Special relativity offers the possibility of going on a trip to the center of our galaxy or even to the end of our universe within a lifetime. On the basis of the well known twin paradox, we discuss uniformly accelerated motion and emphasize the local perspective of each twin concerning the interchange of light signals between both twins as well as their different views of the stellar sky. For this purpose we developed two Java applets that students can use to explore interactively and understand the topics presented here.
REFERENCES
1.
2.
C. E.
Dolby
and S. F.
Gull
, “On radar time and the twin ‘paradox
,'” Am. J. Phys.
69
, 1257
–1261
(2001
).3.
S. P.
Boughn
, “The case of the identically accelerated twins
,” Am. J. Phys.
57
, 791
–793
(1988
).4.
R. P.
Gruber
and Richard H.
Price
, “Zero time dilation in an accelerating rocket
,” Am. J. Phys.
65
, 979
–980
(1997
).5.
W.
Rindler
, Relativity: Special, General, and Cosmology
(Oxford University Press
, New York
, 2001
).6.
R. U.
Sexl
, and H. K.
Urbantke
, Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics
(Springer
, Wien/New York
, 2000
).7.
R.
Perrin
, “Twin paradox: A complete treatment from the point of view of each twin
,” Am. J. Phys.
47
, 317
–319
(1979
).8.
9.
R. H.
Good
, “Uniformly accelerated reference frame and twin paradox
,” Am. J. Phys.
50
, 232
–238
(1982
).10.
H. und M.
Ruder
, Die Spezielle Relativitätstheorie
(Vieweg Studium
, Braunschweig/Wiesbaden
, 1993
).11.
See EPAPS Document No. E-AJPIAS-76-002803 for the Java applets, RelSkyApplet and TwinApplet, and accompanying help manuals. This document can be reached through a direct link in the online article’s HTML reference section or via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html).
12.
One light year equals the distance which is covered by light with velocity , 792, in one year.
13.
The negative acceleration of Tina away from the Sun equals the situation where Tina has positive acceleration but looks contrary to her direction of motion.
14.
The limit can be calculated using l’Hospital’s rule.
15.
G. D.
Scott
and H. J.
van Driel
, “Geometrical appearances at relativistic speeds
,” Am. J. Phys.
38
, 971
–977
(1970
).16.
The Hipparcos star catalogue consists of about 118,000 stars most of which are at a distance of roughly . The data we are interested in are right ascension (H3), declination (H4), magnitude in Johnson V (H5), trigonometric parallax (H11), Johnson B-V color (H37), and Henry Draper (HD) catalogue number (H71). The digital catalogue I/239 can be found at ⟨cdsarc.u-strasbg.fr/viz-bin/Cat?I/239⟩ 1997HIP…C……0E-European Space Agency SP-1200.
17.
B. C.
Reed
, “The composite observational-theoretical HR diagram
,” J. R. Astron. Soc. Can.
92
, 36
–37
(1998
).18.
The bolometric magnitude of a star is defined by the flux density integrated over all wavelengths. However, a real telescope can only measure the flux density within a small region of wavelengths. The bolometric correction compensates the difference between the bolometric and the visual magnitude (Ref. 19).
19.
Fundamental Astronomy
, edited by Hannu
Karttunen
, Pekka
Kröger
, Heikki
Oja
, Markku
Poutanen
, and Karl J.
Donner
(Springer-Verlag
, Heidelberg
, 2003
).20.
We take the physical constants from the National Institute of Standards and Technology, ⟨physics.nist.gov/cuu/Constants⟩.
21.
R. W.
Lindquist
, “Relativistic transport theory
,” Ann. Phys.
37
, 487
–518
(1966
).22.
The absolute bolometric magnitude of the sun is and the effective temperature is , see also ⟨nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html⟩.
23.
U.
Kraus
, “Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited
,” Am. J. Phys.
68
, 56
–60
(2000
).24.
The constellations were obtained from the free planetarium software Stellarium, ⟨www.stellarium.org⟩.
25.
J. C.
Mather
et al, “Calibrator design for the COBE far-infrared absolute spectrophotometer (FIRAS)
,” Astrophys. J.
512
, 511
–520
(1999
).26.
For more information on x rays and rays we refer the reader to the following missions. X-ray: ROSAT, ⟨wave.xray.mpe.mpg.de/rosat⟩, Chandra, ⟨chandra.harvard.edu⟩, XMM-Newton, ⟨sci.esa.int⟩; -ray: INTEGRAL ⟨www.esa.int/esaMI/Integral/⟩.
27.
The rotational velocity of a star in a galaxy does not follow Newton’s law, but can be read from the rotation curve of the galaxy. See Ref. 28 for the outer rotation curve of the Milky Way.
28.
J.
Brand
and L.
Blitz
, “The velocity field of the outer galaxy
,” Astron. Astrophys.
275
, 67
–90
(1993
).29.
The goal of the GAIA mission is to collect high-precision astrometric data for the brightest one billion objects. A detailed description of the ESA Science missions is available at ⟨www.esa.int/esaSC/⟩.
30.
31.
Data are taken from ⟨map.gsfc.nasa.gov⟩.
32.
Single-precision floating-point numbers are stored in a word, whereas double-precision ones are stored in a word. The word itself is composed of a mantissa, a exponent, and one bit for the sign. More information can be found in the IEEE standard for binary floating-point arithmetic for microprocessor systems (ANSI/IEEE Std 754–1985).
33.
This fact is known as the tristimulus theory. For more information see, for example, Foley (Ref. 35).
34.
Color matching functions can be found at the Institute of Ophthalmology, ⟨cvrl.ioo.ucl.ac.uk/basicindex.htm⟩.
35.
J. D.
Foley
, A.
Van Dam
, S. K.
Feiner
, and J. F.
Hughes
, Computer Graphics: Principles and Practice
(Addison–Wesley
, 1991
).36.
We use the color rendering of spectra by
John
Walker
, ⟨www.fourmilab.ch⟩. The white point is the set of chromaticity values that serve to define the color white.37.
An excellent science fiction book on this topic is given in Ref. 30 which might be a useful basis for discussion.
© 2008 American Association of Physics Teachers.
2008
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
