Physicshttp://hdl.handle.net/1903/22692016-05-18T22:06:05Z2016-05-18T22:06:05ZThe Stability of the Schwarzschild MetricVishveshwara, C. V.http://hdl.handle.net/1903/174492016-04-02T02:30:20Z1968-01-01T00:00:00ZThe Stability of the Schwarzschild Metric
Vishveshwara, C. V.
The stability of the Schwarzschild exterior metric against small perturbations is investigated. The exterior extending from the Schwarzschild radius r =2m to spatial infinity is visualized as having been produced by a spherically symmetric mass distribution that collapsed into the Schwarzschild horizon in the remote past. As a preamble to the stability analysis, the phenomenon of spherically symmetric gravitational collapse is discussed under the conditions of zero pressure, absence of rotation and adiabatic flow. This is followed by a brief study of the Kruskal coordinates in which the apparent singularity at r = 2m is no longer present; the process of spherical collapse and the consequent production of the Schwarzschild empty space geometry down to the Schwarzschild horizon are depicted on the Kruskal diagram.
The perturbations superposed on the Schwarzschild background metric are the same as those given by Regge and Wheeler consisting of odd and even parity classes, and with the time dependence exp(-ikt), where k is the frequency. An analysis of the Einstein field equations computed to first order in the perturbations away from the Schwarzschild background metric shows that when the frequency is made purely imaginary, the solutions that vanish at large values of r, conforming to the requirement of asymptotic flatness, will diverge near the Schwarzschild surface in the Kruskal coordinates even at the initial instant t = 0. Since the background metric itself is finite at this surface, the above behaviour of the perturbation clearly contradicts the basic assumption that the perturbations are small compared to .the background metric. Thus perturbations with imaginary frequencies that grow exponentially with time are physically unacceptable and hence the metric is stable. In the case of the odd perturbations, the above proof of stability is made rigorous by showing that the radial functions for real values of k form a complete set, by superposition of which any well behaved initial perturbation can be represented so that the time development of such a perturbation is non-divergent, since each of the component modes is purely oscillatory in time. A similar rigorous extension of the proof of stability has not been possible in the case of the even perturbations because the frequency (or k2) does not appear linearly in the differential equation.
A study of stationary perturbations (k = 0) shows that the only nontrivial stationary perturbation that can exist is that due to the rotation of the source which is given by the odd perturbation with the angular momentum £ = 1. Finally, complex frequencies are introduced under the boundary conditions of only outgoing waves at infinity and purely incoming waves at the Schwarzschild surface. The physical significance of this situation is discussed and its connection with phenomena such as radiation damping and resonance scattering, and with the idea of causality is pointed out.
1968-01-01T00:00:00ZAlmost Symmetric Spaces and Gravitational RadiationMatzner, Richard Alfredhttp://hdl.handle.net/1903/174482016-04-02T02:31:47Z1967-01-01T00:00:00ZAlmost Symmetric Spaces and Gravitational Radiation
Matzner, Richard Alfred
1967-01-01T00:00:00ZGravitational Radiation in the Limit of High FrequencyIsaacson, Richard Allenhttp://hdl.handle.net/1903/174472016-04-02T02:31:45Z1967-01-01T00:00:00ZGravitational Radiation in the Limit of High Frequency
Isaacson, Richard Allen
This dissertation deals with a technique for obtaining approximate radiative solutions to the Einstein equations of general relativity in situations where the gravitational fields of interest are quite strong. In the first chapter, we review the history of the problem and discuss previous work along related lines. In the second chapter, we assume the radiation to be of high frequency and expand the field equations in powers of the small wavelength this supplies. This assumption provides an approximation scheme valid for all orders of 1/r, for arbitrary velocities up to that of light, and for arbitrary intensities of the gravitational field. To lowest order we obtain a gauge invariant linear wave equation for gravitational radiation, which is a covariant generalization of that for massless spin-two fields in flat space, This wave equation is then solved by the W.K.B. approximation to show that gravitational waves travel on null geodesics with amplitude and frequency modified by gravitational fields in exactly the same way as are those of light waves, and with their polarization parallel transported along the geodesics, again as is the case for light. The metric containing high frequency gravitational waves is shown to be type N to lowest order, and some limits to the methods used are discussed. In the third chapter we go beyond the linear terms in the high frequency expansion, and consider the lowest order non-linear terms. They are shown to provide a natural, gauge invariant, averaged effective stress tensor for the energy localized in the high frequency radiation. By assuming the W.K.B. form for the field, this tensor is found to have the same structure as that for an electromagnetic null field. A Poynting vector is used to investigate the flow of energy and momentum in the gravitational wave field, and it is seen that high frequency waves propagate along null hypersurfaces and are not backscattered off by the curvature of space. Expressions for the total energy and momentum carried by the field to flat null infinity are given in terms of coordinate independent integrals valid within regions of strong field strength. The formalism is applied to the case of spherical gravitational waves where a news function is obtained, and where the source is found to lose exactly the energy and momentum contained in the radiation field.
1967-01-01T00:00:00ZAn All-Sky, Three-Flavor Search for Neutrinos from Gamma-Ray Bursts with the IceCube Neutrino ObservatoryHellauer, Robert Eugenehttp://hdl.handle.net/1903/173212016-03-29T02:57:11Z2015-01-01T00:00:00ZAn All-Sky, Three-Flavor Search for Neutrinos from Gamma-Ray Bursts with the IceCube Neutrino Observatory
Hellauer, Robert Eugene
Ultra high energy cosmic rays (UHECRs), defined by energy greater than 10^18 eV, have been observed for decades, but their sources remain unknown. Protons and heavy ions, which comprise cosmic rays, interact with galactic and intergalactic magnetic fields and, consequently, do not point back to their sources upon measurement. Neutrinos, which are inevitably produced in photohadronic interactions, travel unimpeded through the universe and disclose the directions of their sources.
Among the most plausible candidates for the origins of UHECRs is a class of astrophysical phenomena known as gamma-ray bursts (GRBs). GRBs are the most violent and energetic events witnessed in the observable universe.
The IceCube Neutrino Observatory, located in the glacial ice 1450 m to 2450 m below the South Pole surface, is the largest neutrino detector in operation. IceCube detects charged particles, such as those emitted in high energy neutrino interactions in the ice, by the Cherenkov light radiated by these particles. The measurement of neutrinos of 100 TeV energy or greater in IceCube correlated with gamma-ray photons from GRBs, measured by spacecraft detectors, would provide evidence of hadronic interaction in these powerful phenomena and confirm their role in ultra high energy cosmic ray production.
This work presents the first IceCube GRB-neutrino coincidence search optimized for charged-current interactions of electron and tau neutrinos as well as neutral-current interactions of all neutrino flavors, which produce nearly spherical Cherenkov light showers in the ice. These results for three years of data are combined with the results of previous searches over four years of data optimized for charged-current muon neutrino interactions, which produce extended Cherenkov light tracks. Several low significance events correlated with GRBs were detected, but are consistent with the background expectation from atmospheric muons and neutrinos. The combined results produce limits that place the strongest constraints thus far on models of neutrino and UHECR production in GRB fireballs.
2015-01-01T00:00:00Z