Jared Greenwald and Dr. Eric Hirschmann, Physics and Astronomy
There has been a tremendous research effort in trying to understand observed astrophysical phenomena. The difficulties in studying the formation of and interactions between astrophysical objects have led to the use of computational methods to solve these problems. From a theoretical point of view, one difficulty is that many of these problems have no known exact solution and therefore approximations have to be made. Computers can handle these approximations well and since they can perform many operations quickly, we can find greater accuracy by utilizing them. Once a good mathematical model is generated for some astrophysical object (say black holes, galaxies or stars) researchers can investigate stability, formation or collision of these objects with other interstellar objects. These models can be used to understand how a solar system, galaxy or even the whole universe was formed and how it will continue to change.
Our research involves calculating possible initial equilibrium states for magnetized, differentially rotating neutron stars. Once these states have been generated they will be used to consider how neutron stars evolve in time. One can also use these models to see how such possible neutron stars interact with other massive astrophysical bodies during collisions. Research up to this point has been conducted only for magnetized, rigidly rotating neutron stars[1] and we wish to extend this work to a more general and realistic situation. In particular, differential rotation allows for more general current flows to exist throughout the neutron star whereas rigid rotation allows currents to flow only in circles perpendicular to the rotation axis [see figures 1 & 2].
The first step in developing this model is to assume axisymmetry and stationarity in the spacetime. This means that we have assumed two kinds of symmetries to exist in our spacetime: axisymmetry is related to spherical and cylindrical symmetry while stationarity is a symmetry in time that allows us to create an equilibrium model. This means that our model should be resistant to mathematical perturbations and should eventually ‘settle’ down to a stable final state after being disturbed.
Next we projected out the spacetime symmetries from the defining Einstein equations, effectively creating a two dimensional problem. The Einstein equations are ten coupled partial differential equations that relate the curvature of spacetime to the matter and energy present in the spacetime. This process of projecting out dimensions is similar to creating a ‘shadow’ of the 4-D object on a 2-D surface. We can project out these dimensions because we have the freedom to choose a coordinate system where the dimensions that are being projected out correspond directly with the above symmetries. An analogous (though not quite related) example of this is the electric field generated by an unmoving electron. We know that the electric field generated is spherical in nature because the measured electric field at any point depends only on the distance between that point and the electron. In other words, we only need to consider the radial dimension of the spherical symmetry and can ignore the other two angular dimensions, thus reducing a 3-D problem to 1-D.
We are currently using a suitable gauge choice to simplify the Einstein equations and are comparing with previous work done by Cook et. al.[1]. The choice of gauge is related to the freedom to work in any coordinate system and by choosing a convenient coordinate system one can greatly simplify the problem at hand. We can see this, for example, by choosing cylindrical coordinates to solve problems involving magnetic fields of straight, long wires or of solenoids.
Once we have verified that our work reduces to the work done by Cook[1], we will begin to write the computational code to numerically solve the partial differential equations which result from the Einstein equations coupled to the equations of magneto-hydrodynamics. Magneto-hydrodynamics is the physics of electromagnetics coupled with fluid dynamics and can be used to study objects like stars. As described above, the curvature of spacetime is directly related to what matter is present in our spacetime through the Einstein equations. For our work the matter present is a plasma (or highly energized fluid) of neutrons. Like the Einstein equations, we also need to project the magneto-hydrodynamic equations from a 4-D spacetime to 2-D and after doing this we will be able to simulate the complete magnetized, differentially rotating neutron star via computer modeling.
This work is important to understanding currently observed yet not well understood astrophysical phenomena. We expect to use our results to help us better understand the formation and evolution of neutron stars and can also extend it to the study of other objects like black holes in similar settings of formation, evolution and collision.
Reference
- G. Cook, S. Shapiro and S. Teutolsky Spin-Up of a Rapidly Rotating Star by Angular Momentum Loss: Effects of General Relativity, 1992.