This review provides an introduction to the modeling of fluids in General Relativity. Our main target audience is graduate students with a need for an understanding of relativistic fluid dynamics. Hence, we have made an effort to keep the presentation pedagogical. The article will (hopefully) also be useful to researchers who work in areas outside of General Relativity and gravitation per se (e.g. a nuclear physicist who develops neutron star equations of state), but who require a working knowledge of relativistic fluid dynamics.
Throughout most of the article we will assume that General Relativity is the proper description of gravity. Although not too severe, this is a restriction since the problem of fluids in other theories of gravity has interesting aspects. As we hope that the article will be used by students and researchers who are not necessarily experts in General Relativity and techniques of differential geometry, we have included some discussion of the mathematical tools required to build models of relativistic objects. Even though our summary is not a proper introduction to General Relativity we have tried to define the tools that are required for the discussion that follows. Hopefully our description is sufficiently self-contained to provide a less experienced reader with a working understanding of (at least some of) the mathematics involved. In particular, the reader will find an extended discussion of the covariant and Lie derivatives. This is natural since many important properties of fluids, both relativistic and non-relativistic, can be established and understood by the use of parallel transport and Lie-dragging. But it is vital to appreciate the distinctions between the two.
Ideally, the reader should have some familiarity with standard fluid dynamics, e.g. at the level of the discussion in Landau and Lifshitz , basic thermodynamics , and the mathematics of action principles and how they are used to generate equations of motion . Having stated this, it is clear that we are facing a real challenge. We are trying to introduce a topic on which numerous books have been written (e.g. [112, 66, 72, 9, 123]), and which requires an understanding of much of theoretical physics. Yet, one can argue that an article of this kind is timely. In particular, there have recently been exciting developments for multi-constituent systems, such as superfluid/superconducting neutron star cores1. Much of this work has been guided by the geometric approach to fluid dynamics championed by Carter [17, 19, 21]. This provides a powerful framework that makes extensions to multi-fluid situations quite intuitive. A typical example of a phenomenon that arises naturally is the so-called entrainment effect, which plays a crucial role in a superfluid neutron star core. Given the potential for future applications of this formalism, we have opted to base much of our description on the work of Carter and his colleagues.
Even though the subject of relativistic fluids is far from new, a number of issues remain to be resolved. The most obvious shortcoming of the present theory concerns dissipative effects. As we will discuss, dissipative effects are (at least in principle) easy to incorporate in Newtonian theory but the extension to General Relativity is problematic (see, for instance, Hiscock and Lindblom ). Following early work by Eckart , a significant effort was made by Israel and Stewart [58, 57] and Carter [17, 19]. Incorporation of dissipation is still an active enterprise, and of key importance for future gravitational-wave asteroseismology which requires detailed estimates of the role of viscosity in suppressing possible instabilities.
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