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>

Cosmological Milestones and Gravastars - Topics in General Relativity

<p>In this thesis, we consider two different problems relevant to general relativity. Overthe last few years, opinions on physically relevant singularities occurring in FRWcosmologies have considerably changed. We present an extensive catalogue of suchcosmological milestones using generalized power series both at the kinematical anddynamical level. We define the notion of “scale factor singularity” and explore its relationto polynomial and differential curvature singularities. We also extract dynamicalinformation using the Friedmann equations and derive necessary and sufficient conditionsfor the existence of cosmological milestones such as big bangs, big crunches, bigrips, sudden singularities and extremality events. Specifically, we provide a completecharacterization of cosmological milestones for which the dominant energy conditionis satisfied. The second problem looks at one of the very small number of seriousalternatives to the usual concept of an astrophysical black hole, that is, the gravastarmodel developed by Mazur and Mottola. By considering a generalized class of similarmodels with continuous pressure (no infinitesimally thin shells) and negative centralpressure, we demonstrate that gravastars cannot be perfect fluid spheres: anisotropcpressures are unavoidable. We provide bounds on the necessary anisotropic pressureand show that these transverse stresses that support a gravastar permit a higher compactnessthan is given by the Buchdahl–Bondi bound for perfect fluid stars. We alsocomment on the qualitative features of the equation of state that such gravastar-likeobjects without any horizon must have.</p>

Cosmological Milestones and Gravastars - Topics in General Relativity

<p>In this thesis, we consider two different problems relevant to general relativity. Overthe last few years, opinions on physically relevant singularities occurring in FRWcosmologies have considerably changed. We present an extensive catalogue of suchcosmological milestones using generalized power series both at the kinematical anddynamical level. We define the notion of “scale factor singularity” and explore its relationto polynomial and differential curvature singularities. We also extract dynamicalinformation using the Friedmann equations and derive necessary and sufficient conditionsfor the existence of cosmological milestones such as big bangs, big crunches, bigrips, sudden singularities and extremality events. Specifically, we provide a completecharacterization of cosmological milestones for which the dominant energy conditionis satisfied. The second problem looks at one of the very small number of seriousalternatives to the usual concept of an astrophysical black hole, that is, the gravastarmodel developed by Mazur and Mottola. By considering a generalized class of similarmodels with continuous pressure (no infinitesimally thin shells) and negative centralpressure, we demonstrate that gravastars cannot be perfect fluid spheres: anisotropcpressures are unavoidable. We provide bounds on the necessary anisotropic pressureand show that these transverse stresses that support a gravastar permit a higher compactnessthan is given by the Buchdahl–Bondi bound for perfect fluid stars. We alsocomment on the qualitative features of the equation of state that such gravastar-likeobjects without any horizon must have.</p>

Counterexamples to the Maximum Force Conjecture

Dimensional analysis shows that the speed of light and Newton’s constant of gravitation can be combined to define a quantity F*=c4/GN with the dimensions of force (equivalently, tension). Then in any physical situation we must have Fphysical=fF*, where the quantity f is some dimensionless function of dimensionless parameters. In many physical situations explicit calculation yields f=O(1), and quite often f≤1/4. This has led multiple authors to suggest a (weak or strong) maximum force/maximum tension conjecture. Working within the framework of standard general relativity, we will instead focus on idealized counter-examples to this conjecture, paying particular attention to the extent to which the counter-examples are physically reasonable. The various idealized counter-examples we shall explore strongly suggest that one should not put too much credence into any truly universal maximum force/maximum tension conjecture. Specifically, idealized fluid spheres on the verge of gravitational collapse will generically violate the weak (and strong) maximum force conjectures. If one wishes to retain any truly general notion of “maximum force” then one will have to very carefully specify precisely which forces are to be allowed within the domain of discourse.

Onset of Inertial Magnetoconvection in Rotating Fluid Spheres

The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied in this paper. Thermo-inertial convection is a distinctive flow regime on the border between rotating thermal convection and wave propagation. It occurs in astrophysical and geophysical contexts where self-sustained or external magnetic fields are commonly present. To investigate the onset of motion, a perturbation method is used here with an inviscid balance in the leading order and a buoyancy force acting against weak viscous dissipation in the next order of approximation. Analytical evaluation of constituent integral quantities is enabled by applying a Green’s function method for the exact solution of the heat equation following our earlier non-magnetic analysis. Results for the case of thermally infinitely conducting boundaries and for the case of nearly thermally insulating boundaries are obtained. In both cases, explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. It is found that an imposed azimuthal magnetic field exerts a stabilizing influence on the onset of inertial convection and as a consequence magneto-inertial convection with azimuthal wave number of unity is generally preferred.