Sarracino, Robert Sinclair2024-08-152024-08-1519771977https://hdl.handle.net/1828/19571In this thesis we advance a new definition of incompressibility in general relativity, and develop and analyse the equation of state for a uniform proper-density static, perfect fluid. Schwarzschild published the first exact interior solution to Einstein's equations: the field inside a static, spherically symmetric perfect fluid with equation of state ρ = ρₒ = constant, where ρ is the energy-density component of the energy-momentum tensor. He interpreted the ρ-constant condition to be the equation of state for a 'uniform density', hence 'incompressible', perfect fluid, an interpretation that has been generally accepted. As a consequence, the ρ-constant equation of state has been taken to be the limiting equation of state for superdense nonrotating stellar bodies. While there is no general agreement over what constitutes a 'limiting' equation of state for a physically realistic star, to date, to the best of our knowledge, there has been no argument that the Schwarzschild fluid represents the absolute limit for any static system. In this thesis we argue that the Schwarzschild fluid is not an incompressible fluid, and advance a new definition of incompressibility. An incompressible fluid would be one for which the limit of infinite central pressure, and the static limit 2m/rₒ = 1 at the surface, where mis the mass and rₒ the radius of the body, would coincide. In addition we develop the equation of state for a uniform proper-density perfect fluid, and, using the new definition of incompressibility, compare this with the Schwarzschild fluid, and with a third proposed by Eddington. We find that the constant proper-density equation of state more closely approximates incompressibility than either the Schwarzschild or Eddington equations of state, and briefly explore a few of its properties, such as a greater maximum gravitational red-shift, z = Δλ/λ = 2.48, compared with z = 2.00 for the Schwarzschild case, and note that ·the constant proper­ density equation of state would lead to a larger upper limit to the mass of a neutron star.87 pagesAvailable to the World Wide WebThesis