^{1,a)}and Sebastian Boblest

^{2,b)}

### Abstract

An observer who moves on a circular orbit around a Schwarzschild black hole with a constant but arbitrary velocity must compensate for the gravitational and centrifugal acceleration to stay on this orbit. The local reference frame of the observer undergoes a geodesic precession, which depends on the radius of the orbit and the velocity. We describe the details of this circular motion and an interactive program that shows what the observer would see.

The authors would like to thank Daniel Weiskopf for carefully reading the article.

I. INTRODUCTION

II. EXAMPLE

III. ROTATION OF LOCAL REFERENCE FRAME

IV. INTERACTIVE VISUALIZATION

A. The graphical user interface

B. Defining a torus

C. Lookup tables

D. Frequency shift and aberration

E. Visualizing the torus

V. DISCUSSION

A. Comparison of torus definitions

B. Appearance of the torus for different

C. projection and different spectra

D. projection and precession

E. Tidal acceleration

VI. SUMMARY

### Key Topics

- Black holes
- 23.0
- Doppler effect
- 9.0
- Cameras
- 7.0
- Kinematics
- 7.0
- Optical aberrations
- 6.0

## Figures

The observer moves on a circular orbit with radius around a Schwarzschild black hole. The torus has radii and . The bold solid line represents a typical light ray. The black hole horizon with radius is indicated by the dashed line.

The observer moves on a circular orbit with radius around a Schwarzschild black hole. The torus has radii and . The bold solid line represents a typical light ray. The black hole horizon with radius is indicated by the dashed line.

Screenshot of the torus application. The Milky Way background image is taken from ESO (Ref. 22). The left overlay shows the observer’s current position, the orientation of the local reference frame, and the viewing direction. The right overlay shows the tidal accelerations. For a bigger view of the overlays, see Fig. 17.

Screenshot of the torus application. The Milky Way background image is taken from ESO (Ref. 22). The left overlay shows the observer’s current position, the orientation of the local reference frame, and the viewing direction. The right overlay shows the tidal accelerations. For a bigger view of the overlays, see Fig. 17.

The projection shows the complete sky at a glance. The distortions close to are due to the projection itself. In this inverse-color image, the white region corresponds to the black hole. The direction of motion is , .

The projection shows the complete sky at a glance. The distortions close to are due to the projection itself. In this inverse-color image, the white region corresponds to the black hole. The direction of motion is , .

Cross section of a torus in the -plane with outer radius and inner radius .

Cross section of a torus in the -plane with outer radius and inner radius .

To obtain a real circular shape of the inner circle of the torus with respect to proper distance (solid line) or with respect to constant light travel time (dashed line), must be a function of . Here, , , and .

To obtain a real circular shape of the inner circle of the torus with respect to proper distance (solid line) or with respect to constant light travel time (dashed line), must be a function of . Here, , , and .

Initial wave vector with respect to the local reference frame of the observer.

Initial wave vector with respect to the local reference frame of the observer.

Gamuts representing Planck spectra for temperatures from to . (a) Scaled to maximum luminance value for each temperature; (b) scaled to luminance value for temperature .

Gamuts representing Planck spectra for temperatures from to . (a) Scaled to maximum luminance value for each temperature; (b) scaled to luminance value for temperature .

Torus in flat Minkowski space for comparison with the later examples in curved space. The observer is either (a) at rest or (b) moves with half the speed of light. In both cases, the observer is currently located at . The torus has an outer radius .

Torus in flat Minkowski space for comparison with the later examples in curved space. The observer is either (a) at rest or (b) moves with half the speed of light. In both cases, the observer is currently located at . The torus has an outer radius .

Different looks of the three torus definitions: Pseudo-Cartesian, length-scaled, and time-scaled (from left to right). The vertical field of view is 20°, and the observer is located at .

Different looks of the three torus definitions: Pseudo-Cartesian, length-scaled, and time-scaled (from left to right). The vertical field of view is 20°, and the observer is located at .

Torus with , , and (from top to bottom). The observer moves with half the speed of light and the vertical field of view of its camera is 20°. The viewing direction is parallel to the direction of motion. On the left side of the top image, the torus appears to extend into an Einstein ring.

Torus with , , and (from top to bottom). The observer moves with half the speed of light and the vertical field of view of its camera is 20°. The viewing direction is parallel to the direction of motion. On the left side of the top image, the torus appears to extend into an Einstein ring.

Schematic depiction of geodesics for different . Only the -plane is shown. In each picture, only the one geodesic that reaches the observer tangential to the -direction is considered. Depending on , the light ray (solid line) that the observer (black dot) sees as coming from directly ahead originates from the outer part of the torus (, outer dotted circle) or from the inner part (, inner dotted circle). This increasing bending of light rays with decreasing leads to the apparent change in the direction of bending of the torus for .

Schematic depiction of geodesics for different . Only the -plane is shown. In each picture, only the one geodesic that reaches the observer tangential to the -direction is considered. Depending on , the light ray (solid line) that the observer (black dot) sees as coming from directly ahead originates from the outer part of the torus (, outer dotted circle) or from the inner part (, inner dotted circle). This increasing bending of light rays with decreasing leads to the apparent change in the direction of bending of the torus for .

If the torus encompasses the photon orbit, geodesics originating from a point can either reach the observer directly or orbit the black hole one or several times . Therefore, the observer sees the point with the corresponding observation angles , , etc. The gray area marks all points with .

If the torus encompasses the photon orbit, geodesics originating from a point can either reach the observer directly or orbit the black hole one or several times . Therefore, the observer sees the point with the corresponding observation angles , , etc. The gray area marks all points with .

Torus with as in Fig. 10 but with a vertical field of view of only 2°.

Torus with as in Fig. 10 but with a vertical field of view of only 2°.

Torus with . The observer moves with (from top to bottom) and is currently at . The scaled proper times for one complete orbit are . The center of this projection is the direction of motion. (a) The spectrum scaled to -luminance. (b) The spectrum scaled to maximum luminance.

Torus with . The observer moves with (from top to bottom) and is currently at . The scaled proper times for one complete orbit are . The center of this projection is the direction of motion. (a) The spectrum scaled to -luminance. (b) The spectrum scaled to maximum luminance.

Parallel-transport of the frame vector on the innermost stable circular timelike geodesic with (see also Ref. 29); and correspond to the natural local tetrad at that position.

Parallel-transport of the frame vector on the innermost stable circular timelike geodesic with (see also Ref. 29); and correspond to the natural local tetrad at that position.

Geodesic precession for an observer on the timelike circular geodesic with and . The observer’s current position is (from top to bottom). The white disk represents the black hole. The colors are inverted in this projection to enhance the visual impression.

Geodesic precession for an observer on the timelike circular geodesic with and . The observer’s current position is (from top to bottom). The white disk represents the black hole. The colors are inverted in this projection to enhance the visual impression.

(a) The orientation overlay shows the and frame vectors and the viewing direction of the moving observer. (b) The tidal accelerations overlay shows the direction and strengths of the tidal accelerations with respect to the observer’s reference frame. The central thick line indicates the direction to the black hole.

(a) The orientation overlay shows the and frame vectors and the viewing direction of the moving observer. (b) The tidal accelerations overlay shows the direction and strengths of the tidal accelerations with respect to the observer’s reference frame. The central thick line indicates the direction to the black hole.

Tidal acceleration for motion on the last stable timelike circular orbit with and . The central thick arrow shows the direction to the black hole.

Tidal acceleration for motion on the last stable timelike circular orbit with and . The central thick arrow shows the direction to the black hole.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content