Christopher Verhaaren and Dr. Eric Hirschmann, Department of Physics and Astronomy
General relativity, our current theory of gravity, has been complicating physics since 1916. After replacing Newton’s comparatively simple theory of gravity, relativity has predicted one nonintuitive gravitational effect after another.
As one example, gravitational radiation is an effect unique to general relativity. The theory predicts that some gravitational systems will lose energy by way of gravitational radiation. While this radiation has not yet been detected directly, considerable indirect evidence supports its existence.
Currently, several facilities are trying to detect gravitational waves. Specifically we would like these detectors is to conduct gravitational wave astronomy. To use gravitational waves to observe massive astrophysical objects which do not give off light, such as black holes and neutron stars. Before such observations can become routine we must first become adept at detecting such waves.
One likely source of detectable gravitational radiation is the class of so-called extreme mass binaries. These binary systems consist of two orbiting objects, one of which is much more massive than the other. An example might be a neutron star orbiting a supermassive black hole. Since these systems are thought to be common, a study of their dynamics may improve our ability to detect gravitational radiation emitted from such a source.
In astrophysical systems like an extreme mass binary it is common for the orbiting body to be rotating, similar to the Earth’s rotation as it orbits the sun. If Newtonian gravity were used to analyze such a system the rotation of the orbiting particle would have no effect on the orbit. In contrast, general relativity predicts that a spinning body orbiting another object may behave quite differently from a nonspinning body.
It has been shown that the orbits of spinning particles can be chaotic in certain circumstances. This work predicts that only unphysically large amounts of spin can make a spinning particle orbit chaotic. The purpose of our research is to explore this claim in the simplest orbital system, that of a spinning test particle orbiting a massive object like a black hole. This model is very similar to an extreme mass binary in that we take the mass of the test particle to be much less than the mass of the black hole.
Knowing whether or not chaotic orbits exist impacts how we detect gravitational radiation. A chaotic orbit may lead to a gravitational wave signal that appears to be random. Any detector which is unprepared for such a signal would likely discount it as noise or some other error. So, a better understanding of such a chaotic system will prepare us if chaotic signals are present.
Our use of the term chaotic is to be understood in the technical sense. Chaotic systems are deterministic, but can exhibit apparently random behavior. When such systems are analyzed correctly then a pattern or order can be found. Similarly, if a detector is prepared for chaotic gravitational waves, information about the source can be retrieved from the data.
Searching for physical chaotic orbits required the completion of two tasks. First, we needed to model the behavior of a spinning particle in general relativity. Second, we wanted a reliable way of deciding whether or not a particular orbit was chaotic. This first task was accomplished by making use of the Papapetrou equations. These equations govern the motion of a spinning test particle in a given spacetime. In our modeling we used the Schwarzschild spacetime. This spacetime is the simplest system which allows particle orbits. However, the resulting equations of motion for even this simplest spacetime are complicated enough to require a computer to solve them.
Determining which orbits were chaotic was an even more complicated problem. We used two different methods to check whether a particular orbit was chaotic. First, we used the phase space trajectory of the particle. In orbital systems like the one we are studying the phase space has properties that indicate whether or not an orbit is chaotic. In particular nonchaotic orbits are confined to move on a surface in the phase space. When the orbits are chaotic the phase space trajectory remains close to the surface, but can wander off it. Thus, we were able to look at sections of the phase space and look for evidence of these wandering trajectories, or KAM trajectories. When KAM trajectories were present we took the orbit to be chaotic.
Our second check for chaos was to calculate the so-called Lyapunov exponent. The value of this exponent defines a notion of how much phase space trajectories are moving away from each other. In short, if the value of the exponent is greater than zero then the orbit is chaotic. This method is preferred because we can compare the Lyapunov exponents of different orbits and get a sense of one or the other being “more chaotic” based on the relative size of the Lyapunov exponent.
Along with these methods we developed a new way to compare Lyapunov exponents of different orbits. In previous work those who used the Lyapunov exponent were unable to distinguish between small and zero exponents. We have developed a way to make these different values clearly distinct. This allows our results to extend the bounds on chaos set by previous work.
Our results confirm much of the previous results. However, we also find many chaotic orbits with spin values lower than those previously reported. In particular we found an orbit with positive Lyapunov exponent, and KAM trajectories with a physical spin value. We believe this orbit to be representative of a class of chaotic orbits with physical spin values.
These results show that there are some, albeit very specific, physical orbits traversed by spinning test particles which are chaotic. This will likely affect the gravitational radiation emitted by the particle. Further study of these systems should give additional insight into how gravitational radiation from these types of sources can be effectively detected.