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I took up a critical review of the state of causal and acausal features in the class of theoretical frameworks of relativistic hydrodynamics falling under the category of theories of the truncated Müller–Israel–Stewart type, and then subjected them to Heller's causality conditions. The purpose was to see what, if the theorems based on Heller Causality constraints would rule out the obvious acausal theories, and how this can be tested against, and compared with asymptotic causality constraints.
The overall aim of my research work was the systematic re-investigation of causal structure for the hydrodynamical theories, mostly truncated MIS type theories. In a more concrete way, I tried to check that the constraints on causality, as proposed by Heller et al., would work nice in constraining "acausality" of those theories, particularly when such theories are quite generally truncated at different orders.
I discussed two fundamental theorems of Heller et al. Theorem 1 is an ultra general statement about the condition of linear stability in any reference frame: the imaginary part of frequency has to be smaller or equal than the modulus of imaginary part of wavenumber. The second theorem gives us bounds to the Transport Coefficients, which the coefficients have to fulfill for causality to hold true. A detailed presentation of acausality of the truncated MIS theory was then studied. Notably included here was a derivation of the dispersion relations in both the shear and sound channels of the truncated MIS theory.
here, n represents the order or truncation.
More importantly, it was shown that truncation at any finite order is acausal for the MIS theory. In particular, the theory displayed a blatant breach of Heller's causality condition when truncation took place at N=0, 1, and so the theory in this form would not be causal. Likewise, truncations at N=2 and higher displayed ruptures of causality in a most glaring manner, once again illustrating the character of the result that truncated MIS theories are fundamentally acausal. Contrasted with this, the numerical analysis focused on the testing of the causality of the MIS theory in a plethora of truncations. It was further demonstrated that the application of causality conditions for different orders of truncation makes the violations more pronounced at higher orders of truncation. I commented further on the role of transport coefficients in deciding the causal or acausal nature of the theory, and how even small deviations in these coefficients could result in large acausal behavior.
The three roots for untruncated MIS, i.e. N tending to infinity.
I worked on another alternative hydrodynamic theory, that of BDNK, to make the first step in solving some of the problems for the mentioned-above octures of MIS theory. Some extra terms and corrections to the usual MIS approach are added in the BDNK theory, with the obvious aim of recovering causality while giving a more correct description of the relativistic fluid behaviors. I compared its predictions with those of the truncated MIS theory, getting better causal properties for the former.
My thesis, thus, demonstrated that MIS type truncated theories are acausal as manifested through the violation of the Heller causality conditions. The above-mentioned theorems due to Heller et al. proved to be powerful tools in the analysis of the causal or acausal character of hydrodynamic theories. While on theorem gave absolute test for the causality, the second theorem was more trusted only for inherently causal theories, proving to be not a good test for causality. The causality constraints found through theorem 1 matched the asymptotic causality constraints, and we could conclude the project.
The mathematics behind the thesis and slides presented can be shared if wanted. Contact me if interested.
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