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Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 2 2000 Geom., Cont. and Micros., II H.J. Herrmann - W. Muschik - G. R ¨uckner ∗ CONSTITUTIVE THEORY IN GENERAL RELATIVITY: BASIC FIELDS, STATE SPACES AND THE PRINCIPLE OF MINIMAL COUPLING Abstract. A scheme is presented how to describe material properties under the in- fluence of gravitation. The relativistic dissipation inequality is exploited by LIU‘s procedure. As an example an ideal spinning fluid is considered in the given frame- work.

1. Introduction

We investigate how constitutive properties can be introduced into Einstein‘s gravitation theory. Starting out with the balances of particle number density, spin and energy - momentum, Ein- stein‘s field equations and the relativistic dissipation inequality we consider constitutive equa- tions and state spaces in 3-1- decomposition determining classes of materials. The set of possible constitutive equations compatible with the balances, the state space and the dissipation inequality is found out by LIU‘s exploitation of the dissipation inequality [1], [2].

2. Balances

We start out with the balances of particle number density, energy - momentum and spin in Ein- stein‘s gravitation theory, that means in Riemann geometry of a curved space without torsion: N α ;α = 0, 1 equation, 1 T αβ ; β = 0, 4 equations, 2 S αβ ;β = 0, 3 equations. 3 Here the particle flux is defined by N α = nu α with the particle density n and the 4-velocity u α . First of all the energy - momentum tensor is proposed to be not symmetric T αβ 6= T βα . The spin density S αβ is antisymmetric S αβ = −S βα and satisfies the so-called Frenkel condition u α S αβ = 0 = S αβ u β which expresses that the spin tensor is purely spatial. Because we want to describe material under the influence of gravitation in Riemann geometry we need Einstein‘s field equations ˜ R αβ − 12g αβ ˜R = κT αβ , 10 equations. 4 ∗ We would like to thank H.-H. von Borzeszkowski and Thoralf Chrobok for interesting discussions and introduction to geometries with curvature and torsion. We would also like to thank the VISHAY Company, D-95085 Selb and the Deutsche Forschungsgemeinschaft for financial support. 133 134 H.J. Herrmann - W. Muschik - G. R¨uckner Here are ˜ R αβ the Ricci tensor and ˜ R the curvature scalar due to the Riemann geometry. They are marked with a tilde because we although examine material under the framework of other geome- tries, so we have to distinguish between different geometric quantities. Due to the symmetry of the left hand side only the symmetric part of the energy- momentum tensor appears in the field equations. Finally we have to take into account the dissipation inequality S α ;α = σ ≥ 0. 5 Here S α = su α + s α , introducing the entropy density s and the entropy flux density s α . The 18 equations 1 to 4 and the dissipation inequality contain more fields than equations are. Therefore the set of equations is underdetermined. This is due to the fact, that 1 to 5 are valid for all materials and up to now no special material was taken into consideration. Hence we have to split the 37 fields appearing in 1 to 5 into the basic fields which we are looking for and into the constitutive equations describing the considered material or the class of materials. In more detail the 37 fields are: N α , 4 fields, T αβ , 16 fields, S αβ , 3 fields, g αβ , 10 fields, S α , 4 fields. From the energy - momentum tensor we can see, that parts of it belong to the constitutive equa- tions, namely the 3 - stress tensor, and other parts, namely the energy density, belong to the basic fields. Therefore we perform as usual the following 3-1 decomposition ǫ : = T αβ 1 c 2 u α u β , energy density, t αβ : = h αγ T γ σ h σβ , stress tensor, q α : = −h ασ T γ σ u γ , heat flux density, p β : = h ασ T σ γ u γ , momentum density. Here h αβ is the projection tensor perpendicular to the 4-velocity: h αγ : = g αγ + 1 c 2 u α u γ = h γ α . Now we introduce the 18 basic fields: {ǫ, n, u α , g αβ , S αβ }x α , and the remaining 19 constitutive equations: {t αβ , q α , p β , S α }x α . Dealing with Riemann geometry one finally have to satisfy some constraints: u α u α = −c 2 , normalisation of the the 4-velocity, g αβ = g βα , symmetry of the metric, g αβ;γ = 0, vanishing of the non-metricity, { α βγ } = Fg αβ , g αβ,γ , symmetric connection as a function of the metric and the first partial derivative of the metric, A α;β = A α,β − { σ αβ }A σ , covariant derivative according to the geometry. Constitutive theory in general relativity: basic fields 135

3. State Space