Geodesic Motion of Test Particles in Kerr Spacetime

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Geodesic Motion of Test Particles in Kerr
Spacetime Alethea S. Bair
Orbits
Motivation
Numerical Computation
Newtonian
Schwarzschild
Once the equations of motion have been derived
and are in the correct form, they can be
numerically integrated to give the orbits. Using
variable substitution, N second order equations
were turned into 2N first order equations. Then
a fourth-order, Runge-Kutte, variable step size
algorithm was used to find solutions given an
initial position and velocity. In any numerical
solution, a test of the accuracy is
important. Accuracy was gauged by the constancy
of the constants of motion. Constants of motion
such as energy and z-angular momentum were found
using the Euler-Lagrange equation. If L is
independent of x, then When the variable step
size was used, the integration was accurate to
10-15 decimal places.
Gravitational Waves, if detected by experiments
such as LIGO, could serve as an accurate test of
General Relativity. A computer program which
computes and displays orbits around super-massive
spinning black holes was written with the
intention of using it as a basis to add a
radiation reaction force. The program could then
accurately compute the time evolution of
constants of the motion, such as energy. This
evolution, could then be compared to observed
gravitational waves. An interactive display of
general relativistic orbits serves as an
excellent teaching tool. The mathematics involved
in general relativity is well known to be
difficult. Many phenomena that are difficult to
visualize given the spacetime metric are much
clearer given an animation of the orbit.
The Schwarzschild metric was the first solution
found to Einsteins field equations. It
describes a spherically symmetric, non-charged
and non-rotating black hole.
The Newtonian theory of gravity is an
approximation to general relativity. It is valid
for small masses and large distances. Orbits can
be circles, ellipses, parabolas or hyperbolas.
A view of an elliptical Newtonian orbit. The
orbit is planar and closes upon itself
Kerr
Equations of Motion
The Kerr metric models a spinning,
axially-symmetric, non-charged black hole. The
spin of the hole warps the spacetime around it
such that particles are dragged in the direction
of the holes spin. This dragging of inertial
frames becomes more extreme the closer a
particle gets to the rotating black hole.
Graphics
OpenGL A portable application programming
interface (API) for rendering sophisticated 2D
and 3D graphics. GLUT The OpenGL Utility
Toolkit provides an interface for creating
windows, and interfacing with the mouse and
keyboard on both Win32 and X11 workstations. Inte
rface The user can choose starting conditions
via an input file, start and stop the orbit, and
change the viewing position using the keyboard
and mouse. The animation is done using the GLUT
idle function.
Radiation Reaction
Line Element The differential distance in a
given spacetime. Flat spacetime can be described
in polar coordinates as The Kerr line element,
which describes a spinning black hole
is Geodesic Equation Derived by parallel
transporting a vector along itself, the geodesic
equation produces equations of motion for a given
metric (line element). Euler-Lagrange
Equation A method to minimize a function L with
respect to path. If L is the path length ,
a minimum with respect to path is exactly the
requirement for a geodesic. This method produces
the same equations of motion as the geodesic
equation, but the derivation is easier to grasp
without previous knowledge of general relativity.
An accelerated charged particle loses energy in
the form of electromagnetic radiation.
Similarly, General Relativity predicts that an
accelerated massive object will radiate energy in
the form of gravitational waves (a warping of the
spacetime that propagates outward from the
source).
Lense-Thirring Effect
Future
Extremely close to the black hole, the dragging
of inertial frames becomes so strong that given a
slight initial upward velocity a particle is
actually dragged several times around by the
black
As a teaching tool, the program needs an improved
and simpler interface for entering initial
conditions. As a scientific tool, it needs an
accurate radiation reaction force for true
computation of the time evolution of the
constants of motion.
holes spin before it reaches its maximum height,
and similarly on the way down. This causes a
spiral orbit.
I am extremely grateful to Alan Wiseman, my
advisor for this project.
In the Kerr orbit, a mass energy of 1.0 is
subtracted from the total energy to make the
curve clearer.