General Relativity Simulations

Using Open Source PhysicsSharon E. Meidt and Wolfgang Christian

Physics Department, Davidson College

PO Box 6910 Davidson, NC 28035-6910

IntroductionOur 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. Taylor’s recent book, Exploring Black Holes: Introduction to General Relativity [1].

Gravitational RedshiftThe 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 Explorer’s Beacon creates a series of events shown as pulses when the Go button is pressed. The user sets the Beacon’s location in space and its pulse period dτ 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 CurvatureThe 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 dσ . 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 dσ 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 dσ . Both figures show several instances of the Interval, which consists of two events. The Interval calculates dσ 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 dσ depends on the radial position for the same separation dr.

Particle TrajectoriesThe ‘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 TrajectoriesThe ‘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 Explorer’s 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’.

Impact Parameter vs. Scattering AngleThe 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 WorkWe 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.

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