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Abstract
CAVEGlom Revisions - Hypergraphics Final Report
Patrick Draper 2002
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Background and Purpose
Glom began as simulation by Matt Hall of Newtonian orbital mechanics. The user had control over where and with what velocity moons were introduced into the system, as well as the gravitational strength of the central mass. This is useful from a visual learning perspective, as well as a good exercise in computational physics. With implementation of far more advanced object control, it could also serve as a valid tool for astrophysics research.


Revisions
The updated Glom now contains several additional components useful for the modeling of astrophysical phenomena. The first is the ability to toggle the wand projectile from moons to beams of light. With a strong enough gravitational field from the center, the light undergoes gravitational lensing. In this effect, the beam will appear to bend in a nonlinear fashion; however, it is still following the null geodesic predicted by general relativity. Spacetime itself has been bent by the gravitational field. To simulate this, Euler's method of numerical integration is used to solve the following differential equation determined by Einstein's General Theory of Relativity:


u'' + u = 3M * u^2


where u''=d^2  u/dtheta^2 and u=1/r (r is the radius from central mass M). Once a new r is obtained over some small time interval dt, the beam location is converted from (r,theta) to (x,y) and projected onto the firing plane. This plane is constant for the entire beam traversal and is determined from the beam's point of origin and a normal vector. The normal is the cross product of the initial velocity vector and the vector joining the initial position with the center mass.

The beam begins with wavelength 500nm, a rough shade of blue, and experiences a gravitational redshift which causes the wavelength to change according to

                             G*M*lambda
delta lambda = -----------------
                            fromcent*c^2

where G is the Gravitational constant, M is the central mass, fromcent is the distance from the beam to the center at a given point, and lambda is the wavelength. The effect is observable in the changing beam color.


Another modification is the orbit toggle, which allows the user to leave the initial stationary vantage point and enter orbit alongside a moon. This is of less physical interest than it is mathematical; even now not all the bugs of changing coordinate systems have been worked out. There is some discrepancy between how the CAVE handles camera position and translation/rotation thereof and how this process occurs in OpenGL.

The moons can also be selected one at a time, and statistics about the current selection appear on the righthand side of the front CAVE wall. These include the components of the moon's velocity as well as its speed percentage of c (299792458m/s) and the local time at its surface. Because the moon speeds are arbitrarily set very high, they experience a degree of time dilation, which is notable by comparing the local time of a selected moon to the global time displayed in the upper right. Local times are calculated by scaling the global time with a factor of

(1 - s^2/c^2)^(1/2)

which is an example of a Lorentz contraction.

Another effect of the high velocities is interpreted through the moon colors. When fired, they emit light of wavelength 500nm, or a rough shade of blue. Initially they are traveling away from the wand, so the user observes a doppler red-shift phenomena- the wavelength lengthens according to

                                               (1 + awayvelocity/c)^(1/2)
Lambda_new = Lambda_old * --------------------------
                                               (1 - awayvelocity/c)^(1/2)

This causes the color of the moon to shift towards the red end of the spectrum. The color is shifted further still by the same gravitational redshift experienced by the beam.

Future
Glom is anything but finished. The beauty of the simulation is not in what it does, but how it can be scaled to do more. There are years of projects of varying difficulty left to be mined from Glom, from the easy (better data output) and the medium (Runge-Kutta replacement of Euler's method for increased accuracy) to the very challenging (N-body simulation) and the genius-inspired (compact-object relativistic effects). Anyone with an interest in holistic physics has something to gain and offer to Glom.

Patrick Draper CAVEGlom 2002