Static Solutions of Einstein's Field Equations for Spheres of Fluid

Static Solutions of Einstein's Field Equations for Spheres of Fluid

February 15, 1939 | Richard C. Tolman
The text discusses the absorption of radiation in a fluid and its relation to the half-width of the radiation. It explains that the intensity of the additional absorption is much less than the normal absorption, with the ratio being the square of $ G/4\pi\gamma^3 $. When $ \gamma $ is much smaller than $ \Gamma $, the approximation of considering only transitions A to B becomes invalid. The double transition A to B to A can be analyzed using the given method, with $ \gamma + \Gamma $ replacing $ \gamma $ in the square root. This leads to no shift in the absorbed line, consistent with Weisskopf's result. The paper then presents a detailed analysis of Einstein's field equations for static spheres of fluid, aiming to find explicit solutions. It introduces a method to solve these equations, leading to several new solutions. The paper discusses the general form of solutions for an equilibrium distribution of fluid, and presents specific solutions, including the Einstein universe, Schwarzschild-de Sitter solution, and Schwarzschild interior solution. These solutions are derived under different assumptions, such as constant pressure or density, and are analyzed for their physical properties. The paper also discusses the connection between interior and exterior solutions, and the implications of these solutions for the structure of stars. It concludes by emphasizing the importance of stability and the physical interpretation of static solutions.The text discusses the absorption of radiation in a fluid and its relation to the half-width of the radiation. It explains that the intensity of the additional absorption is much less than the normal absorption, with the ratio being the square of $ G/4\pi\gamma^3 $. When $ \gamma $ is much smaller than $ \Gamma $, the approximation of considering only transitions A to B becomes invalid. The double transition A to B to A can be analyzed using the given method, with $ \gamma + \Gamma $ replacing $ \gamma $ in the square root. This leads to no shift in the absorbed line, consistent with Weisskopf's result. The paper then presents a detailed analysis of Einstein's field equations for static spheres of fluid, aiming to find explicit solutions. It introduces a method to solve these equations, leading to several new solutions. The paper discusses the general form of solutions for an equilibrium distribution of fluid, and presents specific solutions, including the Einstein universe, Schwarzschild-de Sitter solution, and Schwarzschild interior solution. These solutions are derived under different assumptions, such as constant pressure or density, and are analyzed for their physical properties. The paper also discusses the connection between interior and exterior solutions, and the implications of these solutions for the structure of stars. It concludes by emphasizing the importance of stability and the physical interpretation of static solutions.
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[slides and audio] Static Solutions of Einstein's Field Equations for Spheres of Fluid