# Detailed study of null and time-like geodesics in theAlcubierre Warp spacetime

Thomas Müller and Daniel Weiskopf

The geodesic equation of the Alcubierre warp spacetime [1] is converted into its non-affinely parametrized form for a detailed discussion of the motion of particles and the visual effects as observed by a traveller inside the warp bubble or a person looking from outside. To include gravitational lensing for point-like light sources, we present a practical approach using the Jacobi equation and the Sachs bases. Additionally, we consider the dragging and geodesic precession of particles due to the warp bubble.

The following images and movies were rendered using the four-dimensional ray tracing system GeoViS [2].

### Expansion of the normal volume elements

If you have a web browser that can handle WebGL context (e.g. google-chrome), you can manipulate this view interactively here.

### View from the bridge

View from the bridge in the direction of motion for different velocities.

### View of a static observer towards the approaching warp bubble

The static observer is located at x = 10, y = 0 and looks in the negative x-direction towards the approaching warp bubble.

### View of a static observer

The static observer is located at x = 0, y = -4 and has panorama camera with 180°×60° field of view.

The Milky Way panorama is by ESO/S. Brunier.

### Movies

 View of a static observer located at x = 0, y = -4 for observation times t = -1 .. 11.Panorama camera has 180°×60° field of view.warp_passing.1440x480.mp4 (mpeg4, 80 MB)warp_passing.300x100.mp4 (mpeg4, 1.6 MB) View of a static observer located at x = 0, y = -4 for observation times t = -1 .. 11.warp_passing_grid.1440x480.mp4 (mpeg4, 37 MB)warp_passing_grid.300x100.mp4 (mpeg4, 6.2 MB) View of a static observer located at x = 0, y = -50 for observation times t = 39 .. 75.Pinhole camera has 50°×20° field of view.warp_passing_far.1000x400.mp4 (mpeg4, 5.7 MB)warp_passing_far.250x100.mp4 (mpeg4, 737 kB) View of a static observer located at x = 0, y = -50 for observation times t = 39 .. 75.Pinhole camera has 50°×20° field of view.warp_passing_far_grid.1000x400.mp4 (mpeg4, 4.8 MB)warp_passing_far_grid.250x100.mp4 (mpeg4, 1.1 MB) View of a static observer located at x = 0, y = -50 for observation times t = 39 .. 75.Pinhole camera has 50°×20° field of view.(combined)warp_passing_far_both.1000x800.mp4 (mpeg4, 6.3 MB)

 Particles from the bridge with initial velocity v=0.5c. The blue-shaded region represents the warp bubble.particleFront_bridge_v2p0.700x500.mp4 (mpeg4, 1.7 MB) Particles injected from outside at t=-6 with initial velocity v=0.5c. The blue-shaded region represents the warp bubble.particleFront_lateral_tm6.700x350.mp4 (mpeg4, 1.3 MB) Particles injected from outside at t=-4 with initial velocity v=0.5c. The blue-shaded region represents the warp bubble.particleFront_lateral_tm4.700x350.mp4 (mpeg4, 1.2 MB)

### Java Application

The Java application was developed using OpenJDK 1.6.0_20 (IcedTead6 1.9.2), NetBeans 6.8, and NetBeans OpenGL Pack 0.5.5.

 Linux amd64 gzip-compressed tar-file (16 MB) md5: 5e7e4d6160d4ce4237493d03bb8317fe Linux i586 gzip-compressed tar-file (16 MB) md5: ee9760a8f328dd6ddeef0f9d38fa830f Windows 64 zip-archive (16 MB) md5: 57f1ba8b3dc367dd01b1f275cd5e8f65 Windows 32 zip-archive (16 MB) md5: d9a7d5c9ab9fa0d19692cc9e34d9e7a3

Technical details and a brief description of the user interface can be found here.

System requierements:

• You need a graphics board that supports the shader language GLSL; e.g. NVidia GeForce 8 Series or higher.
• Java Runtime Library must be installed.

### Geodesic Viewer

The following configuration files can be used for studying geodesics with the GeodesicViewer.

 warp_bridge.zip md5sum: ce47ca94886987b8a6f6e789921d989a warp_towards.zip md5sum: 57b83b0b8f42f67463db9ff153ba219b warp_lateral.zip md5sum: 57b83b0b8f42f67463db9ff153ba219b

### References

• [1] M. Alcubierre, "The warp drive: hyper-fast travel within general relativity,"
Class. Quantum Grav. 11, L73-L77 (1994).
• [2] T. Müller, Dissertation: "Visualisierung in der Relativitätstheorie" (in german),
Eberhard-Karls-Universität Tübingen (2006), [tobias-lib].
• [3] T. Müller, D. Weiskopf, "Detailed study of null and time-like geodesics in the Alcubierre Warp spacetime,"
Gen.Rel.Grav.(online first)
DOI: 10.1007/s10714-011-1289-0.