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.

1.
K.
Schwarzschild
, “
On the gravitational field of a mass point according to Einstein’s theory
,”
Gen. Relativ. Gravit.
35
,
951
959
(
2003
),
K.
Schwarzschild
, translated by S. Antoci from “
Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
,”
Sitzungsber. K. Preuss. Akad. Wiss.
1
,
189
196
(
1916
).
2.
W.
Rindler
,
Relativity: Special, General and Cosmology
(
Oxford U. P.
,
Oxford
,
2001
).
3.
R. M.
Wald
,
General Relativity
(
U. of Chicago Press
,
Chicago, London
,
1984
).
4.
C. W.
Misner
,
K. S.
Thorne
, and
J. A.
Wheeler
,
Gravitation
(
Freeman
,
New York
,
1973
).
5.
M. A.
Abramowicz
, “
Black holes and the centrifugal force paradox
,”
Sci. Am.
268
(
3
),
26
31
(
1993
).
6.
Information about OpenGL and the OpenGL Shading Language can be found at ⟨www.opengl.org⟩.
7.
R. J.
Rost
,
OpenGL Shading Language
(
Addison-Wesley
,
Boston
,
2004
).
8.
Open MPI: A high performance message passing library, ⟨www.open-mpi.org/⟩.
9.
K.
Sakina
and
J.
Chiba
, “
Parallel transport of a vector along a circular orbit in Schwarzschild spacetime
,”
Phys. Rev. D
19
,
2280
2282
(
1979
).
10.
B. R.
Iyer
and
C. V.
Vishveshwara
, “
Frenet–Serret description of gyroscopic precession
,”
Phys. Rev. D
48
,
5706
5720
(
1993
).
11.
D.
Bini
,
C.
Cherubini
,
A.
Geralico
, and
R. T.
Jantzen
, “
Physical frames along circular orbits in stationary axisymmetric spacetimes
,”
Gen. Relativ. Gravit.
40
,
985
1012
(
2008
).
12.
M. A.
Abramowicz
and
A. R.
Prasanna
, “
Centrifugal force reversal near a Schwarzschild black-hole
,”
Mon. Not. R. Astron. Soc.
245
,
720
728
(
1990
).
13.
L.
Lerner
, “
A simple calculation of the deflection of light in a Schwarzschild gravitational field
,”
Am. J. Phys.
65
,
1194−1196
(
1997
).
14.
T.
Müller
and
F.
Grave
, “
GeodesicViewer—A tool for exploring geodesics in the theory of relativity
,”
Comput. Phys. Commun.
181
,
413
419
(
2010
).
15.
T.
Müller
and
D.
Weiskopf
, “
Distortion of the stellar sky by a Schwarzschild black hole
,”
Am. J. Phys.
78
,
204
214
(
2010
).
16.
D.
Weiskopf
,
M.
Borchers
,
T.
Ertl
,
M.
Falk
,
O.
Fechtig
,
R.
Frank
,
F.
Grave
,
P.
Jezler
,
A.
King
,
U.
Kraus
,
T.
Müller
,
H. -P.
Nollert
,
I.
Rica Mendez
,
H.
Ruder
,
T.
Schafhitzel
,
C.
Zahn
, and
M.
Zatloukal
, “
Explanatory and illustrative visualization of special and general relativity
,”
IEEE Trans. Vis. Comput. Graph.
12
,
522
534
(
2006
).
17.
C.
Savage
,
D.
McGrath
,
T. J.
McIntyre
,
M.
Wegener
, and
M.
Williamson
, “
Teaching physics using virtual reality
,”
AIP Conf. Proc.
1263
,
126
129
(
2010
).
18.
D.
McGrath
,
M.
Wegener
,
T. J.
McIntyre
,
C.
Savage
, and
M.
Williamson
, “
Student experiences of virtual reality: A case study in learning special relativity
,”
Am. J. Phys.
78
,
862
868
(
2010
).
20.
The photon orbit at rp=3rs/2 is the only circular lightlike geodesic within the Schwarzschild spacetime. A detailed discussion can be found in Ref. 2.
21.
We use differential geometric notation, where μ=/xμ is the coordinate base vector along xμ. For a detailed discussion of this topic see, for example, Ref. 32.
22.
The Milky Way panorama, ESO/S. Brunier, ⟨www.eso.org/public/images/eso0932a⟩.
23.
The Qt library can be found at ⟨qt.nokia.com⟩.
24.
T.
Müller
and
F.
Grave
, “
Motion4D—A library for lightrays and timelike worldlines in the theory of relativity
,”
Comput. Phys. Commun.
180
,
2355
2360
(
2009
).
25.
D.
Weiskopf
,
U.
Kraus
, and
H.
Ruder
, “
Searchlight and Doppler effects in the visualization of special relativity: A corrected derivation of the transformation of radiance
,”
ACM Trans. Graphics
18
,
278
292
(
1999
).
26.
J. D.
Foley
,
A.
Van Dam
,
S. K.
Feiner
, and
J. F.
Hughes
,
Computer Graphics: Principles and Practice
(
Addison-Wesley
,
New York
,
1991
).
27.
G.
Wyszecki
and
W.
Stiles
,
Color Science: Concepts and Methods, Quantitative Data and Formulae
(
Wiley Classics Library
,
New York
,
2000
).
28.
A.
Einstein
, “
Lens-like action of a star by the deviation of light in the gravitational field
,”
Science
84
,
506
507
(
1936
).
29.
T.
Müller
and
F.
Grave
, “
Catalogue of spacetimes
,” arXiv.org:0904.4184 [gr-qc].
30.
W.
de Sitter
, “
On Einstein’s theory of gravitation and its astronomical consequences
,”
Mon. Not. R. Astron. Soc.
77
,
155
184
(
1916
).
31.
H.
Stephani
and
J.
Stewart
,
General Relativity: An Introduction to the Theory of the Gravitational Field
(
Cambridge U. P.
,
Cambridge
,
1990
).
32.
M.
Nakahara
,
Geometry, Topology and Physics
(
IOP
,
Bristol
,
1990
).
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