,
entropy four-vector,
σ entropy production,
φ entropy supply). The second law of thermodynamics is taken into account by the demand that the entropy production has to be non-negative at each event and for arbitrary materials after having inserted the constitutive equations into the expression of the entropy production:
The (3 + 1)-splits of the tensors in Equations (
1) and (3) are:
Here, the divergence-free particle number flux density
is chosen according to Eckart [
1], and the projector perpendicular to the four-velocity
, respectively
, is introduced:
and Equation (9) results in:
By splitting the stress tensor into its diagonal and its traceless parts:
we introduce the pressure
p and the friction tensor
.
According to Equations (6) and (7), we obtain:
Starting out with Equation (8), it holds:
resulting in:
Taking Equation (12) into account, we obtain:
The (3 + 1)-split of tensors is a usual tool in relativistic continuum physics. The (3 + 1)-components, generated by the split, have physical significance, which originally is hidden in the unsplit tensor Equations (5) to (10). Thus, we generate by (3 + 1)-split the following covariant quantities: the particle number density n, the energy density e, the momentum flux density , the energy flux density , the stress tensor , the spin density , the spin density vector , the couple stress , the spin stress , the entropy density s and the entropy flux density .
It is clear that these quantities are not independent of each other. Especially here, we are interested in expressions for the entropy density and the entropy flux density, which we are going to construct in the next section by use of a special procedure starting out with the entropy identity [
2]. Some further hints can be found in the
Appendix.
2.2. The Entropy Identity
For defining the entropy density
s and the entropy flux density
, which determine the entropy four-vector according to Equation (10)
, we add suitable zeros to the entropy four-vector, a procedure that paves the way for defining
s and
later on. The entropy identity results by multiplying Equations (5)
, (14) and (16) with the present arbitrary quantities
κ,
λ and
, which are suitably chosen below; Equation (10)
becomes:
This entropy identity does not take the entire energy-momentum and spin tensor into account, but only their contractions with according to Equation (18).
That means that the contractions with are not included in the entropy identity, resulting in the consequence that more than one entropy identity can be established, if other secondary conditions are taken into consideration. A generalization of the entropy identity is obtained by replacing κ, λ and by tensors of higher order.
An entropy vector regarding the spin of the system is also constructed by kinetic approaches of the equilibrium theory of spin systems [
5,
6]. Interestingly, in the latter paper, a spin vector was derived that satisfies the entropy identity Equation (19). This spin vector is that one that is also mostly used in phenomenological considerations.
Because the entropy identity is not unique, the entropy production and later on also s and are not unique, either: changing the entropy identity results in changing the material. Here, we start out with the special entropy identity Equation (18).
Because the part of
that is parallel to
does not contribute to the last term of Equation (18) and, consequently, not to the entropy identity, we can demand:
without restricting the generality. The identity Equation (19) becomes another one by differentiation:
This identity changes into the entropy production, if according to Equations (3) and (10), and φ are specified. For achieving that, we now rearrange the five terms of Equation (21).
Introducing the covariant time derivative (
is the relativistic analogue of the non-relativistic material time derivative
, which describes the time rates of a rest-observer; therefore,
is observer independent and zero in equilibrium [
7,
8]):
and using the balance Equations (3)
and (1)
, the entropy identity Equation (21) becomes:
The covariant time derivative Equation (22) can be replaced by the Lie derivative
because:
is valid, and we apply the covariant time derivative only on scalars according to the first row of Equation (23). Consequently, concerning the time derivative appearing in the entropy identity, we can use the covariant time derivative Equation (22) or the Lie derivative, both along
.
Taking Equations (6), (13) and (14) into account, the fourth term of Equation (23) becomes:
We now transform the sixth term of the entropy identity Equation (23) by taking Equations (8) and (20) into account:
Inserting Equations (25) and (26) into Equation (23) results in:
As already mentioned, the entropy identity has to be transferred into the expression for the entropy production by specifying the entropy flux , the entropy density s, the entropy supply φ and the three for the present arbitrary quantities κ, λ and .
Obviously, Equation (27) contains terms of different kinds: a divergence of a vector perpendicular to (the last term of Equation (27)), time derivatives of intensive quantities (the first row of Equation (27)), two terms stemming from the field equations (last terms of the third and fourth row of Equation (27)), three terms containing spin (3 + 1)-components (in the fourth row of Equation (27)) and three further terms containing (3 + 1)-components of the energy-momentum tensor (in the third row of Equation (27)). This structure of the entropy identity allows one to choose a state space and, by virtue of it, to define the entropy density, the entropy supply, the entropy flux, the gr-Gibbs equation and the gr-Gibbs–Duhem equation, which all are represented in the next sections.
2.4. State Space, Gr-Gibbs Equation and Entropy Flux
We now choose a state space that belongs to a one-component spin system in local equilibrium and which is spanned by the particle number
n, the energy density
e and the spin density vector
:
Local equilibrium means: the state at each event is described by a set of equilibrium variables, which change from event to event, generating gradients of equilibrium variables, causing irreversible processes. According to Equations (15) and (16), the three-indexed spin is only partly taken into account, namely by and . Here, is an independent state variable, whereas represents a constitutive property according to Equation (36).
The gr-Gibbs equation is given by the covariant time derivative of the entropy density
s, which is composed of covariant time derivatives belonging to the chosen state space. Such covariant time derivatives appear only in the first row of Equation (27) (the acceleration
is not a material property, but one of the kinematical invariants). Consequently, we define:
and the split of the entropy identity into the gr-Gibbs equation and entropy density later on depends on the choice of the time derivative, although the entropy identity is independent of this choice.
Up to here, the quantities
introduced into the entropy identity Equation (18) are unspecified. Taking the gr-Gibbs Equation (32) into consideration, such a specification is now possible:
λ is the reciprocal rest-temperature:
κ is proportional to the chemical potential:
and
is analogous to Equation (34) proportional to a spin potential:
These quantities, as all of the others that do not belong to the state space variable Equation (31), are constitutive quantities describing the material by constitutive equations. These constitutive quantities are:
They all, including the entropy density
s and the entropy flux density
, are functions of the state space variables:
These constitutive equations are out of scope of this paper (for how to use the constitutive equations in connection with the field equations, see [
9]).
Because of Equation (33), the term
is as in CT a part of the entropy flux. Consequently, we define the entropy flux density according to the last row of Equation (27):
Taking Equation (36) into consideration, the entropy flux density is also a constitutive quantity.