Gravitational Wave Source Simulations

 

Jeff Jauregui

Harvey Mudd College

May 1, 2003

 

Abstract

A simplified model of a collapsing, rotating star is used to simulate the emission of gravitational radiation.  In particular, a two-dimensional polar grid containing local mass density information is implemented in C++.  Appropriate equations from general relativity predict what type of waves this time-evolving system should emit.  The model is significantly limited by its two-dimensional nature and lack of a realistic mass distribution.  Associated computational issues are discussed.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Introduction

A classical result of physics is that a changing electric dipole induces an oscillating electromagnetic field.  In other words, it emits electromagnetic radiation.  Einstein’s theory of general relativity makes an analogous prediction for mass – a mass quadrupole moment that changes with time emits energy-carrying gravitational radiation.  Although most aspects of general relativity have been experimentally verified, gravitational waves have not yet been directly detected.  Intensive research efforts are currently underway to do so.  For example, LIGO (Laser-Interferometry Gravitational Wave Observatory), is the largest endeavor funded by the NSF.

Because incoming gravitational wave signals would be so weak and buried in noise, researchers must have a clear understanding of qualitative and quantitative characteristics of the waveforms.  For example, in what frequency range and with what amplitude will the waves be generated?  What general shape will they have?  Some experts believe that experimental detection will prove to be impossible unless “the template used in the detection is a faithful representation of the true signal” (Damour 1). One way to approach this problem is to perform simulations of gravitational radiation sources.

 

Formulation of the Model

            A star (perhaps a neutron star) will be represented in a two-dimensional polar grid, with the axis of rotation perpendicular to the grid.  Each discrete  point contains a decimal value that represents mass density that particular local region.  An illustration is shown as an example below:

In this case, the density (represented by darkness of color) is greatest near the core, and decreases proportionally to the radius.  Two diametrically opposite slices, however, have a relatively greater density, which will create an asymmetry.  Ultimately, it is this feature of the object’s distribution that will lead to the emission of gravitational.

            To assign values to each place in the polar grid, a density function is needed.  A simple preliminary case is that density is proportional to the negative exponent of radius:

           

Here, the density is constant in time and is radially symmetric.

            With the physical object now defined, we are interested in a certain physical characteristic: its mass quadrupole moment tensor Qkl, which is given by:

            ,

where the indices k and l range from 0 to 2, x0 = x, x1 = y, x2 = z, and r2 is the square of the radius (Ohanian 11).  However, in this two-dimensional model, only x and y are of interest (so Qkl can be represented by a 2 x 2 matrix).

Qkl will be determined through numerical integration.  Specifically, for each radius and angle in the grid, the density times the area element will be multiplied by the appropriate integrand, and the individual values will be summed.  The result will approximate Qkl.

            Now that the quadrupole moment may be calculated for a static mass distribution, the next step is to include time dependence.  In other words: let the density function depend on time, and find Qkl over a range of time steps.  This implementation, of course, will make the program more computationally intensive.

            Recall that quadrupole radiation depends on the second time derivative of Qkl.  In particular, the gravitational field for the oscillating body looks like:

                      (Ohanian 152).

Thus, given an array of values of, say, Q11 over a range of time intervals, we can numerically approximate the second derivative of Q11:

            ,

assuming the time step is 1 unit.  For this investigation, only the relative wave amplitude is of interest, so proportionality constants are ignored.

 

Initial Results

            As a validation of the program, a simulation was performed with a radially symmetric, static mass density (so that we would expect no gravitational radiation).  Indeed, these results were observed:

    

 

Taking a step forward, we may introduce an asymmetry into the model.  A mass distribution given by:

            ,

may be graphically represented as:

 

            Of course, if such an object remained static, it would not emit any gravitational radiation –a changing quadrupole moment is required (this was numerically confirmed).  This model may thus be improved by inserting time dependence.  To rotate the object, a phase factor depending on time can be placed in the argument of the sine function:

,

where w is the angular frequency of rotation of the object.

            Now we have an object with an oscillating Qkl, so we expect radiation to be released.  Furthermore, the oscillation is perfectly periodic, so the waves should be sinusoidal in nature.  Indeed, the simulation produced these results, displayed below.

 

            Another extension of the model is to consider the same basic object, but with a decreasing radius (this could physically correspond to a collapsing neutron star).  Here, a density function is:

            .

Note that as the mass becomes more concentrated near the core, the star will spin faster in order to conserve angular momentum.  After introducing this modification into the simulation program, the wave signature shown below was produced.

Several new gravitational wave features are present in this case.  First, the wave amplitude is not constant: it begins small and increases once the rotation rate becomes appreciable.  However, after more time has elapsed, the small size of the star overpowers the effect of faster spin rate: the quadrupole moment does not shift as much in magnitude.  Consequently, the wave amplitude proceeds on a decreasing trend.  Second, the wave frequency does not remain constant; the above plot shows that frequency increases with time.  Undoubtedly, this is due to the star spinning faster as decreases in volume.  Finally, the absolute amplitude of the waves in this plot differs by orders of magnitude with the previous plot.  This, however, is due to nuances in the code and is not representative of any physical phenomenon.

 

Conclusions

            Several important shortcomings of the model must be addressed.  First, the fact that only two dimensions were considered is a serious issue; gravitational radiation is not emitted in the same form at all angles with the plane of rotation.  (Also, a three dimensional system would have a more complex quadrupole moment.)  Second, the simple density functions that were used are not characteristic of any actual physical system.  In reality, collapsing stars have enormously complicated mass distributions that should produce highly unique gravitational wave signatures.  Thus, one result from this simulation is that extensive information on the makeup of a body would be needed to accurately model its potential wave emission pattern.  Third, parameters for mass, radius, and time were dimensionless and somewhat arbitrary.  Thus, no information about true, dimensional values for frequency and amplitude of the waves could be produced.  However, proceeding in such an investigation would only be useful if detailed data on a star’s makeup were available.

            One suggestion for a highly more sophisticated model is to provide the system with only initial parameters.  If the program were also “equipped” with the laws of physics, it could describe a realistic system evolving through time.  With this addition, the same type of calculations could be performed (determine Q at each time step, then calculate second derivative).  Simulations of gravitational waves should strive to emulate this approach.

            Perhaps one of the most relevant conclusions of this investigation is that a vast amount of computational power would need to be used to perform a meaningful simulation.  A simple trial in the 2D model would take several minutes to run on a relatively small grid over only 100 time steps (on a 1.7 GHz machine).  Implementing the aforementioned modifications and a 3D grid, over a longer time scale, would raise this time considerably.  The main piece of knowledge gained from these simulations has been that simple models produce simple results.  Even the most complicated trial that was performed was vastly simplified compared to the real world.  Consequently, none of the results are quantitatively very meaningful.  This set of simulations acts more as a building block towards future progress in the field.

 

References

Damour. T., et al. “Modeling Gravitational Waves from Inspiralling Compact Binaries”

<http://arxiv.org/abs/gr-qc/9801014>

 

Ohanian, Hans C. Gravitation and Spacetime. New York: W.W. Norton and Co., 1976.