General Relativity Simulations

1
General Relativity Simulations Using Open Source
Physics Sharon E. Meidt and Wolfgang
Christian Physics Department, Davidson College PO
Box 6910 Davidson, NC 28035-6910
Impact Parameter vs. Scattering Angle The Seeing
program introduces the user to what is seen by an
observer. This program calculates and shows the
scattering angles of light from a cross-section
of impact parameters b. The user can set some b
around which n rays over width w are
launched. Figs. 7a-c M 1. a.
Cross-section of stone trajectories given
initial conditions as listed in the properties
table shown in Fig 7b. c. A tables of
calculations of scattering angle and travel time
in Schwarzschild coordinates to reach a radius
of r 100M, for each impact parameter. Figs.
8a-c Same as in Figs. 7a-c, but for light rays.
Fig. 7a shows the trajectories of stones,
which, when given a launch speed of unity appear
like the light trajectories launched with
identical impact parameters in Fig. 8a.
Figs. 9a-c M 1. Same as Figs.
8a-c. Cross-section of light trajectories over
a thin slice of impact parameters. Almost all
have scattered at least 180 degrees, and the
largest scattering angle is just over 372
degrees. Fig. 9 shows light scattered from a
small width of impact parameters. Light coming
from all around the black hole, then, reaches an
observer at impact parameter b. Future Work We
will continue the Seeing program, next mapping a
square grid onto the eye of a far-away
observer. We plan to do this through linear
interpolation using a table of scattering angles
generated for a set of impact parameters to
determine the location of a source that
corresponds to viewing angle theta. Once this is
achieved for a grid, we will enhance this
algorithm to map a pixel image, such as that in
the figures below. Figs. 10a and b a.
The image in the far field, here, Saturn. b.
The generated image. Note the diamond
necklace, the faint ring in the middle
2. The final task is to present what is
actually seen by an observer. We hope, for
example, to show an orbit of a stone about a
black hole in real time. Bibliography 1 Edwin
F. Taylor and John Archibald Wheeler, Exploring
Black Holes Introduction to General Relativity,
San Francisco, Addison Wesley Longman, 2000. 2
From the cover of Black Holes, see Ref. 1.
Partial funding for this work was obtained
through NSF grant DUE-0126439.
Particle Trajectories The S button on the
toolbar of Orbit Explorer creates a Stone and a
Cannon within the Schwarzschild map and adds a
plot of the effective potential of the stone as a
tabbed panel. Tabbed panels can be released
later by double clicking on the tab.
Double-clicking on the Cannon allows the user to
set the initial position, launch angle, and
launch speed for a Stone whose trajectory is
plotted. Figs. 3a-e M 1 a.
Trajectories of four stable circular orbits at r
6M, 7M, 8M, and 9M with speeds v 0.408248,
0.377964, 0.353553, and 0.33333, respectively,
where the speed for stable circular orbits is
. b-e Plots of effective potential for the
stones at the four radii shown in Fig 3a. Fig.
3a shows the trajectories of four stones launched
from cannons at four different radii. The
initial theta and speed for each stone is set in
order to produce a stable circular orbit. Figs.
3b-e show the corresponding plots of effective
potential for each of the stones. The plots show
that circular orbits occur at minima of the
effective potential. Figs. 4a and b M 1.
Trajectory and effective potential for a
non-uniform circular orbit where r 6.3M and v
0.408248, initially. The red line in the
potential plot can be dragged to change the
energy. Fig. 4 is an example of non-uniform
circular orbit where the stone is given the same
speed as the stone at r 6M in Fig. 3, but is
launched from a larger radius. Fig. 4b shows the
effective potential of the stone. In this case,
the energy of the stone is greater than the
minimum of the potential, and so the stone
oscillates within the potential and follows the
non-uniform trajectory in Fig. 4a. Light
Trajectories The LR button on the Metric
Explorer toolbar creates a Light Ray, which is
also launched from a cannon, where the user can
set the initial position and launch angle of the
light ray. Fig. 5 below shows an example of two
light rays about a black hole of mass M 1.
Compared to the trajectories in Fig. 6 of the
same two light rays without a black hole (M 0),
the light rays in Fig. 5 are clearly bent in the
presence of the gravitationally attracting body.
When viewed such that the intersection of the
light rays is a light source, Fig. 5 becomes an
example of gravitational lensing.
Figs. 5a and b Two Light Rays
bending in the presence of a black hole. (In Fig.
5a, M 1 and in Fig. 5b, M 0). Metric
Explorers Multi-Ray is a source of evenly
distributed light rays. The user can change the
position of the Multi-Ray and the number of rays
by double-clicking on the object. Figs.
6a and b M 1. a Multi-Ray at r 5M, 16
rays. b Multi-Ray at r 3M, 16 rays. Figs. 6a
and b are examples of the Multi-Ray. In Fig. 6b,
the light actually completes at least one orbit
about the black hole at r 3M if the Multi-Ray
were to mark the location of an observer, light
from all around the black hole would be incident
there, at r 3M. This causes the effect known
as the diamond necklace.
Introduction Our Java-based general relativity
simulations explore the Schwarzschild metric and
the role of the observer. The Metric Explorer
program allows the user to investigate space-time
curvature, gravitational red-shift, and the
trajectories of light. The Orbit Explorer
program visualizes the trajectories of particles
and their effective potentials in the vicinity of
non-spinning black holes, for a Schwarzschild
observer. These simulations are developed as
part of the Open Source Physics project and are
based, in part, on material from Edwin F.
Taylors recent book, Exploring Black Holes
Introduction to General Relativity 1.
Gravitational Redshift The tables below
identify the buttons on the left-side toolbar of
our programs that create objects, such as
particles and light rays, within the program.
For example, Metric Explorers Beacon
creates a series of events shown as pulses when
the Go button is pressed. The user sets the
Beacons location in space and its pulse period
dt in proper time and observes a far away pulse
period dt in Schwarzschild coordinates.
According to the time-like form of the
Schwarzschild metric below

, (1) for two events at
the same location in space( ),
but different locations in time,  

. (2) As the user
moves the beacon toward the black hole, the pulse
period dt in Schwarzschild coordinates decreases
(and at r 2M, ). Far away from the
black hole, the period of pulses according to the
Schwarzschild observer increases, and at
. Spacetime Curvature The user
can also create Meter Sticks and Intervals within
Metric Explorer to investigate separations in
space according to the space-like form of the
Schwarzschild metric,

. (3) Figs. 1a and b show
five Meter Sticks as measured by the
Schwarzschild observer. Each meter stick is moved
by clicking the center, and rotated by clicking
either end, with the mouse. Figs.
1a and b M 1. Schwarzschild Map showing
circles of constant separation ds . a. Several
instances of the Meter Stick. b. Zoomed in on
meter sticks. The Meter Stick invariably
maintains its proper length of 1 meter, but
according to the Schwarzschild observer, the
length dr changes with its radial position about
the black hole. The meter stick shrinks in the
radial direction and yet is unaffected in the
tangential direction as it approaches the black
hole. This behavior is purely a manifestation of
the Schwarzschild coordinate system a local
observer would see the meter stick
unchanged.   The view shown in Fig. 1 is such
that the proper length ds between rings is
constant. The Schwarzschild coordinate separation
dr, however, decreases with approach to the black
hole. In Fig. 2a we draw the rings with constant
dr. Figs. 2a and b M 1.
Two representations of the radial coordinate. a.
Schwarzschild Map showing circles of constant
separation dr. b. Schwarzschild Map showing
circles of constant separation ds . Both figures
show several instances of the Interval, which
consists of two events. The Interval calculates
ds between the two events, given a separation dr
set by the user. The Interval at the top, for
instance, shows that the proper separation
between the events is greater than 5, the
separation we expect in Schwarzschild
coordinates. The other instances of the Interval
show that ds depends on the radial position for
the same separation dr.
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