Gravitational Waves and Inspiraling Compact Binaries in Alternative Theories of Gravity

This dissertation consists of four parts. In Part I, we briefly review fundamental theories of gravity, performed experimental tests, and gravitational waves. The framework and the methods that we use in our calculations are discussed in Part II. This part includes reviewing the methods of the Parametrized Post-Newtonian (PPN) framework, Direct Integration of Relaxed Einstein Equations (DIRE), and Matched Filtering. In Part III, we calculate the explicit equations of motion for non-spinning compact objects (neutron stars or black holes) to 2.5 post-Newtonian order, or $O(v/c)^5$ beyond Newtonian gravity, in a general class of alternative theories to general relativity known as scalar-tensor theories. For the conservative part of the motion, we obtain the two-body Lagrangian and conserved energy and momentum through second post-Newtonian order. We find the contributions to gravitational radiation reaction to 1.5 post-Newtonian and 2.5 post-Newtonian orders, the former corresponding to the effects of dipole gravitational radiation. For binary black holes we show that the motion through 2.5 post-Newtonian order is observationally identical to that predicted by general relativity. In Part IV, we construct a parametrized dispersion relation that can produce a range of predictions of alternative theories of gravity for violations of Lorentz invariance in gravitation, and investigate their impact on the propagation of gravitational waves. We show how such corrections map to the waveform observable by a gravitational-wave detector, and to the "parametrized post-Einsteinian framework", proposed to model a range of deviations from General Relativity. Given a gravitational-wave detection, the lack of evidence for such corrections could then be used to place a constraint on Lorentz violation.Comment: PhD Dissertation, Submitted to the Graduate School of Arts and Sciences, Washington University in St. Louis (August 2013). arXiv admin note: text overlap with arXiv:1209.0667, arXiv:gr-qc/0201001, arXiv:gr-qc/0510072, arXiv:gr-qc/9502040, arXiv:1304.3473 by other author

Carter-like constants of motion in the Newtonian and relativistic two-center problems

In Newtonian gravity, a stationary axisymmetric system admits a third, Carter-like constant of motion if its mass multipole moments are related to each other in exactly the same manner as for the Kerr black-hole spacetime. The Newtonian source with this property consists of two point masses at rest a fixed distance apart. The integrability of motion about this source was first studied in the 1760s by Euler. We show that the general relativistic analogue of the Euler problem, the Bach-Weyl solution, does not admit a Carter-like constant of motion, first, by showing that it does not possess a non-trivial Killing tensor, and secondly, by showing that the existence of a Carter-like constant for the two-center problem fails at the first post-Newtonian order.Comment: 11 pages; version to be published by Classical and Quantum Gravit

Constraining Lorentz-violating, Modified Dispersion Relations with Gravitational Waves

Modified gravity theories generically predict a violation of Lorentz invariance, which may lead to a modified dispersion relation for propagating modes of gravitational waves. We construct a parametrized dispersion relation that can reproduce a range of known Lorentz-violating predictions and investigate their impact on the propagation of gravitational waves. A modified dispersion relation forces different wavelengths of the gravitational wave train to travel at slightly different velocities, leading to a modified phase evolution observed at a gravitational-wave detector. We show how such corrections map to the waveform observable and to the parametrized post-Einsteinian framework, proposed to model a range of deviations from General Relativity. Given a gravitational-wave detection, the lack of evidence for such corrections could then be used to place a constraint on Lorentz violation. The constraints we obtain are tightest for dispersion relations that scale with small power of the graviton's momentum and deteriorate for a steeper scaling.Comment: 11 pages, 3 figures, 2 tables: title changed slightly, published versio

Energy-momentum Density of Gravitational Waves

In this paper, we elaborate the problem of energy-momentum in general relativity by energy-momentum prescriptions theory. Our aim is to calculate energy and momentum densities for the general form of gravitational waves. In this connection, we have extended the previous works by using the prescriptions of Bergmann and Tolman. It is shown that they are finite and reasonable. In addition, using Tolman prescription, exactly, leads to same results that have been obtained by Einstein and Papapetrou prescriptions.Comment: LaTeX, 9 pages, 1 table: added reference

On Energy Distribution of Two Space-times with Planar and Cylindrical Symmetries

Considering encouraging Virbhadra's results about energy distribution of non-static spherically symmetric metrics in Kerr-Schild class, it would be interesting to study some space-times with other symmetries. Using different energy-momentum complexes, i.e. M{\o}ller, Einstein, and Tolman, in static plane-symmetric and cylindrically symmetric solutions of Einstein-Maxwell equations in 3+1 dimensions, energy (due to matter and fields including gravity) distribution is studied. Energy expressions are obtained finite and well-defined. calculations show interesting coincidences between the results obtained by Einstein and Tolamn prescriptions. Our results support the Cooperstock hypothesis about localized energy.Comment: LaTex, 9 pages: corrected typos, added reference

Energy-momentum Distribution in Static and Non-static Cosmic String Space-times

We elaborate the problem of energy-momentum in general relativity by energy-momentum prescriptions theory. In this regard, we calculate M\oller,Landau-Lifshitz, Papapetrou, Einstein, Bergman, Tolman, and Weinberg's energy-momentum complexes in static and nonstatic cosmic string space-times. We obtain strong coincidences between the results. These coincidences can be considered as an extension of Virbhadra's viewpoint that different energy-momentum prescriptions may provide some basis to define a unique quantity. In addition, our results disagree with Lessner's belief about M\oller's prescription and support the Virbhadra's conclusion about the power of Einstein's prescription.Comment: LaTeX, 5 page: added reference

Energy-momentum Prescriptions in General Spherically Symmetric Space-times

Einstein, Landau-Lifshitz, Papapetrou, Weinberg, and M{\o}ller energy-momentum prescriptions in general spherically symmetric space-times are investigated. It is shown that for two special but not unusual classes of general spherically symmetric space-times several energy-momentum prescriptions in Schwarzschild Cartesian coordinates lead to some coincidences in energy distribution. It is also obtained that for a special class of spherically symmetric metrics M{\o}ller and Einstein energy-momentum prescriptions give the same result for energy distribution if and only if it has a specific dependence on radial coordinate.Comment: LaTeX, 7 pages: added reference

Testing General Relativity with Present and Future Astrophysical Observations

One century after its formulation, Einstein's general relativity has maderemarkable predictions and turned out to be compatible with all experimentaltests. Most of these tests probe the theory in the weak-field regime, and thereare theoretical and experimental reasons to believe that general relativityshould be modified when gravitational fields are strong and spacetime curvatureis large. The best astrophysical laboratories to probe strong-field gravity areblack holes and neutron stars, whether isolated or in binary systems. We reviewthe motivations to consider extensions of general relativity. We present a(necessarily incomplete) catalog of modified theories of gravity for whichstrong-field predictions have been computed and contrasted to Einstein'stheory, and we summarize our current understanding of the structure anddynamics of compact objects in these theories. We discuss current bounds onmodified gravity from binary pulsar and cosmological observations, and wehighlight the potential of future gravitational wave measurements to inform uson the behavior of gravity in the strong-field regime