Measuring emission coordinates in a pulsar-based relativistic positioning system

A relativistic deep space positioning system has been proposed using four or more pulsars with stable repetition rates. (Each pulsar emits pulses at a fixed repetition period in its rest frame.) The positioning system uses the fact that an event in spacetime can be fully described by emission coordinates: the proper emission time of each pulse measured at the event. The proper emission time of each pulse from four different pulsars---interpolated as necessary---provides the four spacetime coordinates of the reception event in the emission coordinate system. If more than four pulsars are available, the redundancy can improve the accuracy of the determination and/or resolve degeneracies resulting from special geometrical arrangements of the sources and the event. We introduce a robust numerical approach to measure the emission coordinates of an event in any arbitrary spacetime geometry. Our approach uses a continuous solution of the eikonal equation describing the backward null cone from the event. The pulsar proper time at the instant the null cone intersects the pulsar world line is one of the four required coordinates. The process is complete (modulo degeneracies) when four pulsar world lines have been crossed by the light cone. The numerical method is applied in two different examples: measuring emission coordinates of an event in Minkowski spacetime using pulses from four pulsars stationary in the spacetime; and measuring emission coordinates of an event in Schwarzschild spacetime using pulses from four pulsars freely falling toward a static black hole. These numerical simulations are merely exploratory, but with improved resolution and computational resources the method can be applied to more pertinent problems. For instance one could measure the emission coordinates, and therefore the trajectory, of the Earth.Comment: 9 pages, 2 figures, v3: replaced with version accepted by Phys. Rev.

Tracking Black Holes in Numerical Relativity

This work addresses and solves the problem of generically tracking black hole event horizons in computational simulation of black hole interactions. Solutions of the hyperbolic eikonal equation, solved on a curved spacetime manifold containing black hole sources, are employed in development of a robust tracking method capable of continuously monitoring arbitrary changes of topology in the event horizon, as well as arbitrary numbers of gravitational sources. The method makes use of continuous families of level set viscosity solutions of the eikonal equation with identification of the black hole event horizon obtained by the signature feature of discontinuity formation in the eikonal's solution. The method is employed in the analysis of the event horizon for the asymmetric merger in a binary black hole system. In this first such three dimensional analysis, we establish both qualitative and quantitative physics for the asymmetric collision; including: 1. Bounds on the topology of the throat connecting the holes following merger, 2. Time of merger, and 3. Continuous accounting for the surface of section areas of the black hole sources.Comment: 14 pages, 16 figure

Adaptive Event Horizon Tracking and Critical Phenomena in Binary Black Hole Coalescence

This work establishes critical phenomena in the topological transition of black hole coalescence. We describe and validate a computational front tracking event horizon solver, developed for generic studies of the black hole coalescence problem. We then apply this to the Kastor - Traschen axisymmetric analytic solution of the extremal Maxwell - Einstein black hole merger with cosmological constant. The surprising result of this computational analysis is a power law scaling of the minimal throat proportional to time. The minimal throat connecting the two holes obeys this power law during a short time immediately at the beginning of merger. We also confirm the behavior analytically. Thus, at least in one axisymmetric situation a critical phenomenon exists. We give arguments for a broader universality class than the restricted requirements of the Kastor - Traschen solution.Comment: 13 pages, 20 figures Corrected labels on figures 17 through 20. Corrected typos in references. Added some comment