Scherungsfreie Fluide in der Allgemeinen Relativitätstheorie
Scherungsfreie Fluide in der Allgemeinen Relativitätstheorie
Fak. 2 Mathematik und Naturwissenschaften
Die vorliegende Arbeit behandelt scherungsfreie Fluide in der Allgemeinen Relativitätstheorie. Dazu wird zunächst ein allgemeiner Fluidformalismus eingeführt. Dieser wird dann für Raumzeiten mit divergenzfreiem Weyltensor benutzt, um Modelle der linearen und erweiterten Thermodynamik zu untersuchen. Es zeigt sich, dass die verbleibenden möglichen Lösungsklassen sowohl eine einfache Quell- als auch eine einfache kinematischen Struktur besitzen. Es stellt sich heraus, dass die Aussagen der erweiterten Thermodynamik detaillierter als die der linearen Thermodynamik sind. Im Folgenden werden Untersuchungen zur shear-free fluid conjecture (SFC) durchgeführt. Zunächst sind eine Reihe von Eigenschaften von rotierenden, expandierenden Raumzeiten mit nicht verschwindender Beschleunigung, deren Feldquelle eine ideale Flüssigkeit ist, abgeleitet. Dann werden zwei neue Spezialfälle der SFC bewiesen und eine Reihe weiterer Eigenschaften von nicht explizit bewiesenen Fällen bestimmt. Es wird gezeigt, dass die Evolution der Zwangsgleichungen mit den aus der Ricci-Identität folgenden Evolutionsgleichungen und den als Bedingung für die Divergenz des Weyltensors geschriebenen Bianchi-Identitäten im Widerspruch steht.
This work considers shear-free fluids in general relativity. The main reason for such an elaboration can be seen in cosmology, where strong limits on a possible shear of the galaxy flow exist, which can be obtained by observation of the highly isotropic microwave background radiation. Moreover also for the description of special phases of stellar evolution - especially the collapse - such considerations are of special interest. After introducing a general fluid formalism, which besides Einstein's field equations uses the Ricci- and Bianchi-identity, a consideration of shear-free spacetimes with vanishing divergence of the Weyl-tensor for different thermodynamical models is performed. The main focus is here a comparison of linear and extended thermodynamics. This shows that the remaining possibilities for the spacetime configuration are mostly simple in the sense that rotation and acceleration in the kinematical level are suppressed as well as the heat-flow in the source terms of the dynamical equations. Therefore these spacetimes are highly homogeneous and isotropic. Moreover most of them are conformally flat. Some singular cases appear where kinematical vector fields are not suppressed but the equation of state has to be that of a cosmological constant, i.e. $ ho=-p$. Surprisingly the predictions of extended thermodynamics are stronger than the predictions of the linear thermodynamics. Motivated by the above results these considerations are extended to spacetimes without any assumption on the Weyl-curvature, though with a priori more specialized source configuration. These considerations are aiming at perfect, shear-free fluids mainly with respect to the question of the shear-free fluid conjecture (SFC). This conjecture states that there are no solutions of Einstein's field equations with an ideal fluid as source, which are shear-free, while rotating and expanding. After summarizing the results published in the literature, a system of basic vector fields is constructed. With the help of this vector fields at first a number of general results on the structure of the spacetimes under consideration are re-derived. Then it is shown that a part of the divergence equations of the Weyl-tensor can be formulated as a set of covariant wave equations acting on the basic vector fields. Moreover it is proven that, using the results from the previous section, the equations of time development of the magnetic part of the Weyl-tensor is identically fulfilled independent of the form of the electric and magnetic part of the Weyl-tensor. On the other hand the time propagation equations of the electric part leads to a difficult set of differential equations. The analogy to Maxwell's equations is discussed. To make further considerations the propagation of the heat flow equation is considered and other reformulations are given. In the following several new special cases of the SFC are proven. The strategy for the proofs is to assume that one is able to construct solutions with non-vanishing expansion, rotation and acceleration and a perfect fluid with barotropic equation of state as a source. Hereupon using a propagation technique one can derive that these assumptions leads to a contradiction, and therefore the assumptions are incorrect and the SFC is confirmed. With the help of the above derived tools and the described strategy it is proven that, for a vanishing gradient of the expansion, the SFC is correct. For the proof the time development of the constraint equations is considered, which leads to new constraint equations. These can be used to simplify the remaining part of the Maxwell-like field equations. By using again the propagation technique one is led to a set of differential equations for the equation of state, which finally excludes the possibility of a consistent solution. Finally the case where the acceleration and the rotation field are perpendicular is considered. Here the time development of the constraint equation leads also to new constraints. However the statement of these constraint equations is unclear, because they include higher-order derivative terms, such that they are not solvable. Making an assumption about the representation of the gradient of the expansion it is possible to find some simplifications and predictions. Fixing the equation of state to $p=alpha ho$ with $alpha=const$, a complicated case study is done, which rules out a number of models for special values of $alpha$. Thus a number of special cases which supports the SFC is found. Moreover it seems to be plausible that the gradient of expansion has to have only components in the time direction and only one special spatial direction. It is proven that this class of models must have a restricted Weyl-curvature and may allow the construction of solutions of Einstein's field equations which contradict the SFC. Also it is argued that the cases studied in this part are very similar to the general case. The work concludes with summarizing remarks and some comments on the relation between thermodynamics and gravitational theory.