Strong gravitational fields and radiation from neutron stars

© 2017 Dr. Arthur George SuvorovPart 1 of this thesis is dedicated to the study of the gravitational field in the strong field regime. In particular, we focus on modified theories of gravity, exploring the implications of high-energy corrections to Einstein's equations in the context of black holes and gravitational waves. Chapter 1 offers a literature review of general relativity and modified theories of gravity. In general relativity, multipole moment constructions allow one to decompose a general metric tensor in order to analyse the physical properties of the solution in detail. For example, the Kerr metric has two free parameters in it, which can be identified as the mass and angular momentum of the black hole through multipole analysis. In Chapter 2 we generalise these multipole moment constructions, with the goal of classifying black holes in modified theories of gravity. One can attempt to design experiments aimed at searching for `non-Einstein' black hole parameters. We explore whether the frequencies of quasi-periodic oscillations from galactic microquasars are consistent with a particular generalised-Kerr black hole solution, after a suitable interpretation of non-Kerr parameters has been made. We use X-ray measurements of the microquasars GRS $1915+105$ and GRO J$1665-40$ to constrain the properties of their black holes. The Ernst formalism of general relativity is a tool used to reduce the tensorial Einstein equations to a single, non-linear differential equation for a complex-valued scalar function. The formalism can be used to generate exact solutions to the Einstein equations. In Chapter 3 we present a generalisation to the $f(R)$ theory of gravity, which allows us to find new exact solutions. We find some exact solutions which represent the gravitational field outside of ellipsoidal, compact objects, and we find some solutions representing exact, solitonic gravitational waves which propagate with arbitrary phase velocity $v \neq c$. We show that neutron stars can be arbitrarily `hairy' in some modified theories of gravity, in the sense that the metric coefficients can depend on an arbitrary number of free parameters. Using the Ernst formalism developed in Chapter 3, we focus on the nature of causality in modified gravity. In Chapter 4, we show that some theories predict the existence of exact, faster-than-light gravitational waves, even when the linearised theory forbids them. The implication is that gravitational information may propagate at a different speed to electromagnetic information, which would have astrophysical consequences for highly relativistic systems such as black holes surrounded by plasma. We show that some theories permitting tachyonic gravitational waves are consistent with data coming from solar system and pulsar timing experiments. In Chapter 5 we investigate black hole structure in modified theories of gravity. We ask the question of what restrictions are placed on a modified theory of gravity if one assumes that event horizons are spherical; a feature of black holes expected from our intuitive understanding of gravitational collapse. We show that spherical horizons are guaranteed if a certain differential inequality is satisfied. This inequality can be used to place constraints on any given theory of gravity. In the context of $f(R)$ gravity, we show that the spherical-horizons condition is consistent with the `no-ghosts' (no negative energy modes) condition. Chapters 2 through 5 are therefore geared towards trying to reduce the pool of possible modified theories of gravity through various theoretical and empirical tests. When considering modified theories of gravity, one faces a degeneracy problem of sorts; non-Einstein gravity with ordinary matter might look like Einstein gravity but with exotic matter, or vice-versa. Part 2 of this thesis is concerned with the other half of the field equations; the matter portion. We focus in particular on the physics of neutron stars with asymmetric mass-density and fluid-velocity distributions, i.e. on continuous gravitational radiation from neutron stars. Our goal is to quantitatively explore the astrophysical properties of neutron stars from gravitational and electromagnetic observations. Chapter 6 serves as a literature review aimed at exploring the physics of neutron stars in various settings. Time-dependent multipole moments lead to gravitational radiation as the star spins. A detection of this radiation would then give us some information, otherwise invisible in the electromagnetic spectrum, about how the star is behaving. In Chapter 7 we consider quantum spin effects of the neutrons, protons, and electrons which comprise the stellar fluid. We find that the resulting quantum force terms may be large $(\chi \gtrsim 1)$ or small $(\chi \lesssim 10^{-3})$ in a neutron star environment depending on the degree of saturation physics, such as the formation of magnetic domains, limiting the value of the magnetic susceptibility $\chi$. We show that the terms are likely to be small except perhaps in magnetar $\left[|\boldsymbol{B}| \gtrsim 10^{13} \left(T_{e}/10^{9} \text{ K}\right) \text{ G}\right]$ environments, where paramagnetic forces can amplify (up to a factor $\sim 10$) the strength of any and all gravitational radiation. In some neutron stars, there is an observational discrepancy between magnetic field strengths inferred from cyclotron lines and spin-down estimates. It has been suggested that strong, non-dipole components may be present in the magnetic field structure, which would account for the apparent inconsistency. In Chapter 8 we investigate a non-ideal magnetohydrodynamic phenomena, known as Hall drift, in the context of neutron star models. In particular, we show that strong, non-dipole components can emerge naturally through Hall drift on timescales of $t \gtrsim 10^{4} \text{ yr}$. The Hall drift necessarily modifies the equilibrium structure of the star, which bolsters the expected gravitational wave luminosity. We find that, if Hall drift is responsible for the magnetic field strength discrepancy, old $(\gtrsim 10^{5} \text{ yr})$ radio pulsars should emit continuous gravitational wave signals which are large enough $(h_{0} \gtrsim 10^{-26})$ to be measurable in the near future with ground-based interferometers such as the Laser Interferometer Gravitational-Wave Observatory (LIGO). In accreting neutron star systems like low-mass X-ray binaries (LMXBs), the time-averaged spin-up torque from accretion is expected to add angular momentum to the neutron star at a rate of $N_{a} \approx \dot{M} \left( G M_{\star} R_{\star} \right)^{1/2}$, where $\dot{M}$ is the accretion rate. It is therefore puzzling that we observe no neutron stars spinning faster than $\nu_{\text{observed}} \lesssim 700 \text{ Hz}$, which is much less than the theoretical maximum set by the break-up limit $\nu_{\text{max}} \gtrsim 1.5 \text{ kHz}$. In Chapter 9 we consider mass-density asymmetries in the form of physical mounds (`magnetic mountains') which build up on neutron star surfaces, when matter is accreted from a main sequence companion. The built-up mountains reduce the global dipole moment of the neutron star by forcing magnetic field lines to buckle underneath infalling matter, and also excite gravitational radiation in the process. The gravitational radiation reaction associated with the magnetic mountain may be responsible for capping the spin frequency of the star. We investigate the characteristics of magnetic mountains when thermal conduction is treated for the first time in self-consistent, numerical simulations. We find that thermal conduction has the effect of softening the equation of state of accreted material on timescales $t \gtrsim 10 \text{ s}$, thereby amplifying the gravitational wave signal (factor $\sim 2$). This strengthens the argument for targeting LMXBs such as Sco X-1 in searches with facilities like LIGO. We show that the clumping of accreted matter leads to the formation of hot regions $(T \gtrsim 10^{8} \text{ K})$ throughout the mountain body, which has implications for type I X-ray burst activity. The excitation of inertial $(r-)$ modes within neutron stars leads to a fluid-velocity asymmetry, which generates a time-dependent current quadrupole moment. Depending on the equation of state, the resulting gravitational radiation may cause $r$-mode amplitudes to grow through an instability known as the Chandrasekhar-Friedman-Schutz (CFS) instability. In particular, it has been suggested that the gravitational radiation reaction resulting from the CFS instability may explain the observation that LMXBs have a narrow range of spin frequencies $0.1 < \nu / \left( \text{ kHz}\right) < 0.7$. However, for barotropic neutron stars, $r$-modes are often described by hyperbolic boundary-value problems (HBVPs). HBVPs are known to admit singularities and other pathological properties in certain circumstances, indicating that some aspects of the model may be unphysical. In Chapter 10 we present models for $r$-modes in nonbarotropic (stratified) neutron stars, and demonstrate that the HBVP nature of the problem cannot arise in certain, well-defined circumstances. We find that nonbarotropic $r$-modes are subject to the CFS instability for magnetic field strengths satisfying $|\boldsymbol{B}| \lesssim 10^{12} \text{ G}$, and spin frequencies in the range $3.0 \times 10^{-2} \leq \nu / \left( \text{ kHz} \right) \leq 1.5$