Some Exact Solutions in General Relativity

<p><b>In this thesis four separate problems in general relativity are considered, dividedinto two separate themes: coordinate conditions and perfect fluid spheres. Regardingcoordinate conditions we present a pedagogical discussion of how the appropriateuse of coordinate conditions can lead to simplifications in the form of the spacetimecurvature — such tricks are often helpful when seeking specific exact solutions of theEinstein equations. Regarding perfect fluid spheres we present several methods oftransforming any given perfect fluid sphere into a possibly new perfect fluid sphere.</b></p> <p>This is done in three qualitatively distinct manners: The first set of solution generatingtheorems apply in Schwarzschild curvature coordinates, and are phrased in termsof the metric components: they show how to transform one static spherical perfectfluid spacetime geometry into another. A second set of solution generating theoremsextends these ideas to other coordinate systems (such as isotropic, Gaussian polar,Buchdahl, Synge, and exponential coordinates), again working directly in terms of themetric components. Finally, the solution generating theorems are rephrased in termsof the TOV equation and density and pressure profiles. Most of the relevant calculationsare carried out analytically, though some numerical explorations are also carriedout.</p>

Some Exact Solutions in General Relativity

<p><b>In this thesis four separate problems in general relativity are considered, dividedinto two separate themes: coordinate conditions and perfect fluid spheres. Regardingcoordinate conditions we present a pedagogical discussion of how the appropriateuse of coordinate conditions can lead to simplifications in the form of the spacetimecurvature — such tricks are often helpful when seeking specific exact solutions of theEinstein equations. Regarding perfect fluid spheres we present several methods oftransforming any given perfect fluid sphere into a possibly new perfect fluid sphere.</b></p> <p>This is done in three qualitatively distinct manners: The first set of solution generatingtheorems apply in Schwarzschild curvature coordinates, and are phrased in termsof the metric components: they show how to transform one static spherical perfectfluid spacetime geometry into another. A second set of solution generating theoremsextends these ideas to other coordinate systems (such as isotropic, Gaussian polar,Buchdahl, Synge, and exponential coordinates), again working directly in terms of themetric components. Finally, the solution generating theorems are rephrased in termsof the TOV equation and density and pressure profiles. Most of the relevant calculationsare carried out analytically, though some numerical explorations are also carriedout.</p>

Weyssenhoff Fluid Sphere Models in Einstein-Cartan Theory

We obtain three models for Geodesic flows and three models for Non-Geodesic flows of Weyssenhoff fluid considering it as the source of gravitation and spin in the Einstein-Cartan field equations. Influence of spin on the pressure, density, equation of state and the kinematical parameters is observed in both geodesic and non-geodesic models.

A new model for dark matter fluid sphere

We develop a new model for a spherically symmetric dark matter fluid sphere containing two regions: (i) Isotropic inner region with constant density and (ii) Anisotropic outer region. We solve the system of field equation by assuming a particular density profile along with a linear equation of state. The obtained solutions are well-behaved and physically acceptable which represent equilibrium and stable matter configuration by satisfying the Tolman–Oppenheimer–Volkoff (TOV) equation and causality condition, condition on adiabatic index, Harrison–Zeldovich–Novikov criterion, respectively. We consider the compact star EXO 1785-248 (Mass [Formula: see text] and radius R[Formula: see text]8.8 km) to analyze our solutions by graphical demonstrations.